Difference Equations, Discrete Dynamical Systems and Applications: IDCEA 2022, Gif-sur-Yvette, France, June 18–22 (Springer Proceedings in Mathematics & Statistics, 444) 3031510488, 9783031510489

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Springer Proceedings in Mathematics & Statistics

Sorin Olaru Jim Cushing Saber Elaydi René Lozi   Editors

Difference Equations, Discrete Dynamical Systems and Applications IDCEA 2022, Gif-sur-Yvette, France, June 18–22

Springer Proceedings in Mathematics & Statistics Volume 444

This book series features volumes composed of selected contributions from workshops and conferences in all areas of current research in mathematics and statistics, including data science, operations research and optimization. In addition to an overall evaluation of the interest, scientific quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most exciting areas of mathematical and statistical research today.

Sorin Olaru · Jim Cushing · Saber Elaydi · René Lozi Editors

Difference Equations, Discrete Dynamical Systems and Applications IDCEA 2022, Gif-sur-Yvette, France, June 18–22

Editors Sorin Olaru CentraleSupélec Gif-sur-Yvette, France

Jim Cushing University of Arizona Tucson, AZ, USA

Saber Elaydi Trinity University San Antonio, TX, USA

René Lozi University Côte d’Azur Nice, Parc Valrose, Cedex 2, France

ISSN 2194-1009 ISSN 2194-1017 (electronic) Springer Proceedings in Mathematics & Statistics ISBN 978-3-031-51048-9 ISBN 978-3-031-51049-6 (eBook) https://doi.org/10.1007/978-3-031-51049-6 Mathematics Subject Classification: 37, 39, 93 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Foreword

This book includes selected papers in direct relationship with the 27th International Conference on Difference Equations and Applications (ICDEA 2022), held at CentraleSupelec, University Paris-Saclay, France, in July 2022. ICDEA is the major event of the International Society of Difference Equations (ISDE) with its more than 1000 members, and brings together researchers and scientists from around the world, to present, discuss, and offer solutions in the fields of Difference Equations, Discrete Dynamical Systems, and their applications to various sciences as mathematical biology, epidemiology, evolutionary game theory, economics, physics, and engineering. In 2022, despite the uncertain situation worldwide, 170 researchers participated physically or online. The conference was organized around scientific talks distributed in 50 Regular/Special Sessions complemented by the program committee with 12 invited plenary lectures from experts in the field. The special sessions covered diverse topics as follows: • “Bifurcation in Invertible and Noninvertible Maps: Theory and Applications”— Organizers: Laura Gardini, Gian Italo Bischi and Iryna Sushko • “Stochastic and Non-Autonomous Difference Systems”—Organizers: Elena Braverman and Conall Kelly • “Nonlinear difference and differential problems, transformations, homogenization techniques and applications”—Organizers: Sandra Carillo, Galina Filipuk and Federico Zullo • “Difference Equations and their Applications in Biology”—Organizers: Azmy S. Ackleh and Amy Veprauskas • “Qualitative Behaviour of Nonautonomous Discrete Dynamical Systems”— Organizers: Davor Dragicevic, Adina Luminita Sasu, Weinian Zhang • “New Trends in Dynamic Geometry”—Organizers: Dorin Andrica, Ovidiu Bagdasar. All ICDEA 2022 participants were invited to submit papers for inclusion in this proceedings volume. After a rigorous peer review evaluation, 19 contributions were accepted by the editorial team. These papers are organized into four categories:

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Foreword

• General Theory of Difference Equations with five chapters dedicated to different recursive sequences, the characterization of solutions for difference equations and geometrical aspects of the iterative processes; • Discrete Dynamical Systems with five chapters dedicated to the topological studies, realization of difference equations over different restrictions but also to stability results for discrete dynamical systems; • Discrete-time models applied to engineering, biology, and economics which gathers four chapters underlying the importance of discrete-time models and the mathematical insight brought by their use in various fields of applications; • Control design techniques and numerical methods in relationship with discretetime models include five chapters with different contributions toward the practical use of discrete-time dynamics and structures in the modeling and control design and simulation. Control design based on discrete-time models represented a major topic for ICDEA 2022 since the conference was jointly organized with the Workshop on Control Applications of Optimization (CAO 2022). Some chapters in this volume have a tutorial style. They cover historical notes and recent developments for a particular topic. Other chapters concentrate on the latest research contributions of the author(s). These chapters range from the more technical articles on abstract systems to those that discuss the application of difference equations to real-world problems. Overall, the book is a valuable resource for young researchers entering the field of difference equations as well as for established scientists who wish to keep abreast of the latest developments in difference equations and discrete dynamical systems. Gif-sur-Yvette, France San Antonio, USA Tucson, USA Nice, France

Sorin Olaru Saber Elaydi Jim Cushing René Lozi

Contents

General Theory of Difference Equations Families of 6-Cycles of Third Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Antonio Linero Bas and Daniel Nieves Roldán About a System of Piecewise Linear Difference Equations with Many Periodic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inese Bula and Agnese S¯ıle Boundedness of Solutions of xn+1 = and yn+1 =

an +bn xn +cn yn An +Bn xn +Cn yn

3

29

an' +bn' yn Cn' xn

with Non-constant

Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zachary A. Kudlak and R. Patrick Vernon

51

On the Dynamic Geometry of Kasner Polygons with Complex Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dorin Andrica and Ovidiu Bagdasar

69

Linear Time-Varying Dynamic-Algebraic Equations of Index One on Time Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Svetlin G. Georgiev and Sergey Kryzhevich

85

Discrete Dynamical Systems Differentiable Conjugacies for One-Dimensional Maps . . . . . . . . . . . . . . . . 115 Paul Glendinning and David J. W. Simpson Topological Entropy of Generalized Bunimovich Stadium Billiards . . . . . 131 Michał Misiurewicz and Hong-Kun Zhang Global Manifolds of Saddle Periodic Orbits Parametrised by Isochrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 James Hannam, Bernd Krauskopf, and Hinke. M. Osinga

vii

viii

Contents

On Uniform Dichotomies for the Growth Rates of Linear Discrete-time Dynamical Systems in Banach Spaces . . . . . . . . . . . . . . . . . . 175 Rovana Boruga (Toma) Stability and Realization of Difference Equations Over Z and R . . . . . . . 189 Erik I. Verriest Discrete-Time Models Applied to Engineering, Biology and Economics Discrete Dynamical Systems in Economics: Two Seminal Models and Their Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 Gian Italo Bischi On a Class of Applications for Difference Equations in Continuous Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 Vladimir R˘asvan The Interplay Between Dispersal and Allee Effects in a Two-Patch Discrete-Time Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 Azmy S. Ackleh and Amy Veprauskas Krause Mean Processes Generated by Off-Diagonally Uniformly Positive Nonautonomous Stochastic Hyper-Matrices . . . . . . . . . . . . . . . . . . 303 Mansoor Saburov and Khikmat Saburov Control Design Techniques and Numerical Methods in Relationship with Discrete-Time Models Passivity Techniques and Hamiltonian Structures in Discrete Time . . . . . 327 Dorothée Normand-Cyrot, Salvatore Monaco, Mattia Mattioni, and Alessio Moreschini Explicit MPC Solution Using Hasse Diagrams: Construction, Storage and Retrieval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 Stefan ¸ S. Mihai, Florin Stoican, and Bogdan D. Ciubotaru Tube Model Predictive Control for Flexible Satellite Dynamics . . . . . . . . . 371 Sabin Diaconescu, Florin Stoican, and Bogdan D. Ciubotaru Numerical Modeling and Some Optimal Control Problems of Dynamic Systems Describing Contact Problems with Friction in Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 Nicolae Pop, Tudor Sireteanu, Luige Vladareanu, Mihaiela Iliescu, Ana-Maria Mitu, and Vicentiu Marius Maxim A Particular Solution for Higher Order Non-homogeneous Discrete Cauchy-Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 Miloud Assal and Skander Belhaj

General Theory of Difference Equations

Families of 6-Cycles of Third Order .

Antonio Linero Bas and Daniel Nieves Roldán

Abstract The present paper deals with the existence of .6-cycles of a certain type of difference equations of third order. In concrete, it focuses on equations of the form.xn+3 = xi g(x j )h(xk ), where.i, j, k ∈ {n, n + 1, n + 2} are pairwise distinct and .g, h : (0, ∞) → (0, ∞) are continuous. The main result of the paper assures that the ( )2 unique .6-cycle displaying such form is given by the potential one .xn+3 = xn xxn+2 . n+1 Keywords . p-cycle · Functional equations · Equilibrium points · Homeomorphism · Monotonicity

1 Introduction In general, for a set . X , a map . f : Ω ⊂ X k → X defined on some subset of a finite Cartesian product of . X , and .x1 , . . . , xk elements of . X (so-called initial conditions), we say that . x n+k = f (x n+k−1 , . . . , x n+1 , x n ), n ≥ 1, (1) is an autonomous difference equation of order.k. Notice that, once we have introduced the initial conditions, the recurrence constructed by. f gives a unique solution.(xn )∞ n=1 if that map. f is well(defined for ) any element.(xn+k−1 , . . . , xn+1 , xn ). When we obtain ¯ x, ¯ . . . we say that .x¯ is an equilibrium point of the difference a constant solution . x, ¯ . . . , x). ¯ equation. Notice that .x¯ satisfies the equation .x¯ = f (x, is a periodic solution if . x = x for all .n ≥ 1 and some Recall that .(xn )∞ n+m n n=1 positive integer .m. The smallest of such values is called the period of .(xn )∞ n=1 . If, additionally, all the solutions are periodic and . p is the least common multiple of their A. Linero Bas · D. Nieves Roldán (B) Departamento de Matemáticas, Universidad de Murcia (Spain), Murcia, Spain e-mail: [email protected] A. Linero Bas e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 S. Olaru et al. (eds.), Difference Equations, Discrete Dynamical Systems and Applications, Springer Proceedings in Mathematics & Statistics 444, https://doi.org/10.1007/978-3-031-51049-6_1

3

4

A. Linero Bas and D. Nieves Roldán

periods, we say that the difference equation is a . p-cycle or is globally periodic of period . p. n , a .5-cycle. Maybe, the most popular . p-cycle is the Lyness’ cycle .xn+1 = 1+x xn−1 It was already known, in an implicit way, by Gauss when working in the spherical geometry of the pentagramma mirificum, a spherical pentagram formed by five successively orthogonal great-circle arcs. Its construction and the relation with the .5-cycle can be consulted in [9]. Let us comment that this.5-cycle receives the name of Lyness because R.C. Lyness accounted for it in a series of papers dealing with the existence of cycles, see [14–16]. Related to the study of global periodicity, different approaches are employed. For instance, techniques of discrete dynamical systems [3, 6, 7]; the resolution of functional equations [2, 17]; and even direct arguments of real analysis [18]. Moreover, we must highlight the problems of existence of families of . p-cycles and the classification of . p-cycles by conjugation. The main families of . p-cycles that can be found in the literature are rational cycles [8, 10]; and potential cycles [5]. For the classification by conjugation let us recall that, given a metric space . X , we can associate to the difference Eq. (1), the following map . F : X k → X k given by .

( ) F(x1 , . . . , xk ) = x2 , . . . , xk , f (xk , . . . , x2 , x1 ) .

(2)

So, if . X 1 , X 2 are two metric spaces and .g1 : X 1 → X 1 , .g2 : X 2 → X 2 , are two continuous maps, we way that .(X 1 , g1 ) and .(X 2 , g2 ) are topologically conjugate if there exists a homeomorphism .h : X 1 → X 2 such that .h ◦ g1 = g2 ◦ h. So, if Eq. (1) homeomorphism .α defined from is a . p-cycle of order .k, and we consider an arbitrary ( ) .(0, ∞) into itself, then . F(x 1 , x 2 , . . . , x k ) = x 2 , . . . , x k , f (x k , . . . , x 1 ) is topologically conjugate to .

( ) G(x1 , x2 , . . . , xk ) = x2 , . . . , xk , α −1 ( f (α(xk ), . . . , α(x1 ))) .

Therefore if .xn+k = f (xn+k−1 , . . . , xn ) is a . p-cycle, then x

. n+k

= α −1 ( f (α(xn+k−1 ), . . . , α(xk )))

is also a . p-cycle. For instance, all the . p-cycles of second order .xn+2 = f (xn+1 , xn ), with . p ≥ 3, are topologically conjugate to rotations of the plane [3]. )2 ( In [13], we proved that .xn+3 = xn xxn+2 is the unique potential .6-cycle of the n+1 form .xn+3 = xi f (x j , xk ), with .i, j, k ∈ {n, n + 1, n + 2} pairwise distinct and . f : (0, ∞)2 → (0, ∞) continuous. This means that the cycle must have the form .xn+3 = α xi x j j xkαk , where .α j and .αk are real numbers. Here, we go further and prove that, in fact, is the unique .6-cycle of third order exhibiting the form x

. n+3

= xi g(x j )h(xk ),

(3)

Families of 6-Cycles of Third Order

5

with .i, j, k ∈ {n, n + 1, n + 2}, pairwise distinct, and .g, h : (0, ∞) → (0, ∞) continuous. The main goal of this paper is the proof of the following result, which will follow directly from below Theorems 2, 3 and 4. Theorem 1 Let .g, h : (0, ∞) → (0, ∞) be continuous maps. Then, the unique .6cycle of the form . x n+3 = x i g(x j )h(x k ), where .i, j, k ∈ {n, n + 1, n + 2} are pairwise distinct, is given by ( x

. n+3

= xn

xn+2 xn+1

)2 .

It is worth mentioning that this gives a new proof for the location of .6-cycles different from that presented in [4, Theorem 3], where the proof was based on an incorrect result. In concrete, Lemma 1 [1, Lemma 3.2.] Suppose that every solution of Eq. (1) is periodic of prime period . p. If . S : D p → D is continuous and symmetric, where . D is a non-degenerate interval of real numbers, then the function I (x1 , x2 , . . . , xk ) = S(y p−k , y p−1−k , . . . , y1−k ),

. S

where .(yn )∞ n=−k+1 is a solution of Eq. (1) with . y0 = x 1 , . . . , y−k+1 = x k , is an invariant of Eq. (1) such that . I S (xk , . . . , x2 , x1 ) = I S (x1 , x2 , . . . , xk ). The problem with such statement resides in the fact that the equality I (xk , . . . , x2 , x1 ) = I S (x1 , x2 , . . . , xk )

. S

is not consistent with the form which . I S is defined. For instance, let us consider a .3-cycle of order .k = 2, .xn+2 = f (xn+1 , xn ), and take the symmetric function . S(x1 , x2 , x3 ) = x1 + x2 + x3 . Notice that we can choose .x1 , x2 so that .x3 = f (x2 , x1 ) /= f (x1 , x2 ) = x˜3 , since . f is not symmetric. Hence, . I S (x 1 , x 2 )

= S(x1 , x2 , x˜3 ) = x1 + x2 + x˜3 / = x1 + x2 + x3 = S(x1 , x2 , x3 ) = I S (x2 , x1 ).

The paper is organized as follows. In Sect. 2 we present some general considerations concerning the set of equilibrium points of Eq. (3) and the value of .g(1) and .h(1). These considerations will apply for each of the possible combinations of indices .i, j, k. In Sect. 3 we discuss the case .i = n + 2, j = n + 1, k = n and we show that there are no .6-cycles, see Theorem 2. The same occurs for the combination .i = n + 1, j = n + 2, k = n, as we will see in Theorem 3 of Sect. 4. Nevertheless, in Sect. 5 we will find in Theorem 4 the unique .6-cycle of the form enunciated in

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A. Linero Bas and D. Nieves Roldán

Theorem 1. In each of the sections, the used tools will be the analysis of the sets of equilibrium points and the solution of appropriate functional equations. Finally, in Sect. 6 we state some open problems and forthcoming lines of research related to the topic.

2 General Considerations In this section we are going to present two general considerations about the set of equilibrium points of Eq. (3) which will be useful in the development of each one of the possible combinations for such equation. Unless otherwise stated, we will assume that Eq. (3) is a .6-cycle. Observe that .x is an equilibrium point of this equation if and only if.x = xg(x)h(x), i.e..g(x)h(x) = 1. This allows us to describe its set of equilibrium points by the closed set F = {x > 0 : g(x)h(x) = 1}.

.

We prove here the non-emptiness character of .F. In the proof, .Id|(0,∞) represents the identity map from .(0, ∞) into itself. Lemma 2 .F /= ∅. Proof Notice that belonging to .F is equivalent to being a fixed point of the discrete dynamical system .φ associate to the difference equation, namely ( ) Φ : (0, ∞)3 → (0, ∞)3 , Φ(x1 , x2 , x3 ) = x2 , x3 , xi g(x j )h(xk ) .

.

Since the difference equation is a . p-cycle, . p = 6, then .Φ 6 ≡ Id|(0,∞) , so .Φ is a periodic homeomorphism. In that case, being .Φ defined in a space homeomorphic to .R3 , it is well-known (see [11], where the reader will find a brief explanation on the fact that any periodic homeomorphism on .Rn has fixed points if .n ≤ 4) that .Φ possesses necessarily a fixed point, which means that the initial difference equation ◻ has an equilibrium point, and .F /= ∅. Remark 1 The emptiness character of .F in the previous lemma is crucial for our study. In general, a periodic homeomorphism, . F : Rk → Rk , does not have necessarily fixed points (for instance, for an example in .R7 , consult [11]). Nevertheless, all the known . p-cycles in the literature present equilibrium points or, equivalently, its associate discrete dynamical systems have fixed points. It is still an open problem to determine whether a . p-cycle has necessarily fixed points. For a scope of the problem, see [12]. Concerning the value of the period . p of the cycle, if the order .k of the difference equation .xn+k = f (xn+k−1 , . . . , xn+1 , xn ) is .k = 1, necessarily . p = 1 or . p = 2 (the identity or an involution in the real line, for instance, .xn+1 = xn or .xn+1 = x1n for .(0, ∞)); when .k ≥ 2, the period . p can reach the value of any positive integer,

Families of 6-Cycles of Third Order

7

to see it, take for instance a linear homogeneous difference equation of order .k whose characteristic polynomial have . pth-roots of the unity as eigenvalues, and select them by conjugated ( 2π ) if necessary .1 or .−1 (for instance, for ( ) pairs, adding + i · sin , .λ2 = λ1 , we form the difference equathe .7th-roots .λ1 = cos 2π 7 ( ) 7 tion .xn+2 = 2 cos 2π x − x , a . 7-cycle). . ◻ n+1 n 7 Remark 2 Observe that, without loss of generality, we can assume that .1 ∈ F and g(1) = h(1) = 1. Indeed, since .F / = ∅, let .z 0 ∈ F and, firstly, write Eq. (3) as

.

( x

. n+3

= xi

) ) g(x j ) ( g(z 0 )h(xk ) =: xi g(x j )h(xk ). g(z 0 )

In this case, we have .g(z 0 ) = 1 = h(z 0 ) since .g(z 0 )h(z 0 ) = 1. Secondly, apply the transformation .x = z 0 y to obtain z y

. 0 n+3

yn+3

( ) = z 0 yi g z 0 y j h (z 0 yk ) , ) ( ) g(z 0 y j ) ( = yi g(z 0 )h(z 0 yk ) =: yi γ (y j )η(yk ). g(z 0 )

It is easily seen that this new equation continues to be a.6-cycle, and.γ (1) = η(1) = 1. Thus, unless otherwise stated, in the next sections we will assume that.g(1) = h(1) = ◻ 1, .1 ∈ F. .

3 The Case . xn+3 = xn+2 g(xn+1 )h(xn ) In order to prove that the difference equation x

. n+3

= xn+2 g(xn+1 )h(xn )

(4)

cannot be a .6-cycle, firstly we present some general necessary properties for the equation to be a .6-cycle, related with the fact that .h must be a homeomorphism and with the set of equilibrium points .F associate to the equation. Secondly, we will divide our study in several steps, depending on the cardinality of .F (.≥ 2 or .1) and the monotonic character of .h. The complete analysis will be collected in Theorem 2. Lemma 3 Suppose that Eq. (4) is a .6-cycle. Then .h is a homeomorphism. Proof First, we prove that .h is surjective. Given a value . y > 0, take the initial conditions .x1 = 1, x2 = 1, x3 = yg(1). By the global periodicity, we obtain a periodic sequence of period a divisor of .6. In particular, . yg(1) = x3 = x9 = x8 g(x7 )h(x6 ) = x2 g(x1 )h(x6 ) = g(1)h(x6 ). Then, .h(x6 ) = y which proves the surjectivity of .h. To prove the injectivity, suppose that .h(x) = h(y) for some values .x, y > 0. Consider, on the one hand, the initial conditions .x1 = x, x2 = x3 = 1, and on the

8

A. Linero Bas and D. Nieves Roldán

other hand, . y1 = y, y2 = y3 = 1. If .h(x) = h(y) it is easily seen that .xn = yn for all n ≥ 2. In particular, .x = x1 = x7 = y7 = y1 = y, so .x = y. ◻

.

Remark 3 Notice that Lemma 3 implies that .h is strictly monotonic. In the development of our discussion, the following properties on the set of equilibrium points .F will be essential for the case in which .Card (F) ≥ 2. Lemma 4 Assume that Eq. (4) is a .6-cycle. Let .u, x ∈ F, with .u /= x, u < x. Then: ( ) (a) .x = ug(u)h xg(x)h(u) . (b) .ug(u)h(x) ∈ F. In fact, if we put .x0 = x, then the sequence .(xk )k defined recur/ {x, u}. sively as .xk+1 = ug(u)h(xk ), k ≥ 0, is contained in .F. Moreover, .x1 ∈ (c) .xg(x)h(u) ∈ F. In fact, if we put .u 0 = u, then the sequence .(u k )k defined recur/ {x, u}. sively as .u k+1 = xg(x)h(u k ), k ≥ 0, is contained in .F. Moreover, .u 1 ∈ h(xn+1 ) g(xn ) = for all .n ≥ 0. (d) It holds . xun = g(x h(xn ) n+1 ) (e) If .h is increasing, .x1 , u 1 ∈ (u, x). (f) If .h is decreasing, then .x1 < u < x2 < x and .u < u 1 < x < u 2 , where .x1 , x2 , u 1 , u 2 are defined in Part (b) and Part (c). (g) If .h is decreasing, then .inf F = 0 and .sup F = ∞. In fact, .F = (0, ∞). Proof (a) Set .z 1 = u, z 2 = u, z 3 = x to obtain (recall that .g(z)h(z) = 1 if .z ∈ {x, u}): .z 4 = x, z 5 = xg(x)h(u), z 6 = xg(x)h(u). From the global periodicity, .

( ) x = z 3 = z 9 = z 8 g(z 7 ) h(z 6 ) = z 2 g(z 1 ) h(z 6 ) = u g(u) h xg(x)h(u) .

(b) Take .z 1 = x, z 2 = x, z 3 = u. Then ug(u)h(x). Again, by global periodicity we find .

z = u, z 5 = ug(u)h(x), z 6 =

. 4

( ) ( ) x = z 2 = z 8 = z 7 g(z 6 )h(z 5 ) = z 1 g(z 6 )h(z 5 ) = xg ug(u)h(x) h ug(u)h(x) .

( ) ( ) Then .1 = g ug(u)h(x) h ug(u)h(x) , which implies .ug(u)h(x) ∈ F. If we repeat the process with the points .x1 := ug(u)h(x) and .u, we get an element of .F, namely .x2 := ug(u)h(x1 ). By induction it is a simple matter to see that .xk+1 defined recursively as .xk+1 = ug(u)h(xk ), k ≥ 0, is a point in .F. / {x, u}. Suppose that .ug(u)h(x) = x1 = x. ReturnWe proceed to prove that .x1 ∈ ing to the above sequence generated (by .z 1 = x,)z 2 = x, z 3 = u, we would find .u = z 3 = z 9 = z 8 g(z 7 )h(z 6 ) = xg(x)h ug(u)h(x) = xg(x)h(x) = x, so .u = x, a contradiction. Suppose that .ug(u)h(x) =(x1 )= u. Now, the initial conditions .z j , . j = 1, 2, 3, would generate the sequence . z j j≥1 = (x, x, u, u, u, u, . . .), consequently .u = x by global periodicity, a new contradiction. (c) Its proof is completely analogous to Part (b). Now it suffices to consider initial conditions .u, u, x. (d) Let.x0 = x and.xn+1 = ug(u)h(xn ) for all.n ≥ 0. We have seen in (a) that, from the initial terms .x, x, u, we deduce that .u = xg(x)h(x1 ). Thus, .x1 = ug(u)h(x) =

Families of 6-Cycles of Third Order

9

(

) ( ) xg(x)h(x1 ) g(u)h(x) = xh(x1 )g(u) g(x)h(x) = xh(x1 )g(u) and .uh(x) = h(x1 ) u xh(x1 ). Therefore, . x = h(x) . Since .x j ∈ F, . j = 0, 1, then .h(x j ) = g(x1 j ) , j = 0, 1, and we obtain u h(x1 ) g(x0 ) = . = . x0 h(x0 ) g(x1 )

Now, if we do the same reasoning with the initial conditions .xn , xn , u, .n ≥ 1, analogously u h(xn+1 ) g(xn ) . = = . xn h(xn ) g(xn+1 ) (e) Suppose that .h is increasing. Since .u < x, we have .x1 = ug(u)h(x) ≥ ug(u)h(u) = u. Since .x1 /= u, we find .x1 > u. To see now that .x1 < x, assume on the contrary that .u < x < x1 = ug(u)h(x) (notice that .x1 /= x). In this case, by Part (a), interchanging the roles of .(u and .x, and ) using that .h is increasing with .u < x < x 1 , we obtain .u = xg(x)h ug(u)h(x) = xg(x)h(x 1 ) ≥ xg(x)h(x) = x, so .u > x (recall .u /= x), a contradiction. Therefore, .u < x1 < x. The proof of .u < u 1 < x is completely analogous and we omit it. (f) Suppose that .h is decreasing. Firstly, we claim that .x2 /= u. Indeed, otherwise, .u = x 2 = ug(u)h(x 1 ) would imply .g(u)h(x 1 ) = 1, and from Part (d) we would find 1

g(u) 1) = h(x = h(x) , that is, .ug(u)h(x) = x. Being .ug(u)h(x) = x1 , we would obtain h(x) . x 1 = x = u, a contradiction. Next, since.u < x, the value.x1 satisfies.x1 = ug(u)h(x) ≤ ug(u)h(u) = u, hence . x 1 < u < x (we use that . x 1 / = u). Next, take into account that . x 2 = ug(u)h(x 1 ), with . x 1 < u < x to deduce that . x 2 ≥ ug(u)h(u) = u (recall that . x 2 / = u). Moreover, by force .x2 < x, because otherwise, if .x ≤ x2 , from Part (d) and the monotonicity of .h h(x1 ) 2) ≤ h(x = 1, a contradiction. we would obtain .1 < xu1 = h(x h(x1 ) 1) (g) Firstly, let us see that .inf F = 0 and .sup F = ∞. Arguing by contradiction, if .inf F = a > 0, due to the continuity of the functions .g and .h, .a = min F ∈ F. So, we could apply the same reasoning as in part (f) with .u = a < x and we would get values of .F that are smaller than .a, arriving to a contradiction. A similar reasoning applies to see that .sup F = ∞. Therefore, for any two values in .F, by .( f ) with suitable values .u, x, there exists another value of .F between them. Thus, by density ◻ we obtain .F = (0, ∞).

.

u x

3.1 The Case .Card (F ) ≥ 2 To prove that this case does not provide.6-cycles of the form of Eq. (4), we distinguish two cases, according to the monotonicity of .h. The case .Card (F) ≥ 2, .h increasing: In the increasing case, we will use the general properties of .F established in Lemma 4 to derive the non-existence of .6cycles. Proposition 1 Consider Eq. (4) and assume that .h is an increasing homeomorphism with .Card(F) ≥ 2. Then, the difference equation cannot be a .6-cycle.

10

A. Linero Bas and D. Nieves Roldán

Proof Let .x, y be two different points in .F, with .x < y. By Lemma 4-(b)-(c)-(e), respectively, we know that .xg(x)h(y) and . yg(y)h(x) belong to .F, they are different from .x, y, and they are in the interior of the subinterval having endpoints .x, y. Since .h is increasing and < y, from the equality( established) in ( .x < yg(y)h(x) ) Lemma 4-(a),. y = x g(x) h yg(y)h(x) , we deduce that. y = x g(x) h yg(y)h(x) < x g(x) h(y), therefore . y < xg(x)h(y), which contradicts our previous observation ◻ on the location of the value .xg(x)h(y) in the open interval .(x, y). The case .Card (F) ≥ 2, .h decreasing: When .h is decreasing, we will present a more elaborated proof lying in the study of certain functional equation. Firstly, by Lemma 4-(g), .F = (0, ∞). This means that Eq. (4) can be formulated as .xn+3 = h(xn ) n+1 ) or .xn+3 = xn+2 h(x . We are going to prove that there are no .6-cycles xn+2 g(x g(xn ) n+1 ) displaying this form. To see it, we argue by contradiction, by supposing that the difference equation is a.6-cycle. Recall that we are supposing that.1 ∈ F, with.g(1) = h(1) = 1. Lemma 5 The functional equation ( .

x =ϕ

) x , x > 0, ϕ(x)

(5)

has no solutions in the family of continuous maps defined from .(0, ∞) into itself. Proof Let .ϕ : (0, ∞) → (0, ∞) be a solution of (5). If .ϕ satisfies the functional equation, obviously .Im(ϕ) = (0, ∞). In particular, .1 ∈ Im(ϕ). Next, we claim that −1 .ϕ(1) = 1, so.ϕ(x) / = x if. x / = 1. To see it, being.ϕ ({1}) /= ∅, suppose that.ϕ(z) = 1 ) ( ( ) z z for some .z > 0. Then, .z = ϕ ϕ(z) = ϕ 1 = ϕ(z) = 1, which implies that .z = 1. This ends the claim. According to the above properties, we can distinguish three possibilities for the solution of (5) (the cases .ϕ(x) < 1 for all .x /= 1 and .ϕ(x) > 1 for all .x /= 1 are rejected due to the fact that .Im(ϕ) = (0, ∞)): x (i) .x < ϕ(x) < 1 (see Fig. 1). Here, . ϕ(x) < 1 if .x < 1 and, according to the case, ) ( x x x . Therefore, .ϕ(x) > 1 for all with .x˜ = ϕ(x) , we deduce .x = ϕ ϕ(x) > ϕ(x) . x < 1, a contradiction. (ii) .ϕ(x) < x < 1 (see Fig. 1). Now, necessarily .ϕ(z) > z if .z > 1, otherwise we would ( have, ) from the initial claim, that .ϕ(w) < w for all .w /= 1 and then .w = w w ϕ ϕ(w) < ϕ(w) , so .ϕ(w) < 1 for all .w /= 1, a contradiction, since Im.(ϕ) = ) ( z z < 1, (0, ∞). Then, if .z > 1 we have . ϕ(z) < 1 and, by hypothesis, .z = ϕ ϕ(z) a contradiction. x < 1 for all .x < 1, and (iii) .ϕ(x) > 1 for all .x < 1 (see Fig. 1). In that situation, . ϕ(x) ) ( x consequently, .x = ϕ ϕ(x) > 1, a new contradiction. ◻

Families of 6-Cycles of Third Order

11

Fig. 1 The solution .ϕ, Cases (i)–(ii) .ϕ(x) < 1 and Case (iii) .ϕ(x) > 1 if .x < 1

Corollary 1 The functional equation ( .

x =ϕ

) x , x > 0, ϕ(x)

(6)

has no solutions in the family .H of homeomorphisms .ϕ from .(0, ∞) into itself. Proposition 2 If .Card (F) ≥ 2 and .h is a decreasing homeomorphism, then Eq. (4) is not a .6-cycle. 1 Proof Recall that .F = (0, ∞), .g(x) = h(x) for all .x > 0 and .g(1) = h(1) = 1. Consider the initial conditions .x1 = 1, x2 = 1, x3 = x, with .x > 0 arbitrarily x and .x6 = taken. Then, .x4 = x3 g(x2 )h(x1 ) = x, .x5 = x4 g(x3 )h(x2 ) = xg(x) = h(x) x x x5 g(x4 )h(x3 ) = h(x) g(x)h(x) = h(x) . From the global periodicity and the fact that ) ( x , .F = (0, ∞), we obtain . x = x 3 = x 9 = x 8 g(x 7 )h(x 6 ) = x 2 g(x 1 )h(x 6 ) = h h(x) that is, ) ( x for all x > 0. (7) .x = h h(x)

Finally, Corollary 1 gives the non-existence of solutions for this equation, therefore ◻ Eq. (4) cannot provide a .6-cycle. Remark 4 We emphasize that the last result also works if .h is increasing and F = (0, ∞). Nevertheless, we have only payed attention to the decreasing case, because in the increasing case we would need to demonstrate previously that the existence of two different points in .F imply that really .F = (0, ∞). Another reason to separate the increasing and decreasing cases into two propositions, Proposition 1 and Proposition 2, is to present two perspectives to attack the problem, the second one related with the resolution of functional equations. Maybe, the decreasing case could be studied with the same strategy applied to the increasing case, following the ideas in Lemma 4.

.

12

A. Linero Bas and D. Nieves Roldán

3.2 The Case .Card (F ) = 1 Suppose that .F is a singleton, say .F = {z 0 }. Hence, .g(z 0 )h(z 0 ) = 1. Recall that, without loss of generality, we can assume that .z 0 = 1 and .g(1) = h(1) = 1, as we stated at the end of Sect. 2. Lemma 6 Under the above conditions, if Eq. (4) is a .6-cycle, then for all .x > 0 it holds: ( )2 (a) .h −1 (x) = x g(x) h(x). (b) .h −1 (x) = ( 1 ) . ) ( g xg(x) h(x) ( ) 1) ( h xg(x) . (c) .1 = g g xg(x) h(x)

Proof Since .h is a homeomorphism by Lemma 3, given an arbitrary value .x > 0, we take the initial conditions .x1 = h −1 (x), x2 = 1, x3 = 1 and apply Eq. (4) to obtain . x 4 = x, x 5 = x, x 6 = xg(x). Then by the global periodicity, .

( )2 h −1 (x) = x1 = x7 = x6 g(x5 )h(x4 ) = x g(x) h(x),

( ) 1 = x2 = x8 = x7 g(x6 )h(x5 ) = x1 g(x6 )h(x5 ) = h −1 (x)g xg(x) h(x), ( ) ( ) 1 = x3 = x9 = x8 g(x7 )h(x6 ) = x2 g(x1 )h(x6 ) = g h −1 (x) h xg(x) ,

which are precisely the three formulas of the statement.

◻

Lemma 7 Let Eq. (4) be a .6-cycle. Suppose that .F = {1}. Then: ( )2 1 (a) .x ∈ Fix(h) if and only √ if . g(x) = x . (b) If .x ∈ Fix(h), then . x ∈ Fix(h). Moreover, .x = 1 is the unique fixed point of .h. Proof (a) This follows as a direct consequence of Lemma 6-(a). (b) Let .x ∈ Fix(h), so .h(x) = x = h −1 (x). By Lemma 6-(b), we deduce that 1 ) , or .x = ( g xg(x) x ( ) 2 . x g xg(x) = 1. (8) / ( / ) By Part (a), .g(x) = x1 . Replacing this equality into (8) yields .x 2 g x x1 = 1, that √ is, .g( x) = x12 . Next, we use Lemma 6-(c) to obtain (

) ) ( (√ ) 1 1 ( ) (√ ) .1 = g h x h (xg(x)) = g g xg(x) h(x) g x h(x) ( ) / (√ ) (√ ) 1 1 (√ ) h x . =g 1 h x = g (x) h x = x x x2

Families of 6-Cycles of Third Order

13

(√ ) √ √ Therefore, .h x = x and . x ∈ Fix(h). Finally, suppose that .x ∈ Fix(h) and prove that .x = 1. By Part (a), .g(x) = √1x , √ √ and also . x ∈ Fix(h). In particular .g( x) = √1√ . Furthermore, we have seen that x √ 1 1 1 8 .g( x) = 2 . We then deduce that . √√ = 2 , or simplifying, . x = x . The unique x x x

positive real solution of this equation is .x = 1.

◻

Lemma 8 Let Eq. (4) be a .6-cycle. Suppose that .F = {1}. Then, for all .x > 0, 1 .h(x) = g(x)

/

1 ( ), xg xg(x)

(

and .

xg(x) = h

−1

1

(9)

)

( ) . g h −1 (x)

(10)

( )2 Proof Set .x1 = 1, x2 = 1, x3 = x. Then .x4 = x, x5 = xg(x), x6 = x g(x) h(x). By global periodicity, ( )2 ( ) 1 = x1 = x7 = x6 g(x5 )h(x4 ) = x g(x) h(x)g xg(x) h(x),

.

hence, we conclude that .h(x) = On the other hand,

1 g(x)

/

1 . xg(xg(x))

( ) ( ) 1 = x2 = x8 = x7 g(x6 )h(x5 ) = x1 g(x6 )h(x5 ) = g x(g(x))2 h(x) h xg(x) ,

.

( and since .h is a homeomorphism, .xg(x) = h −1 ( ) Lemma 6-(a), we obtain .xg(x) = h −1 g h −11 (x) . ( )

(

1

) ) . Finally, by

g x(g(x))2 h(x)

◻

From now on, since .1 is the unique fixed point of .h, we distinguish three cases, namely: (a) .h is increasing, with .h(x) > x if .0 < x < 1. (b) .h is increasing, with .h(x) < x if .0 < x < 1. (c) .h is decreasing. In the above cases, we will use that .Card (F) = 1, with .F = {1}, therefore for the rest of values .x /= 1 it must be either .g(x)h(x) < 1 for all .x ∈ (0, 1) or the reverse inequality .g(x)h(x) > 1 for all .x ∈ (0, 1). Case (a): .h increasing, .h(x) > x if .0 < x < 1 (see Fig. 2). Suppose that Eq. (4) is a .6-cycle.

14

A. Linero Bas and D. Nieves Roldán

Fig. 2 Case .h increasing, > x if .0 < x < 1

.h(x)

(a.1) Let .h(x)g(x) > 1, for all .x ∈ (0, 1). According to Lemma 6-(a), .h −1 (x) = )2 x g(x) h(x), and the symmetry of .h −1 and .h with respect to the diagonal . y = x, we have (

.

( ) x > h −1 (x) = x (g(x))2 h(x) = xg(x) g(x)h(x) > xg(x),

that is, .g(x) < 1 if .0 < x < 1. This implies that .h(x) > 1 for all .x ∈ (0, 1), a contradiction. So, we reject this case. (a.2) Let .h(x)g(x) < 1, for all .x ∈ (0, 1). Reasoning as in Case (a.1), we find ( )2 −1 .h (x) = x g(x) h(x) < xg(x). On the other hand, being .h(x) > x in .(0, 1) we ( )2 ( )2 also deduce that .h −1 (x) = x g(x) h(x) > x 2 g(x) . From the two inequalities for .h(−1 (x) we ) conclude that .xg(x) < 1 in .(0, 1). ( Since,) in addition, .xg(x) = h −1 g(h −11 (x)) (see Eq. (10)), we deduce that .h −1 g(h −11 (x)) < 1 for all .x ∈ (0, 1), which implies that . g(h −11 (x)) < 1 or .g(h −1 (x)) > 1 in .(0, 1). Being .h −1 |(0,1) a homeomorphism from .(0, 1) into itself, to obtain .g(w) > 1 for all .w ∈ (0, 1). Thus, . x < xg(x) < 1 in .(0, 1). −1 .h (x) < 1, if .0 < x < 1 we have By Lemma 6-(b), and our ( hypothesis ) 1 . < 1, that is .1 < g xg(x) h(x). Finally, from the monotonicity of .h g(xg(x))h(x) ( ) ( ) ( ) applied to .x < xg(x) < 1 we obtain .1 < g xg(x) h(x) < g xg(x) h xg(x) < 1 due to our assumption on the product .h(z)g(z) < 1 in .(0, 1). We derive a contradiction, thus we must also reject Case (a.2). We summarize our above study in this result:

Proposition 3 In Eq. (4), suppose that .h is increasing, with .h(x) > x if .0 < x < 1, and that the set of equilibrium points reduces to .F = {1}. Then, Eq. (4) cannot be a .6-cycle. Case (b):.h increasing,.h(x) < x if.0 < x < 1. In this part, we suppose that Eq. (4) is a .6-cycle, with .F = {1}, .h increasing and .h(x) < x < h −1 (x) < 1 if .0 < x < 1 (interchange the roles of .h and .h −1 in Fig. 2). Again, since .F ∩ (0, 1) = ∅, we have to discuss two cases, namely, (b.1) .h(x)g(x) > 1 for all .x ∈ (0, 1) or (b.2) .h(x)g(x) < 1 for all . x ∈ (0, 1).

Families of 6-Cycles of Third Order

15

(b.1) Let.g(x)h(x) > 1 for all.x ∈ (0, 1). Since.h(x) < 1, we deduce that.g(x) > 1 in .(0, 1). Also, by Lemma 6-(a) and the monotonicity of .h, .

h −1 (x) x 1 x > h(x) = ( )2 , )2 > ( )2 = ( g(x) x g(x) x g(x)

( )2 hence .x g(x) > 1. At the same time, we also find .xg(x) > h(x)g(x) > 1, and consequently by Eq. (10), ( .

xg(x) = h

−1

1

( ) g h −1 (x)

) > 1,

( ) that implies .g h −1 (x) < 1 for all .0 < x < 1. Since .h −1 is an increasing homeomorphism with .h(1) = 1, putting .w = h −1 (x) we conclude that .g(w) < 1 for all .w ∈ (0, 1), contrary to our initial hypothesis. Therefore, Case (b.1) does not generate .6-cycles of the form of (4). (b.2) Suppose that .g(x)h(x) < 1 for all .x ∈ (0, 1). Since .x < h −1 (x) = ( ) ( )2 x g(x) h(x) = xg(x) g(x)h(x) < xg(x) (we have applied Lemma 6-(a)), we obtain that .g(x) > 1 in .(0, 1). If,( additionally, . xg(x) > 1 for some . x < 1, then from Eq. (10) we get that ) 1 −1 > 1. Hence we obtain .g(h −1 (x)) < 1. Setting .w = h −1 (x), with .h g(h −1 (x)) .w < 1, we deduce that .g(w) < 1, a contradiction with the fact that .g(w) > 1 in .(0, 1). On the other hand, if .xg(x) < 1 for some .x < 1, then we have .x < xg(x) < 1 and from Lemma 6-(b), i.e. .1 > h −1 (x) = ( 1 ) , we deduce that g xg(x) h(x) ( ) .g (xg(x)) h(x) > 1. Then apply that .h is increasing to arrive to .1 < g xg(x) h(x) < ( ) ( ) g xg(x) h xg(x) < 1, a contradiction. We conclude that .xg(x) = 1 for all .x ∈ 1 [0, 1]. Then, Lemma 6-(b) implies .h −1 (x) = h(x) , and therefore, .h(x)h −1 (x) = 1 for all .x ∈ [0, 1], which is impossible since .h(x) < 1 and .h −1 (x) < 1 for all .x ∈ [0, 1]. To sum up, Case (b.2) does not produce .6-cycles of the form (4). Our discussion can be collected in this result: Proposition 4 In Eq. (4), suppose that .h is increasing, with .h(x) < x if .0 < x < 1, and that the set of equilibrium points reduces to .F = {1}. Then, Eq. (4) cannot be a .6-cycle. Case (c):.h decreasing. Next, we suppose that Eq. (4) is a.6-cycle, with.F = {1}, .h decreasing (see Fig. 3). Since.F ∩ (0, 1) = ∅, it is necessary to distinguish two cases, (c.1) .h(x)g(x) < 1 for all .x ∈ (0, 1) or (c.2) .h(x)g(x) > 1 for all .x ∈ (0, 1). As a preliminary, notice that if .h is decreasing, by the symmetry of the inverse map .h −1 with respect to the diagonal, we know that .h −1 is also decreasing with .h −1 (x) > 1 if .x < 1.

16

A. Linero Bas and D. Nieves Roldán

Fig. 3 Case .h decreasing, > 1 if .0 < x < 1 and .h(x) < 1 if . x > 1 .h(x)

(c.1) Here, by force, .g(x) < 1 for all .x ∈ (0, 1). Then .xg(x) < x < 1 in .(0, 1). We claim that .g(z) < 1 for all .z /= 1. It only remains to see that .g(z) < 1 for all . z > 1. By ( Eq. (10)) and the fact that .xg(x) < 1 in .(0, 1), we have .1 > xg(x) = h −1

(

1

)

for all .0 < x < 1. Then, being .h decreasing,

g h −1 (x)

.

1 g(h −1 (x))

> 1 or

g(h −1 (x)) < 1 for all .x ∈ (0, 1). Since .h −1 |(0,1) is a homeomorphism from .(0, 1) to .(1, ∞), setting .w = h −1 (x) we have .g(w) < 1 for all .w > 1. This ends the claim. Next, we distinguish the following cases:

.

– .h(z)g(z) > 1 in .(1, ∞). Then, according to the above claim, necessarily .h(z) > 1 for all .z > 1, a contradiction (see Fig. 3). −1 – .h(z)g(z) < 1 in (.(1, ∞). ) Use Lemma 6-(b) to any .z > 1 to obtain( .1 > h) (z) = 1 ( ) , so .g zg(z) h(z) > 1. As .h(z) < 1, we deduce that .g zg(z) > 1 for g zg(z) h(z)

each .z > 1, which is impossible because .g(u) ≤ 1 for all .u ∈ (0, ∞). Therefore, Case (c.1) derives in a contradiction and we reject this case. (c.2) Let .h(x)g(x) > 1 for all .0 < x < 1. Notice that if .ug(u) = 1 for some 1 −1 .u < 1, then from Lemma 6-(b) we obtain .h (u) = ( 1 ) = h(u) , therefore g ug(u) h(u)

h(u) · h −1 (u) = 1, which is impossible. As a consequence, either .xg(x) > 1 for all .0 < x < 1 or . xg(x) < 1 for all .0 < x < 1. .

(.★) Suppose that .xg(x) > 1 for all .0 < x < 1. By force, ( .g(x) >) 1 in .(0, 1). −1 ( 1 ) , it follows On the other hand, from Eq. (10), .1 < xg(x) = h g h −1 (x) ( ) that . ( −11 ) < 1, so .g h −1 (x) > 1 for all .x < 1. Thus, being the restricg h

(x)

tion .h −1 |(0,1) a homeomorphism from .(0, 1) onto .(1, ∞), it implies that .g(w) > 1 for all .w > 1 and consequently .g(u) ≥ 1 for all .u > 0. Finally, use Lemma 6-(b), .h −1 (x) = ( 1 ) , to an arbitrary point .x < 1 to deduce that ( g xg(x) ) h(x) −1 .1 = h (x) · h(x) · g xg(x) > 1, a contradiction. (.★★) Suppose that .xg(x) < 1 for all .0 < x < 1. Firstly, we claim that .g(w) < 1 for all .w > 1. To see it, use Eq. (10) to an arbitrary point .x < 1, .xg(x) =

Families of 6-Cycles of Third Order

( h −1

(

)

17

)

< 1, and apply that .h −1 is decreasing with .h −1 (1) = 1 to ( ) deduce that . ( −11 ) > 1 or .g h −1 (x) < 1. Since .x < 1 was arbitrarily taken 1

g h −1 (x)

g h

(x)

and .h −1 is a homeomorphism from .(0, 1) onto .(1, ∞) we obtain that .g(w) < 1 for all .w > 1, as we claimed. .

Additionally, .g(x) /= 1 for all .x < 1, since otherwise by Lemma 6-(b), for some x /= 1, we would find .

h −1 (x) =

1 1 1 = = , g (xg(x)) h(x) g(x)h(x) h(x)

therefore .h(x)h −1 (x) = 1, with .x < 1, a contradiction. Thus, we can divide our study in two additional subcases, namely, .g(x) < 1 for all .x < 1, or .g(x) > 1 for all .x < 1. (.∗) If .g(x) < 1 for all .x < 1, then, according to the initial claim of this case, 6-(b) we have .g(u) ≤ 1 for all .u > 0. Hence, by Lemma ( ) and the decreasing character of .h, h −1 , for .w > 1 we have .1 = h −1 (w) · g wg(w) h(w) < 1, a contradiction. (.∗∗) Suppose that .g(x) > 1 for all .x < 1. Moreover, .g(w) < 1 for all .w > 1, and recall that .xg(x) < 1 and .h(x)g(x) > 1 for all .x < 1. Moreover .h(w)g(w) < 1 if .w > 1, since .h(w) < 1 and .g(w) < 1 if .w > 1. We finally distinguish two possibilities: ( ) .• If .wg(w) > 1 for some .w > 1, then .g wg(w) < 1 and Lemma 6-(b) yields 1 1 −1 .h (w) = > > 1, g (wg(w)) h(w) h(w) hence .h −1 (w) > 1, which is impossible for values .w > 1. .• If .wg(w) ≤ 1 for some .w > 1, by Eq. (10), we have .1 ≥ wg(w) = ( ) ( ) 1 −1 ( ) , therefore . ( 1 ) ≥ 1 and consequently .g h −1 (w) ≤ 1, which h −1 −1 g h

(w)

g h

(w)

is impossible because .h −1 (w) < 1 and we had supposed that .g(x) > 1 in .(0, 1). This concludes the discussion of (.∗∗). So, Case (c.2) does not provide any .6-cycle of type (4) and we can gather the discussion of Cases (c.1) and (c.2) in the following result: Proposition 5 In Eq. (4), suppose that.h is decreasing and that the set of equilibrium points reduces to .F = {1}. Then, Eq. (4) cannot be a .6-cycle. As a result of putting together Propositions 1 and 2, for the case in which Card (F) ≥ 2, and Propositions 3, 4 and 5, for the case .Card (F) = 1, we obtain the main result of Sect. 3.

.

18

A. Linero Bas and D. Nieves Roldán

Theorem 2 There are no .6-cycles displaying the form x

. n+3

= xn+2 g(xn+1 ) h(xn ),

with .g, h : (0, ∞) −→ (0, ∞) continuous.

4 The Case . xn+3 = xn+1 g(xn+2 )h(xn ) In this section, we consider the third order difference equation x

. n+3

= xn+1 g(xn+2 )h(xn ),

(11)

with .g, h : (0, ∞) → (0, ∞) continuous. We are going to prove that there are no 6-cycles of this form, see Theorem 3. Our first observation is to recall that, according to that established at the end of Sect. 2, we can assume without loss of generality that .1 ∈ F, with .g(1) = h(1) = 1. As a first necessary condition to obtain a.6-cycle, we prove that.h must be bijective.

.

Lemma 9 Suppose that Eq. (11) is a .6-cycle, with .1 ∈ F, .g(1) = h(1) = 1. Then, h is a homeomorphism.

.

Proof The proof is completely analogous to the one developed in Lemma 3. Now, it suffices to take again the initial conditions .x1 = 1, x2 = 1, x3 = y to prove that .h is surjective, and if .h(x) = h(y), to use the pair of initial conditions .x1 = x, x2 = ◻ x3 = 1 and . y1 = y, y2 = y3 = 1 to deduce that .x = y. Lemma 10 Suppose that Eq. (11) is a .6-cycle, with .1 ∈ F, .g(1) = h(1) = 1. If g(u) = 1 then .u = 1.

.

Proof Set .x1 = 1, x2 = 1 and .x3 = u. Then, by Eq. (11), .x4 = 1, x5 = u, x6 = h(u), and .1 = x2 = x8 = x6 g(x7 )h(x5 ) = x6 g(x1 )h(x5 ) = h(u)h(u). Therefore, .h(u) = 1 and since .h is a homeomorphism with .h(1) = 1, we conclude that .u = 1. ◻ Lemma 11 Suppose that Eq. (11) is a .6-cycle, with .1 ∈ F, .g(1) = h(1) = 1. Then h −1 (x) = ( 1 ) for all .x > 0.

.

h xg(g(x))

Proof Let.x > 0 be arbitrary. Take.x1 = h −1 (x), ( x2)= 1, x3 = 1 and apply the recurrence to obtain .x4 = x, x5 = g(x), x6 = xg g(x) . By global periodicity, ( ) 1 = x3 = x9 = x7 g(x8 )h(x6 ) = x1 g(x2 )h(x6 ) = h −1 (x)h xg(g(x)) .

.

Therefore, .h −1 (x) = (

1

).

h xg(g(x))

◻

Families of 6-Cycles of Third Order

19

According to the last result, we have .

where

1 ) , for all x > 0, h −1 (x) = ( h ψ(x)

(12)

( ) ψ(x) := x g g(x) , for all x > 0.

(13)

.

Notice that from (12) and the fact that .h, h −1 are homeomorphisms, we can also write ( ) 1 −1 , for all x > 0. (14) .ψ(x) = h h −1 (x) Lemma 12 Suppose that Eq. (11) is a .6-cycle, with .1 ∈ F, .g(1) = h(1) = 1. Then, ψ : (0, ∞) → (0, ∞) is a decreasing homeomorphism.

.

Proof Since .h −1 is a homeomorphism, it is immediate to deduce that .ψ is a homeincreasing), omorphism from .(0, ∞) into itself. If .h −1 is decreasing (respectively, ( ) then . h1−1 is increasing (decreasing), and consequently .h −1 h1−1 is decreasing, both ◻ in the cases .h −1 increasing and decreasing. We conclude Sect. 4 with its main result: Theorem 3 There are no .6-cycles displaying the form x

. n+3

= xn+1 g(xn+2 ) h(xn ),

with .g, h : (0, ∞) −→ (0, ∞) continuous.

( ) Proof Suppose that Eq. (11) is a .6-cycle. Then, by Lemma 12 and .g g(x) = ψ(x) , x ( ) we know that the map .g g(x) is a decreasing homeomorphism from .(0, ∞) into itself. In this case, it is a simple matter to show that .g : (0, ∞) → (0, ∞) is also a homeomorphism, either increasing or decreasing. In both cases, however, we find ◻ that .g ◦ g is increasing, a contradiction.

5 The Case . xn+3 = xn h(xn+1 )g(xn+2 ) Contrarily to the study developed in Sects. 3 and 4, in the remaining case, namely the difference equation . x n+3 = x n h(x n+1 ) g(x n+2 ), (15) where .g, h : (0, ∞) → (0, ∞) are continuous maps, we are able to discover the potential .6-cycle ) ( xn+2 2 . (16) . x n+3 = x n xn+1

20

A. Linero Bas and D. Nieves Roldán

Then the question arises whether there exist or not another .6-cycles displaying the form of (15), and different from the potential one above mentioned. We are going to prove that the answer is negative, the unique .6-cycle of type (15) is precisely the potential one given by Eq. (16). Recall that by Remark 2, since .F /= ∅, we can assume that .1 ∈ F jointly with .g(1) = h(1) = 1. The first result exposes a sufficient condition on the maps .g, h in order that Eq. (15) be a .6-cycle. Proposition 6 Let (15) be a.6-cycle. Suppose that .g(x)h(x) = 1 for all.x > 0. Then, the .6-cycle must be ) ( xn+2 2 . . x n+3 = x n xn+1 Proof Observe that the condition .g(x)h(x) = 1 for all .x > 0 is equivalent to state that .F = (0, ∞). Let .x > 0 be arbitrary. If we take initial conditions .x1 = 1, x2 = x, x3 = x and proceed ( )to apply the difference equation, we obtain .x4 = 1, x5 = xh(x), x6 = x g xh(x) . By the .6-periodicity of the solutions, .

Hence,

( ) x = x3 = x9 = x6 h(x7 )g(x8 ) = x6 h(x1 )g(x2 ) = x g xh(x) g(x). (

x .g g(x)

) =

1 , for all x > 0. g(x)

(17)

On the other hand, we generate a new solution from the(initial) conditions .x1 = x, x2 = 1, x3 = 1. Now, .x4 = x, x5 = g(x), x6 = h(x) g g(x) , and .1 = x3 = ( ) 1 x9 = x6 h(x7 )g(x8 ) = x6 h(x1 )g(x2 ) = h(x) g g(x) h(x), so since .h(x) = g(x) , we deduce that ( ) 2 . [g(x)] = g g(x) , for all x > 0. (18) In [4] it was proved that the unique solutions to the system of functional Eqs. (17) and (18) are .g(x) = 1 or .g(x) = x 2 for all .x > 0. The first one provides the .3cycle .xn+3 = xn , whereas the second one generates the potential .6-cycle .xn+3 = ( )2 , which ends the proof. ◻ xn xxn+2 n+1 In the following, we use the notation.ϕ (k) ,.k ≥ 2, to denote the iterate.ϕ ◦ . . . ◦ ϕ. k times

We continue the study to prove the result in the general case, that is, without assuming .g(x)h(x) = 1 for all . x > 0. Lemma 13 Let Eq. (15) be a .6-cycle. Assume that .h(1) = g(1) = 1. Then, for all x > 0 it holds:

.

Families of 6-Cycles of Third Order

21

/ .

h(x) = /

1 , g(g(x))

(19)

) (√ 1 (2) (x) , g · g g (3) (x) / (√ ) 2 . [g(x)] = g (2) g (2) (x) , ┌ (/ ) | (√ ) | (3) g (2) g (2) (x) , .g (x) = √g (2) 1=

(20)

.

g (3) (x) =

/

.

(21)

(22)

( ) g (2) [g(x)]2 .

(23)

Proof From the proof of Proposition 6, where we have considered the initial / condi( ) 1 , tions.x1 = x,.x2 = x3 = 1, we know that.1 = h(x)g g(x) h(x). So,.h(x) = g(g(x)) which proves (19). ( ) ( ) Hence ( Also, ) ( .x = x1 =) x7 = x4 h(x5 )g(x6 ) = xh g(x) g h(x)g(g(x)) . .h g(x) g h(x)g(g(x)) = 1, and using (19) twice, the first one with .g(x) instead of .x, we obtain / .

) (√ 1 ( ) ·g g (2) (x) = 1, or 1 = g g(g(x))

/ 1 g (3) (x)

·g

(√

) g (2) (x) .

We have proved (20). Again by the global periodicity, ( ) 1 = x2 = x8 = x5 h(x6 )g(x7 ) = x5 h(x6 )g(x1 ) = g(x)h h(x)g(g(x)) g(x),

.

( ) and in this case we deduce that.1 = [g(x)]2 h h(x)g(g(x)) and if we use (19) twice, in fact we obtain (21): (/ 1 = [g(x)] h

.

2

┌ | | = [g(x)] √ 2

) ) (√ ( ) 1 ( ) · g g(x) = [g(x)]2 h g (2) (x) g g(x)

1 (√ ). g (2) g (2) (x)

)]2 [ (√ √ g (2) (x) , and using (21) with . g (2) (x) Now, by (20), we have .g (3) (x) = g instead of .x, we get Eq. (22):

22

A. Linero Bas and D. Nieves Roldán

┌ (/ ) | (√ ) | (3) √ (2) (2) (2) g g (x) . .g (x) = g ◻

To obtain Eq. (23), simply combine Eqs. (21) and (22). It is worth pointing out that if we define ϕ(x) :=

.

√

x, x > 0,

Equation (22) can be written as ( ( ( ( ](3) ( ))))) [ g (3) (x) = ϕ g (2) ϕ g (2) ϕ g (2) (x) ⇐⇒ g (3) = ϕ ◦ g (2) .

.

(24)

Lemma 14 Let Eq. (15) be a .6-cycle. Assume that .h(1) = g(1) = 1. Then, the set of fixed points of .g is the singleton .{1}, .Fix(g) = {1}. ( ) Proof Suppose that .g(z) = z for some .z > 0. Then .g g(z) = z and by (19) we √ have.h(z) = √1z . Consider the initial conditions.x1 = z, x2 = z, x3 = 1. Then.x4 = √ zh(z) = 1, x5 = z, x6 = g(z) = z. Using the global periodicity, on the one hand √ √ we find that .g( z) = z, since .

√ 1 √ z = x2 = x8 = x5 h(x6 )g(x7 ) = x5 h(x6 )g(x1 ) = zh(z)g( z) = z √ g( z), z

√ and, on the other hand, we obtain .h( z) =

1 z2

because

√ √ 1 = x3 = x9 = x6 h(x7 )g(x8 ) = x6 h(x1 )g(x2 ) = zh( z)g(z) = z 2 h( z).

.

/ / √ √ √ ( 1√ ) = √1 . By (19) and the fact that .g( z) = z, we also find .h( z) = z g g( z) / √ √ ◻ Hence, .h( z) = z12 and .h( z) = √1z , so we deduce that . z18 = 1z and .z = 1. Lemma 15 Let Eq. (15) be a .6-cycle. Assume that .h(1) = g(1) = 1. ( ) (a) If .g g(x) = 1 for some .x > 0, then .g(x) = 1. (b) If .g(x) = 1 for some .x > 0, then .x ∈ F. (c) If .h(x) = 1 for some ( .x) > 0, then .x ∈ F. 1 x = g(x) . (d) If .x ∈ F, then .g g(x) Proof (a) It suffices to apply Eq. (21) to conclude that .[g(x)]2 = 1. (b) If.g(x) = 1, by (19) and.g(1) = 1, we deduce that.h(x) = 1. Thus,.h(x)g(x) = 1 and .x ∈ F. ( ) (c) If .h(x) = 1, use (19) to get .g g(x) = 1 and apply Part (a) to deduce that .x ∈ F.

Families of 6-Cycles of Third Order

23

(d) Take the initial conditions .1, x, x. In the proof of Proposition 6 (notice that . x ∈ F) we have seen that we arrive to Eq. (17). ◻ ) ( (3) From Eq. (24) we already know that .g (3) = ϕ ◦ g (2) . If .g = ϕ ◦ g (2) , then by / ( ) 1 , that is, .h(x)g(x) = 1 for all .x > 0 or Eq. (19) we have .g(x) = g g(x) = h(x) equivalently .F = (0, ∞). Then, by Proposition 6 we deduce that either .g(x) = 1 or 2 .g(x) = x for all . x > 0, and we obtain the potential cycle (16) in the second case. We have thus proved: Proposition 7 Let Eq. (15) be a .6-cycle, with .g, h : (0, √ ∞) → (0, ∞). Assume that h(1) = g(1) = 1 and that .g = ϕ ◦ g (2) , where .ϕ(x) = x, x > 0. Then the difference equation must be the potential one given by

.

( x

. n+3

= xn

xn+2 xn+1

)2 .

In view of the previous result, we can ask whether it is possible or not the case g /= ϕ ◦ g (2) in order to find new cycles exhibiting the pattern of (15). To do it, once we have deduced the general properties that Eq. (15) must satisfy in order to be a .6-cycle, our strategy will be to reduce the problem to a suitable functional equation whose solutions will provide us the candidates to .6-cycles following the pattern of Eq. (15). Recall that if Eq. (15) is a .6-cycle, then according to Eqs. (20) and (22) we have .

.ϕ ◦ g ◦ g ◦ g = g ◦ ϕ ◦ g ◦ g, g ◦ g ◦ g = ϕ ◦ g ◦ g ◦ ϕ ◦ g ◦ g ◦ ϕ ◦ g ◦ g,

.

where .ϕ(x) =

√

(25) (26)

x, for all .x > 0. From here we deduce that g (3) = (ϕ ◦ g)(3) ◦ g (3) ,

.

(27)

since from (25) and (26) we have g ◦ g ◦ g = ϕ ◦ g ◦ (g ◦ ϕ ◦ g ◦ g) ◦ ϕ ◦ g ◦ g = ϕ ◦ g ◦ (ϕ ◦ g ◦ g ◦ g) ◦ ϕ ◦ g ◦ g = (ϕ ◦ g)(2) ◦ g ◦ (g ◦ ϕ ◦ g ◦ g) = (ϕ ◦ g)(2) ◦ g ◦ (ϕ ◦ g ◦ g ◦ g) = (ϕ ◦ g)(2) ◦ (g ◦ ϕ ◦ g ◦ g) ◦ g = (ϕ ◦ g)(2) ◦ (ϕ ◦ g ◦ g ◦ g) ◦ g = (ϕ ◦ g)(3) ◦ g (3) .

.

On the other hand, we can also gather the compositions into the form

24

A. Linero Bas and D. Nieves Roldán .g

◦ g ◦ g = ϕ ◦ g ◦ (g ◦ ϕ ◦ g ◦ g) ◦ ϕ ◦ g ◦ g = ϕ ◦ g ◦ (ϕ ◦ g ◦ g ◦ g) ◦ ϕ ◦ g ◦ g = ϕ ◦ (g ◦ ϕ ◦ g ◦ g) ◦ g ◦ ϕ ◦ g ◦ g = ϕ ◦ (ϕ ◦ g ◦ g ◦ g) ◦ g ◦ ϕ ◦ g ◦ g = ϕ ◦ (ϕ ◦ g ◦ g ◦ g) ◦ (g ◦ ϕ ◦ g ◦ g) = ϕ ◦ ϕ ◦ g ◦ g ◦ g ◦ (ϕ ◦ g ◦ g ◦ g) = ϕ ◦ ϕ ◦ g ◦ g ◦ (g ◦ ϕ ◦ g ◦ g) ◦ g = ϕ ◦ ϕ ◦ g ◦ g ◦ (ϕ ◦ g ◦ g ◦ g) ◦ g = ϕ ◦ ϕ ◦ g ◦ (g ◦ ϕ ◦ g ◦ g) ◦ g ◦ g = ϕ ◦ ϕ ◦ (g ◦ ϕ ◦ g ◦ g) ◦ g ◦ g ◦ g = ϕ ◦ ϕ ◦ (ϕ ◦ g ◦ g ◦ g) ◦ g ◦ g ◦ g = ϕ(3) ◦ g (3) ◦ g (3) .

Therefore,

g (3) = ϕ (3) ◦ g (3) ◦ g (3) .

.

(28)

√ Taking into account that .ϕ (3) (x) = 8 x and that its inverse is .Φ(x) = x 8 , Eq. (28) can be written as (3) .Φ ◦ g = g (3) ◦ g (3) . (29) If we put .g (3) = f we can think .g (3) as a solution of the functional equation Φ ◦ f = f ◦ f.

.

(30)

In our case, .Φ(x) = x 8 is given and the unknown is the map . f . Inspired by [4], we consider the set ( ) S := Im g (3) = {g (3) (x) : x > 0}. ( ) Lemma 16 Under the above conditions, if . y ∈ S = Im g (3) , then .g (3) (y) = y 8 . ( ) (3) 8 Proof If . y ∈ Im g (3) , and .g (3) satisfies the functional Eq. ( (30), then ) .g ((y) = y) because there exists .z > 0 such that .g (3) (z) = y, so .Φ g (3) (z) = g (3) g (3) (z) , (3) .Φ(y) = g (y), . y 8 = g (3) (y). ◻ .

From Lemma 16, according to the intermediate value property applied to the continuous map .g (3) we deduce that . S must be one of the following sets (take into account that .1 ∈ S as long as .g(1) = 1): . S = {1}, . S = (0, ∞), . S = (0, 1] or . S = [1, ∞). Case (a): . S = {1}. If .g (3) (x) = 1 for all .x > 0, by Lemma 15 we deduce that .g(x) = 1 for all. x > 0, and in fact.h(x) = 1 for all. x > 0 according to (19). Therefore we obtain the difference equation .xn+3 = xn , that is a .3-cycle. Case (b): Suppose that . S = (0, ∞). Then, given an arbitrary .w ∈ (0, ∞), there exists .z = z(w) such that .g (3) (z) = w. By (29) we deduce that .w 8 = g (3) (w). From here, it is easy to see that .g is an increasing homeomorphism (.g is surjective since (3) . S = (0, ∞); it is obviously injective because .g so is due to Lemma (16); and .g is increasing because, otherwise, we would arrive to the decreasing character of .g (3) , contrary to Lemma (16)). We claim that, in fact, we have .g(w) = w2 for all .w > 0. To see it, use (27) 8 to obtain .w 8 = (ϕ ◦ g)(3) (w 8 ), or if we do √ the change of variables .x = w , .x = (3) (ϕ ◦ g) (x) for all .x > 0. Since .ϕ(x) = x, it is straightforward to deduce that

Families of 6-Cycles of Third Order

25

ϕ ◦ g is an increasing homeomorphism, with .(ϕ ◦ g)(3) = Id|(0,∞) , where .Id denotes the identity map. This implies that.ϕ ◦ g = Id|(0,∞) , therefore.g(x) = x 2 for all.x > 0, as we claimed. / / ( 1 ) = g(x1 2 ) = Now, Eq. (19) provides the value of .h(x), namely .h(x) = g g(x) / 1 = x12 , .x > 0. We conclude that the difference Eq. (15) is then the potential x4 ( )2 xn+2 .6-cycle . x n+3 = x n . xn+1 .

Case (c): Suppose that . S = (0, 1]. By Lemma 16 we know that .g (3) (y) = y 8 for all . y ∈ (0, 1]. Reasoning Case (b), we can prove that, in fact, .g(y) = y 2 if / as in / 1 = y14 = y12 , where we have used (19). Consequently, . y ≤ 1. Then .h(y) = g(g(y)) .h(y)g(y) = 1 for all .(0, 1], and .(0, 1] ⊆ F. Now, from Lemma 15-(d), we have ( ( ) ) y 1 1 .g for all . y ∈ (0, 1], that is, .g = = y12 and we deduce that .g(u) = g(y) g(y) y u 2 for all .u = 1y > 1. We have found that .g(v) = v 2 for all .v > 0, and it holds ( (3) ) .Im g = (0, ∞), a contradiction. Case (d): Suppose that . S = [1, ∞). All the steps of Case (c) can be repeated 2 one by one to lead ( )to .g(y) = y if . y ≥ 1. Then, as in Case (c), we find that . y ∈ F if . y ≥ 1, and .g 1y = y12 , therefore .g(v) = v 2 for all .v > 0, again, we arrive to ( (3) ) .Im g = (0, ∞), a contradiction. All our discussion can be gathered in the following result: Proposition 8 Consider Eq. (15), with .g, h : (0, ∞) → (0, ∞) continuous. Assume that .h(1) = g(1) = 1. Then, the unique .6-cycle of type (15) is ( x

. n+3

= xn

xn+2 xn+1

)2 .

To end this section, in order to get rid of the condition .g(1) = h(1) = 1, we must recall the change of variables given at the end of Sect. 2. Therefore, if we return from our obtained .6-cycle ( y

. n+3

= yn

yn+2 yn+1

)2 = yn γ (yn+2 )η(yn+1 )

to the initial one (3), we have to do (here, .i = n, . j = n + 2, .k = n + 1) .

thus

y =

x , γ z0

(

xn+2 z0

) =

g(xn+2 ) , η g(z 0 )

(

xn+1 z0

) = g(z 0 )h(xn+1 ),

26

A. Linero Bas and D. Nieves Roldán ( .g(x n+2 )

= g(z 0 )γ

xn+2 z0

(

) = g(z 0 )

xn+2 z0

)2 , h(xn+1 ) =

η

(

x n+1 z0

g(z 0 )

) =

g(z 0 )

1 (

) x n+1 2 z0

,

therefore the difference equation we are looking for is x

. n+3

= xn h(xn+1 )g(xn+2 ) = xn ·

g(z 0 )

1 (

( xn+1 z0

)2 · g(z 0 )

xn+2 z0

)2

( = xn

xn+2 xn+1

)2 .

We conclude with the main result of Sect. 5. Theorem 4 The unique.6-cycle displaying the form of Eq. (15), with.g, h : (0, ∞) → (0, ∞) continuous, is given by ( x

. n+3

= xn

xn+2 xn+1

)2 .

6 Forthcoming Lines of Research Related to the global periodicity of difference equations of third order, let us mention that in [2] the authors studied the difference equation .xn+3 = xi f (x j , xk ), where 2 .i, j, k ∈ {n, n + 1, n + 2} are pairwise distinct and . f : (0, ∞) → (0, ∞) is continuous, and proved the following: • The unique .3-cycle is .xn+3 = xn . . • The unique .4-cycle is .xn+3 = xn xxn+2 n+1 ( )Φ ( )ψ • There are two .5-cycles: .xn+3 = xn xxn+2 and .xn+3 = xn xxn+2 , where .Φ = n+1 n+1 √ 1+ 5 2

and .ψ = −Φ −1 .

In [13], under the assumption of . f being symmetric, it was proved that there are no .6-cycles exhibiting the form .xn+3 = xi f (x j , xk ). Furthermore, in the present paper, we have seen that in the particular case of separated variables, that is, .xn+3 = xi g(x j )h(xk ), with .i, j, k ∈ {n, n + 1, n + 2} pairwise distinct, and .g, h : (0, ∞) → (0, ∞) continuous, the unique .6-cycle of third order is the potential one .xn+3 = ( )2 xn xxn+2 . n+1 In this sense, a natural question arises: to determine the existence of .6-cycles displaying the form.xn+3 = xi f (x j , xk ), that is, without considering separated variables. Moreover, it would be interesting to study . p-cycles of that form for . p ≥ 6.

Families of 6-Cycles of Third Order

27

Acknowledgements We would like to thank the referees for their useful comments that have improved the paper. This work has been supported by the grant MTM2017-84079-P funded by MCIN/AEI/10.13039/501100011033 and by ERDF “A way of making Europe”, by the European Union.

References 1. Abu-Saris, R.M.: A self-invertibility condition for global periodicity of difference equations. App. Math. Lett. 19, 1078–1082 (2006) 2. Balibrea, F., Linero, A.: On the global periodicity of some difference equations of third order. J. Differ. Equ. Appl. 13, 1011–1027 (2007) 3. Cánovas, J.S., Linero Bas, A., Soler López, G.S.: A characterization of .k-cycles. Nonlinear Anal. 72, 364–372 (2010) 4. Caro, A., Linero, A.: Existence and uniqueness of . p-cycles of second and third order. J. Differ. Equ. Appl. 15, 485–500 (2009) 5. Caro, A., Linero, A.: General cycles of potential form. Int. J. Bifurc. Chaos Appl. Sci. Eng. 20, 2735–2749 (2010) 6. Cima, A., Gasull, A., Mañosa, V.: Global periodicity and complete integrability of discrete dynamical systems. J. Differ. Equ. Appl. 12, 696–716 (2006) 7. Cima, A., Gasull, A., Mañosas, F.: Global linearization of periodic difference equations. Discret. Contin. Dyn. Syst. 32, 1575–1595 (2012) 8. Cima, A., Gasull, A., Mañosas, F.: On periodic rational difference equations of order .k. J. Differ. Equ. Appl. 10, 549–559 (2004) 9. Coxeter, H.S.M.: Frieze patterns. Acta Arith. 18, 297–310 (1971) 10. Csörnyei, M., Laczkovich, M.: Some periodic and non-periodic recursions. Monatsh. Math. 132, 215–236 (2001) 11. Haynes, R., Kwasik, S., Mast, J., Schultz, R.: Periodic maps on .R7 without fixed points. Math. Proc. Camb. Philos. Soc. 132, 131–136 (2002) 12. Linero Bas, A.: Some results on periodicity of difference equations. In: Liz, E., Mañosa, V. (eds.) Proceedings of the International Workshop Future Directions in Difference Equations, pp. 121–143. Servizo de Publicacións da Universidade de Vigo, Vigo (2011) 13. Linero Bas, A., Nieves Roldán, D.: On the existence of.6-cycles for some families of difference equations of third order. Turkish J. Math. 47(7), 2043–2060 (2023). https://doi.org/10.55730/ 1300-0098.3480 14. Lyness, R.C.: Note 1581. Cycles. Math. Gaz. 26, 62 (1942) 15. Lyness, R.C.: Note 1847. Cycles. Math. Gaz. 29, 231–233 (1945) 16. Lyness, R.C.: Note 2952. Cycles. Math. Gaz. 45, 207–209 (1961) n) . J. Differ. 17. Mestel, B.D.: On globally periodic solutions of the difference equation.xn+1 = fx(x n−1 Equ. Appl. 9, 201–209 (2003) 18. Zheng, Y.: On periodic cycles of the Lyness equations. Differ. Equ. Dyn. Syst. 6, 319–324 (1998)

About a System of Piecewise Linear Difference Equations with Many Periodic Solutions Inese Bula and Agnese S¯ıle

Abstract We want to draw attention to the system of first order piecewise linear difference equations of two equations .xn+1 = |xn | − yn − b and . yn+1 = xn − |yn | − d, .n = 0, 1, 2, ..., (x0 , y0 ) ∈ R2 , where the parameters .b and .d are any positive real numbers. We will show that this system has interesting behavior compared to other similar systems. We show that there exists an unstable equilibrium .(d, −b). It has been shown that there are no solutions with period 2 and 3, but depending on the values of parameters .b and .d there are solutions with periods 5, 6, 7, 11, 12, 13, 16, 17, 18, 19, 20, 24, 25, 27, 30, 36. We have a hypothesis that all solutions are eventually periodic solutions. Keywords System of difference equations · Stability · Equilibrium · Periodic solution · Eventually periodic solution

1 Introduction In general, we consider a system of difference equations ( .

xn+1 = f (xn , yn ), yn+1 = g(xn , yn ),

(1)

with initial conditions .(x0 , y0 ) ∈ R2 , .n = 0, 1, 2, .... ∞ If the sequences .(xn )∞ n=0 and .(yn )n=0 satisfy (1) and the given initial conditions for all .n = 0, 1, 2, ..., then the sequence .(xn , yn )∞ n=0 is called a solution of system of difference Eq. (1). I. Bula (B) · A. S¯ıle Department of Mathematics, University of Latvia, Riga, Latvia e-mail: [email protected] I. Bula Institute of Mathematics and Computer Science of University of Latvia, Riga, Latvia © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 S. Olaru et al. (eds.), Difference Equations, Discrete Dynamical Systems and Applications, Springer Proceedings in Mathematics & Statistics 444, https://doi.org/10.1007/978-3-031-51049-6_2

29

30

I. Bula and A. S¯ıle

A pair .(x ∗ , y ∗ ) is said to be an equilibrium point of the system (1) if . f (x ∗ , y ∗ ) = (x , y ∗ ) and .g(x ∗ , y ∗ ) = (x ∗ , y ∗ ). A pair .(x, y) is said to be a periodic point with period .k of the system (1) if . f k (x, y) = (x, y) and .g k (x, y) = (x, y) for some positive integer .k. If in addition . f m (x, y) /= (x, y) or .g m (x, y) /= (x, y) for .0 < m < k, then .k is called the minimal period or prime period of system (1). If there exist real numbers.x ∗ and. y ∗ and an integer. N such that.(xn , yn ) = (x ∗ , y ∗ ) for all .n ≥ N , then we say that the solution .(xn , yn )∞ n=0 is eventually constant, and equal to an equilibrium point of system (1). If . p is the smallest positive integer such that .(xn+ p , yn+ p ) = (xn , yn ) for all .n ≥ N , then we say that the solution .(xn , yn )∞ n=0 is eventually periodic with prime period . p. We say that an equilibrium point .(x ∗ , y ∗ ) of system (1) is stable if for every .ε > 0, there exists .δ > 0 such that if .(xn , yn )∞ n=0 is a solution of (1) with ∗

|x0 − x ∗ | + |y0 − y ∗ | < δ,

.

then

|xn − x ∗ | + |yn − y ∗ | < ε for all n ≥ 0.

.

The Lozi map is the system of difference equations ( .

xn+1 = −α|xn | + yn + 1, n = 0, 1, 2, ..., (x0 , y0 ) ∈ R2 , α, β ∈ R, yn+1 = βxn ,

introduced by Lozi [11] in 1978 as a piecewise linear analogue of the Henon map (see [8]). A two-dimensional piecewise linear map defined by ( .

xn+1 = |xn | − yn + 1, n = 0, 1, 2, ..., (x0 , y0 ) ∈ R2 , yn+1 = xn ,

is called a Gingerbreadman map which was investigated by Devaney [3] in 1984 and was shown to be chaotic in certain regions and stable in others. It has been proven that) the orbit ) of the solution of Gingerbreadman map with initial condition 1 , 0 is periodic with period 126 ([4]). .(x 0 , y0 ) = − 10 Both maps are special cases of a system of difference equations ( .

xn+1 = |xn | + a yn + b, n = 0, 1, 2, ..., (x0 , y0 ) ∈ R2 , yn+1 = xn + c |yn | + d,

(2)

where the parameters .a, .b, .c and .d are any real numbers. If parameters are from the set .{−1, 0, 1}, there are 81 systems. These systems have been intensively studied by E. Lapierre and W. Tikjha and their colleagues ([5, 6, 9, 10, 12, 13, 17]). Some other special cases of (2) with other parameters are considered in [1, 7, 14, 15, 18, 19]. In [16] the global behavior of the system of first order piecewise linear difference equations

About a System of Piecewise Linear Difference …

( .

31

xn+1 = |xn | − yn + b, n = 0, 1, 2, ..., (x0 , y0 ) ∈ R2 , yn+1 = xn − |yn | − d,

where the parameters .b and .d are any positive real numbers is considered. It has been proven that for any initial conditions .(x0 , y0 ) ∈ R2 the solutions to the system is eventually the equilibrium, .(2b + d, b). Moreover, the solutions of the system will reach the equilibrium within six iterations. In all of the above-mentioned articles, if there are cycles, they are with periods of 3, 4 or 5. We consider the global behavior of the system of first order piecewise linear difference equations: ( .

xn+1 = |xn | − yn − b, n = 0, 1, 2, ..., (x0 , y0 ) ∈ R2 , yn+1 = xn − |yn | − d,

(3)

where the parameters .b and .d are any positive real numbers. We show that there exist an unstable equilibrium .(d, −b). We have a hypothesis that all solutions of (3) are eventually periodic. It has been shown that there are no solutions with period 2 or 3, but depending on the values of parameters .b and .d there are solutions with periods 5, 6, 7, 11, 12, 13, 16, 17, 18, 19, 20, 24, 27, 30, 36. The proof of the existence of solutions of (3) with other periods remains open. System with .b = 1 and .d = 0 at first was studied in [12]. It has been shown that the system has exactly two prime period 5 solutions and a unique equilibrium point .(0, −1) and every solution of the system is eventually one of the two prime period 5 solutions or else the unique equilibrium point. We show that if .0 < d < 0.125b, then cycles with periods 5 and 7 exist for system (3).

2 Equilibrium Points and Their Stability Theorem 1 For each parameter .b > 0 and .d > 0 of system (3) there is only one equilibrium .(d, −b) which is unstable. Proof Let us first establish that system (3) has only one equilibrium point .(d, −b). There are 4 cases to consider. Case 1: let .x < 0 and . y < 0, then ( .

x = −x − y − b, y = x + y − d.

From the second equality it follows that .x = d > 0. This contradicts the assumption that .x < 0.

32

I. Bula and A. S¯ıle

Case 2: .x ≥ 0 and . y < 0, then ( .

x = x − y − b, y = x + y − d.

We obtain . y = −b < 0 and .x = d > 0 and thus the equilibrium point is .(d, −b). Case 3: .x ≥ 0 and . y ≥ 0, then ( .

x = x − y − b, y = x − y − d.

From the first equality it follows that . y = −b < 0. This contradicts the assumption that . y ≥ 0. Case 4: .x < 0 and . y ≥ 0, then ( .

x = −x − y − b, y = x − y − d.

From the second equality, it follows that .x = 2y + d however, the left side of this equality is negative, while the right side is positive. So the only equilibrium point is .(d, −b). Next our aim is to prove that all points that are near the equilibrium .(d, −b) and belongs to the fourth quadrant with iterations of system (3) are moving to the other quadrant. That is, we will prove that the equilibrium .(x ∗ , y ∗ ) = (d, −b) is unstable: .∃ε > 0 ∀δ > 0 ∃(x 0 , y0 ) ∃K ∈ {0, 1, 2, ...} |d − x0 | + | − b − y0 | < δ & |d − x K | + | − b − y K | ≥ ε.

.

We fix .ε = min{d, b}. We choose .δ > 0 freely. We consider point .(d + δ1 , −b + δ2 ), .δ1 , δ2 ∈ R, such that this point belongs to the fourth quadrant, that is, .d + δ1 > 0 and .−b + δ2 < 0, and .|d − (d + δ1 )| + | − b − (−b + δ2 )| = |δ1 | + |δ2 | < δ. We want that the sequence of solutions of this point are near to the equilibrium point (.xn > 0 and . yn < 0, .n = 1, 2, 3, ...): ( .

( .

( .

( .

x1 = |x0 | − y0 − b = d + δ1 + b − δ2 − b = d + δ1 − δ2 , y1 = x0 − |y0 | − d = d + δ1 − b + δ2 − d = −b + δ1 + δ2 ,

x2 = |x1 | − y1 − b = d + δ1 − δ2 + b − δ1 − δ2 − b = d − 2δ2 , y2 = x1 − |y1 | − d = d + δ1 − δ2 − b + δ1 + δ2 − d = −b + 2δ1 , x3 = |x2 | − y2 − b = d − 2δ2 + b − 2δ1 − b = d − 2(δ1 + δ2 ), y3 = x2 − |y2 | − d = d − 2δ2 − b + 2δ1 − d = −b + 2(δ1 − δ2 ),

x4 = |x3 | − y3 − b = d − 2δ1 − 2δ2 + b − 2δ1 + 2δ2 − b = d − 4δ1 , y4 = x3 − |y3 | − d = d − 2δ1 − 2δ2 − b + 2δ1 − 2δ2 − d = −b − 4δ2 ,

About a System of Piecewise Linear Difference …

( .

( .

.

.

x5 = |x4 | − y4 − b = d − 4δ1 + b + 4δ2 − b = d − 4(δ1 − δ2 ), y5 = x4 − |y4 | − d = d − 4δ1 − b − 4δ2 − d = −b − 4(δ1 + δ2 ),

x6 = |x5 | − y5 − b = d − 4δ1 + 4δ2 + b + 4δ1 + 4δ2 − b = d + 8δ2 , y6 = x5 − |y5 | − d = d − 4δ1 + 4δ2 − b − 4δ1 − 4δ2 − d = −b − 8δ1 , (

(

33

x7 = |x6 | − y6 − b = d + 8δ2 + b + 8δ1 − b = d + 8(δ1 + δ2 ), y7 = x6 − |y6 | − d = d + 8δ2 − b − 8δ1 − d = −b − 8(δ1 − δ2 ),

x8 = |x7 | − y7 − b = d + 8δ1 + 8δ2 + b + 8δ1 − 8δ2 − b = d + 16δ1 , y8 = x7 − |y7 | − d = d + 8δ1 + 8δ2 − b − 8δ1 + 8δ2 − d = −b − 16δ2 .

By induction we obtain ( ( .

( (

xn = d + (−4)k (δ1 − δ2 ), yn = −b + (−4)k (δ1 + δ2 ),

n = 1 + 4k, k = 0, 1, 2, ...

xn = d + (−4)k (−2δ2 ), yn = −b + (−4)k 2δ1 ,

n = 2 + 4k, k = 0, 1, 2, ...

xn = d + (−4)k (−2)(δ1 + δ2 ), n = 3 + 4k, k = 0, 1, 2, ... yn = −b + (−4)k 2(δ1 − δ2 ), xn = d + (−4)k+1 δ1 , yn = −b + (−4)k+1 δ2 ,

n = 4 + 4k, k = 0, 1, 2, ...

Therefore .∃K ∈ {0, 1, 2, ...} such that .d K

= max{(−4) K (δ1 − δ2 ), (−4) K (−2δ2 ), (−4) K (−2)(δ1 + δ2 ), (−4) K +1 δ1 } < −d .or

b K = min{(−4) K (δ1 + δ2 ), (−4) K 2δ1 , (−4) K 2(δ1 − δ2 ), (−4) K +1 δ2 } > b

and the corresponding .(xn , yn ) is not in the fourth quadrant. It means that .|d

− (d + d K )| + | − b − (−b + b K )| = |d K | + b K > min{d, b} = ε.

◻ Systems of difference Eq. (2), .a = c = −1, which have other signs before the parameters .b and .d, will behave quite differently. Here we will demonstrate the proof with plus signs. We look at the system ( .

xn+1 = |xn | − yn + b, n = 0, 1, 2, ..., (x0 , y0 ) ∈ R2 , b > 0, d > 0. yn+1 = xn − |yn | + d,

(4)

34

I. Bula and A. S¯ıle

Theorem 2 If .2b − d ≥ 0, then system (4) has one equilibrium point .(2b − d, b) and it is stable. Proof Suppose that .2b − d ≥ 0. It is trivial for one to check that .(2b − d, b) is an equilibrium point of system (4). To show that this equilibrium is unique, we assume ∗ ∗ .(x , y ) is an equilibrium, and check the four cases corresponding to which quadrant ∗ ∗ .(x , y ) belongs. The details are left to the reader. We note that if .2b − d = 0, then equilibrium is .(0, b). At first we consider the situation when .2b − d > 0. We fix .ε > 0 and define the corresponding .δ > 0 as follows ) ε b 2b − d , , . .0 < δ < min 4 2 2 (

Let us consider an arbitrary point .(x0 , y0 ) from the .δ-neighborhood of equilibrium (2b − d, b): . x 0 = 2b − d + δ x , y0 = b + δ y ,

.

where .δx and .δ y are real numbers such that |x0 − (2b − d)| + |y0 − b| = |δx | + |δ y | < δ.

.

From the last inequality it follows that ( |δx | < min

.

ε b 2b − d , , 4 2 2

)

( and |δ y | < min

) ε b 2b − d , , . 4 2 2

It means that .x0 = 2b − d + δx > 0 and . y0 = b + δ y > 0 therefore ( .

x1 = 2b − d + δx − b − δ y + b = 2b − d + δx − δ y , y1 = 2b − d + δx − b − δ y + d = b + δx − δ y .

If .δx = δ y , then .(x1 , y1 ) is equal to the equilibrium. If .δx /= δ y , then |x1 − (2b − d)| + |y1 − b| = 2|δx − δ y | ≤ 2(|δx | + |δ y |) < 2

.

(ε 4

+

ε) = ε. 4

} } { { and.|δ y | < min b2 , 2b−d then.x1 > 0 and. y1 > 0 thereSince.|δx | < min b2 , 2b−d 2 2 fore ( x2 = 2b − d + δx − δ y − b − δx + δ y + b = 2b − d, . y2 = 2b − d + δx − δ y − b − δx + δ y + d = b. Therefore, .(x2 , y2 ) and all other elements .(xn , yn ), .n > 2, of the solutions coincide with the equilibrium. The equilibrium is stable.

About a System of Piecewise Linear Difference …

35

If .2b − d = 0, then we need more iterations to show that the points from the equilibrium.δ-neighborhood remain in the equilibrium.ε-neighborhood and are eventually equilibrium points to .(0, b). Similar to before we fix .ε > 0 and define the corresponding .δ > 0 as follows ) ε b , . .0 < δ < min 8 2 (

Let us consider an arbitrary point .(x0 , y0 ) from the .δ-neighborhood of equilibrium (0, b): . x 0 = δ x , y0 = b + δ y ,

.

where .δx and .δ y are real numbers such that |x0 − 0| + |y0 − b| = |δx | + |δ y | < δ.

.

From the last inequality it follows that ( |δx | < min

.

ε b , 8 2

)

( and |δ y | < min

) ε b , , 8 2

therefore .|δx | < 8ε , .|δ y | < 8ε , .|δx − δ y | < 4ε and .|δx + δ y | < 4ε . These inequalities will further guarantee that the considered iteration points are in the equilibrium .εneighborhood. We need to analyze two cases. Case 1: .δx ≥ 0, then .x0 = δx ≥ 0 and . y0 = b + δ y > 0 and therefore ( .

x 1 = δx − b − δ y + b = δx − δ y , y1 = δx − b − δ y + d = b + δx − δ y > 0 (since d = 2b and |δx − δ y | < b).

Again we have two situations. It is possible that .x1 = δx − δ y ≥ 0, then ( .

x2 = δx − δ y − b − δx + δ y + b = 0, y2 = δx − δ y − b − δx + δ y + d = d − b = b,

an equilibrium point has been obtained. If .x1 = δx − δ y < 0, then ( .

x2 = −δx + δ y − b − δx + δ y + b = 2(δ y − δx ) > 0, y2 = δx − δ y − b − δx + δ y + d = d − b = b > 0, ( .

x3 = 2δ y − 2δx − b + b = 2(δ y − δx ) > 0, y3 = 2δ y − 2δx − b + d = b + 2(δ y − δx ) > 0,

36

I. Bula and A. S¯ıle

( .

x4 = 2δ y − 2δx − b − 2δ y + δx + b = 0, y4 = 2δ y − 2δx − b − 2δ y + δx + d = d − b = b,

an equilibrium point has been obtained. Case 2: .δx < 0, then .x0 = δx < 0 and . y0 = b + δ y > 0 and therefore ( .

x1 = −δx − b − δ y + b = −δx − δ y , y1 = δx − b − δ y + d = b + δx − δ y > 0.

Two situations are possible. At first, .x1 ≥ 0, then ( .

x2 = −δx − δ y − b − δx + δ y + b = −2δx > 0, y2 = −δx − δ y − b − δx + δ y + d = b − 2δx > 0, ( .

x3 = −2δx − b + 2δx + b = 0, y3 = −2δx − b + 2δx + d = b,

again an equilibrium point has been obtained. If .x1 < 0, then ( x2 = δx + δ y − b − δx + δ y + b = 2δ y , . y2 = δx + δ y − b − δx + δ y + d = b + 2δ y > 0, but .δ y can be either positive and negative. If .δ y ≥ 0, then ( .

x3 = 2δ y − b − 2δ y + b = 0, y3 = 2δ y − b − 2δ y + d = b > 0,

we have achieved equilibrium. If .δ y < 0, then ( .

x3 = −2δ y − b − 2δ y + b = −4δ y > 0, y3 = 2δ y − b − 2δ y + d = b > 0, (

.

x4 = −4δ y − b + b = −4δ y > 0, y4 = −4δ y − b + d = b − 4δ y > 0, (

.

x5 = −4δ y − b + 4δ y + b = 0, y5 = −4δ y − b + 4δ y + d = b,

at the fifth iteration we have obtained equilibrium.

◻

In a similar way, we can look at all four systems of difference Eq. (2),.a = c = −1, and get a conclusion about the equilibrium points and their stability (see Table 1).

About a System of Piecewise Linear Difference …

37

Table 1 Equilibrium points and their stability, .b > 0, d > 0 Equilibrium System ( xn+1 = |xn | − yn + b, . .(2b + d, b) yn+1 = xn − |yn | − d, ( if 2b − d ≥ 0, then (2b − d, b) xn+1 = |xn | − yn + b, . . ) ) 2d+b yn+1 = xn − |yn | + d, if 2b − d < 0, then 2b−d 5 , 5 ( ) ) if 2d − b ≥ 0, then −2b−d , 2d−b xn+1 = |xn | − yn − b, 5 5 .

( .

yn+1 = xn − |yn | + d, xn+1 = |xn | − yn − b, yn+1 = xn − |yn | − d,

.

if 2d − b < 0, then (−d, 2d − b)

.(d, −b)

Quadrant

Stability

.1

Stable

.

1 2

.

Stable Unstable

.

2 3

.

Unstable Stable

.4

Unstable

We are interested in the last system of Table 1, whose equilibrium is unstable (Theorem 1). Numerical experiments show that all considered solutions are eventually periodic for some periodic point. By varying the parameters .b and .d, different periodic points of different periods are obtained.

3 Periodicity First, we note that system (3) has no cycles with periods . p = 2 and . p = 3. Theorem 3 There are no solutions of system (3) with cycles of periods cycles with periods . p = 2 or . p = 3. Proof Suppose that there is a period-two solution, ...., (x1 , y1 ), (x2 , y2 ), (x1 , y1 ), .... The proof is performed by looking at all possible cases of the signs of .x1 , .x2 , . y1 , . y2 . For a cycle with period 2, the equalities must hold ( .

x1 = |x2 | − y2 − b, and y1 = x2 − |y2 | − d,

(

x2 = |x1 | − y1 − b, y2 = x1 − |y1 | − d,

It is necessary to look at the four cases depending on the values of .x1 and . y1 . We will demonstrate the proof only for case .x1 ≥ 0 and . y1 ≥ 0. Then ( .

x2 = x1 − y1 − b, y2 = x1 − y1 − d.

Again, four situations are possible. The first situation is .x2 ≥ 0 and . y2 ≥ 0. Then ( .

x1 = x2 − y2 − b = x1 − y1 − b − (x1 − y1 − d) − b = d − 2b, y1 = x2 − y2 − d = x1 − y1 − b − (x1 − y1 − d) − d = −b.

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I. Bula and A. S¯ıle

Since . y1 = −b < 0, then a contradiction is obtained, the cycle does not exist here. The second situation is .x2 ≥ 0 and . y2 < 0. Then ( .

x1 = x2 − y2 − b = x1 − y1 − b − (x1 − y1 − d) − b = d − 2b, y1 = x2 + y2 − d = x1 − y1 − b + (x1 − y1 − d) − d = 2x 1 − 2y1 − 2d − b = 2y2 − b.

Since . y1 = 2y2 − b < 0, then a contradiction is obtained, the cycle does not exist here. The third situation is .x2 < 0 and . y2 ≥ 0. Then .

x1 = −x2 − y2 − b = −x1 + y1 + b − (x1 − y1 − d) − b = −2x1 + 2y1 + d = = −2x1 + 2y1 + 2d − d = −2y2 − d.

Since .x1 = −2y2 − d < 0, then a contradiction is obtained, the cycle does not exist here. The fourth situation is .x2 < 0 and . y2 < 0. Then ( .

x1 = −x2 − y2 − b = −x1 + y1 + b − (x1 − y1 − d) − b = −2x1 + 2y1 + d, y1 = x2 + y2 − d = x1 − y1 − b + (x1 − y1 − d) − d = 2x1 − 2y1 − b − 2d.

Since .0 ≤ x1 + y1 = (−2x1 + 2y1 + d) + (2x1 − 2y1 − b − 2d) = −b − d < 0, then a contradiction is obtained, the cycle does not exist here. Similar simple observations show the non-existence of a cycle with period 2 in other cases. The proof of the non-existence of a cycle with period 3 is even longer. Suppose that there is a period-three solution, ...., (x0 , y0 ), .(x1 , y1 ), .(x2 , y2 ), .(x 0 , y0 ), .... The proof is performed by looking at all possible cases of the signs of .x0 , . y0 , .x1 , .x2 , . y1 , . y2 . For a cycle with period 3, the equalities must hold ( .

x1 = |x0 | − y0 − b, y1 = x0 − |y0 | − d,

(

x2 = |x1 | − y1 − b, and y2 = x1 − |y1 | − d,

(

x3 = |x2 | − y2 − b = x0 , y3 = x2 − |y2 | − d = y0 .

It is necessary to look at the four cases depending on the values of .x0 and . y0 . We start with case .x0 ≥ 0 and . y0 ≥ 0. Then ( .

x1 = x0 − y0 − b, y1 = x0 − y0 − d.

Again, four situations are possible. The first situation is .x1 ≥ 0 and . y1 ≥ 0. Then ( .

x2 = x1 − y1 − b = x0 − y0 − b − (x0 − y0 − d) − b = d − 2b, y2 = x1 − y1 − d = x0 − y0 − b − (x0 − y0 − d) − d = −b.

About a System of Piecewise Linear Difference …

39

Let .x2 = d − 2b ≥ 0. Then ( x3 = d − 2b − (−b) − b = d − 2b = x0 , . y3 = d − 2b − b − d = −3b = y0 . Since .−3b < 0 and . y0 ≥ 0, then a contradiction is obtained, the cycle does not exist here. Let .x2 = d − 2b < 0. Then ( x3 = −d + 2b + b − b = 2b − d = x0 , . y3 = d − 2b − b − d = −3b = y0 . Again, since .−3b < 0 and . y0 ≥ 0, then a contradiction is obtained, the cycle does not exist here. The second situation is .x1 ≥ 0 and . y1 < 0. Then ( .

x2 = x0 − y0 − b − x0 + y0 + d − b = d − 2b, y2 = x0 − y0 − b + x0 − y0 − d − d = 2(x0 − y0 − d) − b = 2y1 − b < 0.

Let .x2 = d − 2b ≥ 0. Then ( x3 = d − 2b − 2y1 + b − b = d − 2b − 2y1 > 0, . y3 = d − 2b + 2y1 − b − d = 2y1 − 3b = y0 . Since .2y1 − 3b < 0 and . y0 ≥ 0, then a contradiction is obtained, the cycle does not exist here. Let .x2 = d − 2b < 0. Then ( x3 = −d + 2b − 2y1 + b − b = −2y1 + 2b − d = x0 , . y3 = d − 2b + 2y1 − b − d = 2y1 − 3b = y0 . Again, since .2y1 − 3b < 0 and . y0 ≥ 0, then a contradiction is obtained, the cycle does not exist here. The third situation is .x1 < 0 and . y1 ≥ 0. Then ( .

x2 = −x0 + y0 + b − x0 + y0 + d − b = −2(x0 − y0 − d) − d = −2y1 − d < 0, y2 = x0 − y0 − b − x0 + y0 + d − d = −b < 0,

(

therefore .

x3 = 2y1 + d + b − b = −x2 > 0, y3 = −2y1 − d − b − d = −2y1 − 2d − b = y0 .

Since .−2y1 − 2d − b < 0 and . y0 ≥ 0, then a contradiction is obtained, the cycle does not exist here. The fourth situation is .x1 < 0 and . y1 < 0. Then

40

I. Bula and A. S¯ıle

( .

x2 = −x0 + y0 + b − x0 + y0 + d − b = −2x0 + 2y0 + d, y2 = x0 − y0 − b + x0 − y0 − d − d = 2x0 − 2y0 − 2d − b = 2y1 − b < 0.

Let .x2 = −2x0 + 2y0 + d ≥ 0. Then ( .

x3 = −2x0 + 2y0 + d − 2x0 + 2y0 + 2d + b − b = −4x0 + 4y0 + 3d = 2x2 + d > 0, y3 = −2x0 + 2y0 + d + 2x0 − 2y0 − 2d − b − d = −2d − b = y0 .

Since .−2d − b < 0 and . y0 ≥ 0, then a contradiction is obtained, the cycle does not exist here. The second case .x0 ≥ 0 and . y0 < 0 is similar, four subcases need to be considered here, but at .x1 ≥ 0 and . y1 < 0 the equilibrium point .(b, −d) will be found. If we consider third situation .x0 < 0 and . y0 ≥ 0, then from system ( .

x1 = −x0 − y0 − b, y1 = x0 − y0 − d < 0

it follows that only two cases should be considered. The first situation is .x1 ≥ 0. Then ( x2 = −x0 − y0 − b − x0 + y0 + d − b = −2x0 − 2b + d, . y2 = −x0 − y0 − b + x0 − y0 − d − d = −2y0 − b − 2d < 0. Let .x2 = −2x0 − 2b + d ≥ 0. Then ( x3 = −2x0 − 2b + d + 2y0 + b + 2d − b = −2x0 + 2y0 − 2b + 3d = x0 , . y3 = −2x0 − 2b + d − 2y0 − b − 2d − d = −2x0 − 2y0 − 3b − 2d = y0 . From this system we obtain . y0 = 5b+12d < 0, it is contradiction to . y0 ≥ 0. −13 Let .x2 = −2x0 − 2b + d < 0. Then ( x3 = 2x0 + 2b − d + 2y0 + b + 2d − b = 2x0 + 2y0 + 2b + d = x0 , . y3 = −2x0 − 2b + d − 2y0 − b − 2d − d = −2x0 − 2y0 − 3b − 2d = y0 . From this system we obtain . y0 = −b < 0, it is contradiction to . y0 ≥ 0. The second situation is .x1 < 0. Then ( x2 = x0 + y0 + b − x0 + y0 + d − b = 2y0 + d > 0, . y2 = −x0 − y0 − b + x0 − y0 − d − d = −2y0 − b − 2d < 0, therefore x = 2y0 + d + 2y0 + b + 2d − b = 4y0 + 3d = x0 .

. 3

Since .4y0 + 3d > 0 and .x0 < 0, then a contradiction is obtained, the cycle does not exist here.

About a System of Piecewise Linear Difference …

41

If we consider last fourth situation .x0 < 0 and . y0 < 0, then ( .

x1 = −x0 − y0 − b, y1 = x0 + y0 − d < 0,

therefore similar to the previous situation only two cases should be considered. The first situation is .x1 ≥ 0. Then ( x2 = −x0 − y0 − b − x0 − y0 + d − b = −2x0 − 2y0 − 2b + d, . y2 = −x0 − y0 − b + x0 + y0 − d − d = −b − 2d < 0. Let .x2 = −2x0 − 2y0 − 2b + d ≥ 0. Then ( .

x3 = −2x0 − 2y0 − 2b + d + b + 2d − b = −2x0 − 2y0 − 2b + 3d = x0 , y3 = −2x0 − 2y0 − 2b + d − b − 2d − d = −2x0 − 2y0 − 3b − 2d = y0 .

From this system we obtain .x0 = 13d > 0, it is contradiction to .x0 < 0. 5 Let .x2 = −2x0 − 2y0 − 2b + d < 0. Then ( .

x3 = 2x0 + 2y0 + 2b − d + b + 2d − b = 2x0 + 2y0 + 2b + d = x0 , y3 = −2x0 − 2y0 − 2b + d − b − 2d − d = −2x0 − 2y0 − 3b − 2d = y0 .

From this system we obtain .x0 = −d < 0 and . y0 = −b < 0. But then ( .

x1 = b + d − b = d > 0, y1 = −b − d − d = −b − 2d < 0,

and .x2 = d + b + 2d − b = 3d > 0 which contradicts the condition that .x2 < 0. Therefore, we have looked at all the cases, the cycle with period 3 does not exist. ◻ It still remains open as to the existence of a solution of system (3) with a cycle of prime period-four. Analytically finding cycles for nonlinear difference equations is a very difficult process, as there are many branching possibilities to be considered (see, for example, [2] for rational difference equation). The first cycle that happened to be observed and described is with period . p = 7. Theorem 4 If .d ≥ 2b, then there is a solution of system (3) with period . p = 7: .{(d, b), .(d − 2b, −b), .(d − 2b, −3b), .(d, −5b), .(4b + d, −5b), .(8b + d, −b), .(8b + d, 7b)}. Proof Let .x0 = d > 0 and . y0 = b > 0. Then ( .

x1 = d − b − b = d − 2b ≥ 0 (by assumption of this theorem d ≥ 2b), y1 = d − b − d = −b < 0,

42

I. Bula and A. S¯ıle

( .

x2 = d − 2b + b − b = d − 2b ≥ 0, y2 = d − 2b − b − d = −3b < 0,

(

x3 = d − 2b + 3b − b = d > 0, y3 = d − 2b − 3b − d = −5b < 0,

.

( .

( .

x5 = 4b + d + 5b − b = 8b + d > 0, y5 = 4b + d − 5b − d = −b < 0,

( .

x6 = 8b + d + b − b = 8b + d > 0, y6 = 8b + d − b − d = 7b > 0,

( .

x4 = d + 5b − b = 4b + d > 0, y4 = d − 5b − d = −5b < 0,

x7 = 8b + d − 7b − b = d = x0 , y7 = 8b + d − 7b − d = b = y0 .

◻

But what happens if .d − 2b < 0? Experimentally, we observed that several cycles of different periods contain point .(d − 2b, −b). Let’s start with this point. ( .

( .

( .

( .

x0 = d − 2b < 0, y0 = −b < 0,

x1 = −d + 2b + b − b = 2b − d > 0, y1 = d − 2b − b − d = −3b < 0,

x2 = 2b − d + 3b − b = 4b − d > 0, y2 = 2b − d − 3b − d = −b − 2d < 0,

x3 = 4b − d + b + 2d − b = 4b + d > 0, y3 = 4b − d − b − 2d − d = 3b − 4d.

(5)

There are two possibilities here: . y3 ≥ 0, if .0 < d ≤ 43 b, or . y3 < 0, if . 43 b < d < 2b. We consider the first possibility: ( .

( .

x3 = 4b + d > 0, y3 = 3b − 4d ≥ 0,

x4 = 4b + d − 3b + 4d − b = 5d > 0, y4 = 4b + d − 3b + 4d − d = b + 4d > 0,

About a System of Piecewise Linear Difference …

43

Fig. 1 Solution of difference Eq. (3) with .b = 1, .d = 1.8747, .x0 = d − 2b = −0.1253 and . y0 = −b = −1. The points on the gray line are the .xn points of the solution, while the black line is the . yn points, .n = 0, 1, .... The solution is periodic with period . p = 27

( .

x5 = 5d − b − 4d − b = d − 2b = x0 , y5 = 5d − b − 4d − d = −b = y0 .

So we have proved a new result. Theorem 5 If .0 < d ≤ 43 b, then there is a solution of system (3) with period . p = 5: .{(d − 2b, −b), .(2b − d, −3b), .(4b − d, −b − 2d), .(4b + d, 3b − 4d), .(5d, b + 4d)}. If we consider in (5) the case . y3 = 3b − 4d < 0 or . 43 b < d, then we can obtain cycle with period . p = 7. But a new branch appears again, as a result, we will get several cases. b ≤ d < 2b, then there is a solution of system (3) with period . p = Theorem 6 If . 15 8 7:.{(d − 2b, −b),.(2b − d, −3b),.(4b − d, −b − 2d),.(4b + d, 3b − 4d),.(5d, 7b − 4d), .(9d − 8b, 7b), .(−16b + 9d, −15b + 8d)}. b = 1.875b, a cycle with starting point Looking at the exact case .d = 1.87b < 15 8 .(d − 2b, −b) with period 20 was found. Considering the behavior of the specific cycle (positive and negative coordinates), it was possible to determine the constraints of the parameters. 2047 b < d < 1092 b, then there is a solution of system (3) with period Theorem 7 If . 256 137 . p = 20: .{(d − 2b, −b), .(2b − d, −3b), .(4b − d, −b − 2d), .(4b + d, 3b − 4d), .(5d, 7b − 4d), .(9d − 8b, 7b), .(−16b + 9d, −15b + 8d), .(d − 2b, −31b + 16d), .(32b − 17d, −33b + 16d), .(64b − 33d, −b − 2d), .(64b − 31d, 63b − 36d), .(5d, 127b − 68d), .(−128b + 73d, 127b − 64d), .(−256b + 137d, −255b + 136d), .(−2b + d, −511b + 272d),.(512b − 273d, −513b + 272d),.(1024b − 545d, −b − 2d), .(1024b − 543d, 1023b − 548d), .(5d, 2047b − 1092d), .(−2048b + 1097d, −2047b + 1096d)}. 2047 256 Since . 137 b < d < 1092 b ≈ 1.874542b < 15 b = 1.875b, let’s look at an example 8 with .b = 1 and .d = 1.8747 with .x0 = d − 2b = −0.1253 and . y0 = −b = −1. We obtain cycle with period . p = 27 (see Fig. 1).

44

I. Bula and A. S¯ıle

The case with period . p = 12 is interesting. Depending on which of the intervals the parameter .d = di , .i = 1, 2, 3, is selected .

112 14 7 8 b ≤ d3 < b ≤ d2 < b ≤ d1 < b, 137 17 8 9

a cycle with a period of . p = 12 is obtained, but the points of the cycle are calculated with different formulas. Theorem 8 If . 87 b ≤ d < 98 b, then there is a solution of system (3) with period . p = 12: .{(d − 2b, −b), .(2b − d, −3b), .(4b − d, −b − 2d), .(4b + d, 3b − 4d), .(5d, 7b − 4d), .(−8b + 9d, −7b + 8d), .(14b − 17d, −b), .(−14b + 17d, 13b − 18d), .(−28b + 35d, −b − 2d), .(−28b + 37d, −29b + 32d), .(5d, −57b + 68d), .(56b − 63d, 57b − 64d)}. Proof Since .d < 98 b, then

( .

( .

( .

( .

( .

( .

( .

x5 = 5d − 7b + 4d − b = −8b + 9d < 0 (since d < 98 b), y5 = 5d − 7b + 4d − d = −7b + 8d ≥ 0 (since 87 b ≤ d),

x6 = 8b − 9d + 7b − 8d − b = 14b − 17d < 0 (since y6 = −8b + 9d + 7b − 8d − d = −b < 0, (

.

x2 = 2b − d + 3b − b = 4b − d > 0, y2 = 2b − d − 3b − d = −b − 2d < 0,

x4 = 4b + d − 3b + 4d − b = 5d > 0, y4 = 4b + d + 3b − 4d − d = 7b − 4d > 0 (since d < 89 b < 47 b), (

(

x1 = −d + 2b + b − b = 2b − d > 0, y1 = d − 2b − b − d = −3b < 0,

x3 = 4b − d + b + 2d − b = 4b + d > 0, y3 = 4b − d − b − 2d − d = 3b − 4d < 0 (since 43 b < 78 b ≤ d),

.

.

x0 = d − 2b < 0, y0 = −b < 0,

x7 = −14b + 17d + b − b = −14b + 17d ≥ 0, y7 = 14b − 17d − b − d = 13b − 18d < 0 (since

14 b 17

13 b 18

< 87 b ≤ d),

< 78 b ≤ d),

7 x8 = −14b + 17d − 13b + 18d − b = −28b + 35d > 0 (since 28 35 b < 8 b ≤ d), y8 = −14b + 17d + 13b − 18d − d = −b − 2d < 0,

x9 = −28b + 35d + b + 2d − b = −28b + 37d > 0 (since 28 b < 78 b ≤ d), 37 29 b), y9 = −28b + 35d − b − 2d − d = −29b + 32d < 0 (since d < 98 b < 32

About a System of Piecewise Linear Difference …

( .

x10 = −28b + 37d + 29b − 32d − b = 5d > 0, 57 b < 7 b ≤ d), y10 = −28b + 37d − 29b + 32d − d = −57b + 68d > 0 (since 68 8

( .

45

x11 = 5d + 57b − 68d − b = 56b − 63d > 0, (since d < 89 b = y11 = 5d + 57b − 68d − d = 57b − 64d > 0 (since d < 89 b < ( .

x12 = 56b − 63d − 57b + 64d − b = d − 2b = x0 , y12 = 56b − 63d − 57b + 64d − d = −b = y0 .

56 b), 63 57 b), 64

◻

If . y5 = −7b + 8d < 0, then we obtain another cycle with period . p = 12, the formulas of the first 6 elements coincide with those obtained in Theorem 8. b ≤ d < 87 b, then there is a solution of system (3) with period . p = Theorem 9 If . 14 17 12: .{(d − 2b, −b), .(2b − d, −3b), .(4b − d, −b − 2d), .(4b + d, 3b − 4d), .(5d, 7b − 4d); .(−8b + 9d, −7b + 8d), .(14b − 17d, −15b + 16d), .(d, −b − 2d); .(3d, −b − 2d), .(5d, −b), .(5d, 4d − b), .(d, b)}. .

In Theorem 9, .x6 = 14b − 17d < 0, otherwise we get another cycle with period p = 12.

b ≤ d < 14 b, then there is a solution of system (3) with period Theorem 10 If . 112 137 17 . p = 12: .{(d − 2b, −b), .(2b − d, −3b), .(4b − d, −b − 2d), .(4b + d, 3b − 4d), .(5d, 7b − 4d); .(−8b + 9d, −7b + 8d), .(14b − 17d, −15b + 16d), .(28b − 33d, −b − 2d), .(28b − 31d, 27b − 36d), .(5d, 55b − 68d), .(−56b + 73d, 55b − 64d), .(−112b + 137d, −111b + 136d)}. b ≈ 0.817518b, then a cycle with period . p = 24 is formed. If .0.817b < d < 112 137 For example, if.b = 1 and.d = 0.8174 with.x0 = d − 2b = −1.1826 and. y0 = −b = −1, we obtain cycle with period . p = 24 (see Fig. 2). If .d = 0.817b is chosen exactly, then the result is a cycle with a period of . p = 36 (see Fig. 3). This is the largest period that has been observed. In case .b = d, a cycle with period . p = 6 is obtained. In the general case, the parameter boundaries are wider.

Fig. 2 Solution of system (3) with .b = 1, .d = 0.8174, .x0 = d − 2b = −1.1826 and . y0 = −b = −1. The solution is periodic with period . p = 24

46

I. Bula and A. S¯ıle

Fig. 3 Solution of system (3) with.b = 10,.d = 8.17,.x0 = d − 2b = −11.83 and. y0 = −b = −10. The solution is periodic with period . p = 36

Theorem 11 If . 89 b ≤ d < 47 b = 1.75b, then there is a solution of system (3) with period . p = 6: .{(d − 2b, −b), .(2b − d, −3b), .(4b − d, −b − 2d), .(4b + d, 3b − 4d), .(5d, 7b − 4d); .(−8b + 9d, −7b + 8d)}. An interesting fact is that these 6 points of the cycle completely coincide with the first 6 points of the cycle with period 12 found in Theorem 8, 9 and 10. The difference is that in the case of period 6 the coordinates of the last point are positive, but in the case of period 12 this is not the case. Let us note that at parameter equality .b = d another cycle with period 6 is found, which is different from the one discussed in Theorem 11. Theorem 12 If .d = b, then there is a solution of system (3) with period . p = 6: b), .( 75 b, −3b), .( 17 b, − 13 b), .(5b, − 15 b), .( 21 b, 19 b)}. {(− 53 b, − 35 b), .( 15 b, − 11 5 5 5 5 5

.

We consider example with .b = d = 10. Then one cycle starts with .(−10, −10) and the other with .(−6, −6). Convex hull of both cycles intersects such that it is difficult to imagine how one can identify sets of initial conditions whose sequences of solutions tends to one of the cycles (Fig. 4). Special case .b = d = 1 is investigated in W. Tikjha dissertation [13], where it is proved that all solutions, except the equilibrium, are eventually periodic for one of the two cycles with period 6 (see [9], P. 132). One of the boundaries for cycle with period . p = 19 have also been found. 1919 b ≤ d ≤ 1092 b ≈ 1.7573b, then there is a solution of Theorem 13 If .1.7518b ≈ 240 137 system (3) with period . p = 19: .{(d − 2b, −b), .(2b − d, −3b), .(4b − d, −b − 2b), .(4b + d, 3b − 4d), .(5d, 7b − 4d), .(9d − 8b, 7b), .(9d − 16b, 8d − 15b), .(−17d + 30b, 16d − 31b), .(−33d + 60b, −b − 2d), .(−31d + 60b, −36d + 59b), .(5d, 119b − 68d), .(73d − 120b, 119b − 64d), .(137d − 240b, 136d − 239b), .(d − 2b, 272d − 479b), .(480b − 273d, 272d − 481b), .(960b − 545d, −b − 2d), .(960b − 543d, 959b − 548d), .(5d, 1919b − 1092d), .(1097d − 1920b, 1096d − 1919b)}.

This interval is relatively narrow. If .d ≤ 1.75b, then cycle with period 6 exist (see b other cycles are possible: with Theorem 11). But between .1.75b and .1.7518b ≈ 240 137 .b = 1 and .d = 1.7505 is other cycle with period . p = 19, with .b = 1 and .d = 1.7504 is cycle with period . p = 25 (see Fig. 5).

About a System of Piecewise Linear Difference …

47

Fig. 4 Solution of difference Eq. (3) with .b = d = 10. Two different cycles with period . p = 6 are observed: gray points form a cycle with initial point.(−6, −6), black points form a cycle with initial point .(−10, −10)

Fig. 5 Solution of difference Eq. (3) with .b = 1, .d = 1.7504, .x0 = d − 2b = −0.2496 and . y0 = −b = −1. The solution is periodic with period . p = 25

1919 If we consider parameters inequality . 1092 b ≈ 1.7573b < d then is possible find cycle with period . p = 13, it is by equality .d = 1.8b (see Fig. 6). We have same special cases obtained if.0.75b < d < 0.817b, then there are cycles b ≈ 0.756b, then. p = 16; with such periods that we did not mention before: if.d = 34 45 58 if .d = 0.8b, then . p = 11; if .d = 0.81b, then . p = 17; if .d = 71 b ≈ 0.8169b, then . p = 18; if .d = 0.81695b, then . p = 30. As the last one, let’s note again the cycle with the period . p = 7, which exists together with the period . p = 5 (Theorem 5), when .d < 0.75b. This cycle does not contain point .(d − 2b, −b). And as we see in Fig. 7 in this case convex hulls of both cycles do not intersect.

Theorem 14 If .0 < d < b8 , then there is a solution of system (3) with period . p = 7: .{(−d, −b), .(d, −2d − b), .(3d, −2d − b), .(5d, −b), .(5d, 4d − b), .(d, 8d − b), .(−7d, 8d − b)}.

48

I. Bula and A. S¯ıle

Fig. 6 Solution of difference Eq. (3) with .b = 10, .d = 18, .x0 = d − 2b = −2 and . y0 = −b = −10. The solution is periodic with period . p = 13

Fig. 7 Solution of difference Eq. (3) with .b = 10 and .d = 1 (.d = 0.1b). Two different cycles with period . p = 5 and . p = 7 are observed: lagest convex hull is for cycle with period . p = 5 with initial conditions .(d − 2b, −b) = (−19, −10), smalest convex hull is for cycle with period . p = 7 with initial conditions .(−d, −b) = (−1, −10)

4 Conclusion Table 2 summarizes all the different cycles found with the given period depending on the values of parameters .b and .d of system (3). All possible values of parameters .b and .d are not considered. Note that there are three larger parameter regions with one period: .0 < d < 0.75b with . p = 5, . 89 b ≤ d ≤ 47 b with . p = 6 and .2b ≤ d with . p = 7. If we look at cycles with a period greater than 7, for example with a period of 25 (Fig. 5), then it looks like it consists of several smaller cycles-loops with a period of 6, 6, 6 and 7 but for cycle with a period of 36 (see Fig. 3) the cycles-loops are with periods of 5, 6, 6, 6, 6 and 7. In all cycles found so far, subcycles-loops consist of periods 5, 6 and 7. This would explain why there are no cycles with period 8 and 9, but this does not explain why there are no cycles with periods 10, 14, 15 and 21, and

About a System of Piecewise Linear Difference …

49

Table 2 Constraints of parameters .b, .d, .b > 0, d > 0, and cycle with period . p Period Constraint Other cycle Constraint .0

m for some .m, M > 0 and for all .n. Then x

. n+1

=

sup{ai' } 1 sup{bi' } a' b' yn a' 1 an' + bn' yn b' = ' n + n' < n' + n' M ≤ + M.. ' ' Cn xn Cn xn Cn xn Cn m Cn inf{Ci } m inf{Ci' } ◻

Lemma 4 Let .xn+1 , . yn+1 be as defined in (1). If .{xi } is bounded above, then .{yi } is bounded above. That is, if .{yi } is unbounded, then .{xi } is also unbounded. a ' +b' y

Proof Suppose .xn < M for some . M > 0 and for all .n. Since .xn+1 = nC ' xnn n , this n means ' 2 ' ' ' 2 ' M sup{Ci } xn+1 Cn xn − an xn+1 Cn xn M Cn .. . yn = ≤ < ≤ bn' bn' bn' inf{bi' } ◻ Lemma 5 Let .xn+1 , . yn+1 be as defined in (1). If .{xi } is bounded below by a positive constant and .{yi } is bounded above, then .{xi } is bounded above. Proof Suppose .xn > m and . yn < M for some .m, M > 0 and for all .n. Then x

. n+1

=

sup{ai' } + sup{bi' }M an' + bn' yn an' + bn' M ≤ .. < Cn' xn Cn' m inf{Ci' }m ◻

We consider numbered families for which both .(8, y) and .(23, y) have the same boundedness character in Sect. 2. Then in Sect. 3 we look at the numbered families where the boundedness character between .(8, y) and .(23, y) are different. We give a conclusion as well as an open question in Sect. 4. Section 5 contains a table of all numbered families contained within (1), with the boundedness character of that numbered family with positively bounded coefficients.

Boundedness of Solutions of …

55

2 Main Results, Part I In this section, we present theorems for the cases when the numbered families .(8, y) and .(23, y) have the same boundedness character. We break the section up into three subsections, for the cases when the boundedness character is (U, U), (U, B), and (B, B) respectively.

2.1 Boundedness Character of .(8, y) and .(23, y) is (U, U) We now present a theorem which shows that the boundedness character of the numbered families .(8, 2) and .(23, 2) is (U, U). Theorem 1 of [7] establishes that with constant coefficients, these two numbered families have boundedness character (U, B). The boundedness character of both of these numbered families changes when the coefficients are relaxed to being positively bounded sequences, and hence, the non-constant coefficient sequences destroy the boundedness character. Theorem 1 Numbered families .(8, 2) and .(23, 2) have boundedness character (U, U). a ' +b' y

Proof For each .n ≥ 0, let .xn+1 = nC ' xnn n and . yn+1 = Cannyn , where .{ai' } is either posn itively bounded or identically zero, and all other given coefficient sequences are C2n y2n , we can choose coefficient sequences such positively bounded. . y2n+2 = aa2n+1 2n C 2n+1 C2n that . aa2n+1 > K for some . K > 1 and for all .n. Then .{yi } will be unbounded, so .{xi } 2n C 2n+1 will also be unbounded by Lemma 4. ◻

We do not explicitly mention those cases for which the boundedness character of the numbered family was shown in [7] to be (U, U) in the constant coefficient case, since any constant coefficient sequence is positively bounded, and hence the proof would be the same.

2.2 Boundedness Character of .(8, y) and .(23, y) is (U, B) We now present several theorems which establish a (U, B) boundedness character for numbered families contained in (3) and (4). In Theorems 2, 3, and 4, we show that the boundedness character with positively bounded coefficient sequences is unchanged from the boundedness character of these numbered families with constant coefficients, as presented in [7]. Moreover, since in those cases we know from [7] that there exist initial conditions and coefficient sequences such that .{xi } is unbounded, it only remains to prove the boundedness of .{yi }. Theorem 2 If.k ∈ {8, 23} and.l ∈ {17, 18, 32, 36, 39, 44, 45, 48, 49}, then the numbered family .(k, l) has boundedness character (U, B).

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Z. A. Kudlak and R. P. Vernon

Proof With the exception of .(x, 48), in each of these families, .{yi } is bounded above n x n +cn yn , we know that .{yi } is by Lemma 1. In the case of .(x, 48), since . yn+1 = anB+b n x n +C n yn bounded below by some positive constant .m. Further, we can show that .{yi } is also +cn yn bounded above, since . yn+1 < Cannm + Bbnn xxnn +C . ◻ n yn Theorem 3 Numbered families .(8, 33) and .(23, 33) have boundedness character (U, B). a ' +b' y

n xn Proof For each .n ≥ 0, let .xn+1 = nC ' xnn n and . yn+1 = Ban nx+b , where .{ai' } is either n +C n yn n positively bounded or identically zero, and all other given coefficient sequences are positively bounded. b' y b' B x y +b' C yn2 +an Cn' xn +bn Cn' xn2 n xn Since .xn+1 + yn+1 ≥ Cn' xnn + Ban nx+b = n n n nBn Cn' xn2 +C , we know ' n +C n yn n C n x n yn n n n .{x i + yi } is bounded below by some positive constant .m from Lemma 1. Therefore sup{ai } an . ≤ min{inf{B is bounded above by a positive constant for all .n. FurBn xn +Cn yn i },inf{C i }}m bn xn thermore, . Bn xn +Cn yn is bounded above by Lemma 1. xn n + Bn xbnn+C . ◻ Therefore .{yi } is bounded above since . yn+1 = Bn xna+C n yn n yn

Theorem 4 Numbered families .(8, 35), .(23, 35), .(8, 47), and .(23, 47) have boundedness character (U, B). a ' +b' y

n x n +cn yn Proof For each .n ≥ 0, let .xn+1 = nC ' xnn n and . yn+1 = an +b , where .{ai' } and An +Bn xn n .{ai } are each either positively bounded or identically zero, and all other given coefficient sequences are positively bounded. Since

.

1 + xn+1 An + Bn xn b' yn + C ' xn ≥ n ' n yn+1 Cn xn an + bn xn + cn yn =

An bn' yn + An Cn' xn + Bn bn' xn yn + Bn Cn' xn2 , an Cn' xn + bn Cn' xn2 + cn Cn' xn yn

i we know .{ 1+x } is bounded below by some positive constant .m from Lemma 1, so ) (yi sup{ci } cn yn 1 . ≤ is bounded above by a positive constant for all .n. An +Bn xn min{inf{Ai },inf{Bi }} m

Therefore, since . yn+1 =

an +bn xn An +Bn xn

+

cn yn , An +Bn xn

we know .{yi } is bounded above.

◻

In Theorem 5, we show that the boundedness character with positively bounded coefficient sequences has changed from the boundedness character of these numbered families with constant coefficients, as presented in [7]. Theorem 5 If.k ∈ {8, 23} and.l ∈ {1, 5, 9, 10, 13, 20, 28}, then the numbered family (k, l) has boundedness character (U, B).

.

Boundedness of Solutions of …

57

Proof First note that, with the exception of .(k, 20), .{yi } is bounded above by n yn , we know that .{yi } is bounded below Lemma 1. For .(k, 20), since . yn+1 = anC+c n yn an by some positive constant .m, so . yn+1 ≤ Cn m + Ccnn , meaning .{yi } is also bounded +cn yn , where each of the coefficient sequences is either above. In all cases, . yn+1 = Aann +C n yn positively bounded or identically zero. Then we can choose appropriate values for the positively bounded sequences such that .{yi } is bounded above and below by some positive constants . M y and .m y , respectively, and that these bounds can be expressed solely in terms of the coefficient sequences for . y. Then . x 2n+2

' C ' (a ' C ' (a ' a' + b' y2n+1 ) + b2n+1 my) + b' y2n+1 x2n ≥ 2n' 2n+1' x2n .. = 2n+1 ' 2n+1 = 2n ' 2n+1 ' 2n+1 ' ' C2n+1 x2n+1 C2n+1 (a2n + b2n y2n ) C2n+1 (a2n + b2n M y )

Choose the remaining coefficient sequences such that. . K > 1 and for all .n. Then .{x i } will be unbounded.

' ' ' C2n (a2n+1 +b2n+1 my) ' ' ' C2n+1 (a2n +b2n My )

> K for some ◻

2.3 Boundedness Character of .(8, y) and .(23, y) is (B, B) We complete this section of the paper by presenting theorems for numbered families whose boundedness character remains (B, B). In these numbered families the boundedness character is unchanged when constant coefficients are replaced by positively bounded coefficients. Theorem 6 Numbered families .(8, 21) and .(23, 21) have boundedness character (B, B). a ' +b' y

n yn Proof For each .n ≥ 0, let .xn+1 = nC ' xnn n and . yn+1 = anB+c , where .{ai' } is either n xn n positively bounded or identically zero, and all other given coefficient sequences are Cn' xn C' +cn yn n+1 n yn = anB+c = Bnn aan' +b , we know by Lemma 1 positively bounded. Since . xyn+1 ' an' +bn' yn n xn n n yn yi that .{ xi } is bounded below, so .{xi } is bounded below by Lemma 2 and .{yi } is bounded

C'

n+1 below since . yn+1 ≥ Bcnn yxnn . Since .{yi } is bounded below and . xyn+1 ≤ Bnn ( ba' nyn + bcn' ), we n n can see that .{ xyii } is bounded above, so .{xi } is bounded above by Lemma 3, implying ◻ that .{yi } is also bounded above by Lemma 4.

Theorem 7 Numbered families .(8, 27), (23, 27), (8, 42) and .(23, 42) have boundedness character (B, B). a ' +b' y

n yn , where .{ai' } and Proof For each .n ≥ 0, let .xn+1 = nC ' xnn n and . yn+1 = an +bBn xn nx+c n n .{ai } are each either positively bounded or identically zero, and all other given coefficient sequences are positively bounded. Then

.

yn+1 cn yn an + bn xn + cn yn Cn' xn Cn' an + bn xn + cn yn Cn' = = > .. ' ' ' ' ' xn+1 Bn xn an + bn yn Bn an + bn yn Bn an + bn' yn

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Z. A. Kudlak and R. P. Vernon

Since .{yi } is bounded below by Lemma 1, we know .{ xyii } is bounded below from the inequality above, so .{xi } is bounded below by Lemma 2 and .{ xyii } is bounded above. C'

n+1 ≤ Bnn ( ba' nyn + bbn' xynn + bcn' ) and .{ y1i } and .{ xyii } are bounded above, we can see Since . xyn+1 n n n that .{ xyii } is also bounded above. Then .{xi } is bounded above by Lemma 3, implying .{yi } is also bounded above by Lemma 4. ◻

Theorem 8 Numbered families .(8, 24) and .(23, 24) have boundedness character (B, B). a ' +b' y

n xn , where .{ai' } is Proof For each .n ≥ 0, let .xn+1 = nC ' xnn n and . yn+1 = anB+b n xn n either positively bounded or identically zero, and all other given coefficient sequences are positively bounded. Since .{yi } is bounded below by Lemma 1 and an' +bn' yn +Cn' xn a ' +b' y +C ' x a ' +b' y C' 1+xn+1 n xn . = anB+b = CBn' n ann+bn n xn n n ≥ CBn' min{ n ann n , bnn }, we know yn+1 C ' xn n xn n

n

n

bn' yn is bounded above. an +bn xn an' bn' yn Bn ( + an +b ). Then .{ xyii } is Cn' an +bn xn n xn

i { 1+x } is bounded below. Therefore, yi

.

.

Then

a ' +b' y a ' +b' y n xn = nC ' xnn n anB+b = CBn' ann +bnn xnn = bounded n xn n n yi above, meaning .{ xi } is bounded below, so .{xi } is bounded below by Lemma 2. But an . yn+1 = + Bbnn , so .{yi } is bounded above. Since .{xi } is bounded below and .{yi } Bn xn

xn+1 . yn+1

is bounded above, we have .{xi } is bounded above by Lemma 5.

◻

3 Main Results, Part II In the following section we present theorems for cases in which numbered families (8, y) and.(23, y) do not have the same boundedness character. It is interesting to note that there are only two scenarios in which this happens. In one case, .(8, y) remains (B, B), but .(23, y) loses boundedness and becomes (U, B). In the other scenario, .(23, y) remains (B, B), but .(8, y) loses boundedness and becomes (U, U). .

3.1 Numbered Families for Which .(8, y) is (B, B) but .(23, y) is (U, B) The next theorems establish cases for which the numbered family will have boundedness character (B, B) when family is as in (3), and will have boundedness character (U, B), when the family is as in (4). Theorem 9 Numbered families .(8, 6), (8, 14), (8, 15), and .(8, 38) have boundedness character (B, B). b' y

Proof For each .n ≥ 0, let .xn+1 = Cn' xnn and . yn+1 = An +Bcnnxynn+Cn yn , where .{Ai } and n .{C i } are each either positively bounded or identically zero, and all other given coefficient sequences are positively bounded. Since

Boundedness of Solutions of …

.

59

Cn' xn Cn' xn yn+1 cn yn cn = = ' ,. ' xn+1 An + Bn xn + Cn yn bn yn bn An + Bn xn + Cn yn

we know that.{ xyii } is bounded above by Lemma 1, which means.{xi } is bounded above since .xn+1 =

bn' yn . Cn' xn

This implies .{yi } is also bounded above by Lemma 4.

◻

Theorem 10 Numbered families .(8, 11), (8, 29), (8, 37), and .(8, 43) have boundedness character (B, B). b' y

an +cn yn , where.{ci } and.{Ci } Proof For each.n ≥ 0, let.xn+1 = Cn' xnn and. yn+1 = An +B n x n +C n yn n are each either positively bounded or identically zero, and all other given coefficient sequences are positively bounded. If .{ci } is identically zero or .{Ci } is positively bounded, then .{yi } is bounded above by Lemma 1. In the case where .{ci } is positively bounded and .{Ci } is identically zero, note that

y

. n+2

+cn yn an+1 + cn+1 Aann +B an+1 + cn+1 yn+1 n xn = = b' y An+1 + Bn+1 xn+1 An+1 + Bn+1 Cn' xnn n

an+1 An Cn' xn + an+1 Bn Cn' xn2 + cn+1 an Cn' xn + cn+1 cn Cn' xn yn = , An+1 An Cn' xn + An+1 Bn Cn' xn2 + Bn+1 An bn' yn + Bn+1 Bn bn' xn yn so .{yi } is bounded above by Lemma 1. Let . M y be an upper bound for .{yi }. y

. n+1

Then .{yi + all .n.

yi xi

+

( ) C ' xn yn+1 an + cn yn 1 + 'n = xn+1 An + Bn xn + Cn yn bn yn bn' yn + Cn' xn an ≥ An + Bn xn + Cn yn bn' yn an bn' yn + an Cn' xn = ' An bn yn + Bn bn' xn yn + Cn bn' yn2 ( ( an Cn' xn an bn' yn ≥ min , Bn bn' xn yn An bn' yn + Cn bn' yn2 ( ( an bn' an Cn' , = min Bn bn' yn An bn' + Cn bn' yn ( ( an bn' an Cn' , ≥ min Bn bn' M y An bn' + Cn bn' M y

} is bounded below, so .bn' Bn yn + bn' An xynn > m for some .m > 0 and for

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Z. A. Kudlak and R. P. Vernon

.

an + cn yn yn+1 C ' xn = 'n xn+1 bn yn An + Bn xn + Cn yn Cn' an xn + Cn' cn xn yn = ' bn An yn + bn' Bn xn yn + bn' Cn yn2 C ' an xn + Cn' cn xn yn ≤ 'n bn An yn + bn' Bn xn yn C ' an + Cn' cn M y C ' an + Cn' cn yn = ' n yn < n ' bn An xn + bn Bn yn m

Therefore .{ xyii } is bounded above, which means .{xi } is bounded above since x

. n+1

=

bn' yn . Cn' xn

◻

Theorem 11 Numbered families .(23, 6), (23, 14), and .(23, 29) have boundedness character (U, B). a ' +b' y

+cn yn Proof For each .n ≥ 0, let .xn+1 = nC ' xnn n and . yn+1 = Aann +B , where .{ai } and .{Ai } n xn n are each either positively bounded or identically zero, .{Ai } is identically zero only if .{ai } is identically zero, and all other given coefficient sequences are positively bounded. If .{Ai } is identically zero, then

y

. 2n+2

=

' ' C2n x2n c2n+1 y2n+1 c2n+1 c2n y2n c2n+1 c2n C2n = < ' ' ' .. B2n+1 x2n+1 B2n+1 B2n x2n a2n + b2n y2n B2n+1 B2n b2n

If .{Ai } is positively bounded, then y

. 2n+2

a2n+1 c2n+1 y2n+1 + A2n+1 + B2n+1 x2n+1 A2n+1 + B2n+1 x2n+1 a2n+1 c2n+1 y2n+1 < + A2n+1 B2n+1 x2n+1 ' C2n x2n c2n+1 a2n + c2n y2n a2n+1 + = ' ' A2n+1 B2n+1 A2n + B2n x2n a2n + b2n y2n ' C2n x2n a2n+1 c2n+1 a2n + c2n y2n = + . ' ' A2n+1 B2n+1 a2n + b2n y2n A2n + B2n x2n =

In either case, by Lemma 1 we know . y2n+2 is bounded above by some . M y for all ' n, and the value of . M y does not depend on the values of .{a2n+1 }. Similarly, we can show that . y2n+1 is bounded above, so .{yi } is bounded above. Next, note that

.

Boundedness of Solutions of …

x

. 2n+2

61 ' ' a2n+1 + b2n+1 y2n+1 ' C2n+1 x2n+1 + b' y2n+1 ) C ' (a ' x2n = 2n ' 2n+1 ' 2n+1 ' C2n+1 (a2n + b2n y2n ) ' ' a2n+1 C2n x2n . > ' ' ' C2n+1 (a2n + b2n My )

=

C ' a'

' 2n+1 Choose .{a2i+1 } such that . C ' (a2n' +b > K for some . K > 1 and for all .n. Then ' 2n+1 2n 2n M y ) .{x i } will be unbounded. ◻

Theorem 12 If.l ∈ {11, 15, 37, 38, 43}, then the numbered family.(23, l) has boundedness character (U, B). Proof In each of these systems, we can see by Lemma 1 that . yn is bounded above by some . M y for all .n, and . M y can be expressed solely in terms of the coefficient sequences of . y. Next, note that x

. 2n+2

=

' ' ' ' ' a2n+1 a2n+1 + b2n+1 y2n+1 C2n C2n x > x2n .. 2n ' ' ' ' ' ' a2n + b2n y2n C2n+1 a2n + b2n M y C2n+1 a'

2n+1 Choose coefficient sequences such that . a ' +b ' 2n 2n M y all .n. Then .{xi } will be unbounded.

' C2n ' C2n+1

> K for some . K > 1 and for ◻

3.2 Numbered Families for Which .(8, y) is (U, U) but .(23, y) is (B, B) The next theorems establish cases for which the numbered family will have boundedness character (U, U) when family is as in (3), and will have boundedness character (B, B) when the family is as in (4). The boundedness characters for numbered families .(8, 30) and .(23, 30) have been established in [12], while the boundedness character of numbered family .(8, 3) has been established in [9]. Theorem 13 Numbered families .(23, 3) and .(23, 12) have boundedness character (B, B). a ' +b' y

n , where .{Ci } is either Proof For each .n ≥ 0, let .xn+1 = nC ' xnn n and . yn+1 = Bn xna+C n yn n positively bounded or identically zero, and all other given coefficient sequences are positively bounded. Since

.

Cn' xn Cn' xn yn+1 an an = = ,. xn+1 Bn xn + Cn yn an' + bn' yn an' + bn' yn Bn xn + Cn yn

62

Z. A. Kudlak and R. P. Vernon

we know .{ xyii } is bounded above by Lemma 1. Since .

xn+1 a ' + b' yn Bn xn + Cn yn = n ' n 1 + yn+1 Cn xn an + Bn xn + Cn yn =

an' Bn xn + bn' Bn xn yn + an' Cn yn + bn' Cn yn2 an Cn' xn + Bn Cn' xn2 + Cn Cn' xn yn


1. Consider (1) for ⎛

⎞ t 0 0 ⎛ ⎜0 1 0 ⎟ t 0 00 ⎜ ⎟ 2 ⎝ 0 t2 0 0 B(t) = . A(t) = ⎜ 0 0 t ⎟ , ⎜ ⎟ ⎝0 0 0 ⎠ 0 0 10 00 0 ⎞ ⎛ 0 0 0 −1 1 ⎜ 0 0 t 1 0⎟ ⎟ ⎜ ⎟ C(t) = ⎜ ⎜ 0 −1 0 0 2 0 ⎟ , t ∈ T. ⎝ −1 1 0 −t 0 ⎠ 1 0 0 0 t2 Here .σ(t) = qt, .t ∈ T, and

⎞ 0 0⎠, 0

Linear Time-Varying Dynamic-Algebraic Equations …

⎛

.

10 R(t) = ⎝ 0 1 00 ⎛

0 ⎜0 ⎜ Q 0 (t) = ⎜ ⎜0 ⎝0 0

0 0 0 0 0

109

⎛

t2 0 2 ⎜ 0 ⎜0 t ⎜ ⎠ 0 , G 0 (t) = A(t)B(t) = ⎜ 0 0 ⎝0 0 1 0 0 ⎞ ⎛ ⎞ 000 10000 ⎜0 1 0 0 0⎟ 0 0 0⎟ ⎟ ⎜ ⎟ ⎟ ⎟ 0 0 0 ⎟ , P0 (t) = ⎜ ⎜0 0 1 0 0⎟, ⎠ ⎝ 010 0 0 0 0 0⎠ 001 00000 ⎞

0 0 t2 0 0

0 0 0 0 0

⎞ 0 0⎟ ⎟ 0⎟ ⎟, 0⎠ 0

t ∈ T.

Hence, ⎛

t2 ⎜0 ⎜ . G 1 (t) = G 0 (t) + C(t)Q 0 (t) = ⎜ 0 ⎜ ⎝0 0 ⎛1 ⎞ 0 0 t ⎜ 0 12 0 ⎟ ⎜ t ⎟ ⎟ B − (t) = ⎜ ⎜ 0 0 1 ⎟ , t ∈ T. ⎝0 0 0⎠ 0 0 0

0 t2 0 0 0

0 0 t2 0 0

−1 1 0 −t 2 0

⎞ 1 0⎟ ⎟ 0⎟ ⎟, 0⎠ t2

Next, ⎛

⎞ 0 0 ⎛ ⎞ ⎜ 0 21 2 0 ⎟ qt 0 0 0 0 ⎜ ⎟ q t σ −σ 2 2 ⎟ ⎝ ⎠ .B (t) = ⎜ ⎜ 0 0 1 ⎟ , B (t) = 0 q t 0 0 0 , ⎝ 0 0 0⎠ 0 0 100 0 0 0 ⎛ ⎛ ⎞ 0 0 0 −1 1 00 ⎜ 0 0 qt 1 ⎜0 0 ⎟ 0 ⎜ ⎜ ⎟ 0 0 ⎟ , Q σ0 (t) = ⎜ C σ (t) = ⎜ ⎜ 0 −1 0 ⎜0 0 ⎟ ⎝ −1 1 0 −q 2 t 2 0 ⎠ ⎝0 0 00 1 0 0 0 q 2t 2 ⎛ ⎞ 10000 ⎜0 1 0 0 0⎟ ⎜ ⎟ ⎟ P0σ (t) = ⎜ ⎜ 0 0 1 0 0 ⎟ , t ∈ T. ⎝0 0 0 0 0⎠ 00000 1 qt

0 0 0 0 0

0 0 0 1 0

⎞ 0 0⎟ ⎟ 0⎟ ⎟, 0⎠ 1

110

S. G. Georgiev and S. Kryzhevich

Note that .

det G 1 (t) = −t 10 /= 0, t ∈ T,

and ⎛ −1 . G 1 (t)

1 t2

⎜0 ⎜ =⎜ ⎜0 ⎝0 0 ⎛ 1

0 0 1 0 t2 0 t12 0 0 0 0

q2t2

(

G

) −1 σ

⎜ 0 ⎜ ⎜ (t) = ⎜ 0 ⎜ ⎝ 0 0

− t14 − t14 1 0 t4 0 0 − t12 0 0 t12

⎞ ⎟ ⎟ ⎟, ⎟ ⎠

0 − q 41t 4 − q 41t 4 1 0 q 41t 4 0 q2t2 1 0 q2t2 0 0 0 0 − q 21t 2 0 1 0 0 0 q2t2 0

⎞ ⎟ ⎟ ⎟ ⎟ , t ∈ T. ⎟ ⎠

Therefore ⎛

⎞ 100 − . B(t)P0 (t)B (t) = ⎝ 0 1 0 ⎠ , (B P0 B − )Δ (t) = 0, 001 ⎞ ⎛ 1 0 0 − q 31t 3 − q 31t 3 qt ( )σ ⎜ 0 ⎟ (t) = ⎝ 0 1 0 q 21t 2 B σ (t)P0σ (t) G −1 ⎠, 1 0 0 q 21t 2 0 0 ⎛ .

0

( )σ ⎜ B σ (t)P0σ (t) G −1 (t)C σ (t)B −σ (t) = ⎝ − q 31t 3 1 0 ⎛

0 ⎜0 ⎜ )σ ( 0 Q σ0 (t) G −1 (t) = ⎜ 1 ⎜ ⎝0 0 ⎛

0 0 0 0 0

⎞ − q 51t 5 0 1 qt ⎟ ⎠, q4t4 − q 41t 4 0

0 0 0 0 0 0 0 − q 21t 2 0 0

0 0 ⎜ 0 0 ⎜ )σ ( σ −σ ⎜ 0 0 Q σ0 (t) G −1 (t)C (t)B (t) = 1 ⎜ 1 ⎝ q 3 t 3 − q 41t 4 1 0 q3t3

0 0 0 0

⎞ ⎟ ⎟ ⎟, ⎟ ⎠

1 q2t2

⎞ 0 0⎟ ⎟ 0 ⎟ , t ∈ T. ⎟ 0⎠ 0

Linear Time-Varying Dynamic-Algebraic Equations …

111

Then, the decoupling (22) takes the form ⎛

0

⎜ u Δ (t) = ⎝ − q 31t 3 0

.

⎞ ⎞ ⎛ 1 − q 51t 5 0 0 0 − q 31t 3 − q 31t 3 qt ⎜ 1 1 σ 0 ⎟ qt ⎟ ⎠ f (t) ⎠ u (t) + ⎝ 0 1 0 q 2 t 2 q4t4 0 0 0 q 21t 2 0 − q 41t 4 0

⎛

0 0 ⎜ 0 0 ⎜ 0 0 v σ (t) = − ⎜ ⎜ 1 ⎝ q 3 t 3 − q 41t 4 1 0 q3t3

⎛ ⎞ 0 0 ⎜0 0⎟ ⎜ ⎟ 0 ⎟ u σ (t) − ⎜ 0 ⎜ ⎟ ⎝0 0⎠ 0 0

0 0 0 0 0

0 0 0 0 0 0 0 − q 21t 2 0 0

0 0 0 0

⎞ ⎟ ⎟ ⎟ f (t), t ∈ T. ⎟ ⎠

1 q2t2

If .T = IR, then the decoupling (22) takes the form ⎛

0 − t15 ' 1 1 .u (t) = ⎝ − 3 t t4 0 − t14 ⎛ 0 0 ⎜0 0 ⎜ v(t) = − ⎜ ⎜ 01 01 ⎝ 3 −4 t t 1 0 t3

⎞ ⎛1 0 0 t t ⎠ u(t) + ⎝ 0 1 00 0 ⎛ ⎞ 00 0 ⎜0 0 0⎟ ⎜ ⎟ ⎜ 0⎟ ⎟ u(t) − ⎜ 0 0 ⎝0 0 0⎠ 00 0

⎞ 0 − t13 − t13 0 t12 0 ⎠ f (t) 1 0 0 t2 ⎞ 0 0 0 0 0 0⎟ ⎟ 0 0 0⎟ ⎟ f (t), t ∈ T. 0 − t12 0 ⎠ 0 0 t12

8 Conclusion The problem of decoupling is very significant in the theory of dynamical systems of any origin. First of all, it simplifies the process of finding the analytic solution and qualitative studies. For instance, it is widely applied in the Stability Theory (e.g., for hyperbolic, partially hyperbolic, non-uniformly hyperbolic, or regular systems). The case of an arbitrary time scale is more sophisticated than that of classical ordinary differential equations for the following reasons: 1. time scale calculus involves more complicated formulas than the classical ones and the classical theory cannot be immediately translated to the time scales framework; 2. the theory of time scale systems is much less developed than that of ordinary difference/differential equations; 3. autonomous systems are much more difficult to study for generic time scales. The main result of the paper is proven in Theorem 6 and the related proofs are constructive. Although the approach of this paper is quite straightforward, special techniques based on projector approach were elaborated. The obtained results were illustrated with specific examples.

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Acknowledgements The work of the second co-author was supported by Gda´nsk University of Technology by the DEC 14/2021/IDUB/I.1 grant under the Nobelium—‘Excellence Initiative— Research University’ program and by Sino-Russian Mathematical Center of Peking University.

References 1. Aulbach, B., Hilger, S.: Linear dynamic processes with inhomogenous time scale. In: Nonlinear Dynamics and Quantum Dynamical Systems (Gaussig, 1990). Mathematical Research, , vol. 59, pp. 9–20. Akademie Verlag, Berlin (1990) 2. Bohner, M., Peterson, A.: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston (2001) 3. Bohner, M., Peterson, A. (eds.): Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston, MA (2003) 4. Bohner, M., Georgiev, S.: Multivariable Dynamic Calculus on Time Scales. Springer (2016) 5. Bohner, M., Wintz, N.: Controllability and observability of time-invariant linear dynamic systems. Math. Bohem. 137(2), 149–163 (2012) 6. DaCunha, J.J.: Stability for time varying linear dynamic systems on time scales. J. Comput. Appl. Math. 176(2), 381–410 (2005) 7. Došl`y, O., Elyseeva, J., Hilscher, R.Š.: Symplectic Difference Systems: Oscillation and Spectral Theory. Pathways in Mathematics, pp. 1–589 (2019) 8. Georgiev, S.: Functional Dynamic Equations on Time Scales. Springer (2019) 9. Georgiev, S.: Dynamic Geometry on Time Scales. Chapman and Hall/CRC (2021) 10. Martynyuk, A.A.: Stability Theory for Dynamic Equations on Time Scales. Birkhäuser (2016)

Discrete Dynamical Systems

Differentiable Conjugacies for One-Dimensional Maps Paul Glendinning and David J. W. Simpson

Abstract Differentiable conjugacies link dynamical systems that share properties such as the stability multipliers of corresponding orbits. It provides a stronger classification than topological conjugacy, which only requires qualitative similarity. We describe some of the techniques and recent results that allow differentiable conjugacies to be defined for standard bifurcations, and explain how this leads to a new class of normal forms. Closed-form expressions for differentiable conjugacies exist between some chaotic maps, and we describe some of the constraints that make it possible to recognise when such conjugacies arise. This paper focuses on the consequences of the existence of differentiable conjugacies rather than the conjugacy classes themselves. Keywords Differentiable conjugacy · Bifurcation · Normal form · Chaos

1 Dynamic Conjugacies Let . A and . B be manifolds (in almost everything we do below they are subsets of the real line). Maps . f : A → A and .g : B → B are conjugate if there exists .h : A → B such that .h ◦ f = g ◦ h. (1) The map .h is called the conjugating function, and the type of conjugacy depends on properties of .h. For example P. Glendinning (B) Department of Mathematics, University of Manchester, Manchester M13 9PL, UK e-mail: [email protected] URL: https://personalpages.manchester.ac.uk/staff/paul.glendinning/ D. J. W. Simpson School of Mathematical and Computational Sciences, Massey University, Palmerston North, New Zealand e-mail: [email protected] URL: https://www.massey.ac.nz/djwsimps/ © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 S. Olaru et al. (eds.), Difference Equations, Discrete Dynamical Systems and Applications, Springer Proceedings in Mathematics & Statistics 444, https://doi.org/10.1007/978-3-031-51049-6_6

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– if .h is a homeomorphism (continuous bijection with continuous inverse) then . f and .g are topologically conjugate; – if .h is a diffeomorphism (continuously differentiable with continuously differentiable inverse) then . f and .g are differentiably conjugate; and – if .h is a .C r homeomorphism (.r ≥ 1) then . f and .g are .C r -conjugate. (Recall that if a diffeomorphism is .C r , meaning its first .r derivatives exist and are continuous, then the first.r derivatives of its inverse also exist and are continuous [4].) The idea of a conjugacy provides a formal way of saying that different dynamical systems have the ‘same’ dynamics. It is essentially a change of coordinates: given a homeomorphism .h and a map . f , in the new coordinates . y = h(x) we have y

. n+1

= h(xn+1 ) = h( f (xn )) = h( f (h −1 (yn ))).

So . yn+1 = g(yn ) where .g = h ◦ f ◦ h −1 , which is an alternative way of writing (1). Conjugacy classes of one-dimensional maps with fixed points can be studied for their own sake, see [23, 24] for example, but they are also used as a tool for solving some larger problem at hand. In this paper we will concentrate on the applications of conjugacies to bifurcation theory and chaotic dynamics. We will be particularly interested in cases where topological conjugacies can be made differentiable, since differentiable conjugacies preserve many more features of the dynamics. To see this suppose that two one-dimensional maps . f and .g are topologically conjugate by a conjugating function.h. Suppose. f has a fixed point.x ∗ , so.x ∗ = f (x ∗ ), and let . y ∗ = h(x ∗ ). Then (1) implies .

y ∗ = h(x ∗ ) = h( f (x ∗ )) = g(h(x ∗ )) = g(y ∗ ),

so . y ∗ is a fixed point of .g—we say it is the corresponding fixed point of .g. Moreover, if . f and .g are differentiable and .h is a diffeomorphism then differentiating (1) and evaluating it at .x ∗ gives .

h ' ( f (x ∗ )) f ' (x ∗ ) = g ' (h(x ∗ ))h ' (x ∗ ).

Since . f (x ∗ ) = x ∗ and .h ' (x ∗ ) /= 0 (since its inverse is .C 1 ), we have . f ' (x ∗ ) = g ' (y ∗ ). Since the derivative determines stability properties of hyperbolic fixed points (those with the modulus of the derivative not equal to one) and in particular the rates of convergence or divergence of nearby orbits, this means that corresponding fixed points of differentiably conjugate maps have the same local quantitative behaviour as well as qualitative behaviour implied by topological conjugacy. This stability analysis is easily extended to periodic orbits, where the stability of a period-. p orbit .{x 1 , . . . , x p } is determined by the multiplier, λ=

p ∏

.

n=1

f ' (xn ).

(2)

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Since corresponding periodic orbits have the same multipliers for differentiably conjugate maps, it is not easy to find families of maps arising in applications that are both chaotic and differentiably conjugate. This would require that the multipliers of an infinite number of corresponding periodic orbits are equal, which is an infinite set of constraints. It is of course easy to reverse engineer such families from a family of differentiable conjugacies, but in Sect. 5 we will describe families derived from geometric or algebraic constructions that are differentiably conjugate (in fact ∞ .C -conjugate). The purpose of this paper is to demonstrate the application of differentiable conjugacies to two areas of dynamical systems theory: bifurcations and chaos. In Sect. 2 we review the two main technical results that will be required. Sternberg’s theorem [26] provides local smooth conjugacies to linear maps, while Belitskii’s theorem [3] shows how this can be extended to local basins of attraction and repulsion. In Sect. 3 we introduce the idea of extended normal forms for bifurcation theory [10, 11]. These are polynomial extensions of the standard truncated normal forms for local bifurcations, where the additional terms are chosen so that the extended forms are smoothly conjugate to the original system locally on basins of attraction and repulsion of fixed points. This possibility is mentioned in [14], but the details were not explored there. In Sect. 4 these results are extended to piecewise-smooth maps, with the initially counter-intuitive result that under certain conditions distinct piecewisesmooth maps can be smoothly conjugate [12]. Section 5 considers results for smooth conjugacies in families of maps, and extends the results of [9] to a class of maps studied by Umeno [28], thus making a connection between smooth conjugacy and exactly solvable chaos. Finally Sect. 6 provides a short conclusion.

2 The Theorems of Sternberg and Belitskii The technical results needed to address the applications to bifurcation theory and chaotic maps in later sections are due to Sternberg [26] and Belitskii [3], with some more recent results to deal with non-hyperbolic [24, 32] and orientation-reversing [23] cases. The differentiable equivalence of hyperbolic fixed points comes from Sternberg [26], but we state the result in a slightly different form since by restricting to .C r functions with .r ≥ 2 the conjugacy is also .C r [32] rather than .C r −1 as in the original statement of [26]. Theorem 1 ([26]) Suppose. f : IR → IR is.C r (.r ≥ 2) and. f (0) = 0 with. f ' (0) = λ and .λ ∈ / {−1, 0, 1}. Then there are open neighbourhoods .U of .x = 0 and .V of . y = 0 such that . f (x) on .U is .C r -conjugate to .g(y) = λy on .V . For .|λ| < 1 Sternberg’s proof [26] of this result is based on the analysis of the behaviour of . f n as .n → ∞ (the argument is essentially the same if .|λ| > 1 in reverse time). In [27] Sternberg gives an alternative proof based on an iterative method for the existence of the conjugacy.

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Fig. 1 A map satisfying the conditions of Lemma 1

Now let . f be a map satisfying the conditions of Theorem 1, and .g be another map satisfying the same conditions. That is, .0 is a fixed point of both maps and . f ' (0) = / {−1, 0, 1}. Then from two applications of Theorem 1 we can conclude that g ' (0) ∈ r . f and .g are locally .C -conjugate. Belitskii’s theorem [3] shows this result can be extended to basins of attraction or repulsion of fixed points. The following lemma indicates the flavour of the general result of Belitskii. Lemma 1 Suppose .a < 0 < b and . f : [a, b] → [a, b] is a strictly increasing .C r (.r ≥ 2) map with . f (a) = a, . f (b) = b, . f (0) = 0, and . f ' (0) = λ with .0 < λ < 1. Further suppose . f has no other fixed points on .[a, b], so appears as in Fig. 1. Also ˜ → [a, ˜ is a strictly increasing .C r map with the ˜ b] ˜ b] suppose .a˜ < 0 < b˜ and .g : [a, same properties at corresponding points. Then . f on .(a, b) is .C r -conjugate to .g on ˜ .(a, ˜ b). The main idea behind the proof of Lemma 1 is to extend Sternberg’s conjugacy from a neighbourhood of .x = 0 to the entire interval .(a, b), which is of course the local basin of attraction of the fixed point. Thus we start with .x0 > 0 and . y0 > 0 ˜ and . f on .[−x0 , x0 ] is conjugated to ˜ b) such that .[−x0 , x0 ] ⊂ (a, b), .[−y0 , y0 ] ⊂ (a, r .g on .[−y0 , y0 ] by a .C conjugating function .h. That is, .

h( f (x)) = g(h(x)),

(3)

for all .x ∈ [−x0 , x0 ]. In order to extend the domain of .h towards .b, we use the backward orbits of .x0 and ˜ with .g(yn ) = yn−1 for . y0 to form sequences . x n → b with . f (x n ) = x n−1 and . yn → b all .n ≥ 1. For any .x ∈ (x0 , x1 ), .h is defined at . f (x), so we can extend its definition with −1 .h(x) = g (h( f (x))), for any x ∈ (x0 , x1 ). (4)

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By construction the conjugacy relation (3) holds on the larger domain and is .C r on r .(x 0 , x 1 ). To complete the proof it is necessary to show .h is .C at . x = x 0 and repeat the construction iteratively to extend .h to .[xn , xn+1 ] for all .n ≥ 1, and hence to the whole of .[0, b). The argument in .(a, 0] is similar. We refer the reader to [3] for details.

3 Extended Normal Forms Bifurcations are critical parameter values at which the dynamics of a family of maps undergoes a fundamental (topological) change. There is a vast theory for bifurcations, and much of it is based on normal forms [20]. The basic idea is that a normal form is a family of maps exhibiting the bifurcation and that can be obtained from any family maps exhibiting the bifurcation through a conjugacy. For example 2 .g(y, ν) = y + ν − y , can be viewed as a normal form for a saddle-node bifurcation because if an arbitrary family of maps. f (x, μ) has a saddle-node bifurcation, there exists a homoemorphism .h that conjugates it to .g locally. As discussed above, we would of course like .h to be differentiable. However, on the side of the bifurcation where . f has two fixed points, this is only possible if we can match the stability multipliers of both fixed points of . f to those of the corresponding fixed points of .g. Unless . f has a special symmetry this cannot be done because we cannot tune the single parameter .ν to satisfy both constraints. However, we can obtain a differentiable conjugacy if we instead consider the extended normal form g(y, ν, a) = y + ν − y 2 + ay 3 ,

.

which has two parameters, .ν and .a. To explain why, suppose . f has a saddle-node bifurcation at .(x, μ) = (0, 0). Then .

f (0, 0) = 0,

∂f (0, 0) = 1, ∂x

∂f (0, 0) > 0, ∂μ

∂2 f (0, 0) < 0, ∂x 2

(5)

after substituting .x |→ −x and/or .μ |→ −μ if necessary to obtain the desired signs. For small .μ > 0, . f has two fixed points near .0, Fig. 2. Via a straight-forward calculation we determine the stability multipliers of these points to be 3 / ( 3) 2 ∂∂μf ∂∂x 3f ∂ f ∂2 f .λ (μ) = 1 ± −2 ∂μ ∂x 2 μ − μ + O μ2 , 2 3 ∂∂x 2f

±

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Fig. 2 A sketch of a map on one side of a saddle-node bifurcation. Specifically, . f satisfies (5) and is drawn for small .μ > 0. Recall, the stability multiplier of a fixed point of a one-dimensional map is the slope of the map at the fixed point

where the derivatives are evaluated at .(x, μ) = (0, 0). Similarly for .ν > 0 the map g has two fixed points locally, with stability multipliers

.

( 3) √ σ ± (ν, a) = 1 ± 2 ν + 2aν + O ν 2 .

.

Thus for . f and .g to be differentiably conjugate we need λ+ (μ) = σ + (ν, a),

.

λ− (μ) = σ − (ν, a).

(6)

As shown in [11], we can use the implicit function theorem to show that (6) can indeed be solved for .ν and .a locally to obtain the following result. Theorem 2 Suppose . f is .C r (.r ≥ 4) and satisfies (5). Then there exists .δ > 0, neighbourhoods . N and . M of .0, and continuous functions . F, G : (0, δ) → IR with

.

F(0) = 0,

G(0) = 3

3 | 2 ∂∂x 3f | ( 2 )2 ||

∂ f ∂x 2

,

(7)

(0,0)

such that, for all .μ ∈ (0, δ), the maps . f (x, μ)| N and .g(y, F(μ), G(μ))| M are .C r −1 conjugate on the basins of their corresponding fixed points. Note that in [11] we also show a conjugacy exists for small .μ ≤ 0. The loss of smoothness from .C r to .C r−1 is due to the fact that the implicit function theorem is applied to a function involving the derivatives of . F and .G which are .C r −1 . As another example, .g(y, ν) = y + ν y − y 3 is a normal form for a pitchfork bifurcation, but again we can usually only obtain a continuous conjugacy. As shown in [11], to obtain a differentiable conjugacy it is sufficient to instead use .g(y, ν, a, b) = y + ν y + bν y 2 − y 3 + ay 5 . Here three parameters are needed because on one side of a pitchfork bifurcation there are three fixed points. The extended normal forms are not unique, other families of maps can do the same job, but if we want the extended normal form to be a polynomial with as few terms as

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possible the options are rather limited. For example, in the saddle-node case we saw that the third derivative of . f appears in the .μ-coefficient of .λ± (μ). This coefficient and its corresponding one for .g are involved in the calculations required to construct 3 . F and . G. For this reason if we replace . y in .g with a higher power of . y we cannot in general solve for . F and .G.

4 Piecewise-Smooth Maps Piecewise-smooth maps can exhibit a range of bifurcations that are not possible for smooth maps. In particular, a fixed point can collide with a switching manifold (where the map is non-smooth) giving rise to new dynamics. Such bordercollision bifurcations have been identified in mathematical models in a wide range of disciplines [25]. The skew tent map family ( g(y, ν, s L , s R ) =

.

ν + s L y, y ≤ 0, ν + s R y, y ≥ 0,

(8)

can in some ways be regarded as a normal form for border-collision bifurcations [7]. As the value of .ν ∈ IR passes through .0, a border-collision bifurcation occurs and the resulting dynamics depends in a complicated but well understood way on the values of .s L , s R ∈ IR [1, 16, 21]. In order to connect a typical piecewise-smooth map to (8) via a topological conjugacy that is valid over an interval of parameter values, equality of the kneading sequences [5, 22] may mean this is only possible if we allow .s L and .s R to vary with .ν. A differentiable conjugacy will usually not exist if there are several periodic solutions (for the reasons discussed above). But what about in simple cases where fixed points are the only invariant sets—can we obtain a differentiable conjugacy? The answer, perhaps surprisingly, since the maps themselves are not differentiable, is yes. We just require that the ratio of the slopes at the kink is the same for both maps [12]. To clarify this constraint and show where it comes from, let us derive it with a brief calculation. Consider a continuous map ( .

f (x) =

f L (x), x ≤ 0, f R (x), x ≥ 0,

(9)

where . f L and . f R are smooth on .(−∞, 0] and .[0, ∞) (so derivatives don’t blow up at .x = 0). As indicated in Fig. 3, suppose .x ∗ > 0 is a fixed point of (9) with .λ = f R' (x ∗ ) ∈ (0, 1) and that . f has no other fixed points in an open interval . I containing ∗ .0 and . x . Let ( g L (y), y ≤ 0, .g(y) = (10) g R (y), y ≥ 0,

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Fig. 3 A sketch of a map (9) discussed in the text

be another map with the same properties, i.e. it has a fixed point . y ∗ > 0 with stability multiplier .λ and no other fixed points in an open interval containing .0 and . y ∗ . By Sternberg’s theorem . f and .g are differentiably conjugate on neighbourhoods of .x ∗ and . y ∗ . By Belitskii’s lemma (Lemma 1) we can extend the conjugacy to the left until reaching either .x = 0 or . y = 0. But in fact we can find a conjugacy .h so that .x = 0 and . y = 0 are reached simultaneously by choosing .h so that it maps the forward orbit of .x = 0 under . f to the forward orbit of . y = 0 under .g. That is .

h( f R (x)) = g R (h(x)),

(11)

for all .0 ≤ x ≤ x ∗ , with .h(0) = 0 and .h(x ∗ ) = y ∗ . Differentiating (11) gives .

h ' (x) =

h ' ( f R (x)) f R' (x) , g 'R (h(x))

so in particular .

lim+ h ' (x) =

x→0

h ' ( f R (0)) f R' (0) . g 'R (0)

(12)

(13)

By repeating the procedure used in Sect. 2, we extend the domain of .h to the left by defining −1 .h(x) = g L (h( f L (x))), (14) for small .x < 0. By construction .h provides a conjugacy from . f to .g on the larger domain and is differentiable for small.x < 0 but possibly non-differentiable at.x = 0. Differentiating (14) and taking .x → 0 from the left gives .

lim− h ' (x) =

x→0

h ' ( f L (0)) f L' (0) . g 'L (0)

(15)

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Fig. 4 A piecewise-smooth map (16) satisfying (19) for small .μ > 0

By matching (13) and (15) and using the fact that.h is differentiable at. f L (0) = f R (0), f ' (0) g ' (0) we conclude that .h is differentiable at .x = 0 if and only if . f L' (0) = g'L (0) . That is, the R R slope ratios at the kinks .x = 0 and . y = 0 are the same for both maps. We now demonstrate the consequences of this to conjugacies for border-collision bifurcations. Consider a family of piecewise-smooth maps ( .

f (x, μ) =

f L (x, μ), x ≤ 0, f R (x, μ), x ≥ 0.

(16)

Continuity at .x = 0 implies . f L (0, μ) = f R (0, μ) for all values of .μ. Suppose .x = 0 is a fixed point of (16) with .μ = 0, i.e. f (0, 0) = f R (0, 0) = 0.

. L

(17)

Let a =

. L

∂ fL (0, 0), ∂x

aR =

∂ fR (0, 0), ∂x

β=

∂ fL ∂ fR (0, 0) = (0, 0), ∂μ ∂μ

(18)

and suppose a > 1,

. L

0 < a R < 1,

β > 0.

(19)

In a neighbourhood of .(x, μ) = (0, 0), for .μ < 0 the map has no fixed points, while for .μ > 0 it has two fixed points, Fig. 4. Locally the map is monotone and as the value of .μ is varied through .0 the border-collision bifurcation mimics a saddle-node bifurcation. To obtain a differentiable conjugacy between . f with .μ > 0 and a similar map, we need to match the stability multipliers of both fixed points and the slope ratio at the kink. Straight-forward calculations reveal that the fixed points have multipliers

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.

λ L (μ) =

∂ fL ∂x

λ R (μ) =

∂ fR ∂x

( + ( +

∂2 f L ∂μ∂x ∂2 f R ∂μ∂x

+

∂2 f L ∂x 2 ∂f 1− ∂xL

+

∂2 f R ∂x 2 ∂f 1− ∂xR

β

β

) )

( ) μ + O μ2 , ( ) μ + O μ2 ,

with derivatives evaluated at .(x, μ) = (0, 0), and the slope ratio is .

S(μ) =

∂ fL (0, μ) ∂x . ∂ fR (0, μ) ∂x

(20)

Thus there are three constraints, and the skew tent map family (8) does indeed have three parameters, but tuning the value of .ν does not help us satisfy these constraints because in (8) the value of .ν > 0 can be scaled to .1 (since .g(γ y, γν, s L , s R ) = γg(y, ν, s L , s R ) for any .γ > 0). So instead we can consider ( g(y, ν, s L , s R , t) =

.

ν + s L y + t y 2 , y ≤ 0, y ≥ 0. ν + s R y,

(21)

If .0 < s R < 1 < s L then for small .ν > 0 the fixed points of .g have multipliers σ L (ν, s L , t) = s L +

.

σ R (s R ) = s R ,

( ) 2t ν + O ν2 , 1 − sL

and the slope ratio at . y = 0 is . ssRL . Thus . f and .g are locally conjugate if λ L (μ) = σ L (ν, s L , t),

.

λ R (μ) = σ R (s R ),

r (μ) =

sL . sR

(22)

It turns out we can solve these to obtain .s L , .s R , and .t as functions of .μ, using also ν = μ due to the above scaling property, leading to the following result [12].

.

Theorem 3 Let. f be a piecewise-.C 3 map (16) satisfying (17)–(19), and.g be given by (21). Then there exists .δ > 0, neighbourhoods . N (μ) and . M(μ) of .0, and continuous functions . FL , FR , G : (0, δ) → IR with .

FL (0) = a L ,

FR (0) = a R ,

G(0) =

β 2

(

∂2 f L ∂x 2

−

a L (1−a L ) ∂ 2 f R a R (1−a R ) ∂x 2

)| | |

(0,0)

,

(23)

such that. f (x, μ)| N and.g(y, μ, FL (μ), FR (μ), G(μ))| M are differentiably conjugate on the basins of their corresponding fixed points for all .μ ∈ (0, δ).

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5 Chaotic Maps The elliptic curves . y 2 = x 3 + ax + b have many beautiful properties. One of these is that any line tangential to an elliptic curve intersects the curve at one other point (after adding the point at infinity). This process can be iterated: start at a point on the curve, find the point determined by the tangent at the original point, and now find the point determined by the tangent at the new point, and so on. This generates a map by (for example) considering the .x-coordinates of successive points determined by this process, which gives .

F(x) =

x 4 − 2ax 2 − 8bx + a 2 , 4(x 3 + ax + b)

(24)

see Fig. 5. Equation (24) is a family of maps parametrized by .a and .b and it is not hard to show that these are all topologically conjugate to a full shift on two symbols and hence to the standard quadratic map (Chebyshev map) T (x) = 1 − 2x 2 ,

. 2

(25)

on the interval .[−1, 1], Fig. 6. Glendinning and Glendinning [9] show that each of the maps . F of (24), or more accurately a compactification of (24), are .C ∞ -conjugate to each other and to .T2 . The proof of this statement is based on two observations. First, Jiang [17–19] has shown that chaotic unimodal maps which are topologically conjugate, have the same types of turning points (in this case quadratic), and for which corresponding periodic orbits have the same multipliers, are .C ∞ -conjugate. The first two criteria are easy to establish for the maps defined by (24) so it is the infinite set of equalities of corresponding multipliers that presents a challenge. This is solved by the following lemma.

Fig. 5 The map (24) on where .r is the vertical asymptote, derived from elliptic curves with .a = b = 1. After compactification by a real Möbius transformation this is conjugate to the Chebyshev map (25) on .(−1, 1) .(r, ∞),

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Fig. 6 The Chebyshev map (25)

Lemma 2 ([9]) Let . F be defined by (24) and . H (x) = x 3 + ax + b. Then .

H (F(x)) =

1 ' [F (x)]2 H (x). 4

(26)

The proof is by brute-force calculation, also verified using the symbolic manipulation packages of Mathematica [31]. This has the immediate consequence that if .{x1 , . . . x p } is a period-. p orbit of . F and . H (xn ) /= 0 for each .n, then the stability multiplier of the periodic orbit is .λ with λ2 =

p ∏

.

[F ' (xn )]2 = 4 p

n=1

p−1 H (x1 ) ∏ H (xn+1 ) = 4p. H (x p ) n=1 H (xn )

(27)

In other words, provided . H /= 0 at periodic points, the modulus of the multiplier of every period-. p orbit is .2 p , and this is independent of the parameters .a and .b. There are special points which need to be checked by hand: in this case the endpoints where the derivative at the fixed point is .4 and not .21 . The usual proof that the same is true for .T2 uses the conjugacy to the tent map with slopes .±2 [6], but here let us demonstrate this using the same idea as Lemma 2. Lemma 3 Let . H2 (x) = 1 − x 2 . Then .

H2 (T2 (x)) =

1 ' [T (x)]2 H2 (x). 4 2

Proof By straightforward calculation .

H2 (T2 (x)) = 1 − (1 − 2x 2 )2 = 4x 2 − 4x 4 ,

and since .T2' (x) = −4x,

(28)

Differentiable Conjugacies for One-Dimensional Maps

.

127

1 ' [T (x)]2 H2 (x) = 4x 2 (1 − x 2 ). 4 2

End of proof. Note that at .x = ±1, . H (x) = 0 and so the argument using (27) does not hold. As before at the fixed point .x = −1 the derivative is different: again it is .4 as can be checked by hand. In the case of the rescaled quadratic map .T˜ (x) = 4x(1 − x) on ˜ (x) = x(1 − x) and (28) holds with .(T2 , H2 ) .[0, 1] the equivalent . H function is . H ˜ ˜ replaced by .( F, H ). This will be useful below. Lemmas 2 and 3 suggest a much stronger principle at work. If both . H (x) and . H (F(x)) are non-zero then (26) can be rewritten as /

/ 1 1 = ' H (F(x)) |F (x)|

1 . 2

1 . H (x)

(29)

/ Thus if .ρ(x) = H 1(x) and . y = F(x) has precisely two preimages, then adding the two Eqs. (29) due to the two preimages gives ρ(y) =

∑

.

x∈F −1 (y)

ρ(x) . |F ' (x)|

(30)

This is the well-know equation for an invariant measure of a map . F as a fixed point of the Perron-Frobenius operator [2], and, moreover, (29) implies that each of the two terms on the right hand side of (30) is equal. This argument is made rigorous and generalised in [8]. The invariant measure is called a balanced measure to reflect the property that each of the terms on the right hand side of (30) contributes equally [15]. As pointed out in [9], the examples of exactly solvable maps of Umeno [28, 29] bear strong similarities to the elliptic curve example (24). These maps are constructed using hypergeometric function theory to have ergodic measures that can be written down explicitly. A first example [28] is the iteration of the one parameter family of Katsura-Fukuda maps, . F : [0, 1] → [0, 1] defined by .

F (x) =

4x(1 − x)(1 − x) , (1 − x 2 )2

(31)

with . ∈ [0, 1), Fig. 7. If . = 0 this is the rescaled quadratic map .T˜ (x) and the maps are constructed so that the invariant measure has density ρ =

.

1 , √ 2K ( ) x(1 − x)(1 − x)

(32)

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Fig. 7 The Katsura–Fukuda map (31) for three different values of .

where the normalization constant is the elliptic integral of the first kind [28]. This provides another example of the balanced measure idea at work [8, 15]. In this case it is possible to go through the same analysis as [9], though the functions are more complicated than those of Lemma 3 and were discovered using Mathematica [31]. Lemma 4 Let . F be defined by (31) and let . H (x) = x(1 − x)(1 − x) then .

H (F (x)) =

1 ' [F (x)]2 H (x). 4

(33)

If. = 0 we recover Lemma 3. Noting that. F ' (0) = 4 to deal with the case. H (0) = 0 we have the equivalent result to that of [9] for the family . F . Theorem 4 For all . , [0, 1] → [0, 1].

'

∈ [0, 1), . F : [0, 1] → [0, 1] is .C ∞ -conjugate to . F ' :

The proof follows the proof in [9] with Lemma 2 replaced by Lemma 4. And of course this means that the Katsura–Fukuda maps are .C ∞ -conjugate to .T2 and to . F (24) for every .(a, b) ∈ IR2 .

6 Conclusion In applications differentiable conjugacies have mostly been used in the following three ways. First, as initial scaling or translations, for example to non-dimensionalize a problem or to simplify the parameterization. Second, as part of a linearization process, making it possible to deduce details of local behaviour, which may later also be used in the study of other phenomena such as global bifurcations, e.g. [30]. Third, in the identification of the important nonlinear terms near a non-hyperbolic fixed point as a precursor to a deformation argument to capture local behaviour at a bifurcation point e.g. [13, 20].

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In this paper we have shown that differentiable conjugacies have broader applications. Bifurcations theorems in most textbooks describe the local dynamics away from the bifurcation point via topological conjugacy [13, 20, 30]. This is because of the difficulty presented by having more than one fixed point locally. The analysis of [10, 11, 14] shows that this can be made into a differentiable conjugacy on basins of attraction and repulsion of the fixed points. Moreover, as shown explicitly in [10, 11], the model equations (extended normal forms) involve the addition of one or two extra terms whose coefficients are appropriate functions of the bifurcation parameter to the standard truncated normal forms, see Sect. 3. This idea can be extended to bifurcations in piecewise-smooth systems as shown in Sect. 4. In Sect. 5 we also showed that differentiable conjugacies could be used to show a closer relationship between some specially constructed examples. This allowed us to reveal commonality between maps based on elliptic curves [9] and the exactly solvable chaotic maps of Umeno [28, 29]. These results suggest that further work on differentiable conjugacies may produce new connections between systems for which only topological conjugacy has been established hitherto. Acknowledgements We are grateful to Michal Misiurewicz for an illuminating conversation at the 27.th ICDEA, 2022. Saclay, Paris. This helped us to find the references [8, 15].

References 1. Avrutin, V., Gardini, L., Sushko, I., Tramontana, F.: Continuous and Discontinuous PiecewiseSmooth One-Dimensional Maps. World Scientific, Singapore (2019) 2. Beck, C., Schlögl, F.: Thermodynamics of Chaotic Systems. Cambridge Nonlinear Science Series 4. CUP, Cambridge (1993) 3. Belitskii, G.R: Smooth classification of one-dimensional diffeomorphisms with hyperbolic fixed points. Sibirskii Matematicheskii Zhurnal 27, 25–27 (1986). (Translated in Siberian Math. J. 27, 801–804) 4. Blackadar, B.: A general implicit/inverse function theorem (2015). arXiv:1509.06025v3 5. Buczolich, Z., Keszthelyi, G.: Equi-topological entropy curves for skew tent maps in the square. Math. Slovaca 67, 1577–1594 (2017) 6. Devaney, R.L.: An Introduction to Chaotic Dynamical Systems, 2nd edn. Addison-Wesley, Redwood (1989) 7. di Bernardo, M., Budd, C.J., Champneys, A.R., Kowalczyk, P.: Piecewise-smooth Dynamical Systems. Theory and Applications. Springer, New York (2008) 8. Friere, A., Lopez, A., Mañé, R.: An invariant measure for reational maps. Bol. Soc. Bras. Mat. 14, 45–62 (1983) 9. Glendinning, P., Glendinning, S.: Smooth conjugacy of difference equations derived from elliptic curves. J. Diff. Equ. Appl. 27, 1419–1433 (2021) 10. Glendinning, P.A., Simpson, D.J.W.: Normal forms for saddle-node bifurcations: takens’ coefficient and applications to climate models. Proc. Roy. Soc. A. 478, 20220548 (2022). https:// doi.org/10.1098/rspa.2022.0548 11. Glendinning, P.A., Simpson, D.J.W.: Normal forms, differentiable conjugacies and elementary bifurcations of maps. SIAM J. Appl. Math. 83, 816–836 (2023) 12. Glendinning, P.A., Simpson, D.J.W.: Differentiable conjugacies for border-collision bifurcations of one-dimensional maps. In preparation (2023)

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13. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Applied Mathematical Sciences, vol. 42. Springer, New York (1983) 14. Il’Yashenko, Y.S., Yakovenko, S.Y.: Nonlinear stokes phenomena in smooth classification problems. Adv. Soviet Math. 14, 235–287 (1993) 15. Lyubich, M.: Entropy properties of rational endomorphisms of the Riemann sphere. Ergod. Thy. & Dynam. Syst. 3, 351–386 (1983) 16. Ito, S., Tanaka, S., Nakada, H.: On unimodal linear transformations and chaos. Proc. Jpn. Acad. Ser. A 55, 231–236 (1979) 17. Jiang, Y.: On Ulam-von Neumann transformations. Comm. Math. Phys. 172, 449–459 (1995) 18. Jiang, Y.: On rigidity of one-dimensional maps. Contemp. Math. AMS 211, 319–431 (1997) 19. Jiang, Y.: Differentiable rigidity and smooth conjugacy. Ann. Acad. Sci. Fenn. Math. 30, 361– 383 (2005) 20. Kuznetsov, Y.A.: Elements of applied bifurcation theory. Applied Mathematical Sciences 112. Springer, New York (1995) 21. Maistrenko, Yu.L., Maistrenko, V.L., Chua, L.O.: Cycles of chaotic intervals in a time-delayed Chua’s circuit. Int. J. Bifurc. Chaos. 3, 1557–1572 (1993) 22. Misiurewicz, M., Visinescu, E.: Kneading sequences of skew tent maps. Ann. I. H. Poincaré, Probab. Stat. 27, 125-140 (1991) 23. O’Farrell, A.G., Roginskaya, M.: Reducing conjugacy in the full diffeomorphism group of .IR to conjugacy in the subgroup of orientation-preserving maps. J. Math. Sci. 158, 895–898 (2009) 24. O’Farrell, A.G., Roginskaya, M.: Conjugacy of real diffeomorphisms. A survey. St. Petersburg Math. J. 22, 1–40 (2011) 25. Simpson, D.J.W.: Border-collision bifurcations in .IR N . SIAM Rev. 58, 177–226 (2016) 26. Sternberg, S.: Local .C n transformations of the real line. Duke Math. J. 24, 97–102 (1957) 27. Sternberg, S.: Local contractions and a theorem of Poincaré. Amer. J. Math. 79, 809–824 (1957) 28. Umeno, K.: Method of constructing exactly solvable chaos. Phys. Rev. E 55, 5280–5284 (1997) 29. Umeno, K.: Exactly solvable chaos and addition theorems of elliptic functions. RIMS Kokyuroku 1098, 104–117 (1999) 30. Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd edn. Texts in Applied Mathematics 2. Springer, New York (2003) 31. Wolfram Research, Inc.: Mathematica, Version 13.2. Champaign, Illinois (2022). https://www. wolfram.com/mathematica 32. Young, T.R.: .C k conjugacy of 1-D diffeomorphisms with periodic points. Proc. AMS 125, 1987–1995 (1997)

Topological Entropy of Generalized Bunimovich Stadium Billiards Michał Misiurewicz and Hong-Kun Zhang

Abstract We estimate from below the topological entropy of the generalized Bunimovich stadium billiards. We do it for long billiard tables, and find the limit of estimates as the length goes to infinity. We also get estimates for some shorter tables. We repeat this for generalized semistadium billiards, including the mushroom ones. Keywords Bunimovich stadium billiard · Topological entropy 2010 Mathematics Subject Classification Primary 37D50 · 37B40

1 Introduction In this paper, we generalize the results of [6] to a much larger class of billiards. They are similar to Bunimovich stadium billiards (see [3]), but the semicircles are replaced by almost arbitrary curves. That is, those curves are not completely arbitrary, but the assumptions on them are very mild. An example of such curves is shown in Fig. 1. We also consider the case when one of the curves is a vertical line segment. This class includes Bunimovich mushroom billiards. We consider billiard maps (not the flows) for two-dimensional billiard tables. Thus, the phase space of a billiard is the product of the boundary of the billiard Research of Michał Misiurewicz was partially supported by grant number 426602 from the Simons Foundation. M. Misiurewicz (B) Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, 402 N. Blackford Street, Indianapolis, IN 46202, USA e-mail: [email protected] H.-K. Zhang Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 S. Olaru et al. (eds.), Difference Equations, Discrete Dynamical Systems and Applications, Springer Proceedings in Mathematics & Statistics 444, https://doi.org/10.1007/978-3-031-51049-6_7

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Γ1 Γ3’

Γ4’ Γ2

Fig. 1 Generalized Bunimovich stadium

table and the interval .[−π/2, π/2] of angles of reflection. This phase space will be denoted as .M. We will use the variables .(r, ϕ), where .r parametrizes the table boundary by the arc length, and .ϕ is the angle of reflection. Those billiards have the natural measure; it is .c cos ϕ dr dϕ, where .c is the normalizing constant. This measure is invariant for the billiard map. However, we will not be using this measure, but rather investigate our system as a topological one. The first problem one encounters with this approach is that the map can be discontinuous, or even not defined at certain points. In particular, if we want to define topological entropy of the system, we may use one of several methods, but we cannot be sure that all of them will give the same result. To go around this problem, similarly as in [6], we consider a compact subset of the phase space, invariant for the billiard map, on which the map is continuous. Thus, the topological entropy of the billiard map, no matter how defined, is larger than or equal to the topological entropy of the map restricted to this subset. Positive topological entropy is recognized as one of the forms of chaos. In fact, topological entropy even measures how large this chaos is. Hence, whenever we prove that the topological entropy is positive, we can claim that the system is chaotic from the topological point of view. We will be using similar methods as in [6]. However, the class of billiards to which our results can be applied, is much larger. The class of Bunimovich stadium billiards, up to similarities, depends on one positive parameter only. Our class is enormously larger, although we keep the assumption that two parts of the billiard boundary are parallel segments of straight lines. Nevertheless, some of our proofs are simpler than those in [6].

2 Assumptions We will think about the billiard table positioned as in Fig. 1. Thus, we will use the terms horizontal, vertical, lower, upper, left, right. While we are working with the billiard map, we will also look at the billiard flow. Namely, we will consider trajectory lines, that is, line segments between two consecutive reflections from the table boundary. For such a trajectory line (we consider it really as a line, not a vector) we define its argument (as an argument of a complex number), which is the angle

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Γ1 Γ3

Γ4 Γ2

Fig. 2 Curves .┌i , .i = 1, 2, 3, 4

between the trajectory line and a horizontal line. For definiteness, we take the angle from .(−π/2, π/2]. We will be also speaking about the arguments of lines in the plane. Moreover, for .x ∈ M, we define the argument of .x as the argument of of the trajectory line joining .x with its image. We will assume that the boundary of billiard table is the union of four curves, .┌1 , ' ' .┌2 , .┌3 and .┌4 . The curves .┌1 and .┌2 are horizontal segments of straight lines, and ' .┌2 is obtained from .┌1 by a vertical translation. The curve .┌3 joins the left endpoints ' of .┌1 and .┌2 , while .┌4 joins the right endpoints of .┌1 and .┌2 (see Fig. 1). We will consider all four curves with endpoints, so they are compact. For .ε ≥ 0, we will call a point . p ∈ ┌i' (.i ∈ {3, 4}) .ε-free if any forward trajectory of the flow (here we mean the full forward trajectory, not just the trajectory line), beginning at . p with a trajectory line with argument whose absolute value is less than ' . or equal to .ε, does not collide with .┌i' before it collides with .┌7−i ' Further, we will call a subarc .┌i ⊂ ┌i .ε-free (see Fig. 2) if: (a) .┌i is of class .C 1 , (b) every point of .┌i is .ε-free, (c) there are points . pi+ , pi− ∈ ┌i such that the argument of the line normal to .┌i is larger than or equal to .ε at . pi+ and less than or equal to .−ε at . pi− (see Fig. 2), (d) .┌i is disjoint from .┌1 ∪ ┌2 . Clearly, if .┌i is .ε-free then it is also .δ-free for all .δ ∈ (0, ε). Our last assumption is that there is .ε > 0 and .ε-free subarcs .┌i ⊂ ┌i' for .i = 3, 4, such that .┌3 ∪ ┌4 is disjoint from .┌1 ∪ ┌2 . We will denote the class of billiard tables satisfying all those assumptions by .H(ε). Observe that there are two simple situations when we know that there is .ε > 0 such that .┌i' has an .ε-free subarc. One is when there is a .0-free point . pi ∈ ┌i' such that there is a neighborhood of . pi where .┌i is of class .C 1 and the curvature of .┌i at . pi exists and is non-zero (see Fig. 3). The other one is when .┌i' is the graph of a non-constant function .x = f (y) of class .C 1 (then we take a neighborhood of a point where . f attains its extremum; this neighborhood may be large if the extremum is attained on an interval), like .┌3' (but not .┌4' ) in Fig. 1. We forget about the other parts of the curves .┌i' and look only at .┌i , .i = 1, 2, 3, 4 (see Fig. 2).

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p4

Fig. 3 Points . p3 and . p4

Let us mention that since we will be using only those four pieces of the boundary of the billiard table, it does not matter whether the rest of the boundary is smooth or not. If it is not smooth, we can include it (times .[−π/2, π/2]) into the set of singular points, where the billiard map is not defined.

3 Coding We consider a billiard table from the class .H(ε). Since transforming the table by homothety does not change the entropy, we may assume that the distance between .┌1 and .┌2 is 1. Now we can introduce a new characteristic of our billiard table. We will say that a billiard table from the class.H(ε) is in the class.H(ε, ) if the horizontal distance between .┌3 and .┌4 is at least . . We can think about . as a big number (it will go to infinity). We start with a trivial geometrical fact, that follows immediately from the rule of reflection. We include the assumption that the absolute values of the arguments are smaller than .π/6 in order to be sure that the absolute value of the argument of .T2 is smaller than .π/2. Lemma 3.1 If .T1 and .T2 are incoming and outgoing parts of a trajectory reflecting at .q and the argument of the line normal to the boundary of the billiard at .q is .α, and .|α|, | arg(T1 )| < π/6, then .arg(T2 ) = 2α − arg(T1 ). We consider only trajectories that reflect from the curves.┌i ,.i = 1, 2, 3, 4. In order to have control over this subsystem, we fix an integer . N > 1 and denote by .K ,N the space of points whose (discrete) trajectories go only through .┌i , .i = 1, 2, 3, 4 and have no . N + 1 consecutive collisions with the straight segments. We can unfold the billiard table by using reflections from the straight segments (see Fig. 4). The liftings of trajectories (of the flow) consist of segments between points of liftings of .┌3 and .┌4 . In .K ,N they go at most . N levels up or down. Now for a moment we start working on the lifted billiard. That is, we consider only .┌3 and .┌4 , but at all levels, as pieces of the boundary from which the trajectories of the flow can reflect. We denote those pieces by .┌i, j , where .i ∈ {3, 4} and . j ∈ Z. Clearly, flow trajectories from some points .(r, ϕ) will not have more collisions, so the lifted billiard map . F will be not defined at such points. We denote by .M the product of the union of all sets .┌i, j and the interval .[−π/2, π/2].

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Fig. 4 Five levels of the unfolding. Only .┌3 and .┌4 are shown instead of .┌3' and .┌4'

Now we specify how large. should be for given. N , ε in order to get nice properties of the billiard map restricted to .K ,N . Assume that our billiard table belongs to .H(ε, ) and fix .i ∈ {3, 4}, . j ∈ Z. Call a continuous map .γ : [a, b] → M, given by γ (t) = (γr (t), γϕ (t)),

.

an .(i, j, ε)-curve if .γr ([a, b]) = ┌i, j and for every .t ∈ [a, b] the absolute value of the argument of the trajectory line incoming to .γ (t) is at most .ε. We can think of .γ as a bundle of trajectories of a flow incoming to .┌i, j . In order to be able to use Lemma 3.1, we will always assume that .ε < π/6. Lemma 3.2 Assume that the billiard table belongs to .H(ε, ) and fix . N ≥ 0, .i ∈ {3, 4}, . j ∈ Z, and .k ∈ {−N , −N + 1, . . . , N − 1, N }. Assume that .

≥

N +1 tan ε

(1)

Then every.(i, j, ε)-curve.γ has a subcurve whose image under. F (that is,. F ◦ γ |[a ' ,b' ] for some subinterval .[a ' , b' ] ⊂ [a, b]) is a .(7 − i, j + k, ε)-curve. Proof There are points .c− , c+ ∈ [a, b] such that .γr (c− ) is a lifting of . pi− and .γr (c+ ) is a lifting of . pi+ . Then, by Lemma 3.1, the lifted trajectory line outgoing from .γ (c− )

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(respectively,.γ (c+ )) has argument smaller than.−ε (respectively, larger than.ε). Since the direction of the line normal to .┌i, j at the point .γr (t) varies continuously with .t, the argument of the lifted trajectory line outgoing from .γ (t) also varies continuously with .t. Therefore, there is a subinterval .[a '' , b'' ] ⊂ [a, b] such that at one of the points .a '' , b'' this argument is .−ε, at the other one is .ε, and in between is in .[−ε, ε]. When the bundle of lifted trajectory lines starting at .γ ([a '' , b'' ]) reaches liftings of .┌7−i , it collides with all points of .┌7−i, j+k whenever .k + 1 ≤ tan ε. By (1), this includes all .k with .k ≤ N . Therefore, there is a subinterval .[a ' , b' ] ⊂ [a '' , b'' ] such that .(F ◦ γ )r ([a ' , b' ]) = ┌7−i, j+k . The arguments of the lifted trajectory lines incoming to .(F ◦ γ )([a ' , b' ]) are in .[−ε, ε], so we get a .(7 − i, j + k, ε)-curve. ◻ Using this lemma inductively we get the next lemma. Lemma 3.3 Assume that the billiard table belongs to .H(ε, ) and fix . N ≥ 0 such that (1) is satisfied. Then for every finite sequence (k−n , . . . , k−1 , k0 , k1 , . . . , kn ), with km ∈ Z, |km | ≤ N , m = −n, . . . , n,

.

there is a trajectory piece in the lifted billiard going alternately between liftings of ┌3 and .┌4 with the differences of levels .k−n , . . . , k−1 , k0 , k1 , . . . , kn .

.

Proof We show by induction on .u that, for every finite sequence (k−n , k−n+1 , . . . , k−n+u )

.

of integers with absolute values at most . N , there is a sequence .(γ −n , γ −n+1 , . . . , γ −n+u ) of curves, such that for.v = 0, 1, . . . , u, the curve.γ −n+v is an.(i ' , k−n + · · · + k−n+v−1 , ε)-curve (with .i ' = 3 or .4, depending on the pairity of .v), and .γ −n+v+1 is the image inder . F of a subcurve of .γ −n+v . Indeed, the induction step (from .u to .u + 1) follows immediately from Lemma 3.2. To start the induction, we choose any −n .┌i,0 (.i = 3 or .4) and the horizontal trajectories incoming to it forming .γ . This is a legitimate choice, because every point of .┌i is .ε-free. Now to get a trajectory piece required in the lemma, we take .u = 2n and any point ◻ of .γ −n as the starting point of the trajectory. Note that in the above lemma we are talking about trajectory pieces of length 2n + 1, without requiring that those pieces can be extended backward or forward to a full trajectory.

.

Proposition 3.4 Under the assumption of Lemma 3.3, for every two-sided sequence (. . . , k−2 , k−1 , k0 , k1 , k2 , . . . )

.

of integers with absolute values at most . N there is a trajectory in the billiard, whose lifting is going between liftings of .┌3 and .┌4 with the differences of levels .

. . . , k−2 , k−1 , k0 , k1 , k2 , . . . .

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Proof For every finite sequence .(k−n , . . . , k−1 , k0 , k1 , . . . , kn ), let . An be the set of points of .┌3 × [−π/2, π/2] or .┌4 × [−π/2, π/2] whose trajectories from time .−n to .n exist and satisfy Lemma 3.3. Then the set . An is compact and nonempty. As .n ∞ goes ∩∞ to infinity, we get a nested sequence of compact sets .(An )n=0 , so the intersection . n=0 An is nonempty. One can check easily that this intersection is the set of points whose trajectories behave in the way we demand. ◻ Consider the following subshift of finite type .(∑ .

,N , σ ).

The states are

− N , −N + 1, . . . , −1, 0, 1, . . . , N − 1, N ,

and the transitions are: from 0 to 0, 1 and .−1, from .i to .i + 1 and 0 if .1 ≤ i ≤ N − 1, from . N to 0, from .−i to .−i − 1 and 0 if .1 ≤ i ≤ N − 1, and from .−N to 0. Each trajectory of a point from .K ,N can be coded by assigning the symbol 0 to .┌3 ∪ ┌4 , and for the parts between two zeros: either .1, 2, . . . , j if the the first point is in .┌1 ; or .−1, −2, . . . , − j if the first point is in .┌2 . This defines a map from .K ,N to .∑ ,N . This map is continuous, because the preimage of every cylinder is open (this follows immediately from the fact that the straight pieces of our trajectories of the billiard flow intersect the arcs .┌i , .i = 1, 2, 3, 4, only at the endpoints of those pieces, and that the arcs are disjoint). It is a surjection by Proposition 3.4. Therefore it is a semiconjugacy, and therefore, the topological entropy of the billiard map restricted to .K ,N is larger than or equal to the topological entropy of .(∑ ,N , σ ).

4 Computation of Topological Entropy In the preceding section we obtained a subshift of finite type. Now we have to compute its topological entropy. If the alphabet of a subshift of finite type is .{1, 2, . . . , n}, then we can write the transition matrix . M = (m i j )i,n j=1 , where .m i j = 1 if there is a transition from .i to . j and .m i j = 0 otherwise. Then the topological entropy of our subshift is the logarithm of the spectral radius of . M (see [1, 5]). In the case of large, but not too complicated, matrices, in order to compute the spectral radius one can use the rome method (see [1, 2]). Let . M = (m i j ) be an .n × n matrix. For a sequence . p = ( p j )kj=0 of elements of ∏k .{1, . . . , n} its width is .w( p) = j=1 m p j−1 p j . If .w( p) / = 0 then . p is called a path and .k = l( p) is its length. A subset . R of .{1, . . . , n} is called a rome if there is no loop outside . R, i.e. there is no path .( p j )kj=0 such that . pk = p0 and .{ p j : 0 ≤ j ≤ k} is disjoint from . R. Notice that this condition is equivalent to saying that all paths lead to rome. For a rome . R a path .( p j )kj=0 is called simple if . pi ∈ R for .i = 0, k and . pi ∈ / R for .i = 1, . . . , k − 1. If . R = {r1 , r2 , . . . , rk } is a rome of a matrix . M, then we define ∑a .k × k matrixvalued real function . M R by setting . M R = (ai j ), where .ai j (x) = p w( p) · x −l( p) , and the summation is over all simple paths originating at .ri and terminating at .r j .

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By . E we denote the .k × k unit matrix. Then the characteristic polynomial of . M is equal to .(−1)n−k x n det(M R (x) − E). For the transition matrices of .(∑ ,N , σ ) this method is especially simple, since .{0} is a rome. Then we only have to identify the lengths of all paths from 0 to 0 that do not go through 0 except at the beginning and the end. ∑The spectral radius of the transition matrix is then the largest zero of the function . x − pi − 1, where the sum is over all such paths and . pi is the length of the .ith path. Lemma 4.1 Topological entropy of the system .(∑ largest root of the equation .

,N , σ )

is the logarithm of the

x 2 − 2x − 1 = −2x −N .

(2)

Proof The paths that we mentioned before the lemma, are: one path of length 1 (from 0 directly to itself), and two paths of length .2, 3, . . . , N + 1 each. Therefore, our entropy is the logarithm of the largest zero of the function .2(x −(N +1) + · · · + x −3 + x −2 ) + x −1 − 1. We have .

( ) x(1 − x) 2(x −(N +1) + · · · + x −3 + x −2 ) + x −1 − 1 = (x 2 − 2x − 1) + 2x −N ,

so our entropy is the logarithm of the largest root of Eq. (2).

◻

Corollary 4.2 Assume that the billiard table belongs to .H(ε, ) and fix . N ≥ 0 such that (1) is satisfied. Then the topological entropy of the billiard map restricted to .K ,N is larger than or equal to the logarithm of the largest root of Eq. (2). A particular case of this corollary gives us a sufficient condition for positive topological entropy. Namely, notice that the largest root of the equation .x 2 − 2x − 1 = −2x −1 is .2. Corollary 4.3 Assume that the billiard table belongs to .H(ε, ) and . tan ε ≥ 2. Then the topological entropy of the billiard map is at least .log 2, so the map is chaotic in topological sense. It is interesting how this estimate works for the classical Bunimovich stadium billiard. In fact, for the estimate we will improve a little the corollary (see Fig. 5). Proposition 4.4 If the rectangular part of a stadium has the length/width ratio larger √ than . 3 ≈ 1.732 (see Fig. 6), the billiard map has topological entropy at least .log 2. Fig. 5 Computations for the stadium billiard

p4+ p4-

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Fig. 6 Stadium billiard with topological entropy at least .log 2

Proof We can take.ε as√ close to.π/6 as we want (see Fig. 5), so we get the assumption in the corollary . > 2 3. However, the factor 2 (in general, . N + 1 in (1)) was taken to get an estimate that works for all possible choices of .┌i , .i = 3, 4. For our concrete choice, it is possible to replace it by the vertical size of .┌i,0 ∪ ┌i,1 (or 3 .┌i,0 ∪ ┌i,−1 , but it is the same in our case). This number is not 2, but . . Thus, we 2 √ really get . > 23 3. But . ' is the length of the rectangular part of the stadium, then √ √ √ ' ◻ . = + 2 · 43 = ' + 21 3. This gives us . ' > 3. Now we can prove the main result of this paper. Theorem 4.5 For the billiard tables from the class .H with the shapes of .┌3 and ┌4 fixed, the lower limit of the topological entropy of the generalized √ Bunimovich stadium billiard, as its length . goes to infinity, is at least .log(1 + 2).

.

Proof In view of Corollary √ 4.2 and the fact that the largest root of the equation x 2 − 2x − 1 = 0 is .1 + 2, we only have to prove that the largest root of the Eq. (2) converges to the largest root of the equation .x 2 − 2x − 1 = 0√as . N → ∞. However, this follows from the fact that in the neighborhood of .1 + 2 the right-hand side ◻ of (2) goes uniformly to 0 as . N → ∞.

.

5 Generalized Semistadium Billiards In a similar way we can investigate generalized semistadium billiards. They are like generalized stadium billiards, but one of the caps .┌3' , ┌4' is a vertical straight line segment. The other one contains an .ε-free subarc. This class contains, in particular, Bunimovich’s Mushroom billiards (see [4]), see Fig. 7. We will be talking about the classes .H1/2 , .H1/2 (ε) and .H1/2 (ε, ) of billiard tables. When we construct a lifting, we add the reflection from the flat vertical cap. In such a way we obtain the same picture as in Sect. 3, except that there is an additional vertical line through the middle of the picture, and we have to count the flow trajectory crossing it as an additional reflection (see Fig. 8). Note that since we will be working with the lifted billiard, in the computations we can take .2 instead of . . In particular, inequality (1) will be now replaced by .

≥

N +1 2 tan ε

(3)

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Γ1 Γ3’

Γ4’ Γ2

Fig. 7 A mushroom

Fig. 8 Unfolding

Computation of the topological entropy is this time a little more complicated. We cannot claim that after coding we are obtaining a subshift of finite type. This is due to the fact that if .┌i' is a vertical segment, we would have to take .┌i = ┌i' , and .┌i would not be disjoint from.┌1 and.┌2 . The second reason is that the moment when the reflection from the vertical segment occurs depends on the argument of the trajectory line. The formula for the topological entropy of the subshift of finite type comes from counting of number of cylinders of length .n and then taking the exponential growth rate of this number as .n goes to infinity. Here we can try do exactly the same, but the problem occurs with the growth rate, since we have additional reflections from the vertical segment. This means that the cylinders of length .n from Sect. 3 correspond not to time .n, but to some larger time. How much larger, depends on the cylinder. However, there cannot be two consecutive reflections from the vertical segment, so

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this time is not larger than .2n, and by extending the trajectory we may assume that it is equal to .2n (maybe there will be more cylinders, but we need only a lower estimate). Thus, if the number of cylinders (which we count in Sect. 3) of length .n 1 log an , that is, the is .an , instead of taking the limit of . n1 log an we take the limit of . 2n half of the limit from Sect. 3. In such a way we get the following results. Proposition 5.1 Assume that the billiard table belongs to .H1/2 (ε, ) and fix . N ≥ 0 such that (3) is satisfied. Then the topological entropy of the billiard map restricted to .K ,N is larger than or equal to one half of the logarithm of the largest root of Eq. (2). Proposition 5.2 Assume that the billiard table belongs to.H1/2 (ε, ) and. tan ε ≥ 1. Then the topological entropy of the billiard map is at least . 21 log 2, so the map is chaotic in topological sense. Theorem 5.3 For the billiard tables from the class .H1/2 with the shape of .┌3 or ┌4 (the one that is not the vertical segment) fixed, the lower limit of the topological entropy of the generalized √ Bunimovich stadium billiard, as its length. goes to infinity, is at least . 21 log(1 + 2).

.

We can apply Proposition 5.2 to the Bunimovich mushroom billiard in order to get entropy at least . 21 log 2. As for the stadium, we need to make some computations, and again, we will make a slight improvement in the estimates. The interior of the mushroom billiard consist of a rectangle (the stalk) and a half-disk (the cap). According to our notation, the stalk is of vertical size 1; denote its horizontal size by ' . . Moreover, denote the radius of the cap by .t. √ Proposition 5.4 If . ' > 21 16t 2 − 1 then the topological entropy of the mushroom billiard is at least . 21 log 2. Proof Look at Fig. 9, where √ the largest possible .ε is used. We have .t sin ε = 1/4. Therefore, .tan ε = 1/ 16t 2 − 1. Similarly as for the stadium, when we use (3) with . N = 1, we may replace . N + 1 by . 23 . Taking into account that we need a strict

Fig. 9 Computations for a mushroom

ε

ε

ε

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Fig. 10 A mushroom with topological entropy at least 1 . log 2 2

√ inequality, we get . > 43 16t 2 − 1. However, . = √ so our condition is . ' > 21 16t 2 − 1.

'

+ t cos ε =

'

+

1 4

√

16t 2 − 1, ◻

Observe that the assumption of Proposition 5.4 is satisfied if the length of the stalk is equal to or larger than the diameter of the cap (see Fig. 10).

References 1. Alsedà, L., Llibre, J., Misiurewicz, M.: Combinatorial Dynamics and Entropy in Dimension One, 2nd edn. Advanced Series in Nonlinear Dynamics, vol. 5. World Scientific, Singapore (2000) 2. Block, L., Guckenheimer, J., Misiurewicz, M., Young, L.-S.: Periodic points and topological entropy of one dimensional maps. In: Global Theory of Dynamical Systems. Lecture Notes in Mathematics, vol. 819, pp. 18–34. Springer, Berlin (1980) 3. Bunimovich, L.A.: On the ergodic properties of nowhere dispersing billiards. Comm. Math. Phys. 65, 295–312 (1979) 4. Bunimovich, L.A.: Mushrooms and other billiards with divided phase space. Chaos 11, 802–808 (2001) 5. Kitchens, B.P.: Symbolic Dynamics. One-sided, Two-sided and Countable State Markov Shifts. Springer, Berlin (1998) 6. Misiurewicz, M., Zhang, H.-K.: Topological entropy of Bunimovich stadium billiards. Pure Appl. Funct. Anal. 6, 221–229 (2021)

Global Manifolds of Saddle Periodic Orbits Parametrised by Isochrons James Hannam, Bernd Krauskopf, and Hinke. M. Osinga

Abstract Global stable and unstable manifolds of a saddle periodic orbit of a vector field carry phase information of how trajectories on the manifold approach the periodic orbit in forward or backward time. This information is encoded on the respective manifold by its foliation by isochrons, which are submanifolds of codimension one comprising all points that are in asymptotic synchrony with a point of a given phase on the periodic orbit. We present a numerical method that finds a two-dimensional stable or unstable manifold of a saddle periodic orbit by computing a representative number of one-dimensional isochrons as arclength-parametrised curves on the manifold. As is demonstrated with examples of both orientable and nonorientable manifolds, this computational approach allows us to determine and visualise the interplay between their topological, geometric as well as synchronisation properties. Keywords Stable oscillation · Asymptotic phase · Isochron foliation · boundary value problem formulation When investigating a given physical system, one will most likely come across some form of oscillation. Commonly used models represent sustained and experimentally observable oscillations by attracting periodic orbits of a system of ordinary differential equations (ODEs). In this context, the asymptotic phase of an observed oscillation relative to some reference will often be significant [7, 11, 32]. Phase behaviour of an oscillation is encoded in ODEs by the isochrons, which are curves or, more generally, (hyper-)surfaces of equal phase. The concepts of asymptotic phase and isochrons were introduced by Winfree [36, 37] in the 1970s in the application context of modelling biological oscillations, such as the circadian rhythm; see the next section for their formal definitions. From a dynamical systems perspective, isochrons are characterised by the fact that any point on a given isochron maps back to the same isochron under the time-.T map, where .T is the period of the periodic orbit. Hence, considering the dynamics on an isochron given by the time-.T map is very similar to J. Hannam · B. Krauskopf (B) · Hinke. M. Osinga Department of Mathematics, University of Auckland, Private Bag 92019, Auckland 1142, New Zealand e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 S. Olaru et al. (eds.), Difference Equations, Discrete Dynamical Systems and Applications, Springer Proceedings in Mathematics & Statistics 444, https://doi.org/10.1007/978-3-031-51049-6_8

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considering the Poincaré map of a given section transverse to the periodic orbit. The difference is that for a chosen local and typically flat Poincaré section the return time to the section is not always the same: it is equal to the period .T at the intersection point of the periodic orbit with the section, which is a fixed point, but the return time varies for nearby points. An isochron through a given point of the periodic orbit can be seen as the ‘ideal section’ on which the return time is always equal to the period . T ; in other words, the return map to the isochron acts like the stroboscopic map of a periodically driven system. The ‘price’ one pays for this nice property is that any isochron is typically curved and may be very complicated geometrically further away from the periodic orbit. The isochrons for all point on the periodic orbit foliate the basin of attraction of the respective periodic orbit and, collectively, their topological and geometric properties encode the resynchronisation behaviour of the system following any external perturbation. Foliations by isochrons are given implicitly by the nonlinear synchronisation dynamics of the given system. The connection with the dynamics of maps is that each isochron is the stable manifold of the corresponding point on the periodic orbit with respect to the time-.T map [10]. As such, they can generally not be determined analytically and must be found with numerical methods. With the limited computational and visualisation capabilities of the time, Winfree realised that already in planar systems the geometry of the isochrons, which are curves in this case, may be very complicated. Specifically, he computed and then sketched such isochrons indirectly from computed asymptotic phases of points in the basin, which he found by computing trajectories that end sufficiently close to a chosen point on the periodic orbit. As Winfree demonstrated in [38], this indirect method is not able to resolve the isochrons as curves in regions of large phase sensitivity. This realisation is behind more recent methods that compute isochrons directly as curves when the system is of dimension two. One approach is to compute onedimensional isochrons by extending, via integration in backward time, a first local approximation of a given isochron [2, 4, 12, 16, 30, 33, 34]. In the presence of strong phase sensitivity the globalisation of isochrons as curves via integration may still be challenging numerically. This motivated the alternative approach, taken here as well, of computing isochrons via setting up a one-parameter family of well-posed two-point boundary value problems (BVPs), one end point of which traces out the isochron associated with a given point on the periodic orbit. This BVP approach was introduced in [28] and refined in [13, 14, 22, 23]. It effectively implements the time-.T map in a numerically stable way and allows one to compute one-dimensional isochrons in planar systems reliably, efficiently and accurately—even in the presence of extreme phase sensitivity—as smooth curves parametrised by their arclength; see also [6, 19]. This has been demonstrated for periodic orbits as well as focus equilibria that are attracting or repelling, and we speak of their forward-time and backwardtime isochrons, respectively. In fact, finding both forward-time and backward-time isochrons in regions where they co-exist enables one to identify changes in the intersection properties of these two foliations that are associated with the emergence of phase sensitivity [13, 14, 22, 23]. Apart from attracting and repelling periodic orbits, one commonly encounters periodic orbits of saddle type in continuous-time dynamical systems with phase

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spaces of dimensions larger than two. Saddle periodic orbits have stable and unstable invariant manifolds, along which they are approached by trajectories in forward and backward time, respectively, and these are important objects for understanding the organisation of the overall dynamics. In particular, manifolds of codimension one act as separatrices of the flow that may form, for example, boundaries between different basins of attraction. These global manifolds cannot be found analytically and, hence, must be computed numerically. Already when it is of dimension two, it is a nontrivial task to compute and visualise a global invariant manifold. Several methods have been developed for this purpose, and they differ in how they represent a given two-dimensional stable or unstable manifold: for example, as a set of trajectory segments, a set of closed geodesics, as an invariant object of a PDE formulation, or as a set of covering boxes; see [18, 19] for overviews of the different computational approaches. In this chapter we present a new way of computing and visualising the twodimensional stable manifold .W s (┌ × ) and/or unstable manifold .W u (┌ × ) of a saddletype periodic orbit .┌ × via their foliations by isochrons. These foliations not only represent .W s (┌ × ) and .W u (┌ × ) as surfaces but, in addition to their intrinsic geometric information, also encode information about phase sensitivity and the relative velocity of dynamics on these manifolds. In our implementation, the isochron manifold algorithm, we adapt the BVP approach for one-dimensional isochrons of planar system to this higher-dimensional context. To this end, we formulate a one-parameter family of BVPs whose solutions are orbit segments that lie both on .W s (┌ × ) or u × . W (┌ ) and have end points that both lie on the same isochron of a prescribed asymptotic phase .θ. The key here is to define and compute a suitable fundamental segment that parametrises the family of orbit segments used to compute a specific isochron on the manifold; subsequently our algorithm proceeds effectively as for isochrons of a planar system. The presentation and implementation is for vector fields in .R3 , but it can also be used for systems in .Rn with .n ≥ 4 as long as the manifold itself is of dimension two. More specifically, we compute a sufficient number of isochrons on the respective manifold, which are uniformly distributed in phase and each computed up to a prespecified maximal arclength . L. From the computed set of phase-uniform isochrons we then generate a mesh on .W s (┌ × ) or .W u (┌ × ). This representation has the benefit of providing a natural two-dimensional parametrisation of the manifold in terms of the asymptotic phase .θ and the arclength . along its isochrons. In particular, the boundary of the computed part of the respective manifold lies at the chosen maximal arclength . L along each computed isochron, and visualising the surface for increasing . L allows us to ‘grow the manifold along isochrons’. Indeed, the geometry of the isochrons provides clues to the phase dynamics on the global invariant manifolds .W s (┌ × ) and .W u (┌ × )—globally and not just near .┌ × . This allows us to illustrate the geometry of the two-dimensional invariant manifold as a surface and, at the same time, represent the synchronisation properties on it as the periodic orbit × .┌ is approached in forward or backward time. We demonstrate, with examples of

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both orientable and nonorientable manifolds, how this representation by isochron foliations can be used to visualise and gain insight into the interplay between the topological, geometric and synchronisation properties of global invariant manifolds of periodic orbits. The chapter is organised as follows. Following the introduction of the necessary notation and definitions in Sect. 1, we present in Sect. 2 the formulation of the isochron manifold algorithm. Here the main focus in Sect. 2.1 is on finding the fundamental segment, and Sect. 2.3 then explains how the mesh is rendered from the set of computed isochrons. We proceed by demonstrating our method with three examples, where we make use of the phase information to render the computed surfaces in novel ways. In Sect. 3 we present a constructed model in .R3 with explicitly defined manifold and isochron geometry, which serves as a test case for the isochron manifold algorithm. The next two examples concern two-dimensional manifolds in .R3 whose geometry is not known beforehand. Firstly, in Sect. 4 we compute and show the foliation by isochrons of the orientable stable manifold of a saddle periodic orbit in the model by Sandstede from [29]. Secondly, in Sect. 5 we give an example of a nonorientable manifold of a saddle periodic orbit in the so-called .ζ 3 -model [3]. A first part of this manifold near the periodic orbit was computed and shown in [27]. We extend this computation here to explore the complicated geometry of this nonorientable manifold in a more global way, including how it relates to the geometry of the isochron foliation. We conclude in Sect. 6 with a brief summary and outlook.

1 Background and Notation We assume familiarity with the basic concepts of dynamical systems theory as they can be found in standard textbooks, such as [11, 21, 32], and focus here on additional notions and objects associated with phase and synchronisation. To set the stage, we consider throughout a sufficiently smooth (at least .C 1 ) vector field .

dx = F(x), x ∈ Rn . dt

(1)

Generally, the function .F : Rn → Rn also depends on parameters, but we do not indicate this here for notational convenience. We denote by .Φ(t; ·) the flow .

Φ(t; ·) : Rn → x |→

Rn Φ(t; x)

(2)

of (1) over time .t ∈ R. Then the trajectory (or orbit) of (1) with initial condition x ∈ Rn is given by . x(t) = Φ(t; x0 ). (3)

. 0

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A periodic orbit .┌ with period .T┌ is a closed trajectory given by ┌ := {Φ(t; γ0 ) | 0 ≤ t < T┌ }

.

(4)

with .Φ(T┌ ; γ0 ) = γ0 , where we assume that .T┌ > 0 is the smallest time with this property. Relative to the point .γ0 ∈ ┌, we construct a notion of phase θ = t/T┌ ∈ [0, 1).

.

(5)

By convention .γ0 is chosen to lie at the maximum with respect to the first coordinate along .┌. Note that, throughout, we define the phase .θ to be from the interval .[0, 1) (rather than the circle .S1 ). Suppose now that the periodic orbit .┌ = ┌ s is attracting, that is, all its Floquet multipliers have modulus smaller than .1, apart from the trivial Floquet multiplier at s .1 corresponding to the tangent direction to .┌ . Then, following [36–38], the phase s on .┌ can be extended for all initial conditions . x0 in its basin of attraction A(┌ s ) := {x0 ∈ Rn | Φ(t; x0 ) → ┌ s for t → ∞}

.

(6)

to a notion of asymptotic phase .Θ(x0 ), given implicitly by .

lim ‖Φ(t; x0 ) − Φ(t + T┌s Θ(x0 ); γ0 )‖ = 0,

t→∞

(7)

where .T┌s is the period of .┌ s . For a given phase .θ ∈ [0, 1) the isochron . Iθ (┌ s ) of s s .┌ consists of the initial conditions in .A(┌ ) that have identical asymptotic phase .θ. Winfree [36–38] defines the isochrons as the level sets of the asymptotic phase function, given by s s . Iθ (┌ ) = {x0 ∈ A(┌ ) | Θ(x 0 ) = θ}. (8) Guckenheimer [10] offers the alternative interpretation that the isochron . Iθ (┌ s ) is the stable manifold of the fixed point.γθ ∈ ┌ s under the time-.T┌s map associated with the periodic orbit .┌ s . The Invariant Manifold Theorem [15] implies that the isochrons are .(n − 1)-dimensional smooth manifolds that are tangent to the linearisation of the time-.T┌s map at .γθ . Moreover, any isochron . Iφ (┌ s ) is the diffeomorphic image of a given . Iθ (┌ s ) under the flow .Φ((φ − θ)T┌s ; ·). It follows that the isochrons . Iθ (┌ s ) with .θ ∈ [0, 1), which we refer to as the forward-time isochrons from now on, foliate the basin .A(┌ s ); we denote this foliation I(┌ s ) := {Iθ (┌ s ) | θ ∈ [0, 1)}.

.

(9)

Suppose now that all non-trivial Floquet multipliers of a periodic orbit .┌ = ┌ u have modulus larger than .1, so that the periodic orbit is repelling. As was pointed out in [14, 23], asymptotic phase and isochrons for.┌ u can be defined entirely analogously in this case, simply by reversing time. Hence, the basin of repulsion

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R(┌ u ) := {x0 ∈ Rn | Φ(t; x0 ) → ┌ u for t → −∞}

.

(10)

is foliated by the backward-time isochrons .Uθ (┌ u ) with .θ ∈ [0, 1), which are the unstable manifolds of the fixed points .γθ ∈ ┌ u under the time-.T┌u map; we denote this foliation u u .U(┌ ) := {Uθ (┌ ) | θ ∈ [0, 1)}. (11) We now come to the central case of a periodic orbit .┌ = ┌ × of saddle type, with non-trivial Floquet multipliers of both modulus less than .1 and modulus greater than .1. We further assume that there are no Floquet multipliers on the unit circle (apart from the trivial one), so that the periodic orbit is hyperbolic. Hence, .┌ × has, say, .k ≥ 1 attracting and .n − k − 1 ≥ 1 repelling directions. According to the Invariant Manifold Theorem for flows [11, 15, 21], the periodic orbit .┌ × has a stable invariant manifold s × n × . W (┌ ) = {x0 ∈ R | Φ(t; x0 ) → ┌ for t → ∞} (12) of dimension .k + 1 that is tangent to its stable Floquet bundle, and an unstable invariant manifold .

W u (┌ × ) = {x0 ∈ Rn | Φ(t; x0 ) → ┌ × for t → −∞}

(13)

of dimension .n − k that is tangent to its unstable Floquet bundle. Note that .┌ × ⊆ W s (┌ × ) ∪ W u (┌ × ). Clearly, trajectories on the stable manifold.W s (┌ × ) spiral towards.┌ × with a given phase. Since we can interpret .W s (┌ × ) as the basin of attraction of .┌ × when the flow is restricted to .W s (┌ × ), we have that the stable manifold is foliated by forward-time isochrons . Iθ (┌ × ) of the foliation .

I(┌ × ) = W s (┌ × ).

(14)

Likewise the unstable manifold .W u (┌ × ), the basin of repulsion of .┌ × of the flow restricted to it, is foliated by backward-time isochrons .Uθ (┌ × ) of the foliation .

U(┌ × ) = W u (┌ × ).

(15)

( ) Note that initial conditions. x0 ∈ Rn \ W s (┌ × ) ∪ W u (┌ × ) do not reach.┌ × in either forward or backward time, which means that the notions of asymptotic phase and isochron of a saddle periodic orbit is special and can only be defined on its stable and unstable manifolds. These foliations provide a new way of considering and computing the invariant manifolds .W s (┌ × ) and .W u (┌ × ). In particular, this is the case when these manifolds are of dimension two, that is, surfaces, and the isochrons are curves. Since this is the direct generalisation of one-dimensional isochrons of stable and repelling periodic orbits of planar systems, the task is to adapt the known methods for their computation

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to ambient spaces of higher dimension. The isochron manifold algorithm introduced in the next section achieves exactly that for the refined BVP method from [14].

2 Computing the Isochron Foliation of a Two-Dimensional Invariant Manifold The planar BVP isochron algorithm from [14, 22, 28] computes one-dimensional isochrons as arclength-parametrised curves in a two-dimensional basin of attraction or repulsion. We now extend this approach, in the refined form presented in [14], to account for the further degrees of freedom available in a phase space of dimension larger than two. While this seems straight-forward conceptually, careful consideration of the nature of two-dimensional invariant manifolds turns out to be important in this endeavour. We proceed by discussing the required BVP setup for the forward-time isochrons . Iθ (┌ × ) on the stable manifold .W s (┌ × ); backward-time isochrons .Uθ (┌ × ) of .W u (┌ × ) are computed in the same way by considering the vector field (1) with reversed time. For ease of exposition, the formulation is for vector fields in .R3 , but we do briefly discussed the more general case of computing a two-dimensional global manifold in .Rn with .n > 3. The isochron manifold algorithm has been implemented within the continuation software package AUTO [5]. We remark that the formulation of the respective BVP in AUTO requires a rescaling of the vector field with the integration time, generally . T┌ × in the present context, so that each orbit segment has its begin point at .t = 0 and its end point at .t = 1. For clarity of exposition, the algorithm is presented below not in its implemented form but in conceptual terms and without this rescaling.

2.1 Construction of the Fundamental Segment The key idea behind our approach is to find for the given the point .γθ ∈ ┌ × a fundamental segment . sθ , which acts as an (approximate) fundamental domain. By this we mean that the orbit segments with end points on . sθ after integration time .T┌× have the property that their begin points trace out an accurate numerical representation of (a first piece of) . Iθ (┌ × ). We focus here on the crucial new element compared to the planar case: the definition of the fundamental segment in .R3 by making use of the two-dimensional linear information, which can be obtained from .┌ × and its stable Floquet vector .wθ at .γθ . The periodic orbit .┌ × and associated stable Floquet bundle can be found with a standard BVP setup [20]. This can be done in such a way that .┌ × is represented by an orbit segment .γ θ with integration time .T┌× subject to the combined periodicity and phase condition .γ(0) = γ(T┌× ) = γθ ; then the stable Floquet vector .wθ (which we take to have unit length) at the point .γθ is also readily available [14, 22, 24]. With

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γ θ and .wθ as given input, we now define an orthonormal basis at .γθ as follows. The stable eigenspace . E θs of .┌ × at .γθ is spanned by .wθ and .F(γθ ), which is the Floquet vector at .γθ of the trivial Floquet multiplier .1, given by the right-hand side of (1). We define .wΦ θ ∈ span(w θ , F(γθ )) perpendicular to .w θ by the orthonormalisation

.

Φ .w θ

) ( F(γθ ) · wθ wθ , = c F(γθ ) − ‖F(γθ )‖

3 where .c > 0 is such that .‖w Φ θ ‖ = 1. The third and only additional basis vector in .R is then simply the orthonormal vector to . E θs given by

wnθ = wθ × w Φ θ,

.

n n yielding the orthonormal coordinate system .(w θ , w Φ θ , nw θ ) at .γθ . Note that in .R j s with .n > 3, the required .n − 2 orthonormal vectors .wθ to . E θ can be constructed one-by-one via Gram–Schmidt orthonormalisation. n Figure 1 illustrates how the coordinate system .(w θ , w Φ θ , w θ ) is used to construct the fundamental segment . sθ for the computation of the isochron . Iθ (┌ × ) on the stable manifold .W s (┌ × ) of .┌ × . Panel (a) shows the saddle-type periodic orbit .┌ × with the s constructed coordinate system based at .γθ . Note that .w θ and .wΦ θ span the plane . E θ , Φ n which also contains .F(γθ ) and its projections onto .wθ and .wθ . The vector .wθ is the vector normal to . E θs . To construct the fundamental segment we consider the one-parameter family of orbit segments .u with integration time .T┌× whose end points lie along the stable Floquet direction from .γθ , which is formalised by the boundary condition

u (T┌× ) = γθ + η w θ

. η

with parameter .η ∈ R+ . Note that the orbit segment .γ representing .┌ × is the solution of the BVP for .η = 0. Hence, we find .uη by continuation in .η > 0 from .u0 = γ. As .η increases, the begin point . uη (0) moves away from .γθ , as well as away from .w θ and the latter is measured by the distance .δ. The continuation is stopped when .δ reaches a prescribed value .δmax for some .ηδmax . Since . Iθ (┌ × ) is tangent to .wθ , we have that 2 .δmax is of orderη δmax . Moreover, for suffiently small .ηδmax , the begin point . umax (0) of the final orbit .umax = uηδmax lies (to good approximation) on . Iθ (┌ × ) and, thus, on the stable manifold .W s (┌ × ). Therefore, the two linear segments connecting the three points .γ0 , .umax (T┌× ) and .umax (0) constitute a good approximation of a first and tiny piece of the isochron . Iθ (┌ × ), provided .δmax is sufficiently small. In the examples of stable manifolds in Sects. 3 and 4, .δmax was set to .5 × 10−4 , and for the nonorientable manifold in Sect. 5 we used .δmax = 5 × 10−6 . The oriented line segment between .umax (T┌× ) and .umax (0), shown in Fig. 1b, is the sought after fundamental segment . sθ ; this choice is a direct generalisation to .R3 of the construction of the fundamental segment for the planar case in [14]. Observe that the distance .δmax of .umax (0) to .wθ is composed of the projected distances .δ Φ in

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(a)

F(γθ )

Eθs

wθn

wθΦ Γ×

γθ

wθ

(b) umax (0) δn

Eθs

δmax umax

Iθ (Γ × ) sθ

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ηmax

wθΦ

wθn

wθ

γθ

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vτ (0)

(c)

Eθs

Iθ (Γ × ) sθ vτ

Γ×

vτ (TΓ ×)

τ wθ

wθΦ

wθn γθ

Fig. 1 Set-up to compute the isochron. Iθ (┌ × ) (light-blue) at.γθ ∈ ┌ × on the two-dimensional stable manifold of a saddle-type periodic orbit .┌ × (green) in .R3 . Panel (a) shows the orthonormal basis Φ n .(w θ , w θ , w θ ) (dark-blue) at.γθ with.F(γθ ). Panel (b) illustrates the construction of the fundamental segment . sθ (orange), and panel (c) the orbit segment .v τ used to grow . Iθ (┌ × ) by continuation in .τ ∈ [0, 1]

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E θs and .δ n normal to . E θs . We refer to the components .δ Φ and .δ n as the phase shear and the lift-off of the isochron . Iθ (┌ × ), respectively. Note that .δ n /= 0 when .W s (┌ × ) is nonlinear, which is the typical situation in practice. We find that the phase shear .δ Φ remains relatively small in all of our examples; this means, in practice, that .δmax ≈ δ n primarily controls the lift-off from . E θs .

.

2.2 Continuation of the Isochron as a Curve The isochron algorithm now proceeds, as for the planar case [14, 23], by considering the one-parameter family of orbit segments .v τ with integration time .T┌× and end point on the fundamental segment . sθ = umax (0) − umax (T┌× ), which is defined by the boundary condition v (T┌× ) = γθ + ηδmax w θ + τ

. τ

umax (0) − umax (T┌× ) , ‖umax (0) − umax (T┌× )‖

where .τ ∈ [0, 1]. Note that .umax satisfies this boundary condition, and with .v 0 = umax we perform a continuation in the family parameter .τ . As Fig. 1c illustrates, the begin point .v τ (0) then traces out the isochron . Iθ (┌ × ) ⊂ W s (┌ × ). When .τ = 1 is reached, which means that .v 1 (T┌× ) has maximal distance .δmax , the algorithm switches to the continuation of an orbit segment with integration time .2T┌× . The required extended orbit segment is obtained by appending to .v 1 the orbit segment .v 0 = umax that defines the fundamental segment . s θ ; note that .v 1 (T┌ × ) = v 0 (0) by construction. This extended orbit segment is again continued in .τ until .τ = 1 when the orbit segment is extended in the same way to one with integration time .3T┌× , and so on until . Iθ (┌ × ) is obtained up to a chosen maximal arclength . L. Note that . Iθ (┌ × ) is represented as an arlength-parametrised curve by a sequence of mesh points; the density of mesh points depends on the size of the continuation steps taken, which is determined by prescribed accuracy settings of the continuation. Figure 1 illustrates the situation for an orientable manifold, but in .Rn with .n ≥ 3, the manifold .W s (┌ × ) may be nonorientable; we will see an example of this in Sect. 5. In this case, the begin and end points of the orbit segments .uη for .η > 0 are n on either side of the plane spanned by .w Φ θ and .w θ ; in particular, the line segment between .umax (0) and .umax (T┌× ) does not define a fundamental segment. This issue is overcome by considering (multiples of) the second return time .2T┌× —defining the double-cover of the time-.T┌× map—in the construction of the fundamental segment . sθ . Finally, we mention that the computational set-up for isochrons of foci in planar systems, as presented in [14, 23], can be generalised in the same way to compute (isochron foliations on) two-dimensional stable or unstable manifolds of saddle-foci in .Rn .

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2.3 Representing and Visualising the Manifold Obtaining a representation of .W s (┌ × ) requires the computation of isochrons × × . Iθi (┌ ) ⊂ I(┌ ) for a suitable finite subset .θi ∈ [0, 1]; here we assume that all × . Iθi (┌ ) are computed up to the same maximal arclength . L. In all our examples, we consider . Nθ phase-uniform isochrons, meaning that .θi = i/Nθ for .i = 0, 1, . . . (Nθ − 1) for a suitable . Nθ ∈ N. We remark that the orbit segment .γ with .γ(0) = γθ and the associated Floquet bundle defining .w θ can be rotated via continuation in the phase .θ to obtain the required input for these . Nθ separate isochron computations; see [14, 24] for details. The computed isochrons on the two-dimensional stable (or unstable) manifold can be used to render and visualise the respective manifold as a surface in different ways. Indeed, the foliation of the two-dimensional manifold of an oscillator by isochrons introduces a natural two-dimensional parametrisation that defines the point . x(θ, ) in terms of its asymptotic phase .θ and arclength . from the base point .γθ along the isochron. It is straightforward to render the computed isochrons with base points .γθ in .R3 as a fan or comb of curves parametrised by .0 < ≤ L. This already gives an impression of the two-dimensional manifold as a surface, as well as the phase dynamics on it. However, it is less straightforward (yet highly desirable) to visualise manifolds directly as surfaces. This requires the construction of a suitable mesh, that is, a triangulation that can be rendered for visualisation; we use MATLAB for this purpose. Depending on the geometry of the manifold and of the isochrons on it, generating a mesh of good quality may be a challenge. In light of the parametrisation property, a natural approach is to use a uniform mesh in .(θ, )-space, that is, a triangulation generated from the points . x(θi , j ) on the computed isochrons . Iθi (┌ × ). Here .0 ≤ j ≤ N for a chosen number . N of mesh points on each of the . Nθ isochrons. We assume further that these mesh points are distributed uniformly in arclength along each . Iθi (┌ × ), that is, . j = j/L; note that this is readily achieved by interpolation of the computed points on each . Iθi (┌ × ) that are generated by the isochron manifold algorithm. Generating a triangulation from this phase- and arclength-uniform mesh to represent the surface .W s (┌ × ) works well near the periodic orbit .┌ × when the vectors .wθi of its linear bundle are far from colinear with the tangent vector .F(γθi ); that is, the angle between .w θi and .F(γθi ) is (reasonably) close to . π2 . However, as we will see with the examples later in the section, already in this quite ideal local situation, this type of mesh may be less suitable further away from .┌ × because of nonlinear effects; specifically, a very uneven distance may arise between neighbouring computed isochrons as a function of their phase. The issue is that line segments between point . x(θi , j ) and . x(θi+1 , j ) may become very large, which leads to very obtuse (narrow) triangles and, hence, a deterioration of the mesh quality. To avoid the issue of obtuse triangles, we consider separate triangulations for the respective strips between each pair of successive isochrons, which we also refer to as phase ribbons. More optimal triangulations for such triangle strips are investigated in [35] and also in [25] in the context of triangulated ruled-surfaces, where triangulated

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strips arise from piecewise-linear interpolation between two discrete curves on a surface under consideration. Specifically, we attempt to connect a point . x(θi , j ) on . Iθi (┌ × ) to the closest point on . Iθi+1 (┌ × ), which we denote by . x(θi+1 , ˜( j)). For each . j, the corresponding point at arclength . ˜( j) is determined by a search on the parametrised computed curve . Iθi+1 (┌ × ). The triangulation of the respective phase ribbon is then generated from the resulting mesh points, and the surface rendered as the union of these ribbons. We may take advantage of the representation of .W s (┌ × ) (or .W u (┌ × )) as the union of phase ribbons to illustrate and interpret the geometry of isochrons, whilst also highlighting the geometry of the manifold. This can be done in different ways, and in the examples later in this chapter we employ two types of visualisation. Firstly, it is informative to colour these phase ribbons according to a palette that depends on the phase .θ; specifically, we use shades of blue for stable manifolds and shades or red for unstable manifolds. Secondly, we represent successive ribbons on the manifold alternatingly transparent and coloured by .θ; the contrast between successive phase ribbons, in combination with a ‘see-through’ effect due to this transparency, provides additional insights into the interplay between the geometry of the manifold and the isochron foliation on it.

3 A Constructed Test-Case Example In the spirit of planar models proposed by Winfree [38], we present the vector field ⎧ x 2 + y2 ⎪ ⎪ x˙ = βx − (1 − κz) ω y + σx , ⎪ ⎪ ⎪ 1 − ζz ⎨ x 2 + y2 . ⎪ y˙ = (1 − κz) ωx + β y + σ y , ⎪ ⎪ 1 − ζz ⎪ ⎪ ⎩ z˙ = αz,

(16)

which can be written in cylindrical-polar coordinates with .x + i y = r e2πiψ as ⎧ ( ) r2 ⎪ ⎪ r ˙ = β + σ r, ⎪ ⎨ 1 − ζz .

⎪ ψ˙ = (1 − κz)ω, ⎪ ⎪ ⎩ z˙ = αz,

(17)

For any values of the parameters, system (16) is invariant under rotation about the z-axis, as can be seen from the fact that the equations for .r˙ and .z˙ of system (17) are ˙ Moreover, the .(x, y)-plane is always invariant and the decoupled from that for .ψ. dynamics on it is given by the first two equations of (16) and (17) with .z = 0, which is the (planar) normal-form of the Hopf bifurcation [11, 21, 32].

.

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We are interested in the supercritical case with .σ < 0, and also take .α > 0 so that the .z-direction is expanding. Then for .β > 0, and any value of .ζ and .κ,√the system has a saddle periodic orbit .┌ × in the .(x, y)-plane, which has radius . |β/σ| and frequency .ω. Due to invariance, the .(x, y)-plane is the stable manifolds .W s (┌ × ), with the exception of the origin (which is a repelling equilibrium); note that the frequency of rotation on .W s (┌ × ) is .ω irrespective of the radius .r since here .z = 0. Hence, each forward-time isochron . Iθ (┌ × ) ⊂ W s (┌ × ) is the straight ray from the origin in the .(x, y)-plane through the point .γθ ∈ ┌ × . Due to the rotational invariance, the unstable manifold .W u (┌ × ) is a surface of revolution, and the isochrons of both × × .I(┌ ) and .U(┌ ) map to one another, respectively, under any rotation about the . z-axis.

3.1 The Standard Saddle Periodic Orbit We first consider the case .ζ = 0, when the equation for .r˙ in (17) is decoupled from those for .ψ˙ and .z˙ . Therefore, the unstable manifold of .┌ × is the orthogonal, straight cylinder .

β W u (┌ × ) = {(x, y, z) ∈ R3 | x 2 + y 2 = − }, σ

and we refer to.┌ × as the standard saddle periodic orbit. The parameter .κ determines how the frequency of rotation, which is constant in planes .z = const and given by ˙ deviates from .ω as a function of .z. For .κ = 0, we have that the angular velocity .ψ, ˙ = ω for any .z, so that the time-.T┌× map preserves any plane with .θ = const. As a .ψ result, each isochron .Uθ (┌ × ) ⊂ W u (┌ × ) is a vertical straight line through the point × .γθ ∈ ┌ . For .κ / = 0, on the other hand, the angular velocity about the . z-axis, as given by the equation for.ψ˙ in (17), changes linearly with.z. For.κ > 0, the effective angular velocity is larger for .z < 0 and smaller for .z > 0; thus, trajectories on .W u (┌ × ) lag behind .┌ × above the .(x, y)-plane, while below it, they move increasingly faster than .┌ × in terms of the polar angle .ψ. As a result of this linear decrease/increase of the angular velocity with .z, the backward-time isochrons .Uθ (┌ × ) ∈ U(┌ × ) are no longer straight lines but spirals—specifically, they are helices with slope . ωκ at × s × .┌ —that together form the straight cylinder . W (┌ ). We now employ the isochron computation algorithm for saddle periodic orbits in 3 × .R to compute the global invariant manifolds of the standard saddle periodic orbit.┌ . In our computations, we fix .β = 4, .σ = −1, .ω = 2, and .α = 1 throughout, so that × .┌ has radius .2. Specifically, we compute and show 25 forward-time isochrons on s × u × . W (┌ ) and 25 backward-time isochrons on . W (┌ ), each up to maximal arclength . L = 10 and uniformly distributed in phase; the number of isochrons is kept quite low here purely for illustration purposes. The results for .κ = 0 and for .κ = 0.5 are shown in .(x, y, z)-space in Fig. 2.

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(a1)

(a2)

W u (Γ × )

Γ×

W s (Γ × )

(b1)

Γ×

I(Γ × ) = W s (Γ × )

(b2)

W u (Γ × )

Γ×

W s (Γ × )

U(Γ × ) = W u (Γ × )

U(Γ × ) = W u (Γ × )

Γ×

I(Γ × ) = W s (Γ × )

θ Fig. 2 Computed isochrons on the global invariant manifolds of the standard periodic orbit .┌ × (green curve) of system (16) with .ζ = 0, for .κ = 0 in row (a) and for .κ = 0.5 in row (b). The left column shows the manifolds .W s (┌ × ) (cyan–blue surface) and .W u (┌ × ) (yellow–red surface) up to maximal arclength . L = 10 and coloured according to their phase .θ as given by the colour bars. The right column shows the same manifolds divided into alternatingly coloured and transparent phase 1 . The same phase colouring is used in all subsequent figures ribbons of phase width .∆θ = 25

Row (a) of Fig. 2 for .κ = 0 shows that the computed isochrons are as expected: the forward-time isochrons . Iθ (┌ × ) ⊂ W s (┌ × ) are straight rays in the .(x, y)-plane, and the backward-time isochrons .Uθ (┌ × ) ⊂ W u (┌ × ) are vertical straight lines. Panel (a1) shows .W s (┌ × ) and .W u (┌ × ) as surfaces rendered from the respective 25 computed isochrons, where colour is used to indicate the associated phase on the two manifolds. Panel (a2) illustrates the geometry of, and the phase information on .W s (┌ × ) and .W u (┌ × ) by showing alternatingly coloured and transparent phase 1 ribbons, each of which has a phase width of .∆θ = 25 . This highlights the computed

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isochrons as curves across which the colour changes and, moreover, creates a ‘seethrough’ effect that enhances the three-dimensional nature of the image. This type of illustration also highlights where isochrons are closer to or further away from each other in Euclidean distance in phase space, which indicates slower or faster dynamics along trajectories in the respective regions on the invariant manifold. Row (b) of Fig. 2 shows, in the same way, the computed isochrons for .κ = 0.5. As predicted, the forward-time isochrons . Iθ (┌ × ) ⊂ W s (┌ × ) are unchanged, but the backward-time isochrons .Uθ (┌ × ) are now spirals on the straight cylinder .W u (┌ × ); this is especially visible in the representation by phase ribbons in panel (b2). We stress that the manifold .W u (┌ × ) has now been rendered as the union of .25 spiralling phase ribbons bounded by neighbouring computed isochrons. Comparison of the rendered surfaces .W u (┌ × ) in row (b) with those in row (a) illustrates that this manifold is computed and rendered accurately as the cylinder of radius.2, in spite of the spiralling nature of the isochrons on it.

3.2 The Funnel Saddle Periodic Orbit For .ζ /= 0, the unstable manifold .W u (┌ × ) is no longer given explicitly and needs to be found numerically. For small .ζ and locally near .┌ × , this invariant manifold is a perturbation of the cylinder for .ζ = 0. However, its global properties are quite different. Notice from the equation for .r˙ of (17) that the flow in the .r -direction blows up for .z → ζ1 , while it remains bounded in the .z-direction. This means that the manifold .W u (┌ × ) approaches the plane with .z = ζ1 . For .z → ±∞, on the other hand, .r˙ is bounded, while .z˙ → ±∞; hence .W u (┌ × ) becomes practicall vertical for large .|z|. These asymptotic properties mean that .W u (┌ × ) has the shape of a funnel, and we refer to .┌ × when .ζ /= 0 as a funnel saddle periodic orbit. As we found for the standard periodic orbit in Sect. 3.1, when .κ = 0, the isochron .Uθ (┌ × ) is the intersection of the funnel with the half-plane of angle .ψ = θ with respect to the positive real axis. For .κ /= 0, on the other hand, the isochrons in .U(┌ × ) spiral around this same funnel. Figure 3 shows the surfaces .W s (┌ × ) and .W u (┌ × ) in .(x, y, z)-space of the funnel 1 saddle periodic orbit.┌ × of system (16) with.ζ = − 11 , as computed with the isochron computation algorithm, for .κ = 0 in row (a) and for .κ = 0.5 in row (b). As before, these surfaces have been rendered from 25 phase-uniform isochrons each, and they √ are shown in the region of interest given by .r = x 2 + y 2 ≤ 15 and .−11 ≤ z ≤ 8. The style of presentation is as in Fig. 2 with panels (a1) and (b1) showing the phase on .W s (┌ × ) and .W u (┌ × ) with a continuous colour, while panels (a2) and (b2) show alternatingly coloured and transparent phase ribbons. Notice in Fig. 3 that, as expected, the stable manifold .W s (┌ × ) is unaffected by the fact that .ζ /= 0. The unstable manifold.W u (┌ × ), on the other hand, is no longer a straight cylinder, but has the predicted characteristic funnel shape: it approaches the asymptotic plane defined

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(a2)

(a1)

W s (Γ × )

Γ×

Γ× I(Γ × ) = W s (Γ × )

W u (Γ × )

(b1)

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U(Γ × ) = W u (Γ × )

(b2)

Γ×

Γ× I(Γ × ) = W s (Γ × )

W u (Γ × )

U(Γ × ) = W u (Γ × )

Fig. 3 Computed isochrons on the global invariant manifolds of the funnel periodic orbit.┌ × (green 1 curve) of system (16) with .ζ = − 11 , for .κ = 0 in row (a) and for .κ = 0.5 in row (b). Rendering and colours are as in Fig. 2

by .z = ζ1 = −11 and does not exist for .z ≤ −11. In row (a) of Fig. 3, for .κ = 0, the backward-time isochrons .Uθ (┌ × ) ⊂ W u (┌ × ) are curves with fixed polar angle .ψ for any .z > −11, as is illustrated nicely by the phase ribbons in panel (a2). In row (b) for .κ = 0.5, the geometry of the surface .W u (┌ × ) is the same, but the backward-time isochrons .Uθ (┌ × ) ⊂ W u (┌ × ) now spiral around the funnel .W u (┌ × ). Notice that the rotation is very pronounced where the funnel is quite narrow, but comes to a halt as the isochrons .Uθ (┌ × ) ⊂ W u (┌ × ) approach the asymptotic plane; see panel (b2). This observation is explained by the fact that .r˙ blows up as .z → ζ1 = −11 (while .z˙ is bounded for .z = −11). Comparison of Fig. 3 with Fig. 2 shows that, locally near .┌ × , the surface .W u (┌ × ) 1 for .ζ = − 11 is indeed a perturbation of the cylinder .W u (┌ × ) for .ζ = 0; moreover, 1 for either value of.κ, the isochrons.Uθ (┌ × ) ⊂ W u (┌ × ) for.ζ = − 11 are close to those

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for .ζ = 0 near .┌ × . As for the standard saddle periodic orbit, we find that .W u (┌ × ) is 1 , irrespective of whether the isochrons spiral or computed accurately also for.ζ = − 11 not. Indeed, there are no discernible differences between the two computed manifolds u × . W (┌ ), for .κ = 0 in row (a) and for .κ = 0.5 in row (b) of Fig. 3, when viewed as a surface; more precisely, the difference between the two triangulations is on the order of the interpolation error between mesh points. This demonstrates that the isochron manifold algorithm is capable of computing two-dimensional global manifolds of a saddle periodic orbit reliably and accurately. Moreover, it allows one to investigate the interplay between the geometry of these surfaces and the phase dynamics on them, as we will now illustrate with two more general examples without special symmetry properties.

4 Orientable Manifold in Sandstede’s Model Sandstede [29] designed the model system, given as the three-dimensional vector field ⎧ ⎪ x˙ = ax + by − ax 2 + x(2 − 3x)(μ˜ − αz), ⎪ ⎪ ⎨ 3 . (18) y˙ = bx + ay − (bx 2 + ax y) − 2y(μ˜ − αz), ⎪ 2 ⎪ ⎪ ( 2 ) ⎩ z˙ = cz + μx + γx z + αβ x (1 − x) − y 2 , to explore different types of homoclinic bifurcations; for example, see [1, 8, 9, 26] for numerical investigations of different global bifurcations in this system. We use (18) here as a convenient example of a three-dimensional system. Based on information from [8, 9], we fix the different parameters to.a = 0.22,.b = 1.0, .c = −2.0, .α = 0.3, .β = 1.0, .γ = 2.0, .μ = 0.004, and .μ ˜ = 0, so that system (18) has a saddle periodic orbit .┌ × that is orientable; this means that .┌ × has positive Floquet multipliers so that s × u × × . W (┌ ) and . W (┌ ) are topological cylinders/annuli near .┌ . The system is quite × close to a homoclinic bifurcation of the origin .0 where .┌ bifurcates (when .μ = 0), and the unstable manifold .W u (┌ × ) is bounded by the one-dimensional unstable manifold .W u (0) on one side and by an attracting periodic orbit .┌ s on the other. Nearby is a second saddle equilibrium . p with a one-dimensional stable manifold s s . W ( p) that is ‘surrounded’ by .┌ . Figure 4 illustrates these invariant objects in .(x, y, z)-space. Here, .┌ × , .0, .W u (0), s s .┌ , . p, and . W ( p) are found with standard continuation techniques as implemented in the package AUTO [5], while the surfaces .W s (┌ × ) and .W u (┌ × ) are computed with the isochron manifold algorithm. Panel (a) does not show .W s (┌ × ) and focuses on the unstable manifold .W u (┌ × ), which is represented by 100 computed phase-uniform backward-time isochrons .Uθi (┌ × ) ⊂ W u (┌ × ). The computed isochrons are seen spiralling very tightly as they extend from .┌ × and accumulate onto the attracting periodic orbit .┌ s . On the other side of .┌ × , the backward-time isochrons accumulate

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W s (p) (a)

W u (0)

0

p

Γs

Γ×

W u (Γ × )

(b)

W s (p)

I(Γ × ) = W s (Γ × )

Fig. 4 Phase portrait of system (18) near the saddle periodic orbit .┌ × . Panel (a) shows .┌ × (green curve), the saddle points .0 (green dot) and . p (green dot), their one-dimensional manifolds .W u (0) (red curve) and .W s ( p) (blue curve), and the stable periodic orbit .┌ s (blue curve); also shown are 100 phase-uniform backward-time isochrons .Uθ (┌ × ) ⊂ W u (┌ × ) (yellow–red curves). Panel (b) is a slightly different view and also shows .W s (┌ × ) represented by 50 alternating phase ribbons

onto the curve .W u (0), which has two branches that both tend to infinity along the same direction to the left of the image. In the process, the isochrons on .W u (0) feature larger and larger excursions towards infinity as they keep spiralling around × s × .┌ . Panel (b) shows a slightly different view, where the stable manifold . W (┌ ) is now also included. This surface was rendered from 200 computed phase-uniform forward-time isochrons . Iθi (┌ × ) ⊂ W s (┌ × ) of arclength . L = 5; it is shown here as 50 alternating ribbons of equal phase width. Notice that . p and the first part of the curve .W s ( p) are obscured by a coloured phase ribbon; however, the lower branch of s . W ( p) bends around and re-enters the image on the right of the image.

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Figure 4 shows that the computed backward-time isochrons.Uθi (┌ × ) are extremely close to each other and already give a good impression of the geometry of the unstable manifold .W u (┌ × ), which has the shape of a tennis racket and lies very close to the .(x, y)-plane of system (18). However, due to the closeness of the bounding curves s u × .┌ and . W (0) to the saddle-periodic orbit .┌ , in combination with the strongly spiralling nature of the backward-time isochrons, it is impractical to render .W u (0) as a triangulated surface from this data. The stable manifold .W s (┌ × ), on the other hand, can be rendered effectively from the computed forward-time isochrons. Iθi (┌ × ). This surface is a cylinder, but panel (b) also hints at some interesting geometry and phase dynamics. Figure 5 shows a more global view of the surface .W s (┌ × ) as rendered from the computed forward-time isochrons of arclength . L = 5; for balance of the images, s × . W (┌ ) has been clipped here at the top, namely at . z = 3. As in Sect. 3, we show s × . W (┌ ) in two ways: rendered with asymptotic phase shown in colour in panel (a), 1 in and as alternatingly coloured and transparent ribbons of phase width .∆θ = 50 s × panel (b). This larger view shows that .W (┌ ) has a distinctive funnel shape near × u × .┌ , which is quite similar to that of the unstable manifold . W (┌ ) of system (16); s × compare with Fig. 3. More specifically, the top part of .W (┌ ) narrows in Fig. 5 and approaches the curve .W s ( p), while its lower part becomes very wide. Notice, however, that .W s (┌ × ) does not approach an invariant plane, as .W u (┌ × ) does in Fig. 3, but rather starts to bend upward on the left of the image, towards the region of positive .z. While Fig. 5a gives a very good impression of the geometry of .W s (┌ × ), the phase ribbon structure of Fig. 5b allows us to point out some interesting aspects of the phase dynamics. First of all, locally near .┌ × the isochrons on .W s (┌ × ) are practically vertical (well aligned with the .z-direction), as is the case for .W u (┌ × ) of system (16) with .κ = 0; compare with Fig. 3a2. In contrast to the constructed example, however, the isochrons . Iθ (┌ × ) ⊂ W s (┌ × ) of system (18) cannot be mapped to one another by a fixed rotation. Moreover, the different widths of the phase ribbons in the lower part of .W s (┌ × ) in Fig. 5a show that this manifold contains regions where the flow is faster as well as where it is slower relative to the mean speed of the dynamics on the manifold. Similarly, the spacing of phase ribbons in the upper part of .W s (┌ × ) becomes quite non-uniform as the surface approaches the curve .W s ( p). Figures 6 and 7 show the invariant objects of system (18) in a much larger part of s × .(x, y, z)-space. The manifold . W (┌ ) is shown as a smooth surface with asymptotic phase information in Fig. 6, while Fig. 7 shows it as represented by alternatingly coloured and transparent phase ribbons. Here,.W s (┌ × ) is rendered from the same 200 phase-uniform forward-time isochrons that have now been computed up to maximal arclength . L = 20, without any clipping. The angle of view in panels (a) is very similar to that in Fig. 5b, and panels (b) show a view from ‘the back’. Figures 6 and 7 illustrate that, further away from .┌ × , the topological cylinder .W s (┌ × ) has quite an intriguing geometry with large changes of its local (Gaussian) curvature. Notice how the upper part of .W s (┌ × ) approaches .W s ( p) very closely as .z increases, forming a ‘spire’, while the isochrons are rotating more and more around .W s ( p); this rotation accounts for the spire being quite short compared to the arclength . L = 20 of the

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(a)

Γ× W s (p)

W s (Γ × )

W u (0)

(b)

Γ× W s (p)

I(Γ × ) = W s (Γ × )

W u (0)

Fig. 5 An expanded view of.W s (┌ × ) and the invariant objects from Fig. 4. The manifold is rendered from 200 phase-uniform forward-time isochrons of arclength . L = 5. This view is clipped at .z = 3, and .W s (┌ × ) is shown phase-coloured in panel (a), and as 50 alternating phase ribbons in panel (b)

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W s (Γ × )

(a) W s (p) W u (0)

(b)

W u (0) W s (Γ × )

W s (p)

Fig. 6 Two more global views of the invariant objects from Fig. 5, where.W s (┌ × ) is rendered from 200 phase-uniform isochrons of arclength . L = 20 and coloured by phase

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(a) W s (p) W u (0) I(Γ × ) = W s (Γ × )

(b)

W u (0)

I(Γ × ) = W s (Γ × )

W s (p)

Fig. 7 The same objects and views as in Fig. 6, where .W s (┌ × ) is now shown as alternating phase ribbons

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isochrons that form it. The lower part of .W s (┌ × ), on the other hand, grows quite a bit with arclength, as is illustrated by the wider spacing between the phase ribbons in Fig. 7. Notice from Fig. 6 that, nevertheless, the 200 computed isochrons still allow us to render .W s (┌ × ) as a surface with a triangulation of good quality. From the far vantage points in Fig. 7, we can see that the isochrons foliating s × . W (┌ ) partially mimic the geometry of this surface by following its folding. There are regions where .I(┌ × ) is nearly orthogonal to the boundary of .W s (┌ × ) computed up to . L = 20, as well as regions where .I(┌ × ) almost appears to be tangent to this boundary. In fact, the isochrons are transverse to the boundary everywhere, but where the surface .W s (┌ × ) has high curvature the respective isochrons tend to have more acute angles with the boundary. This phenomenon can be observed in Fig. 7b for phases near .θ = 0 on the lower portion of .W s (┌ × ). Here, .W s (┌ × ) has an extreme fold, along which the isochrons . Iθ (┌ × ) also fold sharply. The large width of the phase-ribbons here indicates relatively fast dynamics along this part of the manifold. We see a similar, though less extreme, example of this folding on the left-hand side of Fig. 7b. In contrast, clearly transverse intersection of .I(┌ × ) with the boundary of .W s (┌ × ) can be observed along the left and bottom sides of Fig. 7a and along the bottom side of Fig. 7b, where .W s (┌ × ) has low curvature and relatively slower dynamics.

5 Nonorientable Manifold in the .ζ 3 -Model We now show that the isochron manifold algorithm is effective also when the saddle periodic orbit .┌ × has a pair of negative real Floquet multipliers, a stable one inside and an unstable one outside the unit circle. The stable and unstable manifolds.W s (┌ × ) and .W u (┌ × ) are then nonorientable, which means that they are Möbius bands locally near .┌ × . Following previous work on the computation of nonorientable manifolds in [27], we choose to consider such a saddle periodic orbit in the .ζ 3 -model ⎧ ⎪ ⎨ x˙ = y, y˙ = z, . ⎪ ⎩ z˙ = (α − x)x − β y − z,

(19)

which was introduced by Arneodo et al. [3]. For .α = 3.2 and .β = 2.0, the .ζ 3 model (19) has a saddle equilibrium . p with a one-dimensional stable manifold s × . W ( p). This curve is surrounded (close to . p) by a saddle periodic orbit .┌ with s negative Floquet multipliers as well as an attracting period-doubled orbit .┌ . Moreover, the origin .0, which is always an equilibrium of system (19), is a saddle and one branch of its one-dimensional unstable manifold .W u (0) accumulates onto .┌ s . All these invariant objects are readily found with AUTO [5]. The two-dimensional unstable manifold .W u (┌ × ) is a narrow Möbius band that is bounded by .┌ s ; we do not show .W u (┌ × ) here, but see [27] for more details and images. Rather, we focus on

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W s (p) (a)

W s (Γ × )

0

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Γ×

Γs p

W s (p) I(Γ × ) = W s (Γ × )

0

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(b)

Γ×

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Fig. 8 Phase portrait of system (19) near.┌ × (green curve). Shown are the saddle points.0 (green dot) and . p (green dot), their one-dimensional manifolds .W u (0) (red curve) and .W s ( p) (blue curve), the period-doubled periodic orbit .┌ s (blue curve), and the stable manifold .W s (┌ × ) (cyan–blue surface) rendered from 200 computed phase-uniform forward-time isochrons of arclength . L = 8. In panel (a) the surface.W s (┌ × ) is phase coloured, and in panel (b) it is represented by 50 alternating phase ribbons

the nonorientable and unbounded stable manifold .W s (┌ × ), which we render from 200 phase-uniform forward time isochrons computed with the isochron manifold algorithm.

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The phase portrait of system (19) near the saddle periodic orbit .┌ × is shown in Fig. 8. More specifically, all invariant objects mentioned above are shown with s × × . W (┌ ) computed and visualised with forward-time isochrons of.I(┌ ) of arclength s × . L = 8. In panel (a), the computed local part of . W (┌ ) is rendered as a surface coloured by asymptotic phase, which shows that it is, indeed, a Möbius band. The rendering by phase ribbons, shown in panel (b), illustrates how the isochrons of × × .I(┌ ), when continued along .┌ , fold over in the process to connect up with a halftwist—just as is the case when one makes a Möbius band (as a developable surface) from a thin strip of paper by giving it a half-twist along its length and glueing it along its short sides. Notice also that the isochrons are almost perpendicular to .┌ × and that the phase ribbons all have a very similar width. On the other hand, and in contrast to a twisted (flat) strip of paper, the isochrons and the Möbius band are curved quite a bit, especially at the bottom-right of the figure where they fold over quite suddenly. Figure 9 shows .W s (┌ × ) from the same angle and position near .┌ × , but now the manifold.W s (┌ × ) has been rendered from isochrons of.I(┌ × ) of arclength. L = 100; to allow for a direct comparison with Fig. 8, all global objects have been clipped to lie in the cube .[0, 5] × [−4, 4] × [−5, 4] ⊂ R3 . Figure 9 illustrates that the Möbius band .W s (┌ × ) approaches the stable manifold .W s ( p) in a spiralling fashion near the point . p. In the process, the isochrons of .I(┌ × ) perform a half-twist with every rotation around .┌ × . Note that the local curvature where the isochrons fold over becomes larger and larger, which is particularly visible in Fig. 9b where .W s ( p) is rendered by alternating phase ribbons. Figures 10 and 11 show the invariant objects of system (19) in a much larger part of .(x, y, z)-space to give a complete impression of the entire Möbius band .W s (┌ × ) as rendered from 200 phase-uniform isochrons of arclength . L = 100. Each figure shows the invariant objects from the same two view points, with .W s (┌ × ) rendered as a smooth phase-coloured surface in Fig. 10 and as alternating phase ribbons in Fig. 11. These two figures illustrate the geometry of .W s (┌ × ), which for . L = 100 is a ‘much wider’ Möbius band compared to the one for . L = 8 shown in Fig. 8. In fact, the Möbius band in Figs. 10 and 11 is too wide to be made from a flat strip of paper; see [31]. This fact highlights the nonlinear and highly curved nature of .W s (┌ × ). Notice in both views of panels (a) and (b), how the boundary of the Möbius band, when followed around from the outside part of .W s (┌ × ) furthest from .┌ × , starts to spiral around .W s ( p) while transitioning from one side of .┌ × to the other. The rendering in Fig. 11(b) illustrates particularly well how the surface .W s (┌ × ) folds very sharply over itself along the bottom-left side of this panel. The development of this sharper and sharper fold on .W s (┌ × ) is an integral part of the process of making an increasingly wider topological Möbius band. Notice that .W s (┌ × ) is actually ‘infinitely wide’ meaning that it leaves any finite ball around .┌ × when computed and rendered with isochrons in .I(┌ × ) of suffiently large arclength . L. Specifically, and as Figs. 10 and 11 illustrate, the Möbius band .W s (┌ × ) approaches infinity along s . W ( p).

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W s (p) (a) W s (Γ × )

W u (0)

0

Γ×

Γs p

W s (p) (b)

I(Γ × ) = W s (Γ × )

W u (0)

0

Γ×

Γs p

Fig. 9 Phase portrait of system (19) in the cube .[0, 5] × [−4, 4] × [−5, 4] ⊂ R3 with invariant objects and presentation as in Fig. 8, where .W s (┌ × ) (cyan–blue surface) is now rendered from 200 phase-uniform isochrons of arclength . L = 100

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W s (p) (a)

W s (Γ × )

Γs

W u (0)

(b)

Γs

W u (0)

Γ×

Γ× W s (p)

W s (Γ × )

Fig. 10 The invariant objects from Fig. 9 in a larger region of phase space, shown from two view points in panels (a) and (b), where .W s (┌ × ) is rendered from 200 phase-uniform isochrons of arclength . L = 100 and coloured by phase

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W s (p) (a)

I(Γ × ) = W s (Γ × )

Γs

W u (0)

(b)

Γs

W u (0)

Γ×

Γ× W s (p)

I(Γ × ) = W s (Γ × )

Fig. 11 The same objects and views as in Fig. 10, where .W s (┌ × ) is now shown as alternating phase ribbons

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6 Conclusions We presented the isochron manifold algorithm, which computes a representative number of the isochrons that foliate a two-dimensional stable or unstable manifold of a saddle periodic orbit .┌ × . This approach is complementary to other methods that represent and compute a two-dimensional global invariant manifold as one-parameter families of either trajectories or topological circles of constant geodesic distance from .┌ × ; see [17, 19]. The isochron manifold algorithm provides an efficient way of finding and then rendering such global invariant surfaces, with the new feature that it emphasises the phase dynamics of trajectories on the manifold as they approach × .┌ in forward or backward time. In particular, note that the isochrons are transverse to .┌ × , while trajectories on the invariant manifold become tangent to .┌ × . The isochron manifold algorithm is a relatively straightforward generalisation of the isochron algorithm for one-dimensional isochrons of planar systems. It has been implemented for two-dimensional manifolds in .R3 , and the key new ingredient is the construction of a suitable fundamental segment. The set of computed isochrons (each with its own sequence of mesh points) is then turned into a triangulation that represents the invariant manifold as a surface. While the parametrisation by asymptotic phase and arclength along isochrons is natural, simply selecting points on neighbouring isochrons at equal arclength distances generally does not translate into a triangulation of good quality. This is why we generate a triangulation based on smallest Euclidean distance between points on neighbouring isochrons. The practicality of the isochron manifold algorithm for computing and visualising two-dimensional invariant manifolds of saddle periodic orbits was demonstrated with three examples: orientable manifolds of periodic orbits in a constructed vector field and in Sandstede’s model, as well as a nonorientable manifold of the .ζ 3 -model. The foliation of an invariant manifold by its isochrons provides phase information, which we showed to be useful when it comes to understanding the geometric properties of two-dimensional stable or unstable manifolds. Colouring the computed part of an invariant manifold according to the phase already gives a good impression of the synchronisation dynamics. Moreover, rendering such a surface as a set of alternating phase ribbons is an effective way to illustrate geometry and phase dynamics at the same time. In particular, the width of ribbons readily allows one to identify regions on the manifold where the flow is fast (greater distances between isochrons) and regions where it is slow (small distances between isochrons). We remark that the relationship between the curvature of the isochrons and the curvature of the invariant manifold as a surface is an intriguing one. One might expect that regions of high curvature of isochrons naturally correspond to high curvature of the surface, but this is not necessarily the case. As we have seen, a sharp fold of the surface may lead to large curvature of the isochrons when their crossing of the fold is sufficiently transverse; on the other hand, when isochrons effectively run in the direction of such a fold they need not be curved much themselves. Moreover, we have seen that the surface may be rather flat while the isochrons are, nevertheless, strongly curved.

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Overall, the isochron manifold algorithm is a new tool that allows one to investigate the synchronisation and phase properties together with the geometric properties of the respective two-dimensional invariant manifold. While our present implementation is for .R3 , the computational framework works in an ambient space of any dimension. Moreover, an adaptation of the algorithm for computing isochrons of foci in planar systems [14, 23] provides, in the same spirit as for saddle periodic orbits, a BVP setup for finding a two-dimensional stable or unstable manifold of a saddle focus in .Rn , associated with a single pair of stable or unstable complex conjugate eigenvalues. Such an implementation and the investigation of relevant examples remain tasks for the future. Finally, we mention that the isochron manifold algorithm presented here is an intermediate step towards computing two-dimensional isochrons that foliate the basin of an attracting or repelling periodic orbit in .R3 —or, similarly, a three-dimensional stable or unstable manifold of a saddle periodic orbit .┌ × in a higher-dimensional space. Studying the dynamics of the respective time-.T maps on geometrically complicated two-dimensional isochrons would constitute a new way of determining the dynamics of higher-dimensional ODEs that is complimentary to their study by way of Poincaré return maps to a chosen section. Acknowledgements We thank Peter Langfield for helpful discussions.

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On Uniform Dichotomies for the Growth Rates of Linear Discrete-time Dynamical Systems in Banach Spaces Rovana Boruga (Toma)

Abstract The aim of the present paper is to give some characterizations for the growth rates of linear discrete-time dynamical systems in Banach spaces. More precisely, necessary and sufficient conditions of Datko–Zabczyk type as well as characterizations using Lyapunov functions are given using both invariant and strongly invariant projection sequences. Also, as consequences we obtain characterizations for the uniform exponential dichotomy behavior. Keywords Linear Discrete-time systems · Uniform .h-dichotomy · Uniform exponential dichotomy

1 Introduction The study of asymptotic properties for linear discrete time systems has gained an impressive development due to the fact that a large variety of real world phenomena are described by dynamical systems. One of the most important asymptotic behavior in the theory of nonautonomous dynamical systems is represented by the exponential dichotomy which was approached by Perron [19] for differential equations. Later, analogous results were ¨ [5] . These given for difference equations by Li [14] and by Coffman and Sch.affer were followed by the book of Henry [13], who introduced in his study the discrete dichotomy of a sequence .{Tn }n∈Z of bounded linear operators and pointed out the connection between discrete dichotomy and exponential dichotomy of an evolution family. Among the most remarkable papers regarding the asymptotic properties of dynamical systems, it is important to mention the works of Datko [8] and Zabczyk [25] who obtained results for evolution families, respectively for discrete time systems. Later, these results were extended by Przyłuski and Rolewicz [22] for the case of discrete R. Boruga (Toma) (B) West University of Timi¸soara, V. Pârvan Blv. No. 4, 300223 Timisoara, Romania e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 S. Olaru et al. (eds.), Difference Equations, Discrete Dynamical Systems and Applications, Springer Proceedings in Mathematics & Statistics 444, https://doi.org/10.1007/978-3-031-51049-6_9

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nonautonomous systems. A general overview concerning the Rolewicz-Zabczyk type techniques in the stability theory of dynamical systems is presented by Sasu, Megan and Sasu [23]. Also, Lupa [15] obtain a Datko–Zabczyk type characterization of nonuniform exponential stability for a discrete dynamics defined by a sequence of bounded linear operators on a Banach space. A different perspective of researching the exponential dichotomy behavior consists of using the discrete input-output methods. In this sense we mention the work of Megan, Sasu, Sasu [16] who characterize the uniform exponential dichotomy of a discrete evolution family in terms of admissibility in order to express dichotomy using Banach sequence spaces over .N. Moreover, Sasu and Sasu [24] propose a new study concerning the existence of the dichotomies of discrete dynamical systems on the half-line by considering an admissibility concept which consists in a solvability of an associated control system between two distinct spaces which belong to a general class of Banach sequence spaces. Another direction of studying the dichotomic behavior refers to the situation when the asymptotic behaviors are of polynomial type. In this case we discuss about the nonuniform polynomial dichotomy concepts, which were introduced by Barreira and Valls [3] for the continuous case of evolution operators and by Bento and Silva [4] for discrete time systems. In [10] D. Dragiˇcevi´c , A.L. Sasu and B. Sasu obtain for the first time a characterization of polynomial dichotomy with respect to a sequence of norms in terms of ordinary dichotomy and exponential dichotomy of suitable systems with respect to well-chosen sequences of norms. As an application in this area there are many papers that establish some robustness properties of the dichotomy under some small perturbation, for both exponential and polynomial cases. (see [7, 11, 26]). In a natural manner, as a generalization of the concepts mentioned above, we focus on another type of uniform dichotomy. More precisely, we consider the concept of uniform .h-dichotomy where .h is an unbounded nondecreasing sequence of real numbers with .h n ≥ 1. This idea was mentioned for the first time in the work of Pinto [20] who obtained interesting results about stability for a system which is at least weaker than those given by exponential stability under some perturbations. In this paper we obtain different characterizations of Datko–Zabczyk type for the uniform dichotomy with growth rates for linear discrete time systems in Banach spaces with respect to invariant and strongly invariant projection sequences. As consequences we give some characterizations that use Lyapunov functions for the uniform .h- dichotomy and also we deduce necessary and sufficient conditions for the particular case of uniform exponential dichotomy.

2 Preliminaries In this paper we consider . X a real or complex Banach space, .B(X ) the Banach algebra of all bounded linear operator acting on . X and . I the identity operator. The norm on . X and on .B(X ) will be denoted by .‖ · ‖. Also, we will denote by .Δ and .T the following sets

On Uniform Dichotomies for the Growth Rates of Linear Discrete-time …

177

Δ = {(m, n) ∈ N2 : m ≥ n}, T = {(m, n, p) ∈ N3 : m ≥ n ≥ p}.

.

We consider the linear discrete time system (A)

xn+1 = An xn , n ∈ N

.

generated by the sequence in .B(X ) .

A : N → B(X ) A(n) = An .

The solution of the system (.A) is represented by x = Anm xn ,

. m

where .

Anm =

for all (m, n) ∈ Δ,

⎧ ⎪ ⎨ Am−1 · · · An , m > n ⎪ ⎩

m=n

I,

which is the evolution operator associated to the discrete time system (.A). Remark 1 The following properties state: (i) . Ann+1 = An , for all .n ∈ N p p n .(ii) . A m A n = A m , for all .(m, n, p) ∈ T. .

Definition 1 A sequence . Pn on .B(X ) is called projection sequence on . X if .

Pn2 = Pn ,

f or all n ∈ N.

Remark 2 If . Pn is a projection sequence on . X , then the sequence . Q n = I − Pn is also a projection sequence on . X , called the complementary projection sequence of . Pn . Definition 2 The projection sequence. Pn is called invariant with respect to the linear system (.A) if . A n Pn = Pn+1 A n , for all n ∈ N. Remark 3 The projection sequence . Pn is invariant with respect to the linear system (.A) if n n . A m Pn = Pm A m , for all (m, n) ∈ N2 . Definition 3 The projection sequence . Pn is called strongly invariant with respect to the linear system (.A) if it is invariant to (.A) and for every .(m, n) ∈ Δ the restriction n . A m is an isomorphism from . Range Pn to . Range Pm .

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R. Boruga (Toma)

Remark 4 If . Pn is a strongly invariant projection sequence with respect to (.A) then there exists . B : Δ → B(X ) such that . Bmn is an isomorphism from . Range Q m to . Range Q n with the following properties: (i) . Anm Bmn Q m = Q m n n .(ii) . Bm A m Q n = Q n n n .(iii) . Bm Q m = Q n Bm Q m p p n .(iv) . Bm Q m = Bn Bm Q m .

Proof See [2]. Proof It follows immediately using the properties given in Remark 4 (see [1]). In what follows, we denote by .h = h n an unbounded nondecreasing sequence of real numbers with .h n ≥ 1. Definition 4 Let . Pn be an invariant projection sequence with respect to the linear system .(A). We say that the pair .(A, P) has uniform .h-growth if there are . M ≥ 1 and .ω > 0 such that (.uhg1 ) (.uhg2 )

h ω ‖Anm Pn x‖ ≤ Mh ωm ‖Pn x‖ ω ω n .h n ‖Q n x‖ ≤ Mh m ‖A m Q n x‖,

. n

for all .(m, n, x) ∈ Δ × X. Remark 5 Let . Pn be an invariant projection sequence with respect to the linear system .(A). Then the pair .(A, P) has uniform .h-growth if and only if there are . M ≥ 1 and .ω > 0 such that (.uhg1' ) (.uhg2' )

h ω ‖Am Pp x‖ ≤ Mh ωm ‖An Pp x‖ p p ω ω .h n ‖A n Q p x‖ ≤ Mh m ‖A m Q p x‖,

. n

p

p

for all .(m, n, p, x) ∈ T × X. Remark 6 In particular: 1. If .h n = en , for all .n ∈ N we obtain the concept of uniform exponential growth (u.e.g.). 2. If .h n = n + 1, for all .n ∈ N we obtain the concept of uniform polynomial growth (u.p.g.). Definition 5 Let . Pn be an invariant projection sequence with respect to the linear system .(A). The pair .(A, P) is called uniformly .h-dichotomic if there are . N ≥ 1 and .ν > 0 such that (.uhd1 ) (.uhd2 )

h ν ‖Anm Pn x‖ ≤ N h νn ‖Pn x‖ ν ν n .h m ‖Q n x‖ ≤ N h n ‖A m Q n x‖,

. m

for all .(m, n, x) ∈ Δ × X.

On Uniform Dichotomies for the Growth Rates of Linear Discrete-time …

179

Remark 7 Let . Pn be an invariant projection sequence with respect to the linear system .(A). Then the pair .(A, P) is uniformly .h-dichotomic if and only if there are . N ≥ 1 and .ν > 0 such that (.uhd1' ) (.uhd2' )

h ν ‖Am Pp x‖ ≤ N h νn ‖An Pp x‖ p p ν ν .h m ‖A n Q p x‖ ≤ N h n ‖A m Q p x‖,

. m

p

p

for all .(m, n, p, x) ∈ T × X. Remark 8 As particular cases we have: 1. If.h n = en , for all.n ∈ N we recover the concept of uniform exponential dichotomy (u.e.d.). 2. If .h n = n + 1, for all .n ∈ N we obtain the concept of uniform polynomial dichotomy (u.p.d.). 3. If.h n = ln(n + e) we obtain the concept of uniform logarithmic dichotomy (u.l.d.). Remark 9 Let . Pn be a strongly invariant projection sequence with respect to the linear system (.A). Then the pair .(A, P) is uniformly .h-dichotomic if and only if there are . N ≥ 1 and .ν > 0 such that (.uhd1'' ) (.uhd2'' )

h ν ‖Anm Pn x‖ ≤ N h νn ‖Pn x‖ ν n ν .h m ‖Bm Q m x‖ ≤ N h n ‖Q m x‖, . m

for all .(m, n, x) ∈ Δ × X. Theorem 1 ([12]) Let . Pn be an invariant projection sequence with respect to the linear system .(A). If .(A, P) has uniform .h-growth then it is uniformly .hdichotomic if and only if there exists a nondecreasing function . M : R∗+ → (1, ∞) with . lim M(n) = ∞ such that n→∞

(ueM1 ) .(ueM2 ) .

M(m − n)‖Anm Pn x‖ ≤ ‖Pn x‖ n . M(m − n)‖Q n x‖ ≤ ‖A m Q n x‖,

.

for all .(m, n, x) ∈ Δ × X.

3 The Main Results In what follows, we will consider . Pn an invariant projection sequence with respect to the linear system .(A). We denote by .H the set of .h n that satisfy the following properties: (h 1 )

.

(h 2 )

.

∃ H1 > 1 : h m+1 ≤ H1 h m , ∀ m ∈ N. ∞ ∑ .∀ α ∈ (−1, 0) ∃ H2 > 1 : h αj ≤ H2 h αn , ∀ n ∈ N.

.

j=n

(h 3 )

.

∀ α ∈ (0, 1) ∃ H3 > 1 :

m ∑

.

j=0

h αj ≤ H3 h αm , ∀ m ∈ N.

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R. Boruga (Toma)

Remark 10 It is easy to see that if .h n = en , then .h n ∈ H. Theorem 2 Let .h ∈ H. Then the pair .(A, P) is uniformly .h-dichotomic if and only if there are . D > 1 and .d ∈ (0, 1) such that (uh D1 )

.

∞ ∑ .

p

h dj ‖A j Pp x‖ ≤ Dh dn ‖Anp Pp x‖, for all .(n, p, x) ∈ Δ × X.

j=n

(uh D2 )

.

m ∑ .

n −d n h −d j ‖A j Q n x‖ ≤ Dh m ‖Am Q n x‖, for all .(m, n, x) ∈ Δ × X.

j=n

Proof Necessity. We suppose that .(A, P) is uniformly .h-dichotomic. Let .d ∈ (0, ν). Then for .(uh D1 ) we have ∞ ∑ .

p

h dj ‖A j Pp x‖ ≤ N

j=n

≤

∞ ∑

( h dj

j=n

N H2 h dn ‖Anp Pp x‖

=

hn hj

)ν

‖Anp Pp x‖ = N h νn ‖Anp Pp x‖

D1 h dn ‖Anp Pp x‖,

∞ ∑

h d−ν ≤ j

j=n

where D1 = N H2 > 1.

Also, for .(uh D2 ) we do a similar computation and we obtain m ∑ .

n −d n h −d j ‖A j Q n x‖D2 h m ‖Am Q n x‖, where D2 = N H3 > 1.

j=n

So, it follows that the relations .(uh D1 ) and .(uh D2 ) are satisfied for . D = 1 + max{D1 , D2 }. Sufficiency. We suppose that there are . D > 1 and .d ∈ (0, 1) such that .(uh D1 ) and .(uh D2 ) hold. For . j = m in .(uh D1 ) we obtain h d ‖Amp Pp x‖ ≤ Dh dn ‖Anp Pp x‖.

. m

For . j = n in .(uh D2 ) we obtain n h −d ‖Q n x‖ ≤ Dh −d m ‖Am Q n x‖.

. n

Corollary 1 The pair .(A, P) is uniformly exponentially dichotomic if and only if there are . D > 1 and .d ∈ [0, 1) such that (ueD1 )

.

∞ ∑ .

p

ed( j−n) ‖A j Pp x‖ ≤ D‖Anp Pp x‖, for all .(n, p, x) ∈ Δ × X.

j=n

(ueD2 )

.

m ∑ .

e−d( j−m) ‖Anj Q n x‖ ≤ D‖Anm Q n x‖, for all .(m, n, x) ∈ Δ × X.

j=n

Proof For .d ∈ (0, 1) the proof follows immediately form Theorem 2 if we take h = en . For .d = 0 see [21].

. n

On Uniform Dichotomies for the Growth Rates of Linear Discrete-time …

181

Corollary 2 The pair .(A, P) is uniformly .h-dichotomic if and only if there are D > 1, d ∈ (0, 1) and . L : Δ × X → R+ with

.

(uh L 1 ) .(uh L 2 )

p

L(n, p, Pp x) ≤ D‖An Pp x‖, for all .(n, p, x) ∈ Δ × X. n . L(m, n, Q n x) ≤ D‖A m Q n x‖, for all .(m, n, x) ∈ Δ × X. ) m ( ∑ hj d p . L(m, p, Pp x) + ‖A j Pp x‖ ≤ 2L(n, p, Pp x) h n j=n

.

.

(uh L 3 )

.

for all .(m, n, p, x) ∈ T × X. ) m ( ∑ h j −d p ‖A j Q p x‖ ≤ 2L(m, p, Q p x), .(uh L 4 ) . L(n, p, Q p x) + h m j=n for all .(m, n, p, x) ∈ T × X.

Proof Necessity. We suppose that .(A, P) is uniformly .h-dichotomic . We consider the Lyapunov function . L : Δ × X → R+ , defined by .

L(m, n, x) =

) ∞ ( ∑ hj d hm

j=m

‖Anj Pn x‖

+

) m ( ∑ h j −d hm

j=n

‖Anj Q n x‖.

Then .(uh L 1 ) and .(uh L 2 ) follow immediately using Theorem 2. For .(uh L 3 ) we have . L(m,

p, Pp x) +

) m ( ∑ hj d j=n

hn

p

‖A j Pp x‖ =

) ∞ ( ∑ hj d j=m

≤2

hm

p

‖A j Pp x‖ +

) ∞ ( ∑ hj d j=n

hn

) m ( ∑ hj d j=n

hn

p

‖A j Pp x‖ ≤

p

‖A j Pp x‖ = 2L(n, p, Pp x).

For .(uh L 4 ) we obtain . L(n,

p, Q p x) +

) m ( ∑ h j −d j=n

hm

p

‖A j Q p x‖ =

) n ( ∑ h j −d j= p

≤2

hn

) m ( ∑ h j −d j= p

hm

p

‖A j Q p x‖ +

) m ( ∑ h j −d j=n

hm

p

‖A j Q p x‖ ≤

p

‖A j Q p x‖ = 2L(m, p, Pp x)

Sufficiency. We suppose that there are . D > 1, d ∈ (0, 1) and . L : Δ × X → R+ such that the properties .(uh L 1 ) − (uh L 4 ) are satisfied. Then for .m → ∞ in .(uh L 3 ), respectively for . p = n in .(uh L 4 ) and using Theorem 2 we obtain the conclusion. Corollary 3 The pair .(A, P) is uniformly exponentially dichotomic if and only if there are . D > 1, d ∈ [0, 1) and . L : Δ × X → R+ with (ueL 1 ) .(ueL 2 ) .

(ueL 2 )

.

p

L(n, p, Pp x) ≤ D‖An Pp x‖ n . L(m, n, Q n x) ≤ D‖A m Q n x‖ m ∑ p . L(m, p, Pp x) + ed( j−n) ‖A j Pp x‖ ≤ 2L(n, p, Pp x)

.

j=n

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R. Boruga (Toma)

(ueL 3 )

.

.

L(n, p, Q p x) +

m ∑

e−d( j−m) ‖A j Q p x‖ ≤ 2L(m, p, Q p x), p

j=n

for all .(m, n, p) ∈ T, for all .x ∈ X. Proof It follows from Corollaries 1 and 2. Theorem 3 Let .h ∈ H. Then the pair .(A, P) is uniformly .h-dichotomic if and only if there are . D > 1 and .d ∈ (0, 1) such that (uh D1' )

.

(uh D2' )

.

m ∑ .

j=n ∞ ∑ .

h −d j ‖Anj Pn x‖

≤D

h dj p

j=n

‖A j Q p x‖

h −d m , for all .(m, n, x) ∈ Δ × X, Anm Pn x /= 0. ‖Anm Pn x‖

≤D

h dn p , for all .(n, p, x) ∈ Δ × X, An Q p x /= 0. p ‖An Q p x‖

Proof Necessity. We suppose that .(A, P) is uniformly .h-dichotomic. Let .d ∈ (0, ν). Then for .(uh D1' ) we have m ∑

h −d j

j=n

‖Anj Pn x‖

.

≤N

m ∑

h −d j

j=n

(

hj hm

)ν

∑ 1 1 1 ≤N· ν · h ν−d ≤ n n ‖Am Pn x‖ h m ‖Am Pn x‖ j=0 j m

·

1 , where D1 = N H3 > 1. ≤ D1 d n h m ‖Am Pn x‖ Also, for .(uh D2' ) we do a similar computation and we obtain ∞ ∑ .

h dj p ‖A j Q p x‖ j=n

≤ D2 h dn ·

1

, p ‖A p Q p x‖

where D2 = N H2 > 1.

So, it follows that the relations .(uh D1' ) and .(uh D2' ) are satisfied if we take . D = max{D1 , D2 }. Sufficiency. We suppose that there are. D > 1 and.d > 0 such that.(uh D1' ) and.(uh D2' ) hold. For . j = n in .(uh D1' ), respectively for . j = m in .(uh D2' ) we obtain that the relations .(uhd1 ) and .(uhd2 ) are satisfied, so .(A, P) is uniformly .h-dichotomic. Corollary 4 If the pair .(A, P) has uniform exponential growth, then it is uniformly exponentially dichotomic if and only if there are . D > 1 and .d ∈ [0, 1) such that m ∑ D ed(m− j) ≤ , for all .(m, n, x) ∈ Δ × X, Anm Pn x /= 0. n n ‖A P x‖ ‖A P x‖ n n m j j=n

.

(ueD1' )

.

(ueD2' )

.

.

∞ ∑ D ed( j−n) p ≤ , for all .(n, p, x) ∈ Δ × X, An Q p x /= 0. p p ‖A Q x‖ ‖A Q x‖ n p p j j=n

On Uniform Dichotomies for the Growth Rates of Linear Discrete-time …

183

Proof For .d ∈ (0, 1), the proof follows immediately form Theorem 3 if we take h = en . For .d = 0, if .m ≥ n + 1 and . Anm Pn x /= 0 then we have

. n

) n+1 n+1 ( m ∑ ∑ ∑ hj ω 1 1 1 1 . ≤ ≤M ≤ M H1ω ≤ n n ‖Pn x‖ ‖P x‖ h A P x A n n j n j Pn x j=n j=n j=n ≤ M D H1ω We obtained

1 . ‖Anm Pn x‖

‖Anm Pn x‖ ≤ N1 ‖Pn x‖, where N1 = M D H1ω .

.

(1)

If .m ∈ [m, n + 1] and . Anm Pn x /= 0 we use the growth property and we have ( n .‖A m Pn x‖

≤M

hm hn

)ω

‖Pn x‖ ≤ M H1ω ‖Pn x‖ ≤ N1 ‖Pn x‖.

(2)

From (1) and (2) it follows ‖Anm Pn x‖ ≤ N1 ‖Px ‖, for all (m, n, x) ∈ Δ × X.

.

Then ∞

∑ m−n+1 ∑ 1 1 1 = ≤ N1 ≤ N1 D n . n ‖Pn x‖ ‖P x‖ ‖A P x‖ ‖A n n m Pn x‖ j j=n j=n m

.

We obtained (m − n + 1)‖Anm Pn x‖ ≤ N1 D‖Pn x‖, for all (m, n, x) ∈ Δ × X.

.

(3)

Making a similar computation for . Q n we have (m − n + 1)‖Anp Q p x‖ ≤ N1 D‖Amp Q p x‖, for all (m, n, p, x) ∈ T × X.

.

(4)

From (3) and (4) using the majorization criteria given by Theorem 1 we obtain the conclusion. Corollary 5 The pair .(A, P) is uniformly .h-dichotomic if and only if there are D > 1, d ∈ (0, 1) and . L : Δ × X → R+ with

.

(uh L '1 )

.

(uh L '2 )

.

D , for all (m, n, x) ∈ Δ × X, Anm Pn x /= 0. ‖Anm Pn x‖ D p , for all (n, p, x) ∈ Δ × X, An Q p x /= 0. . L(n, p, Q p x) ≤ p ‖An Q p x‖

.

L(m, n, Pn x) ≤

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R. Boruga (Toma)

' .(uh L 3 )

.

' .(uh L 4 )

.

L(n, p, Pp x) +

) m ( ∑ hm d hj j=n ( m ∑

L(m, p, Q p x) +

j=n

hj hn

1

·

p ‖A j Pp x‖

)d ·

≤ 2L(m, p, Pp x)

1 p ‖A j Q p x‖

≤ 2L(n, p, Q p x).

for all .(m, n, p) ∈ T, for all .x ∈ X. Proof Necessity. It follows in a similar manner as the proof of Corollary 2 if we consider the Lyapunov function . L : Δ × X → R+ , defined by .

L(m, n, x) =

) m ( ∑ hm d j=n

hj

·

) ∞ ( ∑ hj d 1 1 + . · n n ‖A j Pn x‖ j=m h m ‖A j Q n x‖

Sufficiency. We suppose that there are . D > 1, d ∈ (0, 1) and . L : Δ × X → R+ such that the properties .(uh L '1 ) − (uh L '3 ) are satisfied. Then for . p = n in .(uh L 2 ), respectively for . p = n in .(uh L 3 ) and using Theorem 3 we obtain the conclusion. Corollary 6 If the pair .(A, P) has uniform exponential growth, then it is uniformly exponentially dichotomic if and only if there are . D > 1, d ∈ [0, 1) and . L : Δ × X → R+ with (ueL '1 )

.

(ueL '2 )

.

(ueL '3 )

.

' .(ueL 4 )

D for all .(m, n, x) ∈ Δ × X, Anm Pn x /= 0. ‖Anm Pn x‖ D p , for all .(n, p, x) ∈ Δ × X, An Q p x /= 0. . L(n, p, Q p x) ≤ p ‖An Q p x‖ m ∑ ed(m− j) ≤ 2L(m, p, Pp x) . L(n, p, Pp x) + p ‖A j Pp x‖ j=n

.

L(m, n, Pn x) ≤

m ∑ ed( j−n) ≤ 2L(n, p, Q p x), . L(m, p, Q p x) + p ‖A j Q p x‖ j=n

for all .(m, n, p, x) ∈ T × X. Proof It follows from Corollaries 4 and 5. In what follows, we will extend the characterizations obtained above, for the case of uniform dichotomies for the growth rates of linear discrete-time dynamical systems with respect to strongly invariant projection sequences. Therefore, next we will consider the linear system (.A) and . Pn a projection sequence which is strongly invariant with respect to (.A). Theorem 4 Let .h ∈ H. Then the pair .(A, P) is uniformly .h-dichotomic if and only if there are . D > 1 and .d ∈ (0, 1) such that (uh D1s )

.

∞ ∑ .

j=n

h dj ‖Anj Pn x‖ ≤ Dh dn ‖Pn x‖, for all .(n, x) ∈ N × X.

On Uniform Dichotomies for the Growth Rates of Linear Discrete-time …

(uh D2s )

.

m ∑ .

185

j −d h −d j ‖Bm Q m x‖ ≤ Dh m ‖Q m x‖, for all .(m, x) ∈ N × X.

j=n

Proof It is obvious using Theorem 2 that we have to prove only the second condition, (uh D2s ).

.

m ∑ .

h −d j ‖Bm Q m x‖ = j

j=n

m ∑

h −d j ‖Q j Bm Q m x‖ ≤ N j

j=n

= N h −ν m

m−1 ∑

j

m−1 ∑

j=n

= N h −ν m ‖Q m x‖

h −d j

(

j=n

h ν−d ‖Am Bm Q m x‖ = N h −ν m j j

m ∑

hj hm

)ν

j

j

‖Am Q j Bm Q m x‖ =

h ν−d ‖Q m x‖ = j

j=n m ∑

h ν−d ≤ Dh −d m ‖Q m x‖, where D = N H2 > 1. j

j=n

For the sufficiency, if we consider . j = n in .(uh D2s ), we obtain −d

.h n

‖Bmn Q m x‖ ≤ Dh −d m ‖Q m x‖.

From Remark 9 it follows the conclusion. Corollary 7 The pair .(A, P) is uniformly exponentially dichotomic if and only if there exists . D > 1 and .d ∈ (0, 1) such that s .(ueD1 )

∞ ∑ .

ed j ‖Anj Pn x‖ ≤ Dedn ‖Pn x‖, for all .(n, x) ∈ N × X.

j=n

(ueD2s )

.

m ∑ .

e−d j ‖Bmj Q m x‖ ≤ De−dm ‖Q m x‖, for all .(m, x) ∈ N × X.

j=n

Proof It follows from Theorem 4 for .h n = en . Corollary 8 The pair .(A, P) is uniformly .h-dichotomic if and only if there are D > 1 and . L : Δ × X → R+ with

.

(uh L s1 )

.

(uh L s2 )

.

(uh L s3 )

.

L(m, n, x) ≤ D(‖Pn x‖ + ‖Q n x‖) ) m ( ∑ hj d p ‖A j Pp x‖ ≤ 2L(n, p, Pp x) . L(m, p, Pp x) + h n j=n ) m ( ∑ hm d . L(m, n, Q m x) + ‖Bmj Q m x‖ ≤ 2L(m, p, Q m x), h j j=n .

for all .(m, n, p, x) ∈ T × X. Proof Necessity. We suppose that.(A, P) is uniformly.h-dichotomic and we consider the Lyapunov function . L : Δ × X → R+ , defined by

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R. Boruga (Toma)

.

L(m, n, x) =

) ∞ ( ∑ hj d hn

j=m

‖Anj Pn x‖

+

) m ( ∑ hm d j=n

hj

‖Bmj Q m x‖.

Then .(uh L s1 ) is follows immediately using Theorem 4. .(uh L s2 ) is the same as .(uh L 3 ) s .(uh L 3 ) . L(m, n,

=2

Q m x) +

) m ( ∑ hm d j=n

hj

) m ( ∑ hm d hj

j=n

j

‖Bm Q m x‖ =

j

) m ( ∑ hm d hj

j=n

‖Bm Q m x‖ ≤ 2

) m ( ∑ hm d j= p

hj

j

‖Bm Q m x‖ +

) m ( ∑ hm d j=n

hj

j

‖Bm Q m x‖ =

j

‖Bm Q m x‖ = 2L(m, p, Q m x).

Sufficiency. We suppose that there are . D > 1 and . L : Δ × X → R+ such that the properties .(uh L s1 ) − (uh L s3 ) are satisfied. Then for . p = n in .(uh L s3 ) and using Theorem 4 we obtain the conclusion. Corollary 9 The pair .(A, P) is uniformly exponentially dichotomic if and only if there are . D > 1, d ∈ (0, 1) and . L : Δ × X → R+ with (ueL s1 )

.

(ueL s2 )

.

(uh L s3 )

.

L(m, n, x) ≤ D(‖Pn x‖ + ‖Q n x‖) m ∑ p . L(m, p, Pp x) + ed( j−n) ‖A j Pp x‖ = L(n, p, Pp x)

.

.

L(m, n, Q m x) +

j=n m ∑

ed(m− j) ‖Bmj Q m x‖ ≤ 2L(m, p, Q m x),

j=n

for all .(m, n, p, x) ∈ T × X. Proof It follows immediately from Corollary 8. Notes and Comments. Other variants of Datko–Zabczyk type characterizations, but using different function sets were obtained in [17] for the general case of trisplitting with different growth rates. Also, in [6, 18] the authors give on one hand some necessary and sufficient conditions of Datko–Zabczyk type for the general property of .(h, k)-trichotomy through some Lyapunov functions and on the other hand they obtain some characterizations for the .(h, k)- dichotomy behavior in terms of Lyapunov type sequences of norms. Examples which illustrate the theory presented and clarify the connections between the concepts approached are given in the papers [1, 6, 21]. Moreover, an interesting application to the study of the uniform exponential dichotomy is presented by Dragiˇcevi´c, Sasu and Sasu [9]. Conclusions. The present work gives generalizations of some well known results for the case of uniform exponential dichotomy to the general case of uniform .hdichotomy. In fact, we obtain characterizations of Datko–Zabczyk type for the general concept of uniform .h-dichotomy, on one hand with respect to invariant projection sequences and, on the other hand with respect to strongly invariant projection

On Uniform Dichotomies for the Growth Rates of Linear Discrete-time …

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sequences. As consequences, necessary and sufficient conditions for the particular case of uniform exponential dichotomy are given. Also, using the idea of the equivalence of Datko’s type characterizations and Lyapunov type characterizations [18] we obtain some Lyapunov characterizations for the general case of uniform .h-dichotomy. As an open problem regarding this type of asymptotic behavior, we mention that we intend to find some function sets that are suitable for giving other characterizations for the uniform .h-dichotomy, in order to provide as consequences necessary and sufficient conditions for the uniform polynomial dichotomy. Also, it would be interesting to generalize all the results obtained in this paper to the nonuniform case. Acknowledgements The author would like to thank Professor Emeritus Mihail Megan for his entire support and the useful suggestions that helped to finalize this paper. Also, the author would like to express her gratitude to the referees for the valuable comments which have improved the final version of the paper.

References 1. Babu¸tia, M.G., Megan, M., Popa, I.-L.: On .(h, k)-dichotomies for nonautonomous linear difference equations in Banach spaces. Int. J. Diff. Equ. 2013, Article ID 761680, 7. https://doi. org/10.1155/2013/761680 2. Babu¸tia, M.G., Megan, M.: Nonuniform exponential dichotomy for noninvertible evolution operators in Banach Spaces. An. Stiint. Univ. Al. I. Cuza Ia¸si Mat. (N.S.) Tomul LXII, f. 2 3, 771–788 (2016) 3. Barreira, L., Valls, C.: Polynomial growth rates. Nonlinear Anal. 71, 5208–5219 (2009) 4. Bento, A.J.G., Silva, C.: Stable manifolds for nonuniform polynomial dichotomies. J. Funct. Anal. 257, 122–148 (2009) 5. Coffman, C.V., Sch.affer, ¨ J.J.: Dichotomies for linear difference equations. Math. Ann. 172, 139–166 (1967) 6. Crai, V., Aldescu, M.: On .(h, k)-dichotomy of linear discrete-time systems in Banach spaces. Difference Equations, Discrete Dynamical Systems and Applications. Springer Proceedings in Mathematics and Statistics, vol. 287, pp. 257–271 (2019). https://doi.org/10.1007/978-3-03020016-9_10 7. Crai, V.: On the robustness of a concept of dichotomy with different growth rates for linear discrete-time systems in Banach Spaces. In: 11th IEEE International Symposium on Applied Computational Intelligence and Informatics, pp. 123–129. Timi¸soara (2016) 8. Datko, R.: Uniform asymptotic stability of evolutionary processes in a Banach space. SIAM J. Math. Anal. 3, 428–445 (1972) 9. Dragiˇcevi´c, D., Sasu, A.L., Sasu, B.: On the asymptotic behavior of discrete dynamical systems -an ergodic theory approach. J. Diff. Equ. 268, 4786–4829 (2020) 10. Dragiˇcevi´c, D., Sasu, A.L., Sasu, B.: On polynomial dichotomies of discrete nonautonomous systems on the half-line. Carpath. J. Math. 38, 663–680 (2022). https://doi.org/10.37193/CJM. 2022.03.12 11. Dragiˇcevi´c, D., Sasu, A.L., Sasu, B.: Admissibility and polynomial dichotomy of discrete nonautonomous systems. Carpath. J. Math. 38, 737–762 (2022). https://doi.org/10.37193/CJM. 2022.03.18 12. G˘ain˘a, A., Megan, M., Popa, C.F.: Uniform dichotomy concepts for discrete-time skew evolution cocycles in Banach Spaces. Mathematics 9, 2177 (2021). https://doi.org/10.3390/ math9172177

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13. Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Springer, New York (1981) 14. Li, T.: Die stabilit.atsfrage ¨ bei Differenzengleichungen. Acta Math. 63, 99–141 (1934) 15. Lupa, N.: A new approach on Datko-Zabczyk method for nonuniform exponential stability. Mathematics 8(7), 1095 (2020). https://doi.org/10.3390/math8071095 16. Megan, M., Sasu, A.L., Sasu, B.: Discrete admissibility and exponential dichotomy for evolution families. Discrete Contin. Dyn. Syst. 9(2), 383–397 (2003) 17. Megan, M., Mihi¸t, C.L., Lolea, R.: On splitting with different growth rates for linear discretetime systems in Banach spaces. Difference Equations, Discrete Dynamical Systems and Applications. Springer Proceedings in Mathematics and Statistics, vol. 287, pp. 351–368 (2019). https://doi.org/10.1007/978-3-030-20016-9 18. Mihi¸t, C.L., Megan, M., Ceau¸su, T.: The equivalence of Datko and Lyapunov properties for .(h, k)-trichotomic linear discrete-time systems. Discrete Dyn. Nat. Soc. 2016, Article ID 3760262, 8. https://doi.org/10.1155/2016/3760262 19. Perron, O.: Die Stabilit.atsfrage ¨ bei Differentialgleichungen. Math. Z. 32, 703–728 (1930) 20. Pinto, M.: Asymptotic integrations of systems resulting from the perturbation of an h-system. J. Math. Anal. Appl. 131, 194–216 (1988) 21. Popa, I.-L., Megan, M., Ceau¸su, T.: Exponential dichotomies for linear discrete-time systems in Banach spaces. Appl. Anal. Discrete Math. 6, 140–155 (2012). https://doi.org/10.2298/ AADM120319008P 22. Przyłuski, K.M., Rolewicz, S.: On stability of linear time-varying infinite-dimensional discretetime systems. Syst. Control Lett. 4, 307–315 (1984) 23. Sasu, A.L., Megan, M., Sasu, B.: On Rolewicz-Zabczyk techniques in the stability theory of dynamical systems. Fixed Point Theory 13(1), 205–236 (2012) 24. Sasu, B., Sasu, A.L.: On the dichotomic behavior of discrete dynamical systems on the halfline. Discrete Contin. Dyn. Syst. 33(7), 3057–3084 (2013). https://doi.org/10.3934/dcds.2013. 33.3057 25. Zabczyk, J.: Remarks on the control of discrete-time distributed parameter systems. SIAM J. Control 12, 721–735 (1974) 26. Zhou, L., Zhang, W.: Admissibility and roughness of nonuniform exponential dichotomies for difference equations. J. Funct. Anal. 271, 1087–1129 (2016)

Stability and Realization of Difference Equations Over Z and R .

.

Erik I. Verriest

Abstract While the Jury and Routh-Hurwitz tests provide necessary and sufficient conditions (NASC) for stability of linear difference equations (DE), for a class of “sparse systems” sufficient conditions of much lower complexity for stability are derived based on a Riccati-equation type condition and its function-theoretic equivalent. Next, we use the Jury test to present an algorithm to get NASC for stability dependent on the delay. The second part deals with difference equations defined over .R. Time-varying and state-dependent delay present new problems, including a potential loss of linearity, and require an appropriate state space to define the notions of trajectory and stability. We show with a simple toy example that a discrete event space-time structure is appropriate, and that iterated functional equations characterize solutions. Keywords Sparse equations · Riccati-condition · Lyapunov theory · Stability · Causality · Functional difference equation

1 Introduction In this chapter, we contrast difference equations (DE) in discrete (.t ∈ Z) and in continuous (.t ∈ R) time. Thus in the sense of the behavioral approach of Willems [26], dynamics governed by difference equations may be characterized by an ordered triple .Σ = (T, W, B), where .T is an ordered time set: .Z or .R. .W is the set where variables of interest take their values and .B ⊂ WT is the law of the dynamics, thus characterizing the behavior. It verifies which signals are allowed by the specified dynamics. Behavioural system theory does not make a distinction between input and output signals. An element of .B will be referred to as a solution or a trajectory (even in the discrete case). E. I. Verriest (B) Georgia Institute of Technology, Atlanta, GA 30332-0250, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 S. Olaru et al. (eds.), Difference Equations, Discrete Dynamical Systems and Applications, Springer Proceedings in Mathematics & Statistics 444, https://doi.org/10.1007/978-3-031-51049-6_10

189

190

E. I. Verriest

A dynamical system .Σ = (T, W, B) is linear if and only if .W is a vector space over .R or .C and the behavior .B is a linear subspace of .WT . In problems of modeling, it is of interest to specify a notion of minimal sufficient information (MSI), which uniquely specifies a particular trajectory of the behavior. In fact, when .T is .Z or .R we shall refer to the MSI as a pinning condition, which identifies a unique element from the behavior. There is not a notion of evolution associated with this behavioral viewpoint. This may differ from the case where .T is .Z+ or .R+ , with the DE then associated with an initial value problem in the sense of Cauchy. In this case there is an obvious notion of the propagation from the given MSI (or initial condition) to the induced new MSI applied at a later time. These sets of MSI’s will then characterize the classical notion of state space. We note that the space of MSI’s should then itself be fixed, and not a function of .T. Once the state space is identified, the evolution of the DE specified system is visualized as the ordered set of points in this space, each point corresponding to an MSI. Clearly this will only make sense if the state space is a stationary construct, and does not depend on .T. In many cases, one may not be interest in the precise form specification of the trajectory. Maybe only its boundedness over a subset of .T or convergence to a fixed point as .T is taken to the limit is of interest. In this case various notions of stability are defined. The rest of this chapter is organized as follows: In Sect. 2 we review the stability criteria for linear tim-invariant (LTI) Difference Equations over .Z: We present the classical results. In Sect. 3 simplifications for sparse systems (defined below) based on Shur-Cohn and Jury tests are derived. We derive sufficient conditions for stability using Lyapunov-Krasovskii methods, Frequency Sweep, and a delay-dependent NASC using Lyapunov-Jury theory. The case of varying delay requires special consideration to maintain causality. In Sect. 4, the stability of LTI Difference Equations over .R is investigated for fixed and variable delay. The structural properties for a toy example with state dependent delay are studied in detail in Sect. 5.

2 Classical Results Consider the difference equation, .xk+1 = Axk , in .Rn , starting from .x0 = ξ. It follows directly that .xk = Ak ξ and a NASC for global asymptotic stability is: def

Spec A ⊂ {z ∈ C | |z| < 1 } = D.

.

A square matrix satisfying this condition is called a Schur-Cohn (stable) matrix. The spectrum is the set of roots of .a(z), its characteristic polynomial: (.a0 = 1) a(z) = det(z I − A) = a0 z n + a1 z n−1 + . . . + an−1 z + an .

.

Stability and Realization of Difference Equations …

191

By mapping the open unit disk, .D to the open left half plane .C− by a bilinear (or Möbius) transform .

z=

z−1 1+w ⇔w= undefined at z = −1 1−w z+1

z=

z+1 w+1 ⇔w= undefined at z = 1. w−1 z−1

or .

def

and performing the Routh-Hurwitz test on .a(z) = a(w(z)), an equivalent NASC for asymptotic stability is obtained in terms of the coefficients of the numerator of .a. A polynomial, all of whose roots lie in .D is called a Schur polynomial. Alternatively, the Schur-Cohn matrix criterion (1918–1922) or any of its variants (Jury Criterion) can be used directly as developed by Jury [6], Bistritz [1] and Keel algorithm for the and Bhattacharyya [8]. Details may be found in [4]. The ∑nmodified ai z n−i , with .a0 /= 0, is general (not necessarily monic) polynomial, .a(z) = i=0 initialized with the forward and reverse row of coefficients .

a0 a1 a2 · · · an−2 an−1 an 1 an an−1 an−2 · · · a2 a1 a0 − aan0

There are .n + 1 entries in these rows. Obtain the next row by subtracting . aan0 (row2 ) from .row1 , (multipliers indicated to the right of the vertical line) which zeroes the (n−1) and constitute .row3 . last entry. The remaining elements are labeled .a0(n−1) to .an−1 Then reverse this row (except for the zero) again to obtain .row4 , as shown below. Thus (n−1) (n−1) a0(n−1) a1(n−1) a2(n−1) · · · an−2 an−1 0 . (n−1) (n−1) (n−1) (n−1) (n−1) a0 an−1 an−2 an−3 · · · a1 Keep iterating this procedure, and note that at each step each pair of rows is shortened by one element. This array of numbers is called the Raible table [10]. The final row is .row2n+1 with the single element .a0(0) . Sufficient conditions for asymptotic stability (Jury’s original form) are a (k) ≥ 0

. 0

k = 0, . . . , n − 1,

and if none of the .a0(k) is zero, the number of negative elements in the sequence .{a0(k) } equals the number of roots outside .D. Necessary conditions for asymptotic stability are as follows: If all .a0(k) > 0 for .k = 1, . . . , n − 1, then ( a(1) > 0 (0) .a >0 ⇔ (−1)n a(−1) > 0.

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It is clear that the bilinear modification of Routh-Hurwitz and the Jury-like algorithms require extensive computations if .n is large for . A ∈ Rn×n . One simplification is to use Gershgorin’s theorem (See [2]) which gives simple bounds (Gershgorin circles) on the location of the eigenvalues of a matrix. A in simple terms of its elements .{ai j }. In the next section we shall focus on sparse DE’s. Whereas Gershgorin’s theorem can still be used here (see [25]), we report on joint research with Ivanov [5] and [24], on alternative sufficient criteria for stability for sparse DE’s.

3 Sparse LTI Systems Consider a difference equation in the sparse form (i.e., many zero matrix coefficients occur) with . A, B ∈ Rn×n , where one can think of .1 < m ∈ N as a delay parameter x

. k+1

= Axk + Bxk−m

(1)

State augmentation represents such a DE with a one-step DE involving a sparse block companion form, hence its name. ⎡

χk+1

.

⎤

⎡

A ⎥ ⎢I ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ ⎥=⎢ =⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ ⎦ ⎣ ⎣ 0 xk−m+1 xk+1 xk .. . .. .

0 ··· 0 ··· .. .. . . .. . ··· ···

⎤⎡ ⎤ xk 0 B ⎢ ⎥ 0 0⎥ ⎥ ⎢ xk−1 ⎥ ⎥ ⎢ .. ⎥ ⎥ ⎢ . ⎥ = A χk . ⎥⎢ ⎥ ⎥⎢ . ⎥ .. . ⎣ ⎦ . . ⎦ I 0 xk−m

(2)

This has a characteristic polynomial of degree.n(m + 1), consequently the Jury test is computationally involved. It is desirable to obtain sufficient conditions for asymptotic stability directly involving only the .n × n matrices.

3.1 Lyapunov-Krasovskii Method Theorem 1 ([24]) The system .xk+1 = Axk + Bxk−m is asymptotically stable .∀m ≥ 0, if either of the following two conditions hold: (i) There exists a triple of positive definite (symmetric) matrices . P, . R and .W such that T −1 T . A P BW B P A + AT P A + W + B T P B + R = P (3)

Stability and Realization of Difference Equations …

193

(ii) There exists a triple of positive definite (symmetric) matrices .Π , . S and . Z such that T −1 T . B Π AZ A Π B + B T Π B + Z + AT Π A + S = Π . (4)

Proof Consider the quadratic function

.

Vk (x) = xkT P xk +

k−1 ∑

xiT Qxi .

(5)

i=k−m

Along solutions of (1) we have (.ΔVk = Vk+1 − Vk ) .

T T ΔVk = xk+1 P xk+1 + xkT Qxk − xkT P xk − xk−m Qxk−m T = xkT [AT P A + Q − P]xk + 2xkT AT Bxk−m + xk−m [B T P B − Q]xk−m .

(6)

Complete the squares in two different ways: First, let ΔVk = −[M xk−m + N xk ]T [M xk−m + N xk ] + xkT [N T N + AT P A + Q − P]xk . (7) Identify terms in (6) and (7) to get. M T M = −B T P B + Q and. N T M = −AT P B. A def necessary condition to have a (real) solution is .W = Q − B T P B ≥ 0. This implies 1/2 .M = W and . N = −W −1/2 B T P A. By Lyapunov’s lemma, global asymptotically stability follows if . AT P BW −1 B T P A + AT P A + Q − P ≤ 0. i.e., if .∃R > 0 such that (3) holds. Criterion (4) is obtained similarly by completing the squares as (noting that . P = Π .) .

T ΔVk = −[ Mˆ xk−m + Nˆ xk ]T [ Mˆ xk−m + Nˆ xk ] + xk−m [ MˆT Mˆ + B T P B − Q]xk−m .

.

◻

.

We can obtain an alternative form of Theorem 1 which resembles the Lyapunov NASC condition for the LTI equation of the form .xk+1 = Axk . Corollary 1 The system (1) is asymptotically stable for all .m ≥ 0, if either of the following is satisfied: 1. There exist positive definite (symmetric) matrices . P, . R, .Λ and .Ω solving the pair ( .

AT Λ−1 A + Ω + R = P BΩ −1 B T + Λ = P −1 .

(8)

2. There exists positive definite (symmetric) matrices .Π , . S, .Γ and .Σ solving the pair ( T −1 B Σ B+Γ +S =Π . (9) AΓ −1 AT + Σ = Π −1

194

E. I. Verriest

Proof Reorganize (3) as . AT [P BW −1 B T P + P]A + [W + B T P B] + R = P. Let −1 def .Λ = P BW −1 B T P + P. The Woodbury (matrix inversion) lemma yields Λ = P −1 − B[W + B T P B]−1 B T .

.

def

Letting also .Ω = W + B T P B, leads to the first equality in (8). Substitution in the original conditions gives then the second equality in (8). To obtain the second set of conditions, start from (2), and proceed analogously with −1 .Σ = Π + Π AZ −1 AT Π, Γ = Z + AT Π A. (10) ◻

.

Remark (i) The two versions of the criteria are equivalent: In Theorem 1 the substitutions . P = Π, R = S, W = B T P AZ −1 AT P B imply . Z = AT Π BW −1 B T Π A, while for Corollary 1, the equivalences are .

P = Π,

R = S, Σ = BΩ −1 B T , Λ = AΓ −1 AT .

(ii) If either . A or . B are invertible, we obtain the following (sufficient) condition, similar to the usual Lyapunov NASC condition for LTI DE’s. Corollary 2 If positive definite matrices . X and . Q exist such that .

p AT X A + q B T X B + Q = X

for scalars . p, q > 0 satisfying the Holder condition, . 1p + q1 = 1, then positive definite matrices . P, Λ, Ω and . R exist satisfying (8) if . B is invertible, and matrices .Π, Σ, S and .Γ exist if . A is invertible for which (9) holds. Proof Set . P = X , .Γ = 1p X −1 , .Ω = q B T X B and . R = Q > 0 if . B is invertible and 1 −1 .Π = X , .Σ = X , . S = Q and .Γ = p AT X A id . A is nonsingular. . ◻ q For example, the choice . X = I leads to a more conservative but useful simple sufficient condition (“. 0 and . L , W , such that .

AT P A + C T C + L T L = P

AT P B + C T D + L T W = 0 B T P B + D T D + W T W = I. An interpretation of these conditions stems from: (ii) Kalman’s extension of Lyapunov’s lemma For .A ∈ Rn×n and .C ∈ R p×n any two of the following statements imply the third: (i) There exists .P = P T > 0 such that AT PA + C T C = P,

.

196

E. I. Verriest

(ii) The matrix .A is Schur-Cohn stable, (iii) The pair .(A, C) is observable. Let’s now apply Kalman’s extension to the matrices [

] AB , 0 0

A=

.

[

] C D C= . L W

and note that . A is Schur-Cohn if .A is. The observability [ of this ] augmented pair AB implies the equivalence of the Schur-Cohn stability of . , hence of . A, with 0 0 [ ] P R the existence of a positive definite matrix .P = satisfying the Lyapunov RT Q equation: [ .

AT 0 BT 0

][

P0 R0 R0T Q 0

][

[ ] ] [ T] [ ] AB P0 R0 C C D = . + DT R0T Q 0 0 0

(12)

[ ] A B If.z = 0 is not a zero of.T (z) the matrix. has full rank. Moreover, if in addition C D the original .(A, C) is observable, then the matrix ⎡

⎤ z In − A −B 0 z Im ⎦ , .⎣ C D has full rank for all .z ∈ C. By the Popov-Belevitch-Hautus criterion (see [7]) this implies the observability of ([ .

] ) ] AB [ , C D , 0 0

and therefore also of the augmented pair ([ .

] [ ]) AB C D , 0 0 L W

for all . L and .W of compatible dimension. Let’s enter another lemma: (iii) Schur’s matrix lemma

[ .

X Z ZT Y

] ≥0

is equivalent to either .{ X>0 ∧ Y − Z ' X −1 Z≥0 } or .{ Y>0 ∧ X − Z Y −1 Z '≥0 }. We can then formulate a sufficient condition in the frequency domain in the following theorem:

Stability and Realization of Difference Equations …

197

Theorem 2 If. A is Schur-Cohn stable, and.‖(e jθ I − A)−1 B‖∞ < 1, then the sparse discrete system (1) is asymptotically stable .∀m ≥ 0. Proof By virtue of the bounded real lemma, for .C = I and . D = 0, the stability of . A, together with the infinity norm condition on .T (z) = (z I − A)−1 B imply the existence of . P = P T > 0, with . L and .W of full rank such that [ T ] [ T] [ ] A P A − P + I AT P B L L W ≤ 0. =− . BT P B − I WT BT P A By Schur’s lemma: .

I − BT P B = W TW = Z > 0

−A P A + P − I − B P A(I − B T P B )−1 AT P B < 0. T

T

W T W =Z

Using the Schur-complement property .

AT P A +

I

+B T P AZ −1 AT P B < P.

B T P B+Z

Thus, . P, S > 0 exist such that .

A' P A + B ' P B + Z + B ' P AZ −1 A' P B + S = P,

and by Theorem 1, the sparse systems is asymptotically stable for all .m ≥ 0. .

◻

Remark The unit circle .|z| = 1 is compact, so this frequency sweep test is simpler to execute (finite sweep) than its continuous time counterpart, .Re s = 0. How far is the condition in Theorem 2 from necessary? In [3] one finds the stronger result for the spectral radius .ρ(M) = max { |λ||λ ∈ Spec (M)} of . M: Theorem 3 System .xk+1 = Axk + Bxk−m is asymptotically stable .∀m > 0 if (i) . A is asymptotically stable, (ii) .∀ θ : ρ((e jθ I − A)−1 B) < 1. If, in addition, there exists a point .z 0 on the unit circle where .|(z 0 I − A)−1 B| < 1, then the conditions are also necessary.

3.3 Delay-Dependent NASC The stability criteria considered so far are independent of the fixed parameter, .m, and therefore are possibly too conservative. It is very likely that the actual stability or instability of the DE will depend on this delay .m. For the scalar sparse difference

198

E. I. Verriest

equation, .xk+1 = axk + bxk−m , where .m ∈ N, the associated characteristic polynomial is m+1 . pm (z) = z − az m − b, (13) and the obvious approach consists in finding the roots of. pm (z) for successively larger values of .m, and test if their modulus exceeds one. A computationally more efficient method is desirable. We show how this follows from a Jury-test based NASC [17]. Let us start with the Raible-table associated with the characteristic polynomial (13) The first two rows are 1 −a 0 · · · 0 0 −b . −b 0 0 · · · 0 −a 1 There are.m + 1 entries in these rows, and both rows have.m − 3 positions in common having the entry 0. The next pair of rows in the array has the form .

1 − b2 −a 0 · · · 0 0 ab ab 0 0 · · · 0 −a 1 − b2

Now there are .m entries per row, and .m − 4 of these are 0 in both rows. One can easily follow this pattern down almost to the end. To this effect relabel the initial pair of rows as, α1 β1 0 · · · 0 0 γ1 . γ1 0 0 · · · 0 β1 α1 and note that the .kth pair will be in the same form (but with one less intermediate zero for each subsequent pair) up to the .(m − 2)nd pair .

αk βk 0 · · · 0 γk γk 0 0 · · · βk αk .

The Raible construction implies a nonlinear iteration for .k = 1, . . . , m −3: αk+1 = αk −

.

γk2 βk γk βk+1 = βk γk+1 = − , αk αk

(14)

initialized with .α1 = 1, β1 = −a, γ1 = −b. The Raible table terminates with the (m − 1)st pair of rows: α−1 βm−1 γm−1 . γm−1 βm−1 αm−1 ,

.

with the triple .(αm−1 , βm−1 , γm−1 ) computed from .(αm−2 , βm−2 , γm−2 ) as before The .mth pair in the Raible table is .

αm γm γm αm ,

Stability and Realization of Difference Equations …

where now .αm = αm−1 − computation of

2 γm−1 αm−1

and .γm =

αm−1 −γm−1 βm−1 . αm−1

αm+1 = αm −

.

199

The final step involves the

γm2 . αm

All zeros of . pm (z) lie inside the unit disk iff .αk > 0, for .k = 1, . . . , m + 1 (Jury test). Let us now analyse these iterations. First, it is obvious that .β1 = β2 = · · · = βm−1 = −a. Hence the three component recursion (14) reduces to αk+1 = αk −

.

γk2 γk , γk+1 = a , αk αk

(15)

for .k = 1, . . . , n − 1, with termination αm = αm−1 −

.

2 γm−1 , γm = a αm−1

(

) γm−1 −1 . αm−1

(16)

Elimination of .γ leads to the nonlinear recursion for .k = 1, . . . , m − 2 αk+2 = αk+1 −

.

a2 (αk − αk+1 ). αk αk+1

(17)

This can be rewritten as αk+2 +

.

a2 a2 = αk+1 + , αk+1 αk

which means that the quantity .αk+1 + a 2 /αk is constant. It follows then from the induced initial conditions and that .α1 = 1 and .α2 = γ2 α1 − α11 = 1 − b2 that ∀k :

.

αk+1 +

a2 = 1 + a 2 − b2 . αk

As .γn−1 is required in the terminal step, we must also propagate the .γ variable. Multiplying the equations a γk .γk+1 = αk for .k = 1 to .k = m − 2, one gets γ

. m−1

=

−a m−2 b . α1 · · · αm−2

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Thus we conclude finally that the nonlinear recursions (15) simplify to αk+1 = (1 + a 2 − b2 ) −

.

a2 , αk

γk+1 =

aγk , αk

(18)

with initialization .α1 = 1, and .γ1 = −b. Now let’s solve (18) by exact linearization (see [16]). Let αk =

.

Nk . Dk

This linear recursion is [ ] [ ] [ ] [ ] ][ Nk+1 1 + a 2 − b2 −a 2 N1 1 Nk = , = . . Dk+1 1 0 Dk D1 1

(19)

Note that this is not a unique possible choice. We also have from (15): γ =

. k

−a k−1 b , Nk−1

(20)

) ( a m−2 b γ Ter = −a 1 + Nm−1

which terminates with

. m

to yield the potential last .α in the ‘Jury’-sequence, for each .k: def

Ter ωk = αk+1 = αk −

.

(γkTer )2 . αk

(21)

The stability test proceeds now as follows: If . N is the largest integer such that α > 0 and .ω > 0 for all . ≤ N + 1, then . pm (z) passes the Jury test, and the (delayparameterized) sparse system is asymptotically stable for all .m ≤ N . We summarize:

.

Theorem 4 Compute the sequences .{αk } and .{ωk } as indicated from the linear recursion (19) of .(Nk , Dk ). If . N is the largest integer such that .∀m ≤ N , .αm > 0, ωm > 0, then the sparse difference system is asymptotically stable for all delays .m < N . Tree Representation: Letting the diamond indicates a nonpositive element, while the ‘o’ denotes a positive quantity, the flow of the algorithm is represented by

Stability and Realization of Difference Equations …

201

m: 1 2 3 4 5 α : −→ o −→ o −→ o −→ o −→ . | | | | ω: o o o o In this example, . N = 4, indicating stability for all .m < N = 4. To perform Jury’s test, it suffices therefore to check the sign of the . Nm sequence and its (nonlinear) terminator, .ωm . This is further reducible to (See [17]) .

| | Nk > |a| | Nk−1 + a k−2 b| .

The tree for this test is of the form:

.

m: Nm :

1 1 = N1 −→

f m (Nm−1 ) :

2 N V2 f 2 (N1 )

−→

3 N V3 f 3 (N2 )

−→

4 N V4

5 −→

f 4 (N3 )

where . f m (N ) = |a||N + a m−2 b|. In this example, . Nm > 0 and . Nm > f m (Nm−1 ) for .m = 1, . . . , 4, but . N5 < 0. The sparse system is asymptotically stable for .m = 1, 2, and 3. System Theoretic Interpretation The recursion (19) and its initialization prompt the question “When is the sequence .{Nk } positive?” This sequence, .(Nk ) is precisely the Markov parameter sequence (or pulse response) for the fixed second order linear system: ] 1 + a 2 − b2 −a 2 , .A = 1 0 [

[ ] 1 B= , 1

C = [1 0].

In addition, . Nm > f m (Nm−1 ) must hold. A further simplification of this condition gives: | | | | b k| | (sgn a) | , . Nk > Nk−1 + | |a| Nk ˆ ˆ where . Nk = |a| k is a normalized version of . N k . The feasible iterates .( N k−1 , N k ) are shown in the Fig. 1. The Markov sequence .(Nk ) is readily found, e.g. using standard Z-transform techniques: Let .ξ1 and .ξ2 be the roots of the characteristic equation .z 2 − (1 + a 2 − b2 )z + a 2 . Then in the generic case of disjoint roots, one finds

[ .

Nk Dk

] =

ξ1k − ξ2k ξ1 − ξ2

[ ] [ ] ξ k−1 − ξ2k−1 1 −a 2 + 1 . b2 − a 2 1 ξ1 − ξ2

A sufficient condition for the . N -sequence to be positive, is that the poles of the second order system are real and positive.

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Fig. 1 Feasible Iterates

These poles are the zeros of the characteristic equation .z 2 − (1 + a 2 − b2 )z + a 2 . It suffices therefore that .(1 + a 2 − b2 )2 > 4a 2 , i.e., (1 + a + b)(1 + a − b)(1 − a + b)(1 − a − b) > 0.

.

This stability domain in the parameter space is a diamond with vertices (0,1), (1,0), (−1, 0) and .(0, −1). Note that this region guarantees stability independent of delay (See [24]).

.

Example The sufficient condition for stability independent of the delay fails for the delay system .xk+1 = 0.4xk − 0.61xk−n . .m

.αm

.ωm

1 2 3 4 5 6 7 8 9

1 0.6279 0.5331 0.4878 0.4599 0.4400 0.4242 0.4108 0.3984

1 0.5891 0.4209 0.3231 0.2584 0.2119 0.1762 0.1477 0.1241

.m 10 11 12 13 14 15 16 17 18

.αm 0.3863 0.3737 0.3597 0.3431 0.3216 0.2904 0.2369 0.1125 .−0.6349

.ωm 0.1039 0.0861 0.0702 0.0557 0.0421 0.0292 0.0168 0.0047 .−0.0074

The .αi and .ωi are positive up to index 17. For .m = 18, the difference equation is unstable. Indeed, a direct check of the eigenvalues of the 18th order system shows that two eigenvalues exist with .|λ| = 1.000056476.

3.4 LTI Difference Equations with Variable Delay (Over .Z) In this section we briefly comment on the autonomous system of the form where m : Z → Z+ . The realization is now more subtle. If one thinks of a feedback structure with a shift register (buffer) of variable length .m, then it is obvious that once data

.

Stability and Realization of Difference Equations …

203

shifts out of the last cell in this shift register, it is lost. This implies that if at the next step the length is increased by more than one, it is not possible to stuff the lost data back in there. See [20] for a discussion on the potential inconsistencies in the case of a functional differential equation, and for the difference equation in [19, 23]. In order to restore causality, first one has to ensure that the state space has a stationary structure. Two causal realizations are suggested: The first is the distributed delay causalization, where the register length is taken as . N = maxk m(k), and the system is interpreted as one with time-variant connection structure. The second is the lossless causalization, which treats the system as a multi-mode multi-dimensional (. M 3 D) system. See [18, 23] for details. Each buffer size corresponds now to a mode, and the state space has the structure of a discrete sheaf over the set of modes [22].

4 Difference Equations Over .R We consider here the sparse difference equation as a behavior,.(R, W, B), with.T = R (or .R+ ). First we consider DE with fixed delays, now denoted by .τ ∈ R+ , then show that for a time-variant delay, a consistency condition for causality is required.

4.1 One Fixed Delay An autonomous sparse equation with commensurate delays over .R is equivalent to a difference equation with a single delay over .R and with time scaling, .τ = 1. Thus consider .x(t) = Ax(t − 1) where . A ∈ Rn×n for some .n. Its most general function space is .(Rn )R . In terms of .Δ, the unit delay operator, the behavioral form for this difference equation is n R .B = { w ∈ (R ) | (1 − AΔ)w = 0 }. We note that the nullspace, .ker Δ, when defined on .(Rn )R is trivial, while .Δ is surjective on this space. Hence, .Δ acts like a nonsingular matrix in a finite dimensional ← − space. The inverse image of .w is . Δ w = { y ∈ (Rn )R | Δy = w }. Clearly, .w ∈ B is completely specified by its values assumed in any interval .[α, α + 1). In particular, one can refer to .{φ(θ) | θ ∈ [−1, 0) } as the pinning data. This generalizes the notion of initial data in the corresponding Cauchy or one-sided problem by providing a sufficient statistic for the entire trajectory .w, but necessarily in a non-causal form. However, this space is too big to be of practical significance. If only continuous solutions are of interest, we may restrict the states to .φ ∈ C([−1, 0), R). However, this is still not right since continuity of .w fails for .t ∈ Z. For continuity in .R of the trajectories, we must impose .φ(0) = Aφ(−1), which implies for all .n ∈ Z that .w(n+) = Ae−A w(n−).

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Equivalently, the subspace, . S1 = { φ ∈ C([−1, 0], Rn ) | φ(0) = Aφ(−1) } is the state space for the difference equation over .R. Indeed, letting .t = + θ be the unique decomposition where . ∈ Z and .θ ∈ [−1, 0), we find that .w ∈ B is obtained from .φ by w(t) = A− φ(t + ) if t ∈ [−( + 1), − ) w(t) = φ(t) if t =∈ [−1, 0)

.

w(t) = A φ(t − )

if t ∈ [ − 1, ).

The corresponding Cauchy (initial value problem (IVP)) is characterized by .B

= { w ∈ C([−1, ∞), R) | (1 − AΔ)w = 0 for t > 0, and w = φ for t ∈ [−1, 0)}.

The operator .Δ is a verification operator on .C([0, ∞), Rn ) in the sense that it is surjective and has a nontrivial kernel .ker Δ = {w ∈ C([0, ∞), Rn ) | Δw = 0 } = (1 − t)C([0, 1), Rn )χ[0,1] (t), where .χΣ denotes the indicator function of .Σ. Analytic Solution Adapting the Floquet-theory, let’s assume that the behaviors are of the form .w(t) = Φ(t) p(t), where . p is periodic (i.e., satisfies . p = Δp) and .Φ is real analytic. Substituting the latter gives for all .t Φ(t) p(t) = AΦ(t − 1) p(t − 1) = AΦ(t − 1) p(t).

.

(22)

As we want .Φ to be independent of . p, the Eq. (22) must hold for all . p. Thus we can lift the equation from the vector to the matrix form Φ(t) = AΦ(t − 1).

.

Invoking the matrix logarithm, it is shown in [12, 13] (Theorem 2) that if . A is nonsingular, this may be expressed in complex form as.Φ(t) = et ln A Φ(0) of the same dimension. If a real representation ∑α is sought, a real augmented system of dimension .n + N is derived, where . N = i=1 νi and .α is the number of different elementary divisors .(s − λi )νi of . A that vanish on the negative real axis and occur an odd number of times. Recall that with each elementary divisor .(s − λi )νi of . A, there corresponds a Jordan block . J (λi , νi ) in the Jordan canonical form of . A. This result extends the scalar behavior we considered in [21], where a representation of the system is given in terms of dynamics on a closed string.

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4.2 Non-commensurate Delays In this section we consider difference equations, for .t ∈ R, of the form .

x(t + 1) = Ax(t) + Bx(t + ϵ), 0 < ϵ < 1.

(23)

If .ϵ ∈ Q, state augmentation reduces the problem to the previous one. Thus we let here .ϵ ∈ R \ Q. Such equations model the effects of delays in computer controlled systems and are discussed in [4, 9]. Let us first consider the DE (23) in the scalar case. This will then serve as a comparison system to investigate the stability of (23), for which a new Lyapunov technique is developed in [15]. First, it is easily shown by induction that the solution satisfies for . ∈ N ∑(n) . x(t + ) = a −k bk x(t + kϵ). k k=0

Consequently, [ |x(t + )| ≤

∑( )

.

k=0

k

] |a|

−k

|b|

max |x(t + kϵ)|.

k

0 t, but ' ' ' ' ' . x(t ) < x(t) − (t − t). Then .t f (t ) = t + x(t ) < t + x(t) = t f (t), which violates causality (Fig. 3). This motivates the extension of .Tt from an operator on .R+ × [t0 , t1 ) to an operator .Tt on . X × [t0 , t1 ], where . X ⊂ C([t0 , t1 ], R+ ) is a suitably defined subset (see further). .Tt (φ(·), t) = Tt (φ(t), t). Definition:.Tt is contiguous on. X ⊂ C([t0 , t1 ), R+ ) if for all.t ' > t in.(t0 , t1 ), it holds that .Tt (φ, t ' ) > Tt (φ, t) for all .φ ∈ X with .φ(t0 ) = x0 . Theorem 7 ([21]) A necessary and sufficient condition for contiguity of .Tt on . X with initial event .(x0 > 0, t0 ): is that the initial data .φ(t) with domain .(t0 , t0 + x0 ) satisfies φ(t ' ) − φ(t) . (37) > −1 ∀t ' > t. t' − t

Stability and Realization of Difference Equations … Fig. 3 Time update, .Tt , not contiguous on initial data .φ

x0

211

x(t)

φ(t)

x(t')

t

Tt(t')

t'

t0

Tt(t)

t1 = t0 + x0

Let .Cctg ([t0 , t1 ], R) denote the subset of contiguous continuous functions in [t0 , t1 ]. For .a > 0 it is seen that the state dependent continuous difference equation starting at the initial event .(x0 , t0 ) will have a continuous solution if the initial data .φ belongs to .Cctg ([t0 , t0 + x0 ], R), and .φ(t0 ) = x0 and .φ(t0 + x0 ) = ax0 . Without the latter constraint, only piecewise continuity of the solution is guaranteed. The minimal sufficient information for this problem is an element of

.

( .

X=

| (t0 , x0 , φ(·)) ∈ R × (R+ \ {0}) × Cctg (R, R+ ) |

(

φ(t0 ) = x0 φ(t0 + x0 ) = ax0

) .

5.3 Analytic Solutions Let’s now look for analytic solutions to the forward SD-DE (34). Transform the state˜ = t + x(t), to dependent delay system by the time-dependent transformation .x(t) ˜ f ) = (a + 1)x(t) ˜ − at and .t f = x(t). ˜ Elimination of .t f leads to an iteratedget .x(t function functional (IFF) equation, .

x( ˜ x(t)) ˜ + a1 x(t) ˜ + a2 t = 0,

(38)

where .a1 = −(a + 1) and .a2 = a. Its solutions, for .a /= 1, parameterized by .x(0) ˜ are (see [21]) . x(t) ˜ = at + x(0), ˜ together with a solution .x(t) ˜ = t. Hence there are two solutions with .x(0) ˜ = 0. In terms of the original space-time coordinates these solutions are .

x(t) = x(0) + (a − 1)t,

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E. I. Verriest

and the trivial solution .x(t) ≡ 0. Then the nontrivial continuous time trajectories coincide with the orbits we found for the discrete iteration. Concretely, this implies that the analytic solutions to the SD-DE .x(t + x(t)) = ax(t) constitute a one-dimensional function space. Each solution is completely specified by .x(0). If.a = 1, the characteristic polynomial.λ2 − (a + 1)λ + a of the IFF has a positive double root, .λ = 1. In general, if .λ is a double root of the characteristic polynomial of a second order IFF, the solution of the IFF through .(t0 , x˜0 ) can be expressed in terms of the Lambert W-function: ) ] [ ( λ ln(λ)t0 T γ γe −γ , γ = . . x(t) ˜ = λt + (x0 − λt0 ) exp W t0 ln(λ) x˜0 − λt0 For .λ = 1, as is the case here when .a = 1, this solution set is the set .x(t) ˜ =t+ x˜0 − t0 , which is again one-dimensional. In the original space-time form, these are all constant solutions, .x(t) ≡ x(0). Remark It is shown in [21] that a higher order IFF with a characteristic polynomial ˜ = λi t passing through .(0, 0). having disjoint real roots .λi has the ray solutions .x(t) ˜ = λi t + It is easily verified that under specific additional conditions, solutions .x(t) exists for arbitrary .x(0), ˜ where .λi is a root of the characteristic equation. This x(0) ˜ extends also to complex .λi .

6 Conclusions We presented a guided tour to stability and realization for difference equations. We reviewed the Jury NASC stability conditions for LTI difference equations, and developed simplified (but only sufficient) stability conditions for sparse LTI DE’s. For the scalar sparse LTI DE the Jury criterion implied a simpler NASC for the delay dependent stability. Some intricacies for the time-variant case were explored. Then we resolved some problems related to the structure of the continuous difference equations with time variant and state-dependent delay. The emphasis is on wellposedness of the problem.

References 1. Bistritz, Y.: Zero location with respect to the unit circle of dscrte-time linear system polynomials. Proc. IEEE 72, 1131–1142 (1984) 2. Gantmacher, F.R.: The Theory of Matrices, vol. 1, Chelsea (1960) 3. Gu, K., Kharitonov, V.L., Chen, J.: Stability of Time-Delay Systems. Birkhäuser (2003) 4. Hinrichsen, D., Pritchard, A.J.: Mathematical Systems Theory I. Springer (2005) 5. Ivanov, A.F., Verriest, E.I.: Robust stability of delay-difference equations. In: Helmke, U., Mennicken, R., Saurer, J. (eds.) Systems and Networks: Mathematical Theory and Applications, pp. 725–726. University of Regensburg Press (1994)

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6. Jury, E.I.: A simplified stability criterion for linear discrete systems. In: Proceedings of the IRE, pp. 1493–1500 (1962) 7. Kailath, T.: Linear Systems. Prentice Hall (1980) 8. Keel, L.H., Bhattacharyya, S.P.: Automatica 35, 251–258 (1999) 9. Logemann, H., Townley, S.: The effect of small delays in the feedback loop on the stability of neutral systems. Syst. Control Lett. 27, 267–274 (1996) 10. Raible, R.R.: A simplification of Jury’s tabular form. IEEE Trans. Autom. Control 19, 248–250 (1974) 11. Singh, V.: A new proof of the discrete-time bounded-real lemma and lossless bounded-real lemma. IEEE Trans. Circuits Syst. 34(8), 960–962 (1987) 12. Verriest, E.I.: The matrix logarithm and the continuization of a discrete process. In: Proceedings of the 1991 American Control Conference, pp. 184–189 (1991) 13. Verriest, E.I.: The continuization of a discrete process and applications in interpolation and multi-rate control. Math. Comput. Simul. 35, 15–31 (1993) 14. Verriest, E.I.: Frequency domain stability criterion for discrete delay systems. IFAC Proc. Vol. 34(23), 107–110 (2001) 15. Verriest, E.I.: New qualitative methods for stability of delay systems. Kybernetika 37(3), 229– 238 (2001) 16. Verriest, E.I.: Systems over projective fields. IFAC Proc. Vol. 34(6), 231–236 (2001) 17. Verriest, E.I.: Delay dependent stability of scalar discrete delay systems. IFAC Proc. 34(23), 135–139 (2001) 18. Verriest, E.I.: Causal behavior of switched delay systems as multi-mode multi-dimensional systems. In: 8th IFAC Workshop on Time-Delay Systems, Sinaia, Romania. IFAC Proceedings Volumes 42(14), 414–419 (2009) 19. Verriest, E.I.: Structure of discrete systems with switched delay. IFAC Proc. 44(1), 1465–1470 (2011) 20. Verriest, E.I.: Inconsistencies in systems with time varying delays and their resolution. IMA J. Math. Control. Inf. 28, 147–162 (2011) 21. Verriest, E.I.: State space realization for a continuous difference equation with varying delay. IFAC Proc. Vol. 45(14), 242–247 (2012) 22. Verriest, E.I.: Pseudo-continuous multi-dimensional multi-mode systems: behavior, structure and optimal control. J. Discrete Event Dyn. Syst. 22(1), 27–59 (2012) 23. Verriest, E.I.: Structure of discrete systems with variable nonlocal behavior. In: Vyhlidal, T., Lafay, J.-F., Sipahi, R. (eds.) Advances in Delays and Dynamics, vol. 1, pp. 185–198. Springer (2014) 24. Verriest, E.I., Ivanov, A.F.: Robust stability of delay-difference equations. In: Proceedings of the 34th IEEE Conference on Decision and Control, vol. 1, pp. 386–391 (1995) 25. Verriest, E.I.: Bounds on the eigenvalues of systems with delay. IFAC-ProceedingsOnLine 56(2), 911–916 (2023) 26. Willems, J.C.: The behavioral approach to open and interconnected systems. IEEE Control Syst. Mag. 27(6), 46–99 (2007)

Discrete-Time Models Applied to Engineering, Biology and Economics

Discrete Dynamical Systems in Economics: Two Seminal Models and Their Developments Gian Italo Bischi

Abstract This survey paper starts from two famous discrete-time dynamic models in economics, namely the Cobweb model to describe price dynamics and the Cournot duopoly model to describe competition between two firms producing homogeneous goods, and shows how their study has stimulated new fruitful streams of literature rooted in the field of qualitative analysis of nonlinear discrete dynamical systems. In the case of the Cobweb model, starting from the standard one-dimensional dynamic model, the introduction of new kinds of expectations and learning mechanisms opened new mathematical research about two-dimensional maps with a vanishing denominator, leading to the study of new kinds of singularities called focal points and prefocal curves. Analogously, in the case of the two-dimensional Cournot duopoly model, some recent developments are described concerning the introduction of nonlinearities leading to multistability, i.e. the coexistence of several stable equilibria, with the related problem of the delimitation of basins of attraction, which requires a global dynamical analysis based on the method of critical curves. Moreover, in the particular case of identical players, some recent results about chaos synchronization and related bifurcations (such as riddling or blowout bifurcation) are illustrated, with extensive reference to the rich and flourishing recent stream of literature. Keywords Economic dynamics · Discrete dynamical systems · Nonlinear dynamics · Stability · Bounded rationality

1 Introduction Discrete time dynamical systems naturally arise in economic and social modeling, because changes in the state of a system occur as a consequence of decisions that cannot be continuously revised (decision-driven time). For example, production deciG. I. Bischi (B) Department of Economics, Society, Politics (DESP), Universià di Urbino, Urbino, Italy e-mail: [email protected] URL: https://www.uniurb.it/persone/gian-bischi © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 S. Olaru et al. (eds.), Difference Equations, Discrete Dynamical Systems and Applications, Springer Proceedings in Mathematics & Statistics 444, https://doi.org/10.1007/978-3-031-51049-6_11

217

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G. I. Bischi

sions can be changed only after the meetings of a governing board, or after the conclusions of contracts, or taking into account production lag (an exemplary case occurs for agricultural productions, whose amount can be changed only at sowing seasons). So, if the state of an economic system at time .t is defined by the values of .n real dynamic variables . x 1 (t),…, . x n (t), given a characteristic time interval taken as a unit of time, the state at the next time period can be obtained by the application of a map .T : Rn → Rn , i.e. a point transformation in a n-dimensional state space into itself. In other words, the time evolution (or trajectory) is inductively obtained by the recurrence .x(t + 1) = T (x(t)) where .x(t) = (x1 (t), . . . , xn (t)) ∈ Rn , starting from a given initial condition .x(0) = (x1 (0), . . . , xn (0)). Sometimes, in standard economic theory, the agents are assumed to be all rational, i.e. they are assumed to have full knowledge of the social and economic environment where they are operating and to possess the computational tools they need to choose their actions through an optimization process. This gives rise to a rational choice leading to an optimal decision. However, humans are generally not so rational or informed, and they follow adaptive methods, such as learning-by-doing or trial-and-error practices. This leads to replacing one-shot optimal decisions with repeated adaptive decisions, i.e. a dynamic process that may or may not converge to a rational equilibrium. Moreover, given a dynamic process governed by a nonlinear map, several attractors may coexist, so the asymptotic (or long run) behavior of a dynamical system becomes path dependent. As a consequence, the step-by-step adaptive process considered may act as a selection device to describe which kind of long run outcome will prevail. In these cases, stability arguments become crucial, together with the role of initial conditions. Therefore, a global study of the basins of attraction can give information about the path dependence, i.e. how the long-run time evolution depends on historical accidents (represented by exogenous shifts of initial conditions). In order to illustrate these points, in this paper we analyze two exemplary models that represent important milestones in the development of economic dynamics, as they have given rise to fruitful and long-lasting developments in several directions. The first one, denoted as the Cobweb model, explains why prices might be subject to periodic fluctuations. It describes cyclical supply and demand in a market where the amount produced must be chosen before prices are observed, due to production lags. Producers’ expectations about prices that will prevail when the produced good will be available in the market, are assumed to be based on observations of current and past prices. The Cambridge economist Nicholas Kaldor (1908–1986) proposed the model in 1934, coining the term “cobweb theorem” (see [47, 55]). The second one is the Cournot duopoly model, which describes an industry where two firms compete on the amount of output they will produce, seeking to maximize their own profit given their competitors’ decisions. It is named after the French philosopher, mathematician and economist Antoine Augustin Cournot (1801–1877) who was inspired by observing competition in a spring water duopoly, see [30]. The mathematical model proposed by Cournot in 1838, assumes that two firms produce homogeneous products, there is no collusion and both firms have market power, i.e., each firm’s output decision affects the good’s price according to a given

Discrete Dynamical Systems in Economics: Two Seminal Models …

219

demand function. In this paper we start from these two basic exemplary models and we propose a survey of some developments in several directions, in particular some recent results which have argued for advanced new mathematical tools. For example, starting from the standard one-dimensional cobweb model, we describe how the introduction of new kinds of expectations and learning mechanisms have stimulated research about two-dimensional maps with a vanishing denominator, leading to the study of new kinds of singularities called focal points and prefocal curves in [21], see also [39]. Analogously, in the case of the two-dimensional Cournot duopoly model we propose a survey of some recent developments concerning the introduction of nonlinearities leading to multistability, i.e. the coexistence of several stable equilibria, see e.g. [17], where the related problem of the delimitation of basins of attraction requires a global dynamical analysis which is often obtained by combining numerical and analytical methods based on the method of critical curves, see [51] and references therein. Moreover, in the particular case of identical players, some recent results about chaos synchronization and related bifurcations (such as riddling or blowout bifurcation) have been employed, see e.g. [5, 9, 10]. The paper is organized as follows. In Sect. 2 we introduce the standard textbook Cobweb model and then we describe some improvements both in the form of supply and demand functions, and in the kind of expectations. In Sect. 3 we introduce the standard textbook Cournot duopoly model and then we describe some improvements both in the form of supply and demand functions and in the kind of adaptive adjustment followed in the repeated choices of players. The symmetric case of identical competitors is considered as well. Section 4 concludes and suggests some further developments.

2 The Cobweb Model The Cobweb Model, described in any economics textbook, takes into account a good which is sold in the market at time .t at a unit price . p(t). The quantity demanded by consumers is a function of the price . Q d = D( p) denoted as demand function, usually a continuous and decreasing function (hence invertible). The supply function expresses the output decided by producers as a function of the price . Q s = S( p e ), where . p e represents the price that producers expect to prevail when the good will be available in the market, on the basis of the information they have when the production is decided. Let .Δt = 1 be the time lag necessary for production. Then the economic equilibrium condition at time .t + 1, . Q d (t + 1) = Q s (t + 1) becomes: .

D( p(t + 1)) = S( p e (t + 1)).

(1)

By applying the inverse of demand function . p = D −1 (q) it assumes the explicit form: −1 . p(t + 1) = D (S( p e (t + 1))) = f ( p e (t + 1)). (2)

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The early assumption given in the literature to close the model, see e.g. [34], is expressed by naïve expectations . p e (t + 1) = p(t), i.e. without reliable information the producers expect that the price at the end of production will be as the one prevailing at the beginning. Under such assumption the model becomes a standard difference equation . p(t + 1) = f ( p(t)). For example, with linear demand and linear supply functions . D( p) = a − bp and S( p) = −c + dp, with .a, .b, .c, .d positive real parameters, the model becomes:

.

.

a+c d . p(t + 1) = f ( p(t)) = − p(t) + b b

(3)

a+c This discrete linear model gives rise to oscillatory convergence to . p ∗ = b+b when .b > d and oscillatory divergence otherwise. The corresponding staircase diagram is a classical picture in economics textbooks, that justifies the name of the model. Of course, divergence is not a suitable outcome for an economic model, hence some authors assumed nonlinear supply function expressing an (economically more realistic) upper limit for production, see e.g. [28], such as

.

S( p) = arctan (λ( p − 1))

(4)

proposed in [44], where .λ represents a measure of producers’ reactivity to profit opportunities, expressed by the slope of the supply at the reference price . p = 1. With the same linear demand function, the cobweb model with naïve expectations and nonlinear supply (4) gives rise to a nonlinear discrete dynamic model .

p(t + 1) = f ( p(t)) =

1 [a − arctan (λ( p(t) − 1))] b

characterized by a decreasing map . f ( p), and by using the supply slope .λ as a bifurcation parameter the equilibrium price . p ∗ undergoes a flip bifurcation for increasing values of .λ, so that an attracting cycle of period 2 replaces the stable equilibrium price. Hence, divergence no longer occurs, and two kinds of possible long run dynamics can be obtained, as usual for a nonlinear decreasing map: oscillatory convergence to the equilibrium price or stable self-sustained oscillations of period 2, due to overshooting effects. A further modification of the classical cobweb model, proposed in [28, 44, 45], consists in the introduction of adaptive expectations .

p e (t + 1) = p e (t) + α( p(t) − p e (t)), 0

α

1,

(5)

that can be described as follows. At any time .t producers observe the discrepancy between the realized price . p(t) and the expected price for the same period e .( p(t) − p (t)) and according to such observed “estimation error” correct the previous price estimate . p e (t) in order to obtain the next one: if the expected price was

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underestimated, i.e. . p e (t) turns out to be less than the observed one . p(t), then they increase the current estimation in order to form the next expected price . p e (t + 1); if the expected price . p e (t) was overestimated, i.e. it turns out to be greater than the one observed by producers, then they decrease it to form the next expected price. The value of the parameter .α modulates the entity of the correction; notice that for .α = 1 adaptive expectations (5) reduce to naïve expectations. p e (t + 1) = p(t). In this sense (5) is a generalization of naïve expectations as these are included as a particular case. In the other limiting case .α = 0 we obtain a complete inertia . p e (t + 1) = p e (t), as producers never change their initial guess . p e (0) on the basis of observed prices. By inserting . p(t) = f ( p e (t)), according to (2), inside (5) we get a law of motion in the space of expected prices: .

( ) p e (t + 1) = g p e (t) = p e (t) + α( f ( p e (t)) − p e (t)) = (1 − α) p e (t) + α f ( p e (t)).

(6)

From the dynamics of expected prices (6) the corresponding dynamics of realized prices (i.e. prices really observed in the market) is obtained again by. p(t) = f ( p e (t)), a transformation from beliefs to realizations. In order to analyze the dynamic behavior of (6) let us notice that the function .g( p) is a convex combination (i.e. a weighted average) of the identity function (whose graph is the diagonal) and the decreasing function . f , so its graph is placed between the two graphs (see the left panel of Fig. 1), being closer to the diagonal as .α → 0 and closer to the graph of . f as .α → 1. b The derivative .g ' ( p) = 1 − α + α f ' ( p) with (4), for .α > b+λ , vanishes in two points, relative minimum and maximum respectively (see Fig. 1). Moreover, it is always stable for sufficiently small values of .α, whereas for a given value of .α the equilibrium becomes unstable through a flip bifurcation for increasing values of .λ. Differently from the model with naïve expectations, where the decreasing map (even if nonlinear) could not have attractors more complex than a cycle of period 2, in this case, being a noninvertible map (i.e. characterized by the presence of turning points, relative maximum and minimum in this case) the first period doubling bifurcation is followed by a sequence of successive period doublings (the period doubling route to chaos) as shown in the bifurcation diagram of Fig. 1.

Fig. 1 Nonlinear cobweb with adaptive expectations

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Cobweb models with different kinds of demand and supply functions have been considered in the economic literature, such as the ones proposed in [53] or [46] in different economic contexts, as well as different kinds of expectations in order to conjecture . p e (t + 1) according to observed prices both in the current and past time periods. Many authors have proposed learning rules which describe how the economic agents make forecasts by using the information gained from the data observed in the past, such as [25] who examined a linear model where agents use a simple arithmetic mean of the prices observed in the past to obtain a “reasonable” forecast of the next price t 1∑ e . p (t + 1) = p(k). (7) t k=1

2.1 Learning with Fading Memory In [43] the learning rule (7) is used in a cobweb model with nonlinear demand and supply, and its stabilizing effect on the long run dynamics is proved. However, it is stressed that the short and intermediate run dynamics can be rather complex and of considerable interest. In the same year [32] considered a cobweb model with quadratic demand endowed with Bray’s learning (7), and raised some questions related to the global behavior, with statements like “...the evolution of the model is indeed very much dependent upon the starting position...”. Indeed, by some numerical simulations, [32] clearly show that the convergence toward the locally stable equilibrium is not ensured even starting from initial conditions very close to the equilibrium. Starting from these remarks, in [19] the same quadratic model is considered with expected prices computed according to a generalization of (7) given by a weighted arithmetic mean

.

p e (t + 1) =

t ∑ k=1

atk p(k),

with atk

0, and

t ∑

atk = 1

(8)

k=1

with weights distributed as the terms of a geometric progression of ratio .ρ ∈ [0, 1], that is ( t 1−ρt ∑ ρt−k if 0 ρ < 1 t−k .atk = , with Wt = ρ = 1−ρ (9) . Wt t if ρ = 1 k=1 This learning rule is a generalization of (7), as for .ρ = 1, (9) gives a uniform distribution of weights, .atk = 1t for each .1 k t, so that (8) reduces to (7). For lower values of the memory ratio .ρ we obtain the more realistic situation of a fading memory, in which earlier observations receive less weight than recent ones. In the other limiting case .ρ = 0 (8) reduces to naïve expectations . p e (t + 1) = p(t). Thus, by

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using .ρ as a varying parameter in the interval .[0, 1], we can explore the effect of a learning rule with different degrees of fading memory, and in particular of Bray’s learning (7) in the limiting case .ρ = 1. It can be considered a realistic assumption given that older observations are less considered by economic agents, as stressed in [37, 61], see also [24, 27, 36, 57, 63], as well as a further generalization with power means in [15]. The learning rule (7) with “geometric weights” (9) becomes

.

p e (t + 1) =

t ∑ ρt−k k=0

Wt

p(k)

(10)

and can be written as .p

t−1 t−1−k ∑ ρ e (t + 1) = ρW (t − 1) p(k) +

W (t)

W (t − 1)

k=0

1 W (t) − 1 e 1 p(t) = p (t) + p(t) W (t) W (t) W (t)

where the recursive relation .W (t + 1) = 1 + ρW (t), with. W (0) = 1 has been used. So, if we define .αt = W1(t) we get .

( ) p e (t + 1) = (1 − αt ) p e (t) + αt p(t) = (1 − αt ) p e (t) + αt f p e (t)

(11)

which is similar to an “adaptive rule” (5) except for the fact that the constant speed .α is replaced by a time-dependent speed of adjustment given by a decreasing sequence .{αt } with .αt ∈ (0, 1) for each .t and .αt → (1 − ρ) as .t → +∞. In [19], see also [24] for a more general analysis, a general method for the analysis of the global properties of the class of models (2) with learning rule (10) is applied to a cobweb model with the quadratic map .

f (x) = μx (1 − x) , μ > 1.

(12)

In this case, the dynamics of the expected prices under the assumption of adaptive expectations (5) is governed by a quadratic map as well, given by .

z(t + 1) = gα (z(t)) = (1 − α) z(t) + α f (z(t))

(13)

where the dynamic variable .z(t) = p e (t + 1) has been introduced for notational convenience. Under the assumption of learning (10) the dynamics of the expected prices is obtained by the equivalent two-dimensional map

.

T :

⎧ ⎨ z (t + 1) = ⎩

ρW (t)z(t)+ f (z(t)) 1+ρW (t)

W (t + 1) = 1 + ρW (t)

with initial condition .z(0) = p0 and .W (0) = 1.

(14)

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The class of maps (14) has been initially studied in [6, 19], which inspired a stream of literature on maps with vanishing denominator, see [21–23], as well several applications, e.g. [36, 40, 63].

2.2 Limit Sets and Basins of Attraction for Cobwebs Fading Memory Any trajectory of (14) starting from an initial condition on the line .W = 1 is con1 , because the second difference equation in (14), fined in the strip .0 < .W < 1−ρ which gives the dynamics of the variable .W , is independent of .z and gives a monotonically increasing sequence (the partial sums of a geometric series of ratio .ρ) and if .0 ρ .< 1 such sequence {.Wt } converges to the sum of the geometric series .

W∗ =

1 . 1−ρ

(15)

For .0 ρ .< 1 the line .W = W ∗ is an invariant and globally attracting line for the map .T , on which the .ω-limit sets of all its trajectories are located. The restriction of . T to this line is given by the one-dimensional map g(z) = ρz + (1 − ρ) f (z) ,

(16)

.

which is called limiting map, since it governs the asymptotic behavior of the map } { T . This implies that any .k-cycle . A = z 1∗ , . . . , z k∗ of the map .g(z) is in one-toone correspondence with a .k-cycle . A = A × {W ∗ }={.(z 1∗ , W ∗ ), . . . , (z k∗ , W ∗ )} of the map .T , located on the line of .ω-limit sets. Moreover, as shown in [6], the basins of attraction of any attracting set located on the line of .ω-limit sets is given by the intersection of the two-dimensional basin of the map (14) with the line of initial conditions .W = 1. This suggests a general procedure to obtain the boundaries of the basins of attraction when two or more coexisting attractors are present, as often occurs in the case of nonlinear models. This is an important issue that cannot be studied on the basis of the limiting map .g(z), because the initial conditions are to be taken on the line .W = 1, whereas .g(z) governs the dynamics near the line of .ω-limit sets .W = W ∗ . This means that only a global knowledge of the two-dimensional map .T allows one to follow the short-run (or transient) behavior, during which the dynamics are not governed by the limiting map .g(z). We recall that the two dimensional basin of attraction of an attractor . A of the map . T is the open set of points which generate trajectories converging to . A: .

{ } B (A) = (z, W ) |T t (z, W ) → A as t → +∞

.

.

(17)

A closed invariant set . A ⊂ {W = W ∗ } is called asymptotically stable (or attracting) if a neighborhood .U of . A exists such that .T (U ) ⊆ U and .T n (x) → A as .n → +∞

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for each .x ∈ U . Then, the basin of . A is obtained by taking all the preimages of the points of .U ∞ ∐ .B (A) = T −n (U ) n=0

where .T −n (x) denotes the set of all the preimages of .x of rank .n, i.e. the set of all the points which are mapped into .x after .n iterations of .T . So, the study of the two-dimensional basin is based on the study of the inverses of .T . In the case of the map (14) we have that the properties and the qualitative changes of its basins are strongly influenced by the presence of the denominator which can vanish along the line .W = − ρ1 and, in particular, by the points in which the first component of . T assumes the form 0/0, as shown in [21], see also [6] for the particular class of triangular maps (14). In these papers it is proved that the presence of points where a component of the map assumes the form 0/0, called focal points, may have an important impact on the structure of the basins and their global bifurcations, because fans of basins boundaries arise from them giving peculiar finger-shaped structures called lobes. The existence of lobes, originating from the focal points, may have important consequences on the structure of the basins of attraction of the model with learning (10) whenever they intersect the line of initial conditions .W = 1. This occurrence causes the creation of basins with a complicated topological structure, such as basins formed by many disjoint intervals, as we shall see in the next section.

2.3 Global Properties of the Triangular Map with a Vanishing denominator In the following we briefly recall some definitions and properties specific to maps with a vanishing denominator (see ) [21–23] for a more complete treatment). Let us ( consider a map .(x, y) → x ' , y ' = T (x, y) of the form ( .

T :

x ' = F(x, y) y ' = G(x, y)

(18)

where .x and . y are real variables and at least one of the components has the form of a fractional rational function, i.e. .

F(x, y) =

N1 (x, y) D1 (x, y)

and/or G(x, y) =

N2 (x, y) D2 (x, y)

(19)

The functions . Ni (x, y) and . Di (x, y), .i = 1, 2, are defined in the whole plane .R2 , so that the set with no definition .δs of the map .T coincides with the locus of points in which at least one denominator vanishes:

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{ } δ = (x, y) ∈ R2 |D1 (x, y) = 0 or D2 (x, y) = 0

. s

(20)

The two dimensional recurrence obtained by the iteration of .T is well defined, i.e. it generates not terminating trajectories, provided that the initial condition belongs to the set . E given by ∞ ∐ 2 .E = R \ T −k (δs ) (21) k=0

so that .T : E → E. We recall here the following Definition. A point . Q=(.x Q ,. y Q ) is a focal point if at least one component of the map T takes the form 0/0 in . Q and there exist smooth simple arcs .γ(t), with .γ(0)=. Q, such that .limt→0 T (γ (t)) is finite. The set of all such finite values, obtained by taking all the arcs .γ(t) through . Q, is the prefocal set .δ Q . Roughly speaking, a prefocal curve is a set of points for which at least one inverse exists which maps (or “focalizes”) the whole set .δ Q into a single point, called focal point. For maps with a vanishing denominator, new kinds of contact bifurcations have been evidenced which involve the singularities defined above. In particular, contacts between basin boundaries and prefocal curves may cause the creation of particular structures of the basin boundaries, denoted as lobes. These particular structures have been observed in the study of discrete dynamical systems of the plane which arise in some different contexts, see e.g. [27, 38, 57, 63]. We now briefly describe the basic mechanism leading to their formation for the map (14). Let us consider the image of an arc crossing through a focal point. According to the general results given in [21], a one-to-one correspondence is obtained between the slopes of the arcs through a focal point and the points in which their images cross the corresponding prefocal curve. In the map (14) only the first component has a denominator, which vanishes in the points of the singular line .δs of equation 1 . W = − , where the numerator becomes . f (x) − x. Hence the numerator vanishes ρ at every fixed point of the function . f (and thus also ( of the ) limiting map .g in (16)). It 1 ∗ follows that a focal point is necessarily of type . x , − ρ , where .x ∗ is a fixed point of . f (x). In order to explain the role of a focal point and the related prefocal set in the geometric and dynamic properties of the map .T , following the arguments given in [21], we consider a smooth simple arc .γ transverse to .δs and how it is transformed by .T . Let (.z 0 ,.−1/ρ) be the point where .γ intersects .δs and assume that the arc .γ is deprived of (.z 0 ,.−1/ρ). If.z 0 /= x ∗ , i.e.. F(z 0 ) /= z 0 , then the image.T (γ) is made up of two disjoint unbounded arcs asymptotic to the line of equation. y = 0, as qualitatively shown in Fig. 2. A different situation may occur if .z 0 = x ∗ , i.e. . f (z 0 ) = z 0 , because in this case the numerator of the first component ) also vanishes, and the limit of .T (γ) ( may take finite values as .(z, W ) → x ∗ , − ρ1 , so that .T (γ) is a bounded arc (as qualitatively sketched in Fig. 2 for the arc .γ2 ). If .m is the slope of the tangent to the

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Fig. 2 Schematic picture of the action of a two-dimensional map on an arc crossing a singular curve .δ S along which a denominator vanishes. Left: The arcs .γ1 and .γ3 cross the singular curve in a generic point of .δ S whereas .γ2 crosses it through a focal point. Right: Two arcs crossing .δ S through a focal point with different slopes are mapped into finite arcs crossing the prefocal curve .δ Q in different points

) ( smooth arc .γ in the focal point . Q = x ∗ , − ρ1 then in [21] it is proved that the image . T (γ) crosses the line . W = 0 in the point .(u m , 0) with u (x ∗ ) = x ∗ +

. m

f ' (x ∗ ) − 1 . ρm

(22)

( ) This means that the images of the arcs crossing through . x ∗ , − ρ1 with slope .m /= 0 are bounded arcs (as qualitatively shown in the right panel of Fig. 2), and as .m varies in .R all the points of the line .W = 0 are obtained, provided that . f ' (x ∗ ) /= 1. Thus the line of equation .W = 0 represents the prefocal set .δ Q for the map (14). The as situation in which . f ' (x ∗ ) = 1 can be considered ) a bifurcation case (see [23]). ( To sum up, for each focal point . Q i = xi∗ , − ρ1 the map T in (18) defines a oneto-one correspondence between the slope .m of an arc .γ through . Q i and the point .(u, 0) in which the image . T (γ) crosses the prefocal curve .δ Q , given by

.

f ' (xi∗ )−1 ρm f ' (xi∗ )−1 ρ(u−xi∗ )

m → (u, 0)

:

u = xi∗ +

(u, 0) → m

:

m=

(23)

Some consequences of this correspondence, which are important for the characterization of the basins boundaries and their bifurcations, are deduced by considering a smooth arc .ω that intersects the prefocal line in two points. In this case, the . N rank-1 preimages of .ω, say .Ti−1 (ω), i = 1, . . . , N , are arcs such that each .Ti−1 (ω) has a loop with knot in the focal point . Q i = (xi∗ , − ρ1 ). This implies that a remarkable contact bifurcation occurs when a smooth curve segment .ω moves towards the prefocal curve .δ Q until it has a contact and then crosses .δ Q (as qualitatively shown in Fig. 3). As .ω moves toward .δ Q , its preimages move towards . Q i , and when .ω becomes

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Fig. 3 Qualitative picture of a preimage of an arc moving towards a prefocal line until crossing it

i tangent to .δ Q then each preimage .ω−1 = Ti−1 (ω) has a cusp point at . Q i . The slope of the common tangent line to the two arcs that join in . Q i is given by .m i (u c ), according to (23). If the curve segment .ω moves further, so that it crosses .δ Q in two points, say .(u 1 , 0) and .(u 2 , 0), then its preimages form loops with double points at the focal points . Q i . This kind of contact bifurcation is important in the study of the boundaries of the basins of attraction, because if .ω is a portion of a basin boundary, a contact between .ω and .δ Q implies that a loop is created along the basin boundary, because a basin boundary is backward invariant, i.e. it includes all the preimages of any portion of it, and the portion of the basin inside the loop is a lobe, as we shall discuss in the following. Now, let us consider the forward iteration of .T . It is easy to see that the image of rank-n of the prefocal line .W = 0 belongs to the line of equation .W = Wn where

.

Wn =

1 − ρn+1 1−ρ

(24)

i.e. a sequence of lines parallel to the prefocal line .δ Q and convergent to the line of the .ω-limit sets .W = W ∗ . This implies that any cycle belonging to the .ω-limit set ∗ . W = W is transversely attracting. This property is important in order to study the boundaries of the basins. In fact, recall that, in general, the boundaries of a basin are obtained by taking the stable sets of some cycles on it. In the case of maps (14) such cycles can only be located on the line of .ω-limit sets .W = W ∗ . To get the stable set .W s of a saddle it is enough to take the preimages of any rank of a local stable

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∞

s s s set .Wloc , that is .W s = ∪ T −n (Wloc ), where .Wloc is transverse to the line .W = W ∗ . n=0

Due to the expansive character of .T −1 along the .W direction, such preimages must necessarily reach, in a finite number of steps, the prefocal line .W = 0. So, all these preimages must necessarily cross the singular line .W = − ρ1 through focal points . Q i . From this observation it follows that the stable set of any saddle cycle of .T , obtained by taking the preimages of a local stable set, is made up of branches issuing from the focal points. In fact, the preimages of any local stable set, transverse to the line of .ω-limit sets .W = W ∗ , necessarily go back to the prefocal line .W = 0 in a finite number of steps. Thus, any stable set must be made up of branches which “cross” the singular line in the focal points, i.e. are “focalized” through the focal points. This argument, applied to the stable set of some saddle cycle belonging to the line of .ω-limit sets, constitutes the global mechanism which causes the formation of the particular structures of the basins which will be shown in the examples. It is worth stressing that as the learning rule (7) can be obtained from the expectations with fading memory (10) in the limiting case .ρ → 1, any equilibrium price . p ∗ with ' ∗ . f ( p ) < 1 is stable, i.e. it can be “learned” by the agents, because the stability | | condition .|g ' ( p ∗ )| < 1 is always satisfied as .ρ → 1− . This confirms, and extends, the stability results obtained, for particular models, by [25, 32, 43]. In other words, in the case of Bray’s learning (7) any complexity is lost, and any trajectory is either divergent or convergent to a stable equilibrium. p ∗ . However, the corresponding basin of attraction can be obtained following the same procedure outlined in this section.

2.4 The Quadratic Cobweb Model with Fading Memory In the case of the quadratic (logistic) map (12) the limiting map which governs the asymptotic behavior is given: .

z ' = g (z) = ρz + (1 − ρ) μz (1 − z) .

(25)

So, for each .μ > 1 there are two equilibria given by s∗ = 0

.

and

p∗ =

μ−1 μ

where .s ∗ is an unstable fixed point of . f , with . f ' (s ∗ ) > 1, hence it is also an unstable fixed point of the limiting map .g for each .ρ ∈ [0, 1). Instead . f ' ( p ∗ ) = 1 − μ, hence the|sufficient | condition for local asymptotic stability under the limiting map .g, given ' by .|g ( p ∗ )| < 1, becomes: 3−ρ . (26) .1 < μ < 1−ρ

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Fig. 4 Numerically generated basins of attraction of the two-dimensional map (14): The white region represents the set of points converging to the fixed point . P ∗ and the gray region represents the set of points which generate diverging trajectories. a .ρ = 0.75 and.μ = 6; b.ρ = 0.75 and.μ = 7

This means that . p ∗ is always stable provided that .ρ is sufficiently close to .1. As stated above, the delimitation of the boundary which separates the basin of . p ∗ from the basin of infinity requires a global analysis of the two-dimensional map taking into account its two focal points, (

1 .Q1 = 0, − ρ

)

( and

Q2 =

) 1 μ−1 ,− . μ ρ

(27)

In Fig. 4a, obtained with .ρ = 0.75 and .μ = 6, the basins of the two-dimensional map are shown: The white ( region )represents the set of points converging to the fixed point μ−1 1 ∗ ∗ , 1−ρ , and the gray region represents the set of points which .P = ( p , W ) = μ generate diverging trajectories. The intersections with the line of initial conditions . W = 1 represent the respective one-dimensional basins of the cobweb model with learning (10) given by the interval .(0, z) with .z = 1.125. By contrast, in the situation shown in Fig. 4b, obtained with .ρ = 0.75 and .μ = 7, the basin . B ( p ∗ ) along the line of initial conditions .W = 1 is formed by two disjoint intervals, because a “hole” formed by points which generate diverging trajectories is nested inside . B ( p ∗ ). The above considerations are even more evident when applied to situations like the one shown in Fig. 5, obtained for .ρ = 0.75 and .μ = 9. In this case the basin of . p ∗ is formed by 4 disjoint intervals, due to the presence of lobes of .B(∞) intersecting the line of initial conditions .W = 1. The global structure of the basin boundaries described above holds for any value of the memory ratio .ρ belonging to the interval .(0, 1). In particular, it also holds in the limiting case .ρ → 1. So, also in the case of Bray learning (7), the complexity in the structure of the basins is conserved, as shown in Fig. 6, obtained with .ρ = 1 and .μ = 12. Similar structures of the basins are obtained for other models represented by unimodal maps, like the model proposed in [32], whose global analysis is given in [19].

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Fig. 5 The same as Fig. 4, with .ρ = 0.75 and .μ = 9

Fig. 6 The case of Bray’s learning .ρ → 1− with .μ = 12. The dots represent a trajectory starting from the initial condition .(z 0 , 1) with .z 0 = 0.3

3 Cournot Duopoly Model Let us consider two firms that in time period .t produce quantities .q1 (t) and .q2 (t) of homogeneous goods respectively, and sell them in the same market characterized by an inverse demand function . p = D −1 (Q) = f (Q), where . Q = q1 + q2 is the total quantity produced. If .Ci (q1 , q2 ), .i = 1, 2, are the respective cost functions (usually the cost function of firm .i only depends on its own production .qi , but in general one cannot exclude positive or negative cost externalities, due to the presence of a competitor) then the profits of the two firms are given by Πi (q1 , q2 ) = pqi − Ci = f (Q)qi − Ci (q1 , q2 ).

.

Hence, the profit of each firm also depends on the production of the other one. At each time .t any firm solves a profit maximizing problem in order to decide the production .qi (t + 1) to be sold after the production lag .Δt = 1. However, at time .t a firm usually

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does not know the production decision of the other one, so an expected value must be considered in the maximization problems .qi (t

( ) ( ) e e + 1) = arg max Πi (t + 1) = arg max[ f qi + q−i (t + 1) ] (28) (t + 1) qi − Ci qi , q−i qi

qi

where the notation .q−i indicates the production of the competitor of firm .i. So, the computation of an optimal production requires that each firm has: (i) knowledge of the demand function . p = f (Q); (ii) knowledge of its own cost function .Ci (qi ); (iii) e (t + 1) = q−i (t + 1); perfect foresight about competitors’ production choices .q−i (iv) computational skills to solve the optimization problem. Under such assumptions, each firm will compute its Best Reply, implicitly defined i = 0, that give by the first order (necessary) conditions . ∂π ∂qi .

∂πi ∂Ci (qi , q−i (t + 1)) ' = qi f (qi + q−i (t + 1)) + f (qi + q−i (t + 1)) + = 0 i = 1, . . . , n ∂qi ∂qi

(29)

together with (sufficient) second order conditions . ∂∂qπ2i < 0. In some cases, a unique i and explicit solution of (29) can be obtained, expressed by the reaction functions 2

q (t + 1) = Ri (q−i (t + 1))

. i

i = 1, 2.

(30)

The solutions of the .2 equations with .2 unknowns .qi = Ri (q−i ), .i = 1, 2, give the Nash equilibria, located at the intersections of the reaction functions (where each firm plays its best response to the other player’s best response strategies) as shown in the schematic picture of Fig. 7.

Fig. 7 Schematic representation of Best Reply (or Reaction) curves

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However, weaker levels of rationality, as well as lower information degree of players, can be considered in order to model more realistic economic situations. Let us first relax the assumption of perfect foresight about the expected production choice of other players, and replace it by assuming naive expectations q e (t + 1) = q−i (t)

(31)

. −i

that is, in the absence of information about competitors’ production decisions, each player assumes that competitors will produce in the next time period the same output as in the current period. Under this assumption the Best Reply gives rise to a discrete dynamical system .qi (t + 1) = Ri (q−i (t)) . (32) For example, if we consider linear demand and linear cost functions, . p = a − b(q1 + q2 ) and .Ci (qi ) = ci qi , then producer 1 faces the optimization problem ∂Π1 2 .maxq1 [(a − c1 )q1 − bq1 q2 (t) − bq1 ], and from the first order condition . = 0, ∂q1 a−c1 1 e we get .(a − c1 ) − bq2 (t + 1) − 2bq1 = 0 from which .q1 (t + 1) = − 2 q2 (t) + 2b . 2 The second order condition . ∂∂qΠ21 = −2b < 0 ensures that it is indeed a maximum. If 1 we solve the same problem for the other firm we obtain the following two-dimensional linear discrete time dynamical system ⎧ ⎨ q1 (t + 1) = R1 (q2 (t)) = − 21 q2 (t) + .

⎩

a−c1 2b

(33) q2 (t + 1) = R2 (q1 (t)) = − 21 q1 (t) +

a−c2 2b

where the unique equilibrium ( .

E=

a + c2 − 2c1 a + c1 − 2c2 , 3b 3b

) (34)

is positive provided that .a + c2 − 2c1 > 0 and .a + c1 − 2c2 > 0, two inequalities that define a nonempty set in the space of marginal costs.(c1 , c2 ) provided that.c1 < a and .c2 < a, the usual condition of unitary production costs less than unitary price. The equilibrium . E is always globally asymptotically stable, as the eigenvalues of the linear model are.λ1,2 = ± 21 . The eigenvector associated to.λ1 = 21 is.v1 = (−1, 1) and with .λ2 = − 21 is .v2 = (1, 1), hence we have an oscillatory convergence with oscillations along .(1, 1) direction (see Fig. 8). Instead, if the inverse demand function is assumed to be . p = Q1 , then the same arguments lead to the following nonlinear discrete dynamical system: ⎧ / ⎨ q1 (t + 1) = −q2 (t) + q2 (t) / c1 . ⎩ q2 (t + 1) = −q1 (t) + q1 (t) . c2

(35)

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Fig. 8 Trajectories for the Cournot model with linear demand and linear cost

Again, a unique Nash equilibrium exists, given by ( .

E=

c2 c1 ; 2 (c1 + c2 ) (c1 + c2 )2

) ,

(36)

whose local stability properties are given in terms of the ratio between the unitary costs .c1 /c2 (see [58, 59]). First of all, feasible (i.e. bounded and non negative) trajectories of the best reply dynamics are obtained provided that .c1 /c2 ∈ the Nash equilibrium (36) is stable if and 6.25]. Moreover, [4/25, 25/4] = [0.16, √ √ only if .c1 /c2 ∈ (3 − 2 2, 3 + 2 2) ≃ (0.17, 5.83). In the left panel of Fig. 9, the white region represents the basin of attraction of the stable Nash equilibrium and the gray region the set of points that generate unfeasible trajectories. If .c1 /c2 exits this interval then the loses stability. Indeed, if .c1 /c2 falls outside the √ Nash equilibrium √ interval .(3 − 2 2, 3 + 2 2) then the asymptotic dynamics may converge at periodic cycles or even exhibit chaotic motion around the Nash equilibrium, as shown in the right panel of Fig. 9, where a chaotic trajectory is shown, together with the reaction curves, obtained with .c1 = 1 and .c2 = 0.161. In the former case we can say that Nash equilibrium is reached as a long run outcome of the repeated game. This may be seen as an evolutionary explanation of the outcome of a Nash equilibrium. Instead, in the latter case (characterized by a greater difference between production costs) Nash equilibrium is not reached in the long run, and players will never “learn to play” the Nash equilibrium. If we assume that the players do not have a complete knowledge of the demand function, we can guess that they try to infer how the market will respond to its production changes by an empirical estimate of their profit variation. This estimate may be obtained by market research or by brief experiments of small production or price variations performed at the beginning of period .t. Following a well known stream of literature, see e.g. [20, 33, 35], we assume that even if the firms are quite ignorant about the market demand, they are able to obtain a correct empirical

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Fig. 9 Reaction curves and Nash equilibrium for the Cournot duopoly model with isoelastic demand

estimate of the partial derivative of their profit (also called marginal profit in economic ( )(e) i i = ∂Π (q1 , q2 ),.i = 1, 2 . This local estimate of expected marginal literature). ∂Π ∂qi ∂qi profits is much easier to obtain than a global knowledge of the demand function (involving values of . Q that may be very different from the current ones). With this kind of information the producers behave as local profit maximizers, the local adjustment process being one where a firm increases its output if it perceives a positive marginal profit and decreases its production if it is negative: q (t + 1) = qi (t) + αi (qi )

. i

∂Πi (q1 , q2 ) ; ∂qi

i = 1, 2

(37)

where .αi (qi ) is a positive function which gives the extent of production variation of firm .i following a given profit signal. An adjustment mechanism similar to (37) has been proposed by some authors, especially in continuous time modeling, with constant .αi (see e.g. [33, 35]), whereas [20] assumes .αi proportional to .qi , i.e. .αi (qi ) = vi qi ,. i = 1, 2, where.vi is a positive constant representing a speed of adjustment for the “relative production change” assumed to be proportional to the estimated profit gradient qi (t + 1) − qi (t) ∂Πi = vi (q1 , q2 ). . qi (t) ∂qi This is the choice usually followed in discrete time oligopoly models (see [14] and references therein), leading to the following discrete dynamical system: [ ] ∂Πi .qi (t + 1) = qi (t) 1 + vi (q1 (t), q2 (t)) ∂qi

i = 1, 2.

(38)

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For example, if we assume a linear demand function . f (Q) = a − bQ, with .a, b positive constants, and linear cost functions .Ci (qi ) = ci qi ; i = 1, 2, we have ∂Πi . = a − ci − 2bqi − bq−i and the dynamic model (38) becomes: ∂qi

.

T :

⎧ ' ⎨ q1 = (1 + v1 (a − c1 ))q1 − 2bv1 q12 − bv1 q1 q2 ⎩

(39)

'

q2 = (1 + v2 (a − c2 ))q2 − 2bv2 q22 − bv2 q1 q2

where .' denotes the unit-time advancement operator, that is, if the right-hand side variables are productions of period .t then the left-hand ones represent productions of 1 , 0) if period .(t + 1). The fixed points of the map (39) are . E 0 = (0, 0), . E 1 = ( a−c 2b a−c2 .c1 < a, . E 2 = (0, ) if .c2 < a, which will be called boundary equilibria, and the 2b same . E given in (34), i.e. the Nash equilibrium located at the intersection between the reaction functions. We consider now “gradient dynamics” (38) with the nonlinear demand already considered before, . p = Q1 . In this case the one-period profit of firm .i is given by qi −i i − ci qi hence the marginal profits are . ∂Π = (q q+q .Πi (q1 , q2 ) = 2 − ci . With q1 +q2 ∂qi 1 2) this assumption, the adjustment process (38) becomes

.

T :

⎧ ) ( ' q2 ⎪ ⎪ ⎨ q1 = q1 1 − c1 v1 + v1 (q1 +q2 )2 ( ⎪ ⎪ ⎩ q ' = q2 1 − c2 v2 + v2 2 (q

q1 2 1 +q2 )

)

(40)

As the map (40) is not defined in .(0, 0), the unique equilibrium point is the Nash equilibrium (36), located at the intersection of the two unimodal reaction curves. As stressed above, if each player is rational and fully informed then everybody will choose a Nash equilibrium. However, several Nash equilibria may exist in the case of nonlinear models, hence a problem of equilibrium selection arises. Moreover, agents are sometimes not so informed, and they behave following adaptive methods, such as learning-by-doing or trial-and-error practices. Sometimes agents do not optimize at all, just tend to follow rough rules of thumb. This leads players to replace one-shot optimal decisions with repeated myopic or adaptive decisions, in other words to a dynamic process that may or may not converge to a Nash equilibrium, provided it is an equilibrium point of the dynamical system as well. Moreover, when a game has several Nash equilibrium points, which are equilibrium points of the dynamical system as well, then the step-by-step dynamic process may act as a selection device, i.e. the stability of the equilibria suggests which of them will prevail in the longrun. In addiction, if several equilibrium points are stable, then the study of their basins of attraction will give information about the path dependence, i.e. how the convergence will depend on historical accidents (represented by exogenous shifts of initial conditions). A Cournot duopoly model with several coexisting Nash equilibria has been proposed in [48] where the reaction curves are quadratic functions in the form of standard

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logistic maps .qi = Ri (q−i ) = μi q−i (1 − q−i ). If, besides best reply, also a certain degree of inertia is introduced, as suggested by the same author in [48], see also [14], the adaptive adjustment with best reply and inertia becomes: q (t + 1) = (1 − λ1 ) q1 (t) + λ1 μ1 q2 (t) (1 − q2 (t)) q2 (t + 1) = (1 − λ2 ) q2 (t) + λ2 μ2 q1 (t) (1 − q1 (t))

. 1

(41)

where the parameters.λi ∈ [0, 1] represent the degree of inertia (or anchoring attitude) of firm .i. Indeed, this model reduces to the repeated game with best reply and naive expectations if .λ1 = λ2 = 1 q (t + 1) = μ1 q2 (t) (1 − q2 (t))

. 1

(42)

q2 (t + 1) = μ2 q1 (t) (1 − q1 (t)) whereas complete inertia is obtained as .λi → 0. Up to four Nash equilibrium points, located at the intersection of the two unimodal reaction curves, may exist, which are also fixed point of the map (41). Moreover, two coexisting stable Nash equilibria can be obtained for certain sets of parameters, as well as other more complicated coexisting attractors, such as stable cycles of chaotic attractors. An exemplary case is shown in Fig. 10, where two stable Nash equilibria coexist with a stable cycle of period 2, each with its own basin of attraction. The basins are multiply connected, i.e. besides the immediate basin several (really infinitely many) non connected portions exist that accumulate along the outer boundary of the phase space, see e.g. [12] for a detailed analysis of (42). In the case of coexisting stable Nash equilibria, an important question concerns the delimitation of their basins of attraction and the global bifurcations that cause qualitative modifications of their boundaries. This is illustrated in Fig. 11, where in the left panel a contact occurs between the boundary of the basin of. E 1 (formed by the stable set of the saddle point . S) and the critical curve . LC that separates . Z 2 from . Z 4 . The portion of the basin of. E 1 that enters. Z 4 after the contact generates new preimages that give rise to a sequence of non-connected portions of the basin, as shown in the central panel of Fig. 11. However, as the equation of the boundary is not known in this case, an analytic computation of the values of the parameters at which the contact occurs is not possible. This is an usual occurrence, as the analytical expressions of the stable sets (that bound the basins) as well as the analytic expressions of the critical curves, are very rarely known. So, in order to study global bifurcations which are typical of two-dimensional noninvertible maps, numerical methods have to be employed. The delimitation of the basins of attraction of coexisting Nash equilibria requires a study of the global dynamical properties of the dynamical system, i.e. a study which is not based on the linear approximation of the map. Hence, the occurrence of contact bifurcations can only be revealed numerically. A detailed study, by using analytical and numerical methods based on the theory of critical curves for noninvertible maps is given in [17].

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Fig. 10 Nash equilibrium points and basins of attraction for the Cournot duopoly model with unimodal reaction functions

Fig. 11 Basins, equilibrium points and attractors for the adaptive Cournot duopoly model with logistic reaction functions

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3.1 Duopoly Games with Identical Players: Chaos Synchronization, Riddling and Intermittency Phenomena In this section, following a recent stream of literature, see e.g. [5, 9–11], we consider a general discrete-time dynamic duopoly game in the form of an iterated map .T : ' ' (q1 , q2 ) → (q1 , q2 ), defined by '

.

q1 = T1 (q1 , q2 ) ' q2 = T2 (q1 , q2 )

(43)

and we assume that the two players are identical, so that map.T presents the following symmetry property: it remains the same if the variables.q1 and.q2 are interchanged, i.e. . T ◦ S = S ◦ T , where . S : (q1 , q2 ) → (q2 , q1 ) is the reflection through the diagonal } { Δ = (q1 , q2 ) ∈ R2 |q1 = q2 .

.

(44)

This symmetry property implies that the diagonal is an invariant submanifold. The trajectories embedded into .Δ, i.e. characterized by .q1 (t) = q2 (t) for every .t, are called synchronized trajectories, and they are governed by the one-dimensional map given by the restriction of .T to the invariant submanifold .Δ q(t + 1) = f (q(t))

.

with

f = T |Δ : Δ → Δ.

(45)

In [8] the one-dimensional model (45) has been considered as the model of a representative agent whose dynamics summarize the common behavior of the two synchronized competitors. A question which naturally arises is whether a trajectory starting out of .Δ, i.e. with .q1 (0) /= q2 (0), will synchronize in the long run, i.e. if .|q1 (t) − q2 (t)| → 0 as .t → +∞, so that the asymptotic behavior is governed by the simpler one-dimensional model (45). This question can be reformulated as follows. Let . As be an attractor of the one-dimensional map (45). Is it also an attractor for the two-dimensional map .T ? Of course, an attractor . As of the restriction . f is stable with respect to perturbations along .Δ, so an answer to the question raised above can be given through a study of the stability of . As with respect to perturbations transverse to .Δ (transverse stability). In fact, if . As is a cycle of period .k embedded into .Δ, say . As = (γ1 , . . . , γk ), an eigenvalue is always associated with the corresponding eigendirection along .Δ and coincides with the∏multiplier of the ' k f (γi ). The restriction (45) computed at the periodic points of the cycle, i.e. . i=1 other eigenvalue, generally associated with an eigendirection transverse to .Δ, can be used for the study of the transverse stability, based on the usual evaluation of the modulus of the eigenvalue of the cycle in the direction transverse to .Δ. The problem becomes more interesting when the dynamics restricted to the invariant submanifold are chaotic. In this case the phenomenon of chaos synchronization may occur (see e.g. [2–4, 26, 56]). The key property for the study of the transverse stability of a chaotic set . As ⊂ Δ is that it includes infinitely many periodic orbits which are unsta-

240

G. I. Bischi

ble in the direction along .Δ. For any of these cycles, it is generally easy to compute the associated eigenvalues. For example, in the case of invariance of the diagonal map, the Jacobian matrix of .T computed at any point of .Δ, say (44) for a symmetric { } . DT (q, q) = Ti j (q) , is such that . T11 = T22 and . T12 = T21 , and the two orthogonal eigenvectors of such a symmetric matrix are one parallel to .Δ, say .v‖ = (1, 1), and one perpendicular to it, say .v⊥ = (1, −1), with related eigenvalues given by λ (q) = T11 (q) + T12 (q)

. ‖

and

λ⊥ (q) = T11 (q) − T12 (q)

respectively. Of course, .λ‖ (q) = f ' (q). Since the product of matrices with the struca.k-cycle .{γ1 , . . . , γk } embedded into ture of. DT (x, x) has the same ∏k structure as well, ∏k k k .Δ has eigenvalues .λ‖ = λ and . λ = λ (γ ) ‖ i ⊥ i=1 i=1 ⊥ (γi ), with eigenvectors .v‖ and .v⊥ respectively. For a chaotic set . As ⊂ Δ, infinitely many transverse Lyapunov exponents can be defined as Λ⊥ = lim

.

N →∞

N 1 ∑ ln |λ⊥ (xi )| N i=0

(46)

{ } where . xi = f|i (x0| ), i 0 is a trajectory embedded in . As . If .x0 belongs to a .k-cycle then .Λ⊥ = ln |λk⊥ |, so that the cycle is transversely stable if .Λ⊥ < 0, whereas if .x0 belongs to a generic aperiodic trajectory embedded inside the chaotic set . As then nat .Λ⊥ is the natural transverse Lyapunov exponent .Λ⊥ , where by the term “natural” we mean the Lyapunov exponent associated to the natural, or SBR (Sinai-BowenRuelle), measure, i.e., computed for a typical trajectory taken in the chaotic attractor . A s . Since infinitely many cycles, all unstable along .Δ, are embedded inside a chaotic attractor . As , a spectrum of transverse Lyapunov exponents can be defined, see [4, 26] min .Λ⊥ · · · Λnat · · · Λmax (47) ⊥ ⊥ . The meaning of the inequalities in (47) can be intuitively understood on the basis of the property that.Λnat ⊥ expresses a sort of “weighted balance” between the transversely repelling and transversely attracting cycles. If .Λmax ⊥ < 0, i.e. all the cycles embedded in . As are transversely stable, then . As is asymptotically stable, in the usual Lyapunov sense, for the two-dimensional map .T . However, it may occur that some cycles embedded in the chaotic set . As become transversely unstable, i.e. .Λmax ⊥ > 0, while nat .Λ⊥ < 0. In this case, . A s is no longer Lyapunov stable, but it continues to be a Milnor attractor [50] i.e. it attracts a positive (Lebesgue) measure set of points of the two-dimensional phase space. The transition from asymptotic stability to attractivity from negative to only in the Milnor sense, marked by a change of sign of .Λmax ⊥ positive, is denoted as the riddling bifurcation in [49] (or bubbling bifurcation in [65]). Even if the occurrence of such bifurcations is detected through the study of the transverse Lyapunov exponents, their effects depend on the action of the nonlinearities far from .Δ, that is, on the global properties of the dynamical system. In fact,

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after the riddling bifurcation two possible scenarios can be observed according to the fate of the trajectories that are locally repelled along (or near) the local unstable manifolds of the transversely repelling cycles: (L) they can be reinjected towards .Δ, so that the dynamics of such trajectories are characterized by some bursts far from .Δ before synchronizing on it, a phenomenon denoted as bubbling1 in [64, 65]; (G) they may belong to the basin of another attractor, in which case the phenomenon of riddled basins (see [2]) is obtained. Some authors call soft bubbling the situation (L) and, by contrast, hard bubbling the situation (G) (see [4]). When also .Λnat ⊥ becomes positive, due to the fact that the transversely unstable periodic orbits embedded into . As have a greater weight as compared with the stable ones, a blowout bifurcation occurs, after which . As is no longer a Milnor attractor, because it attracts a set of points of zero measure, and becomes a chaotic saddle, see [26]. In particular, for.λmin ⊥ > 0 all the cycles embedded into .Δ are transversely repelling, and . As is called normally repelling chaotic saddle see [26]. Also the macroscopic effect of a blowout bifurcation is strongly influenced by the behavior of the dynamical system far from the invariant submanifold .Δ: the trajectories starting close to the chaotic saddle may be attracted by some attracting set far from .Δ or remain inside a two-dimensional compact set located around the chaotic saddle. As , thus giving on-off intermittency. The study of transverse Lyapunov exponents says nothing about the fate of the locally repelled trajectories, and the occurrence of the different scenarios described above is determined by the global properties of the map. When .T is a noninvertible map, these global properties can be described by the method of critical curves, which may be used to obtain an invariant absorbing area inside which intermittency phenomena are confined. Noninvertible map means “many-to-one”, that is, distinct points . p1 /= p2 may have the same image, i.e. .T ( p1 ) = T ( p2 ) = p. Geometrically, the action of a noninvertible map of the plane can be expressed by saying that it “folds and pleats” the plane, so that the two distinct points . p1 and . p2 are mapped into the same point . p. This is formally expressed by saying that . p has several distinct rank-1 preimages, the plane. i.e., several inverses are defined in . p, and that these inverses ) ( “unfold” More formally, a two-dimensional map .T : (q1 , q2 ) → q1' , q2' is said to be noninvertible if the rank-1 preimages .(q1 , q2 ) = T −1 (q1' , q2' ), obtained by solving the system (43) with respect to .q1 and .q2 , may be more than one. In this case, the plane can be subdivided into regions(. Z k , .k ) 0, whose points have .k distinct rank-1 preimages. Generally, as the point . q1' , q2' varies in the plane .R2 , pairs of preimages appear or disappear as it crosses the boundaries separating different regions, hence such boundaries are characterized by the presence of at least two coincident (merging) preimages. This leads to the definition of the critical curves, one of the distinguishing features of noninvertible maps. Following the notations of [41, 51], the critical set . LC (from the French “Ligne Critique”) is defined as the locus of points A very long sequence of such bursts, which can be observed when .Λnat ⊥ is close to zero, has been called on-off intermittency in [54], even if this term is more suitable after the blowout bifurcation, when .Λnat ⊥ becomes positive.

1

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G. I. Bischi

having two, or more, coincident rank-1 preimages, located on a set (set of merging preimages) called . LC−1 . . LC is the two-dimensional generalization of the notion of critical value (when it is a local minimum or maximum value) of a one-dimensional map, . LC−1 is the generalization of the notion of critical point (when it is a local extremum point). Arcs of . LC separate the regions of the phase plane characterized by a different number of real rank-1 preimages (see [41, 51]). For a differentiable noninvertible map of the plane, the study of the sign of the Jacobian determinant can help one to find the critical curves, because the set . LC−1 is included in the set where .det DT (q1 , q2 ) changes sign, since .T is locally an orientation preserving map near points .(q1 , q2 ) such that .det DT (q1 , q2 ) > 0 and orientation reversing if .det DT (q1 , q2 ) < 0, and . LC = T (LC−1 ). Of course, points of . LC−1 in which the map is differentiable are necessarily points where the Jacobian determinant vanishes: in fact in any neighborhood of a point of . LC−1 there are at least two distinct points which are mapped by .T in the same point, hence the map is not locally invertible in points of . LC−1 . This implies, for a differentiable map, that .

{ } LC−1 ⊆ J0 = (q1 , q2 ) ∈ R2 | det DT (q1 , q2 ) = 0 .

(48)

The critical sets of rank .k are the images of rank .k of . LC−1 denoted by . LCk−1 = T k (LC−1 ) = T k−1 (LC), . LC0 being . LC. Segments of critical curves of rank-.k, .k = 0, 1, . . ., can be used in order to define trapping regions of the phase plane. An absorbing area .A is a bounded region of the plane whose boundary is given by critical curve segments (segments of the critical curve . LC and its images) such that a neighborhood .U ⊃ A exists whose points enter .A after a finite number of iterations and then never escape it, i.e. .T (A) ⊆ A (see [51], Ch. 4, or [7], for more details). Following [51], a practical procedure can be outlined in order to obtain the boundary of an absorbing area (although it is difficult to give a general method). Starting from a portion of . LC−1 , approximately taken in the region occupied by the area of interest, its images of increasing rank are computed until a closed region is obtained. When such a region is mapped into itself, then it is an absorbing area .A. The length of the initial segment is to be taken, in general, by a trial and error method, although several suggestions are given in the books referenced above. Once an absorbing area .A is found, in order to see if it is invariant (or strictly mapped into itself) the same procedure must be repeated by taking only the portion γ = A ∩ LC−1

.

(49)

as the starting segment. Then one of the following two cases occurs: (i) the union of .m iterates of .γ (for a suitable .m) covers the whole boundary of .A; in which case .A is an invariant absorbing area, and ∂A ⊂

m ∐

.

k=1

T k (γ)

(50)

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Um (ii) no natural .m exists such that . i=1 T i (γ) covers the whole boundary of .A, in which case .A is not invariant but strictly mapped into itself. An invariant absorbing area is obtained by .∩n>0 T n (A) (and may be obtained by a finite number of images of .A). The minimal invariant absorbing area is the smallest absorbing area that includes the attractor . As on which the synchronized dynamics occur. Indeed, boundaries of trapping regions can also be obtained by the union of segments of critical curves and portions of unstable sets of saddle cycles, and in this case we have a so called absorbing areas of mixed type (see [51]). We will not go into depth here, as in the examples given in this paper only standard absorbing areas (i.e. completely bounded by critical arcs) are used. When (14) is a noninvertible map, as generally occurs in problems of chaos synchronization, the global dynamical properties can be usefully described by the method of critical curves, and the reinjection of the locally repelled trajectories can be described in terms of their folding action. This idea has been proposed in [5] for the study of symmetric maps arising in game theory, in [8, 9] for the study of the effects of small asymmetries due to parameters mismatches, in [7] for the study of coupled chaotic oscillators. In these papers the geometric properties of the critical curves have been used to obtain the boundary of suitable compact trapping sets inside which bubbling and blowout phenomena are confined. In other words, the critical curves are used to bound a compact region of the phase plane that acts as a trapping bounded vessel inside which the trajectories starting near . As are confined. As an example, let us consider the following dynamical system, obtained from (42) with a linear coupling between the two films ( .

T :

q1 (t + 1) = μ1 q2 (t)(1 − q2 (t)) + ε(q2 (t) − q1 (t)) . q2 (t + 1) = μ2 q1 (t)(1 − q1 (t)) + ε(q1 (t) − q2 (t))

(51)

For coupling parameter .ε = 0 it reduces to a map with a structure which is typical of Cournot duopoly games, such that the composition with (itself (i.e. the second ) iterate) is a decoupled map, as .T 2 (q1 , q2 ) = T (T (q1 , q2 )) = T12 (q1 ), T22 (q2 ) , see e.g. [12]. We now consider identical parameters .μ1 = μ2 = μ (i.e. identical players). The restriction .T |Δ = f μ (x) = μx(1 − x), the standard logistic map. The eigenvalues of the symmetric Jacobian matrix . DT (x, x) are λ (x) = a − 2ax , λ⊥ (x) = 2ax − a − 2ε.

. ‖

with eigenvectors which are parallel to .Δ (.v‖ = (1, 1)) and orthogonal to .Δ (.v⊥ = (1, −1)) respectively. The coupling parameter .ε is a normal parameter as it only appears in the transverse eigenvalue .λ⊥ . Hence, we can consider fixed values of the parameter .μ, such that a chaotic attractor . As ⊂ Δ of the map (45) exists, with an absolutely continuous invariant measure on it, and we can study the transverse stability of . As as the coupling between the two components, measured by the param-

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Fig. 12 Natural transverse Lyapunov exponent for fixed.μ = μ2 = 3.5748049387592 as a function of the coupling parameter .ε

eter .ε, varies. Suitable values of the parameter .μ, at which chaotic intervals for the restriction (45) exist, are obtained from the well known properties of the logistic map (see e.g. [29]). For example, at the parameter value .μ2 = 3.5748049387592... the period-4 cycle of the logistic map undergoes the homoclinic bifurcation, at which four cyclic chaotic intervals are obtained by the merging of 8 cyclic chaotic intervals. The corresponding natural transverse Lyapunov exponent .Λnat ⊥ is represented in Fig. 12 for .μ = μ2 as a function of the coupling parameter .ε. By using.μ = μ2 and.ε = 0.24 we get a four-band chaotic set. As along the diagonal max −3 .Δ, as shown in Fig. 13a. In this case, we have .Λ⊥ > 0 and .Λnat < ⊥ = −4.7 × 10 0. Hence, . As is a Milnor attractor and local riddling occurs. The generic trajectory starting from initial conditions taken in the white region of Fig. 13a leads to asymptotic synchronization. In Fig. 13a the asymptotic part of a trajectory is shown, after a transient of .15, 000 iterations has been discarded. Indeed, if also the transient is represented, Fig. 13b is obtained. During the transient, the time evolution of the system is characterized by several bursts away from .Δ before synchronization occurs, as shown in Fig. 13c, where the difference .q1 (t) − q2 (t), computed along the trajectory of Fig. 13b, is represented versus time. It is worth noting the intermittent behavior of the trajectory: sometimes it seems to synchronize for a quite long number of iterations, then a sudden burst occurs. This phenomenon is also called on-off intermittency. The Milnor attractor . As is included inside a minimal invariant absorbing area whose boundary can be easily obtained by five iterations of an arc of . LC−1 , as shown in Fig. 14a. This absorbing area, obtained by the procedure outlined above, constitutes a trapping region inside which the bursts observed during the transient are contained. This means that, even if it is difficult to predict the sequence of times at which asynchronous bursts occur, an estimate of their maximum amplitude can be obtained

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Fig. 13 a The four-band chaotic attracting set along the diagonal. b The transient part of a trajectory converging to the four band chaotic set c versus time .q1 (t) − q2 (t) after the transient with bubbling

Fig. 14 a Minimal invariant absorbing area, obtained by iteration of . LC, including the Milnor attractor. b A trajectory filling up the absorbing area, after the blowout bifurcation, obtained for nat −2 > 0 .ε = 0.245, at which .Λ⊥ = 2.2 × 10

by the construction of the minimal invariant absorbing area which includes the Milnor attractor on which synchronized dynamics take place. In such a situation, a method to obtain trajectories which never synchronize, so that the bursts never stop and the iterated points fill up the whole minimal absorbing area, consists in the introduction of a small parameters’ mismatch (see e.g. [7]), such as .ε1 slightly different from .ε2 or .μ1 slightly different from .μ2 , so that the symmetry is broken. This implies that the invariance of .Δ is lost, and consequently the embedded one-dimensional Milnor attractor no longer exists. The study of the effects of small parameters’ mismatches may be important in economic dynamic modelling, as stressed in [8]. A similar effect is obtained even in the symmetric case, if the value of the coupling parameter.ε is increased so that.Λnat ⊥ increases until it becomes positive, i.e. a blowout bifurcation occurs. After this bifurcation the bursts which characterize the first part of the trajectory never stop, i.e. the firms never synchronize. . As is now a chaotic saddle,

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and on-off intermittency is observed. This is what happens in the situation shown in −2 > 0. Now the point Fig. 14b, obtained for .ε = 0.245, at which .Λnat ⊥ = 2.2 × 10 of a generic trajectory starting from the white region fills the whole absorbing area, still bounded by segments of critical arcs.

4 Conclusions In this paper an overview of some results on discrete time dynamic economic models have been presented, starting from two classical and celebrated models, Cobweb price dynamics and the Cournot duopoly competition model. Both these milestones in economic dynamic modeling have given rise to a flourishing literature and even in the last few years several developments have been made involving new and advanced mathematical methods. For example, we have analyzed the inclusion of the memory of past prices in nonlinear cobweb models with boundedly rational expectations in the form of a weighted average with exponentially decreasing weights. In cases of multistability, the study of the basins of attraction has been approached by considering an equivalent two-dimensional map with a denominator that vanishes along a line whose global dynamic properties are strongly influenced by the presence of singularities denoted as focal points, and these have been recently proposed in the literature. These methods may be usefully applied in Cournot oligopoly games in discrete time, see e.g. [31] or [62]. Such games, endowed with fading memory, may be reduced to equivalent autonomous maps with a denominator of dimension greater than two, a quite challenging mathematical task. Further developments, with respect to those described in this survey, can also be considered for Cournot oligopoly games with bounded rational firms that are free to choose the behavioral rule to adopt on the basis of relative past performances, i.e. through evolutionary processes. In this regard, an evolutionary framework can be used to model oligopolistic markets where a population of firms can employ different heuristics for deciding their next-period production plans. The time evolution of the fractions of firms adopting the different adaptive behavioral rules may be simulated by a profit-driven switching mechanism, for example replicator dynamics, like in [18] or [60]. In this case, some attractors may be located on the invariant planes where pure strategies are played, i.e. all agents adopt the same behavioral rule, and may coexist with other attractors where heterogeneous heuristics coexist within the population of firms, thus providing an example of evolutionary stable heterogeneity. Even more complex situations can be obtained for the attractors embedded inside the invariant planes where homogeneous behavior take place. In fact, these chaotic twodimensional invariant sets inside a three-dimensional phase space can be transversely stable on average, thus giving rise to weaker attractors in the Milnor sense and, consequently, to on-off intermittency phenomena or riddled basins.

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Acknowledgements I want to thank the organizers and the participants of ICDEA (International Conference on Difference Equations and Applications) held at the University of Paris Saclay in July 2022, where I gave a plenary talk concerning the topics of this survey. In particular I received interesting comments and suggestions from Laura Gardini, Iryna Sushko, Davide Radi, Fabio Lamantia and Fabio Tramontana. This work has been developed in the framework of the research project FOR US AN EXP - FOod secuRity and sUStainAbility: Nudging and EXPerimental evidence, financed by the University of Urbino.

References 1. Agliari, A., Bischi, G.I., Gardini, L.: Some methods for the Global Analysis of Dynamic Games represented by Noninvertible Maps. In: Puu, T., Sushko, I. (eds.) Chapter 3 in Oligopoly Dynamics: Models and Tools. Springer, Berlin (2002) 2. Alexander, J.C., Yorke, J.A., You, Z., Kan, I.: Riddled basins. Int. J. Bifurc. Chaos 2, 795–813 (1992) 3. Ashwin, P., Buescu, J., Stewart, I.: Bubbling of attractors and synchronization of chaotic oscillators. Phys. Lett. A 193, 126–139 (1994) 4. Ashwin, P., Buescu, J., Stewart, I.: From attractor to chaotic saddle: a tale of transverse instability. Nonlinearity 9, 703–737 (1996) 5. Bischi, G.I., Stefanini, L., Gardini, L.: Synchronization, intermittency and critical curves in duopoly games. Math. Comput. Simul. 44, 559–585 (1998) 6. Bischi, G.I., Gardini, L.: Basin fractalization due to focal points in a class of triangular maps. Int. J. Bifurc. Chaos 7(7), 1555–1577 (1997) 7. Bischi, G.I., Gardini, L.: Role of invariant and minimal absorbing areas in chaos synchronization. Phys. Rev. E 58, 5710–5719 (1998) 8. Bischi, G.I., Gallegati, M., Naimzada, A.: Symmetry-breaking bifurcations and representative firm in dynamic duopoly games. Ann. Oper. Res. 89, 253–272 (1999) 9. Bischi, G.I., Gardini, L.: Global properties of symmetric competition models with riddling and blowout phenomena. Discret. Dyn. Nat. Soc. 5, 149–160 (2000) 10. Bischi, G.I., Cerboni, Baiardi L.: Bubbling, riddling, blowout and critical curves. J. Differ. Equations Appl. 23(5), 939–964 (2017) 11. Bischi, G.I., Cerboni, Baiardi L.: Fallacies of composition in nonlinear marketing models. Commun. Nonlinear Sci. Numer. Simul. 20, 209–228 (2015) 12. Bischi, G.I., Mammana, C., Gardini, L.: Multistability and cyclic attractors in duopoly games. Chaos, Solitons Fractals 11, 543–564 (2000) 13. Bischi, G.I., Lamantia, F.: Chaos synchronization and intermittency in a duopoly game with spillover effects 195–217. In: Puu, T., Sushko, I. (eds.) Oligopoly Dynamics: Models and Tools. Springer, Berlin (2002) 14. Bischi, G.I., Chiarella, C., Kopel, M., Szidarovszky, F.: Nonlinear Oligopolies: Stability and Bifurcations. Springer, Berlin (2010) 15. Bischi, G.I., Cavalli, F., Naimzada, A.K.: Mann iteration with power means. J. Differ. Equations Appl. 21(12), 1212–1233 (2015) 16. Bischi, G.I., Lamantia, F., Scardamaglia, B.: On the influence of memory on complex dynamics of evolutionary oligopoly models. Nonlinear Dyn. 102(2), 1097–1110 (2020) 17. Bischi, G.I., Kopel, M.: Equilibrium selection in a nonlinear duopoly game with adaptive expectations. J. Econ. Behav. Organ. 46(1), 73–100 (2001) 18. Bischi, G.I. Merlone., U., Pruscini E.: Evolutionary dynamics in club goods binary games. J. Econ. Dyn. Control 91, 104–119 (2018) 19. Bischi, G.I., Naimzada, A.K.: Global analysis of a nonlinear model with learning. Econ. Notes 26(3), 143–174 (1997)

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49. Lai, Y.C., Grebogi, C.: Noise-induced riddling in chaotic systems. Phys. Rev. Lett. 77, 5047– 5050 (1996) 50. Milnor, J.: On the concept of attractor. Commun. Math. Phys. 99, 177–195 (1985) 51. Mira, C., Gardini, L., Barugola, A., Cathala, J.C.: Chaotic Dynamics in Two-Dimensional Noninvertible Maps. World Scientific, Singapore (1996) 52. Mira, C.: Chaotic Dynamics. World Scientific, Singapore (1987) 53. Nerlove, M.: Adaptive expectations and cobweb phenomena. Quart. J. Econ. 73, 227–240 (1958) 54. Ott, E., Sommerer, J.C.: Blowout bifurcations: the occurrence of riddled basins and on-off intermittency. Phys. Lett. A 188, 39–47 (1994) 55. Pashigian, B.P.: Cobweb Theorem. The New Palgrave Dictionary of Economics, 2nd edn (2008) 56. Pecora, L.M., Carrol, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64(8), 821–824 (1990) 57. Pecora, N., Tramontana, F.: Maps with vanishing denominator and their applications. Front. Appl. Math. Stat. (2016). https://doi.org/10.3389/fams.2016.00011 58. Puu, T.: Chaos in duopoly pricing. Chaos, Solitons Fractals 1(6), 573–581 (1991) 59. Puu, T.: Attractors, Bifurcations, & Chaos. Nonlinear Phenomena in Economics, 2nd edn. Springer, Berlin (2003) 60. Radi, D., Lamantia, F., Bischi, G.I.: Offshoring, reshoring, unemployment, and wage dynamics in a two-country evolutionary model. Macroecon. Dyn. 25, 705–732 (2021) 61. Radner, R.: Comment to Convergence to rational expectations equilibrium by M. Bray. In: Friedman, R., Phelps, E.S. (eds.) Individual Forecasting and Aggregate Outcomes. Cambridge University Press (1983) 62. Thorlund-Petersen, L.: Iterative computation of Cournot equilibrium. Games Econ. Behav. 2, 61–95 (1990) 63. Tramontana, F.: Maps with vanishing denominator explained through applications in economics. J. Phys.: Conf. Ser. 692(1) (2016) 64. Venkataramani, S.C., Hunt, B.R., Ott, E., Gauthier, D.J., Bienfang, J.C.: Transition to bubbling of chaotic systems. Phys. Rev. Lett. 77, 5361–5364 (1996) 65. Venkataramani, S.C., Hunt, B.R., Ott, E.: Bubbling transition. Phys. Rev. E 54, 1346–1360 (1996)

On a Class of Applications for Difference Equations in Continuous Time Vladimir R˘asvan

Abstract There are considered certain engineering applications described by nonstandard boundary value problems for .1D hyperbolic partial differential equations. Their qualitative analysis is performed by associating a class of functional differential equations with deviated argument, in most cases of neutral type. At their turn, to these neutral equations it is associated a system of continuous time difference equations. The difference equations are required to be asymptotically stable to ensure the same behavior for the basic system via a “weak” Lyapunov functional of energy type. The explanation is as follows: the derivative of a “weak” Lyapunov function(al) being only negative semi-definite, asymptotic stability is obtained from the application of the Barbashin-Krasovskii-LaSalle invariance principle. In the case of the neutral functional differential equations the invariance principle holds under the aforementioned property of asymptotic stability for the difference equation associated to it. It is then shown how the introduction of additional dissipation in the model can lead to asymptotic stability for both the difference system and the basic one, described by the boundary value problems for .1D hyperbolic partial differential equations. There are also pointed out connections to other mathematical problems such as dissipative/conservative boundary conditions. Keywords .1D hyperbolic partial differential equations · Neutral functional differential equations · Continuous time difference equations · Stable difference operator · Dissipative terms

V. R˘asvan (B) Romanian Academy of Engineering Sciences ASTR, 26 Dacia Blvd. Bucharest Romania, 010413 Bucuresti, Romania University of Craiova, Department of Automatic Control and Electronics, 13 A.I. Cuza Str. Craiova Romania, 200585 Craiova, Romania e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 S. Olaru et al. (eds.), Difference Equations, Discrete Dynamical Systems and Applications, Springer Proceedings in Mathematics & Statistics 444, https://doi.org/10.1007/978-3-031-51049-6_12

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1 Introduction and Motivation According to our view, the development of the numerical methods for dynamics studies contributed to an extended vision of the difference equations as associated to dynamics “in discrete time” i.e with the independent variable—a sequence of integers. As a consequence, an entire class of dynamical systems applications— described by difference equations in “continuous time” i.e. having a genuine real variable as independent one appeared as—mildly said—left aside. The aim of the present contribution is to fill somehow this gap, starting from some applications leading to such mathematical objects and pointing out the importance of establishing some qualitative problems of the dynamics within their framework. The author’s own experience in choosing the topics cannot be underestimated. The chapter starts with a section dedicated to certain applications leading to difference equations in continuous time. It is worth mentioning here the interesting and comprehensive monograph [65] which contains in nuce (i.e. in a nutshell) almost all motivating applications we follow in this chapter. From the simplest difference equation in continuous time, there are next considered the so called “.1D propagation systems” i.e. described by hyperbolic partial differential equations in one space variable: to their two-point boundary value problems one can associate a system of equations with deviated argument which in several cases turns into a system of difference equations in continuous time. Between these mathematical objects i.e. the aforementioned boundary value problem for .1D propagation equations and the functional equations with deviated argument there is established a one-to-one correspondence between their solutions. In this way any mathematical result obtained for one of these objects is automatically projected back on the other one. Next, the importance of the difference equations in continuous time appears with reference to the class of neutral functional differential equations introduced by J. K. Hale in 1966—see [30]: it appears that several qualitative properties of these equations are related to the similar ones for the difference equations associated to them. Consequently, the next section of this contribution deals with qualitative properties for the difference equations with continuous time, more precisely with stability and stable forced oscillations. Afterwards, these qualitative properties are considered precisely in the context of the same qualitative properties for neutral functional differential equations as appear in the cited publications of Hale; particular reference is made to some motivating applications. The chapter continues with the description of certain applications of the same type but leading to continuous time difference equations in critical cases—i.e. marginally stable—from the point of view of the required properties. The first application comes from the field of oilwell engineering and finally displays an associated continuous time difference equation which is in a critical case from the stability point of view— stable but not asymptotically stable. The second application arises from hydraulic engineering—the water hammer modeling. The importance of the hydraulic losses— often neglected by the hydraulic engineers—is pointed out through the critical stability aspects of the associated difference equations. In fact it is pointed out an engi-

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neering way to remove the critical properties—to improve the mathematical models by taking into account various neglected factors. The paper ends with a section of Conclusions and perspective aspects where the discussion on removing critical properties by re-modeling is completed with suggestions on tackling to critical problems as they are i.e. mathematically, without changing the model.

2 Applications Generating Difference Equations in Continuous Time While several quite recent publications on difference equations in continuous time mention some of their applications in brief in order to motivate the research object, we shall start from a rather well known monograph written by experts in the field [65]. The Introduction of this book, after enumerating various sources of difference equations with either discrete or continuous time, gives the remark that even the simplest nonlinear difference equation in continuous time, namely .

x(t + 1) = f (x(t))

(1)

can generate complex behavior and, moreover, suggests usefulness of the difference equations in continuous time for the modeling of turbulence phenomena—rather complex physical ones. Next, it is mentioned there that “a rich source of difference,.qdifference, differential difference and functional differential equations are the boundary value problems for systems of hyperbolic P(artial) D(ifferential) E(quations)— PDEs”. This mention is followed by the simple example of the Nagumo-Shimura electronic circuit [49], but without capacitor. This simplest application had been discussed from the point of view of the forced oscillations in [54], leading to the difference vector equation in continuous time .

x(t) = Ax(t − τ ) − BΦ(v(t − τ )) + f (t) , v(t) = C ∗ x(t)

(2)

where .v = col{ν1 , . . . , νm } and .Φ(v) = col{ϕ1 (ν1 ), . . . , ϕm (νm )}, .x and . f are .ndimensional vectors and . A, . B, .C are matrices of appropriate dimensions. Also the entries of . f were periodic or almost periodic. In the subsequent papers [27, 55] the problems of absolute stability and stability of the forced oscillations were tackled. Introduction (or occurrence) of such difference equations had been a consequence of the applications of an approach associating them or other functional equations to rather general .1D hyperbolic PDEs with very general boundary conditions. This approach had been followed by A. D. Myshkis with his co-workers [1, 46–48] in USSR (and in Russia) and by US scientists as R. K. Brayton at IBM [6–9] and K. L. Cooke in academia [12, 13]. The mathematical structure examined by Brayton and Cooke was simpler and their development—easier to follow and apply. It is from their result that we shall start this contribution.

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3 A Basic Mathematical Result and Some Applications of It We consider in this section a mathematical result displaying the one-to-one correspondence between the solutions of two mathematical objects: a non-standard boundary value problem for .1D hyperbolic partial differential equations written in the Riemann invariants and an associated system of functional differential equations obtained by considering the Riemann invariants along the characteristics. From this result a natural way of defining equations with deviated argument of retarded, neutral or advanced type is deduced—relying on a simple arithmetic inequality. The result is illustrated by a model arising from thermal power engineering. As mentioned in the previous section, the starting point leading to this direction of analysis was provided by the research of Brayton and Miranker at IBM in the 60’s of the previous century—see the references at the end of the previous section. The mathematical synthesis done by Cooke and Krumme [12, 13] led to the following result, completely proven (i.e. in both “directions” of the equivalence) much later in [57]. We reproduce it after [59]. Consider the following nonstandard boundary value problem with initial and derivative boundary conditions ∂u + ∂u + + τ + (λ, t) = Φ + (λ, t) ∂t ∂λ ∂u − ∂u − + τ − (λ, t) = Φ − (λ, t), 0 ≤ λ ≤ 1, t ≥ t0 , ∂t ∂λ ] m [ ∑ dk dk . ak+ (t) k u + (0, t) + ak− (t) k u − (0, t) = f 0 (t) dt dt k=0 [ ] m ∑ dk dk bk+ (t) k u + (1, t) + bk− (t) k u − (1, t) = f 1 (t) dt dt k=0

(3)

u ± (λ, t0 ) = ω± (λ) , 0 ≤ λ ≤ 1 with .τ + (λ, t) > 0, .τ − (λ, t) < 0. Observe that the two equations for the Riemann invariants .u ± : [0, 1] × [t0 , t1 ] |→ R are decoupled; a coupling is realized through the boundary conditions only. Consider the two families of characteristics .

1 dt = ± , τ + (λ, t) > 0 , τ − (λ, t) < 0 dλ τ (λ, t)

(4)

and let .t ± (σ ; λ, t) the two characteristic curves crossing some point .(λ, t) of the strip .[0, 1] × [t0 , t1 ). Define .

T + (t) := t + (1; 0, t) − t , T − (t) := t − (0; 1, t) − t

(5)

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as propagation times along the characteristics or forward and backward propagation time respectively. We write down the “progressive (forward) wave”.u + (λ, t) along the increasing characteristic .t + (σ ; λ, t)—extendable “to the right” up to .σ = 1, and the “reflected (backward) wave” .u − (λ, t) along the decreasing characteristic .t − (σ ; λ, t) and integrate from .λ to 1 and from .λ to 0 respectively, to obtain +

+

∫

+

u (λ, t) = u (1, t (1; λ, t)) − .

u − (λ, t) = u − (0, t − (0; λ, t)) +

Φ + (σ, t + (σ ; λ, t)) dσ τ + (σ, t + (σ ; λ, t))

1

λ

(6)

∫

λ

0

−

−

Φ (σ, t (σ ; λ, t)) dσ . τ − (σ, t − (σ ; λ, t))

For those cases when .t + (σ ; λ, t) can be extended “to the left” up to .σ = 0 and the decreasing characteristic .t − (σ ; λ, t)—“to the right” up to .σ = 1, (6) becomes +

+

∫

+

1

u (0, t) = u (1, t + T (t)) − 0 .

u − (1, t) = u − (0, t + T − (t)) +

Φ + (σ, t + (σ ; 0, t)) dσ τ + (σ, t + (σ ; 0, t)) (7)

∫

1 0

−

−

Φ (σ, t (σ ; 1, t)) dσ , τ − (σ, t − (σ ; 1, t))

with .T ± (t) defined by (5). In this way (7) define certain functional relations between the boundary values of the two waves. Denoting y + (t) := u + (1, t) , 𝚿 + (t) :=

∫

1

Φ + (σ, t + (σ ; 0, t)) dσ τ + (σ, t + (σ ; 0, t))

1

Φ − (σ, t − (σ ; 1, t)) dσ , τ − (σ, t − (σ ; 1, t))

0 .

y − (t) := u − (0, t) , 𝚿 − (t) :=

∫ 0

(8)

we find that .(y + (t), y − (t)) thus defined satisfy the following system of differential equations with deviated argument ] m [ m ∑ ∑ dk dk dk ak+ (t) k y + (t + T + (t)) + ak− (t) k y − (t) = f 0 (t) + ak+ (t) k 𝚿 + (t) dt dt dt k=0 k=0 .

m [ ∑ k=0

bk+ (t)

] m ∑ dk − dk dk + − − y (t) + b (t) y (t + T (t)) = f (t) − bk− (t) k 𝚿 − (t). 1 k k k dt dt dt k=0 (9)

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Its solutions can be constructed by steps for .t > t0 + max{T − (t0 ), T + (t0 )} provided initial conditions are given; these initial conditions can be obtained by considering those characteristics which cannot be extended on the entire segment .[0, 1] since they cross the axis .t = t0 instead of .σ = 0 for .t + (·; λ, t) or of .σ = 1 for .t − (·; λ, t). If .ω± (λ) are the given initial conditions for (3) i.e .u ± (λ, t0 ) = ω± (λ), then the initial conditions for (9) are + + . y0 (t (1; λ, t0 ))

+

∫

= ω (λ) +

1

λ

Φ + (σ, t + (σ ; λ, t0 )) dσ , τ + (σ, t + (σ ; λ, t0 ))

(10)

where .0 ≤ λ ≤ 1 ⇔ t0 ≤ t + (1; λ, t0 ) ≤ t0 + T + (t0 ) and − − . y0 (t (0; λ, t0 ))

−

∫

λ

= ω (λ) − 0

Φ − (σ, t − (σ ; λ, t0 )) dσ , τ − (σ, t − (σ ; λ, t0 ))

(11)

where .0 ≤ λ ≤ 1 ⇔ t0 ≤ t − (0; λ, t0 ) ≤ t0 + T − (t0 ). Next, the converse relations, suggested by (6), namely +

+

+

u (λ, t) = y (t (1; λ, t)) − .

u − (λ, t) = y − (t − (0; λ, t)) +

∫

1

λ

Φ + (σ, t + (σ ; λ, t)) dσ τ + (σ, t + (σ ; λ, t)) (12)

∫

λ 0

−

−

Φ (σ, t (σ ; λ, t)) dσ , τ − (σ, t − (σ ; λ, t))

may be viewed as representation formulae for the solutions of (3). The following result is true Theorem 1 Consider the boundary value problem (3). If .u ± (λ, t) is a solution satisfying the equations as well as the initial and the boundary conditions, then . y ± (t) defined by (8) are a solution of (9) with the initial conditions defined by (10) and (11). Conversely, let . y ± (t) be a sufficiently smooth solution of (9) with some initial conditions . y ± (t) defined on .t0 ≤ λ ≤ t0 + T ± (t0 ). Then .u ± (λ, t) defined by (12) is a solution of (3) with the initial conditions .ω± (λ) defined also by (12) computed at .t = t0 . Theorem 1 ascertains a one-to-one correspondence between the solutions of two mathematical objects describing some dynamic processes. In this way all properties and mathematical results obtained for one of them are valid for the other. Since (9) are equations with deviated arguments, the following considerations are useful.

On a Class of Applications for Difference Equations in Continuous Time

257

Define the integers L + = max{k : ak+ (t) /= 0} , L − = max{k : bk− (t) /= 0} .

K + = max{k : bk+ (t) /= 0} , K − = max{k : ak− (t) /= 0}

(13)

M = L + + L − − (K + + K − ). According to the sign of . M system (9) belongs to one of the three classes of systems with deviated argument: if . M > 0 it is of delayed type; if . M < 0 it is of advanced type; if . M = 0 it is of neutral type. This assertion follows in a straightforward way from the definitions of [4] and is consistent with the classification of [18, 19, 35]. It has to be mentioned that one of the most striking proof of the usefulness of the difference equations with continuous time arises from this fact—NFDEs (Neutral Functional Differential Equations) are the most frequent result of the aforementioned methodology which should be viewed as a research guide more than a specific result. Turning to NFDEs, the following class of such equations was introduced firstly in [30] (reported earlier at the World Mathematical Congress in Moscow, 1966) .

d D(t)xt = L (t)xt + h(t) dt

(14)

2 where .xt (ϑ) := x(t + ϑ), .−r ≤ ϑ ≤ 0, .h ∈ Lloc (0, ∞; Rn ) and .D(t) : C (−r, 0; n n n n R ) |→ R , .L (t) : C (−r, 0; R ) |→ R being linear operators given by

∫ D(t)φ = φ(0) −

0

.

−r

∫ d[μ(t, ϑ)]φ(ϑ) , L (t)φ =

0

−r

d[η(t, ϑ)]φ(ϑ)

(15)

This representation appeared for the first time in [30] and, clearly, standard linear NFDE can be viewed as belonging to (15). Let us observe that denoting . y(t) := D(t)xt , we obtain from (14) y˙ (t) = L (t)xt + h(t) .

D(t)xt = y(t)

(16)

This is a system of coupled delay differential and difference (in continuous time) equations. This structure was introduced firstly in [8] starting from the theory of electrical circuits with lossless (. LC) transmission linear and in [52, 53] starting from some pioneering work in automatic control, done on the eve of the war in USSR, being published later posthumously by the decision of the journal’s Editorial Board led at that time by Galerkin [33, 34, 66]. A synthesis of these models led to the nonlinear (in the most general case) model of the controlled dynamics in energy co-generation as follows [16]

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ψc Tc ∂t ξρ + ∂λ ξw = 0 ψc Tc ∂t ξw + ∂λ ξρ = 0 ( ) ξw (0, t) = πs (t)Φ ξρ (0, t)/πs (t) √ ξw (1, t) = 2ψs ξρ (1, t) .

ds = απ1 + (1 − α)π2 − νg , 0 < α < 1 dt dπ1 = μ1 (t) − π1 , 0 ≤ μ1 (t) ≤ 1 T1 dt dπs = π1 − μ2 (t)πs − (β/ψc )ξw (0, t), 0 < β < 1 Tp dt dπ2 T2 = μ2 (t)πs − π2 , 1 − β ≤ μ2 (t) ≤ 1, dt Ta

(17)

where the function .Φ(z) describing the subcritical/critical steam flow is defined by ⎧√ 2(1 − z), νcr ≤ z ≤ 1 ⎪ ⎪ ⎨ √ . Φ(z) = 2(1 − νcr ), 0 < z ≤ νcr ⎪ ⎪ ⎩ 0, z≥1

(18)

and the critical pressure ratio .νcr results from the Saint Venant formula or from experimental data. Application of the methodology of [12, 57] will associate to (17) the following nonlinear system of coupled nonlinear delay differential and difference (in continuous time) equations as follows ds = απ1 + (1 − α)π2 − νg dt dπ1 = μ1 (t) − π1 T1 dt dπs = π1 − (μ2 (t) + (β/ψc )g p (y − (t − ψc Tc )/πs (t)))πs Tp . dt dπ2 = μ2 (t)πs − π2 T2 dt y + (t) = y − (t − ψc Tc ) + 2g p (y − (t − ψc Tc )/πs (t))πs (t) Ta

(19)

y − (t) = ρ1 y + (t − ψc Tc ), where we have ρ :=

. 1

1− 1+

√ √

2ψs 2ψs

,

(20)

On a Class of Applications for Difference Equations in Continuous Time

259

the function .g p (x) + x being the inverse mapping of the increasing mapping .z − Φ(z).

4 Qualitative Problems for Difference Equations This chapter deals with two basic qualitative properties of the continuous time difference equations with sector restricted nonlinearities—stability and existence of stable oscillations. They rely on the two standard approaches for such systems: quadratic Lyapunov functional and/or frequency domain stability inequalities of V. M. Popov type. Our first results refer to system (2) and are concerned with absolute stability (since the nonlinear functions .ϕk (νk ) are subject to sector restrictions) and with existence and stability of the forced oscillations. The results were published in [27, 55] and each of them has some specific features. We start with the stability result which reads as follows [55]. Theorem 2 Consider the continuous time difference system with a single sector restricted nonlinearity .

x(t) = Ax(t − τ ) − bϕ(c∗ x(t − τ ))

(21)

where .ϕν 2 ≤ ϕ(ν)ν ≤ ϕν ¯ 2 . Suppose that the following conditions are fulfilled: (i) ¯ such that the matrix . A − bϕ0 c∗ has the eigenvalues inside there exists .ϕ0 ∈ [ϕ, ϕ] the unit disk .D1 ⊂ C; (ii) the following inequality 1 + (ϕ + ϕ) ¯ e γ (z) + ϕ ϕ|γ ¯ (z)|2 ≥ 0

.

(22)

holds for all .|z| = 1, det(z I − A) /= 0; here .γ (z) = c∗ (z I − A)−1 b is the transfer function of the linear block of (21); (iii) the characteristic function 1 ¯ (λ) + γ (σ )] + ϕ ϕγ ¯ (λ)γ (σ ) χ (λ, σ ) = 1 + (ϕ + ϕ)[γ 2

.

(23)

is not identically zero; (iv) either the LHS of (22) is strictly positive or the sector condition (22) is strict. Then the zero solution of (21) is exponentially stable for all functions satisfying the sector condition. It is worth mentioning that this result follows as an absolute stability result from the hyperstability theory of Popov [51] for discrete time systems. This “finite dimensional” character of the approach follows from the single delay structure of (21) by denoting . yk (ϑ) := x(t0 + kτ + ϑ), .−τ ≤ ϑ < 0, and considering the discrete time system ∗ . yk+1 = Ayk − bϕ(c yk ) (24)

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for each fixed .ϑ. This approach still holds for a finite number of rationally dependent delays. Exponential stability follows from the generalization of a K. P. Persidskii theorem, due to Halanay [23], pp. 25–29. The problem of the forced oscillations was settled within a different framework. We recall here firstly the result [52]. Theorem 3 Consider system (2) under the following assumptions: (i) matrix . A has all its eigenvalues inside the unit disk .D1 ⊂ C; (ii) each nonlinear function .ϕk (·) of .Φ(v) is subject to the global Lipschitz condition 0≤

.

ϕk (α) − ϕk (β) ≤ μk , ϕk (0) = 0; α−β

(25)

(iii) there exist .τk ≥ 0 and .δ > 0 such that ∗ ıωτ τ μ−1 − A)−1 B ≥ δ I , ∀ω ∈ [−π/τ, π/τ ]. d + e τd C (e

. d

(26)

Then, for any bounded . f (i.e. with bounded on .R entries) there exists a bounded exponentially stable solution of the system; if . f is periodic, this solution is periodic and if . f is almost periodic, this solution is almost periodic. Here we denoted .τd = diag{τ1 , . . . , τm }, .μd = diag{μ1 , . . . , μm }. Now, the conditions of the theorem allow obtaining suitable inequalities on system’s solutions.At their turn these inequalities allow application of Kurzweil-Halanay theorems concerning invariant manifolds for flows on Banach spaces [24, 41] in order to obtain the conclusion of the theorem. In most recent studies on difference equations in continuous time, the state space is chosen to be .PC (−τ, 0; Rn )—the space of piecewise continuous functions on the finite interval .(−τ, 0). Unfortunately this is not a Banach space but, however, it can be embedded in a Banach space of integrable functions. In our case the naturally occurring Banach space is .L 2 (−τ, 0; Rn ) and this will give (almost)-periodicity in the sense of the metrics of the aforementioned space. In particular, the almost periodic solutions of (2) are .S 2 -functions i.e. they are almost periodic in the sense of Stepanov [2, 14, 15, 40, 44]. To end this discussion, we observe that the aforementioned results on forced oscillations had been obtained for coupled delay differential and difference (in continuous time) equations within either frequency domain inequalities [26] or Lyapunov framework [63]. In what follows we shall give more aspects concerning the applications of the stability for continuous time difference equations.

On a Class of Applications for Difference Equations in Continuous Time

261

5 Stability of the Difference Equations as Necessary in Stability Applications In this section we shall discuss the way continuous time difference are connected to the problem of stability of the neutral functional differential equations to which these difference equations are attached. As it will appear in the sequel, the difference operator is strongly connected to asymptotics and also to compactness of the trajectories and their limit sets. In order to recall some basic motivations, we turn to system (14). Observe that if .D(t)φ = φ(0) then (14) becomes a standard equation of retarded type. This simple remark guided J. K. Hale to develop (or, better said, to adapt) Lyapunov stability theory from retarded to neutral equations provided .D is a stable difference operator. To be more specific, for the autonomous (time invariant) system, with linear difference operator, of the form d D xt = f (xt ) (27) . dt with . f : Ω ⊆ C (−r, 0; Rn ) |→ Rn , .Ω being an open subset of .C (−r, 0; Rn ), and .D stable i.e. with .limt→∞ y(t) = 0 for all solutions of .D yt = 0, theorems of Lyapunov and Razumikhin type were given (Theorems 9.8.1 and 9.8.4 of [31]). Also a theorem of the Barbashin-Krasovskii-LaSalle type (Theorem 9.8.2 of [31]) is given, thus ensuring an instrument for asymptotic stability for the case when the derivative of the Lyapunov functional is only non-positive definite (the so called “weak” Lyapunov ˇ function(al)s after Cetaev). As an illustration of the above considerations we turn to (19). From the very beginning—see the early papers of Hale cited in [31] at its Chap. 9 on NFDE, it appeared that a necessary condition for obtaining i.e. rigorously proving theorems on stability (mainly on asymptotic stability) is asymptotic stability of the difference operator in (27). In particular, Theorem 9.8.2 of [31] contains explicitly this assumption. In other references [37–39] this assumption is slightly modified, but it is not clear if these last assumptions are relaxed or equivalent versions of the one of Hale. To explain this assumption on the difference operator, we shall use the notations of [31], Chapter 9, and reproduce in brief the part tackling difference equations and operators—pp. 274–276. Consider the homogeneous and non-homogeneous difference equations .D yt = 0 , t ≥ 0 ; D yt = h(t) , t ≥ 0, (28) where .h ∈ C ([0, ∞); Rn ) and the difference operator .D : C (−r, 0; Rn ) |→ Rn is continuous and atomic at 0 being thus defined as ∫ Dφ = φ(0) −

0

.

−r

d[μ(θ )]φ(θ ).

(29)

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V. R˘asvan

In (29) the kernel .μ : R |→ Rn×n is measurable, normalized such that .μ(θ ) ≡ 0 for .θ ≥ 0 and .μ(θ ) ≡ μ(−r ) for .θ ≤ −r ; the kernel is continuous from the left and of bounded variation. The following assumption is supposed to hold for the kernel .μ Assumption 1 (Assumption (J) of [31], p. 271) The entries .μi j of .μ have an atom before they become constant i.e. there is a .ti j such that .μi j (t) ≡ μi j (ti j + 0) for .t ≥ ti j and .μi j (ti j − 0) / = μi j (ti j + 0). Here the index .i stands for an entry of .μ ranging e.g. from 1 to .n while the index . j accounts for the jump and may range from 1 to .∞. This assumption is particularly true when .μ(θ ) is reduced to a stepwise function as below p ∑ . y(t) = Ak y(t − rk ) , t ≥ 0. (30) 1

Let .Δ0 (λ) defined below be the characteristic function of (29) (

∫

Δ0 (λ) = det I −

0

.

−r

) e d[μ(θ )] , λθ

(31)

which in the case of (30) reads ( Δ0 (λ) = det I −

p ∑

.

) Ak e

−λrk

.

(32)

1

Let .a D := sup{ e(λ)|Δ0 (λ) = 0}. We state firstly Definition 1 (Definition 9.3.1 of [31], p. 275) Suppose .D is linear, continuous and atomic at 0—see (29). The operator .D is said to be stable if the zero solution of the homogeneous equation of (28) with the initial condition .ψ ∈ C (−r, 0; Rn ) subject to .Dψ = 0 is asymptotically stable The following result [31], p. 275, concerns the aforementioned stability property Theorem 4 (Theorem 9.3.5 of [31], p. 275) The following statements are equivalent (i) .D is (asymptotically) stable in the sense of Definition 1. (ii) .a D < 0. (iii) There exists constants.α > 0 and.γ (α) > 0 such that for any.h ∈ C ([0, ∞); Rn ) any solution of the non-homogeneous equation of (28) satisfies |y(ψ, h)(t)| ≤ γ (α)[|ψ|e−αt + sup |h(s)|].

.

0≤s≤t

(iv) If.D is given by (31) with.lims→0 Var [−s,0] μ = 0 and.μ is also subject to Assumption 1—Assumption (J), then there exists a .δ > 0 such that all roots of the characteristic equation

On a Class of Applications for Difference Equations in Continuous Time

(

∫

Δ0 (λ) := det I −

0

.

−r

λθ

263

)

e d[μ(θ )] = 0

satisfy . e(λ) ≤ −δ < 0 Observe that .(iii) shows that (asymptotic) stability of .D in the sense of Definition 1 is equivalent to exponential stability (the principle of K. P. Persidskii). Therefore stability of (28) mens in fact exponential stability. Moreover, .(iv) shows—along the same line—that .a D < 0 ensures that the roots of the characteristic equation (31) have their real parts well delimited from 0. In fact Theorem 4 states equivalence of (apparently) weak properties with other, stronger ones. Turning to (28), its (asymptotic, exponential) stability is equivalent to the location of the roots of the characteristic equation .Δ0 (λ) = 0 with .Δ0 (λ) given by (31) in the open left half plane .C− . But for difference operators there exists another property called strong stability. This property is introduced also in [31], Sect. 9.6, for difference operators occurring in (30) i.e. defined by D(r, A)φ = φ(0) −

p ∑

.

Ak φ(−rk ).

(33)

1

Observe that the difference operator (33) is a special case of (29) with .μ containing only the stepwise component with a finite number of steps. Let .r = col(r1 , . . . , r p ) be the vector of the delays .rk > 0, .∀k. Definition 2 (Definitions 9.6.1 and 9.6.2 of [31], p. 285) The operator .D(r, A) is p said to be stable locally in the delays if there is an open neighborhood . I (r ) ⊂ R+ of .r such that .D(v, A) is stable in the sense of Definition 1 for each .v ∈ I (r ). The operator .D(r, A) is said to be stable globally in the delays (strongly stable) p if it is stable for each .r ∈ R+ . For strong stability the following result is true Theorem 5 (Theorem 9.6.1 of [31], p. 286) The following statements are equivalent p

(i) For some .r ∈ R+ , .r = col(r1 , . . . , r p ) with .rk > 0 rationally independent, .D(r, A) is stable in the sense of Definition 1. .(ii) If .γ (B) is the spectral radius of a matrix . B, then .γ0 (A) < 1 where .

( ( γ (A) := sup γ

p ∑

. 0

) Ak e

ıθk

) |θk ∈ [0, 2π ), k = 1, 2, . . . p .

(34)

1

(iii) .D(r, A) is stable locally in the delays in the sense of Definition 2. .(iv) .D(r, A) is stable globally in the delays (strongly stable) in the sense of Definition 2.

.

To illustrate the aforementioned assumptions we shall consider the model describing co-generation [16] given here by (17) with the associated system of coupled delay

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V. R˘asvan

differential and difference equations (19). The difference subsystem will look as follows y + (t) = y − (t − ψc Tc ) + 2g p (y − (t − ψc Tc )/πs (t))πs (t) . (35) y − (t) = ρ1 y + (t − ψc Tc ), where, according to [16], we shall have ⎧√ √ √ 2(1 − νcr ) , for − 2(1 − νcr ) ≤ x ≤ νcr − 2(1 − νcr ) ⎪ ⎪ ⎪ ⎨ √ √ . . g p (x) = −1 + 3 − 2x , for νcr − 2(1 − νcr ) ≤ x ≤ 1 ⎪ ⎪ ⎪ ⎩ 0 , for x ≥ 1

(36)

√ with the possibility to extend.g p (x) for.x ≤ − 2(1 − νcr ). This function is a decreasing one. In [16] only stability of the linearized system (35) was discussed. Here we shall consider the nonlinear case. Since the physically significant invariant set of (17)—or of (19)—implies .πs (t) > 0, stability of (35) can be studied—for large .t > 0—by considering the single continuous time difference equation ( ) y + (t) y + (t − 2ψc Tc ) y + (t − 2ψc Tc ) = ρ1 + 2g p ρ1 . . πs (t) πs (t) πs (t)

(37)

We shall have the equilibrium of (37) corresponding to constant .πs (t) ≡ πs0 —see the equilibrium of (19)—or (17) in [16]—given by ( ) y¯ + y¯ + y¯ + . = ρ1 0 + 2g p ρ1 0 ⇒ y¯ − = ρ1 y¯ + . πs0 πs πs

(38)

Introducing the deviations from the equilibrium η+ (t) =

.

y¯ + y + (t) − 0 , ζs (t) = πs (t) − πs0 , πs (t) πs

(39)

we shall obtain the difference equation in deviations η+ (t) = ρ1 η+ (t − 2ψc Tc ) − 2[g p (ρ1 y¯ + /πs0 ) − g p (ρ1 η+ (t))].

.

(40)

Observe that for .πs (t) ≡ πs0 , .η+ (t) will reproduce the dynamics of .(y + (t) − y¯ + ). Let + .ϕ p (ρ1 η (t)) := 2[g p (ρ1 y ¯ + /πs0 ) − g p (ρ1 η+ (t))] (41) and (40) thus becomes η+ (t) = ρ1 η+ (t − 2ψc Tc ) − ϕ p (ρ1 η+ (t − 2ψc Tc )),

.

(42)

On a Class of Applications for Difference Equations in Continuous Time

265

with .ϕ p (·) subject to the sector condition 0 ≤ ϕ p (σ )σ ≤ 2σ 2 .

.

(43)

From now on we can follow the methodology in [55] and apply the frequency domain inequality of Tsypkin 1 1 ρ1 . + e = 2 eıθ − ρ1 2

(

2ρ1 (cos θ − ρ1 ) 1+ 1 − 2ρ1 cos θ + ρ12

) =

1 − ρ12 1 > 0 , ∀θ , · 2 1 − 2ρ1 cos θ + ρ12

(44) since .|ρ1 | < 1 and, indeed (40) hence (37) and, therefore (35) are asymptotically stable. Consequently system (19) and, therefore, (17), can be analyzed for stabilization and stability via a suitable Lyapunov functional even for nonlinear difference system (unlike in [16] where only the linearized case was considered).

6 Applications Leading to Critical Cases in the Stability of the Difference Equations As it could be seen in the previous section, the basic requirement for the difference operator associated to a neutral functional differential equation was its asymptotic stability. In this section we shall point out certain applications leading to critical cases (marginal stability) for the difference operator: dynamics for the control of the oilwell drillstring in the case of distributed parameters and water hammer in hydraulic engineering—under various assumptions on the hydraulic losses. 6.1 In [57] there were emphasized several applications from Mechanical Engineering, modeled by applying the generalized variational principle of Hamilton. Their common feature resulted as follows: application of the Hamilton principle led to a model described by a non-standard BVP for hyperbolic PDEs. We call the BVP non-standard because the boundary conditions are coupled through a kind of internal feedback to certain ODEs (Ordinary Differential Equations). In the particular case of lossless or distortionless propagation (in the sense of [10]), it was possible to associate, as described in Sect. 3, a system of FDEs with deviated argument which turned to be of neutral type. We reproduce here one of them—dealing with the dynamics of the oilwell drillstring—after [57] ρ(s)I p (s)θtt + c(s)θt − (G(s)I p (s)θs )s = 0 , 0 < s < L c θ˙m (t) + G(0)I p (0)θs (0, t) = 0 .

Jm θ¨m (t) + c0 θ˙m (t) = τ (t) − c θ˙ (0, t) Jb θ¨ (L , t) + G(L)I p (L)θs (L , t) = −T (θ˙ (L , t))

(45)

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V. R˘asvan

The stabilizing control being applied at the surface only (.s = 0)—as can be seen from (45)—and consisting of controlling the speed of the driving motor .ωm = θ˙m in order to maintain a constant rotating speed of the shaft .θt (s, t), the closed loop dynamics written in deviations from the steady state will read [57] ρ(s)I p (s)θtt (s, t) + c(s)(θt (s, t) − ω) ¯ − (G(s)I p (s)(θs (s, t) − θ¯s (s)))s = 0 c (θ˙m − ω¯ m ) + G(0)I p (0)(θs (0, t) − θ¯s (0)) = 0 .

¯ =0 Jm θ¨m + c0 (θ˙m − ω¯ m ) + g0 (θ˙m (t) − ω¯ m ) + c (θ˙ (0, t) − ω) ¨ , t) + T (θ(L ˙ , t)) − T (ω) ¯ + G(L)I p (L)(θs (L , t) − θ¯s (L)) = 0, Jb θ(L (46)

where the nonlinear controller defined by the torque dynamics is given by τ (t) − τ¯ = −g0 (θ˙m (t) − ω¯ m ) ,

.

g0 (σ ) > −c0 . σ

(47)

To system (46) we associate a system of NFDEs as follows. Introduce first the new variables ω(s, t) := θt (s, t) − ω¯ , w(s, t) := G(s)I p (s)(θs (s, t) − θ¯s (s)),

.

(48)

leading to the symmetric Friedrichs form for the hyperbolic PDEs, which we write down in the vector matrix form ⎞( ) ⎛ ( ) ( )( ) 1 ω ω 1 0 ω 0 c(s) ⎠ ⎝ ρ(s)I p (s) . − + = 0. (49) w s ρ(s)I p (s) 0 1 w t w G(s)I p (s) 0 √ The eigenvalues of the first matrix occurring in (49) are .±λ(s) = ± G(s)/ρ(s). One of the matrices ensuring the diagonalization and introduction of the Riemann invariants .ω± (s, t) is given by ( .

T (s) =

1

a(s)

) a −1 (s) 1

√ , a(s) = I p (s) ρ(s)G(s)

Consequently the following form of the equations of the BVP (46) is obtained

(50)

On a Class of Applications for Difference Equations in Continuous Time

267

1 a ' (s) − c(s) − · ω 2ρ(s)I p (s) a(s) ] [ ' a (s) + c(s) + 1 ω − c(s)ω− ωt− = λ(s)ωs− + 2ρ(s)I p (s) a(s) 1 − c ωm (t) + [ω (0, t) − a(0)ω+ (0, t)] = 0 2 . [ ] d 1 ω− (0, t) Jm ωm + c0 ωm + g0 (ωm ) + c ω+ (0, t) + =0 dt 2 a(0) ) ] [ ] [ ( − 1 + ω (L , t) 1 d Jb (ω (L , t) + ω− (L , t)/a(L)) + ω¯ − T (ω) ¯ + ω+ (L , t) + + T 2 dt a(L) 2 1 − + [ω (L , t) − a(L)ω+ (L , t)] = 0. 2 ωt+ = −λ(s)ωs+ +

(51) Various special choices of .a(s) and .c(s) can lead to various cases of distortionless/ loss-less propagation, analogous to those tackled in [10] or, more recently, in [56, 60]. We shall restrict, however, for methodological reasons, to the case of the homogeneous materials i.e. constant material parameters, and also neglect the internal distributed damping i.e. choose.c(s) ≡ 0. Since.a ' (s) ≡ 0, the equations of the Riemann invariants are thus decoupled and we can follow the approach in [57] to associate to the resulting (51) the following system of coupled delay differential and (continuous time) difference equations ( ) √ c2 dωm c + c0 + √ ωm + √ η− (t − L ρ/G) = 0 dt I p ρG I p ρG √ √ √ dωb + I p ρGωb + [T (ωb + ω) ¯ − T (ω)] ¯ − I p ρGη+ (t − L ρ/G) = 0 Jb dt . √ 1 η+ (t) = √ (2c ωm (t) + η− (t − L ρ/G)) I p ρG √ √ η− (t) = I p ρG(2ωb (t) − η− (t − L ρ/G)). (52) System (52) is of neutral type (see [52, 53], also the remarks at the end of Chap. 9 in [31]; the substitution of .ωm , .ωb from the difference subsystem in the differential one will show this). The difference operator associated to the difference subsystem is √ √ ( + ) ( )( + ) φ (0) φ (−L ρ/G) 0 (I p ρG)−1 (53) .DΦ = , − √ √ 0 −I p ρG φ − (−L ρ/G) φ − (0) Jm

268

V. R˘asvan

Fig. 1 Hydroelectric plant structure. 1. Lake. 2. Tunnel. 3. Surge tank. 4. Penstock. 5. Hydraulic turbine

the matrix . D having its roots .±ı located on the unit circle of .C. The operator is thus critically stable. In fact the characteristic equation of (53) will be √ 1 + exp(−2λL ρ/G) = 0,

.

(54)

having its roots on .ıR: the real parts are 0 and the imaginary ones are given by θ =

. k

±(2k + 1)π . √ 2L ρ/G

(55)

We shall comment in the sequel this critical stability of the difference operator which occurs in all models arising from Mechanics—see [57, 58]. 6.2 We shall discuss in what follows some interesting features of the models of the water hammer in hydraulic engineering. We shall consider the standard configuration of Fig. 1 containing the lake (water reservoir), the tunnel, the surge tank and the penstock with the hydraulic turbine We shall start from the model with distributed parameters for the two water conduits. Its physical variables are rated to certain reference values in order to reduce the numerical ill conditioning and to make the variables compatible from the point of view of their magnitudes (sizes). For both rating and significance of the variables, see [59, 62]. The model in rated variables reads as follows

On a Class of Applications for Difference Equations in Continuous Time

( ) 1 λi L i Twi 1 Twi 2 Twi ∂t qi + ∂ξi h i + qi + qi |qi | = 0 , i = 1.2 2 Ti 2 Di Ti δi2 Twi ∂t h i + ∂ξi qi = 0 ; h 1 (0, t) ≡ 1 ( ) 1 dz 1 Tw1 1 h 1 (1, t) − R1 = q1 (1, t)|q1 (1, t)| = 1 + z(t) + Rd f d H0 2 T1 2 H0 dt . 1 Tw2 = h 2 (0, t) + R2 |q2 (0, t)| 2 T2 √ dz Ts = q1 (1, t) − q2 (0, t) ; q2 (1, t) = (1 − k) f θ h p (1, t) + kφ dt dφ = q2 (1, t)h 2 (1, t) − νg . Ta dt

269

(56)

This model contains distributed (Darcy-Weisbach) losses, local (lumped) hydraulic losses and also the hydraulic losses through the throttling of the surge tank. The dynamic head (also called convective acceleration) is also taken into account in the equations of the conduits. The equations of the turbine speed controller were not given since we discuss water hammer, when the turbine is shut down i.e. “cut” from the upstream structure; the shutdown condition is given by q (1, t) ≡ 0.

. 2

(57)

For the subsequent analysis, certain assumptions on losses and dynamic heads are needed. We start with the dynamic heads: their space variation is considered negligible in comparison to the space variation of the piezometric heads. This assertion is certified in [3] by recorded exploitation data from several hundreds of hydroelectric plants in former USSR. The same assertion appears in more recent references [11, 32]. Another reference [45] states that the dynamic head can be neglected whenever the flow velocity . Q/F is (much) smaller than the velocity of the water wave propagation. Next, the losses through the throttling of the surge tank are quite small and, moreover, surge tanks with throttling are rather few throughout the world; usually these losses are added to the local hydraulic ones. Finally, the remaining losses are also neglected (according to the analyzed cases) or taken into account, relying on the fact that neglecting the losses is covering from the engineering point of view since what matters is the quenching of the water mass oscillations during water hammer. 6.2.1 In what follows the aforementioned aspects will appear clearer. We start by neglecting all losses as well as the dynamic heads thus obtaining a linear model as follows Twi ∂t qi + ∂ξi h i = 0 , δi2 Twi ∂t h i + ∂ξi qi = 0 ; i = 1, 2 .

h 1 (0, t) ≡ 1 , h 1 (1, t) = 1 + z(t) = h 2 (0, t) dz Ts = q1 (1, t) − q2 (0, t) ; q2 (1, t) = 0. dt

(58)

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We check firstly the steady state (equilibrium) after the water hammer. h¯ i (ξi ) ≡ const , h¯ i ≡ 1 ; i = 1, 2 q¯i (ξi ) ≡ const , q¯i ≡ 0 ; i = 1, 2 ; z¯ = 0.

.

Introduce the deviations from the equilibrium: .qi (ξi , t) are already deviations (from 0) as well as z(t); let .χi (ξi , t) := h i (ξi , t) − 1 to obtain from (58) Twi ∂t qi + ∂ξi χi = 0 , δi2 Twi ∂t χi + ∂ξi qi = 0 ; i = 1, 2 .

χ1 (0, t) ≡ 0 , χ1 (1, t) = z(t) = χ2 (0, t)

(59)

dz = q1 (1, t) − q2 (0, t) ; q2 (1, t) = 0. Ts dt To (59) we associate the energy identities, where also we take into account the boundary conditions 1 d Tw1 2 dt .

1 d Tw2 2 dt

∫

1 0

∫

1 0

[q12 (ξ1 , t) + δ12 χ12 (ξ1 , t)]dξ1 + χ1 (1, t)q1 (1, t) ≡ 0, (60) [q22 (ξ2 , t) + δ22 χ22 (ξ2 , t)]dξ2 − χ2 (0, t)q2 (0, t) ≡ 0.

Consider the Lyapunov functional suggested by the energy identities, written as a state functional on .R × L 2 (0, 1; R4 ) 1 .V (z, φi (·), ψi (·)) = 2

( Ts z + 2

2 ∑ 1

∫ Twi 0

1

) [φi2 (ξi )

+

δi2 ψi2 (ξi )]dξi

.

(61)

Its derivative along the solutions of (59) will give .

d dV ★ = V (z(t), qi (·, t), χi (·, t)) ≡ 0. dt dt

(62)

With respect to the energy-based Lyapunov functional, a lossless system appears to be conservative hence its Lyapunov stability in the metrics of the Lyapunov functional itself is ensured. Physically speaking, it is known that stability in this case is non-asymptotic. We check however this aspect within the Lyapunov stability theory. Therefore we consider the application of the Barbashin-Krasovskii-LaSalle invariance principle. The invariance principle being well established for functional differential equations with deviated arguments [31], we follow the procedure of [57] and associate to (59) the following system with deviated argument

On a Class of Applications for Difference Equations in Continuous Time

dz = −(δ1 + δ2 )z(t) − 2w1− (t − 2δ1 Tw1 ) + 2w2+ (t − 2δ2 Tw2 ) dt . w − (t) − w − (t − 2δ T ) = δ z(t) 1 w1 1 1 1

271

Ts

(63)

w2+ (t) + w2+ (t − 2δ2 Tw2 ) = δ2 z(t). Based on the representation formulae qi (ξi , t) = wi+ (t − δi Twi ξi ) − wi− (t + δi Twi (ξi − 1)), .

χi (ξi , t) =

1 + [w (t − δi Twi ξi ) + wi− (t + δi Twi (ξi − 1))], δi i

(64)

the Lyapunov functional is expressed as follows V (z(t), w1− (t + ·), w2+ (t + ·)) = .

1 + δ2

∫

0

−2δ2 Tw2

w2+ (t

1 1 Ts z(t)2 + 2 δ1

∫

0

−2δ1 Tw1

w1− (t + ϑ)2 dϑ+ (65)

+ ϑ) dϑ, 2

with its derivative identically 0. The largest invariant set of (63) contained in the set where the derivative of the Lyapunov functional vanishes—in fact the entire state space—is given by the zero solution of (63). The application of the invariance principle would give asymptotic stability for (65) provided the difference operator of the difference subsystem of (63) is asymptotically stable. This subsystem displays two difference equations in continuous time, with two distinct delays, which are decoupled. Therefore its characteristic equation is a factorized quasi-polynomial given by )( ) ( −2δ1 Tw1 λ 1 + e−2δ2 Tw2 λ . .π(λ) = 1 − e (66) Its roots are given by λ1 =

. k

(2k + 1)π kπ , λ2k = ; k = 0, ±1, ±2, . . ., δ1 Tw1 2δ2 Tw2

(67)

hence the difference system is in a critical case and asymptotic stability of the difference system, also of the system of functional differential equations is not achievable. Neither is that of the BVP for the hyperbolic PDEs (59). 6.2.2 The next application is to consider in (58) the losses of the surge tank throttling in its linear version i.e. adding to (58) the term described by . f d (x) ≡ x. The following linear equations are obtained as in [25, 50]

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Twi ∂t qi + ∂ξi h i = 0 , δi2 Twi ∂t h i + ∂ξi qi = 0 ; i = 1, 2 .

h 1 (0, t) ≡ 1 , h 1 (1, t) = 1 + z(t) + Rd Ts

dz = h 2 (0, t) dt

(68)

dz = q1 (1, t) − q2 (0, t) ; q2 (1, t) = 0. dt

This case has been discussed in several contributions but the most complete is in [62]. The equilibrium is the same as in the previous case while the equations in deviations are obtained from (68) as follows Twi ∂t qi + ∂ξi χi = 0 , δi2 Twi ∂t χi + ∂ξi qi = 0 ; i = 1, 2 .

χ1 (0, t) ≡ 0 , χ1 (1, t) = z(t) + Rd Ts

dz = χ2 (0, t) dt

(69)

dz = q1 (1, t) − q2 (0, t) ; q2 (1, t) = 0 dt

We associate the same Lyapunov functional (61) suggested by the energy identities (60). Differentiating (61) along the solutions of (69) will lead to Rd dV ★ =− . dt Ts

(

dz dt

)2 ≤ 0,

(70)

hence Lyapunov stability in the metrics of the Lyapunov functional itself is obtained. For the asymptotic stability, application of the Barbashin Krasovskii LaSalle invariance principle is required. Consequently we shall take the approach of [57] and, as in [62], we shall associate a system of functional differential equations with deviated argument. Introducing the Riemann invariants by qi (ξi , t) = ri+ (ξi , t) − ri− (ξi , t), .

χi (ξi , t) =

1 + [r (ξi , t) − ri− (ξi , t)], δi i

(71)

considering them along the characteristics and introducing the functions . yi± (t) as yi+ (t) := ri+ (1, t) ⇒ ri+ (0, t) = yi+ (t + δi Twi ), .

yi− (t) := ri− (0, t) ⇒ ri− (1, t) = yi− (t + δi Twi ),

(72)

we define further .wi± (t) := yi± (t + δi Twi ) to obtain the system of functional differential equations with deviated argument

On a Class of Applications for Difference Equations in Continuous Time

dz = w1+ (t − δ1 Tw1 ) − w1− (t) − w2+ (t) + w2− (t − δ2 Tw2 ) dt w1+ (t) + w1− (t − δ1 Tw1 ) = 0 ( ) dz . w − (t) + w + (t − δ T ) = δ 1 w1 1 z(t) + Rd 1 1 dt ( ) dz + − w2 (t) + w2 (t − δ2 Tw2 ) = δ2 z(t) + Rd dt − + w2 (t) − w2 (t − δ2 Tw2 ) = 0.

273

Ts

(73)

Since stability is discussed for large .t > 0, the variables .w1+ (t) and .w2− (t) can be eliminated for .t > max{δ1 Tw1 , δ2 Tw2 }; giving the system the form suitable for the method of steps we obtain dz = −(δ1 + δ2 )z − 2w1− (t − 2δ1 Tw1 ) + 2w2+ (t − 2δ2 Tw2 ) dt ) ( dz − − z(t) + Rd . w1 (t) = w1 (t − 2δ1 Tw1 ) + δ1 dt ( ) dz . w2+ (t) = −w2+ (t − 2δ2 Tw2 ) + δ2 z(t) + Rd dt (74) The construction of the solution by steps is obvious. Re-writing the Lyapunov functional and its derivative goes as in the previous case; for the functional, (65) holds, while for the derivative the equality (62) is modified by adding the term in .dz/dt [Ts + (δ1 + δ2 )Rd ]

Rd dV ★ =− . dt Ts

(

dz dt

)2 ≤0

(75)

(see (70)). The derivative is thus non-positive definite; it is zero provided .dz/dt = 0. Therefore .z(t) ≡ const and also .

z(t) =

2 (w+ (t − 2δ2 Tw2 ) − w1− (t − δ1 Tw1 )), δ1 + δ2 2

which is substituted in the difference subsystem of (73) together with .dz/dt = 0. We obtain δ2 − δ1 − 2δ1 w1 (t − 2δ1 Tw1 ) + w+ (t − 2δ2 Tw2 ) δ1 + δ2 δ1 + δ2 2 . δ2 − δ1 + 2δ2 w2+ (t) = − w− (t − 2δ1 Tw1 ) + w (t − 2δ2 Tw2 ). δ1 + δ2 1 δ1 + δ2 2 w1− (t) =

(76)

The invariant set contained in the set where the derivative of the Lyapunov functional vanishes is composed of constant solutions of (76) and of .z(t) ≡ const. But the

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only constant solution of (76) is zero hence .z(t) ≡ 0. Application of the Barbashin Krasovskii LaSalle invariance principle would give asymptotic stability provided the difference subsystem of (74) were asymptotically stable. Taking simply .z(t) ≡ 0 in (74) would show the same result as for (63)—see (66) and (67). However, (74) contains the derivative .dz/dt, accounting for the dynamic losses introduced by the throttling of the surge tank and this will modify the difference subsystem. Indeed, if .dz/dt is taken from the differential of (74) and some manipulation is done, system (74) becomes dz = −(δ1 + δ2 )z(t) − 2w1− (t − 2δ1 Tw1 ) + 2w2+ (t − 2δ2 Tw2 ) dt (1 + (δ1 + δ2 )Rd' )w1− (t) = δ1 z(t) + (1 + (δ2 − δ1 )Rd' )w1− (t − 2δ1 Tw1 )+

(1 + (δ1 + δ2 )Rd' )Ts

+2δ1 Rd' w2+ (t − 2δ2 Tw2 )

.

(1 + (δ1 + δ2 )Rd' )w2+ (t) = δ2 z(t) − 2δ2 Rd' w1− (t − 2δ1 Tw1 )− −(1 + (δ1 − δ2 )Rd' )w2+ (t − 2δ2 Tw2 ),

(77) where we denoted . Rd' := Rd /Ts . The asymptotic stability of the difference subsystem of (77) was thoroughly and rigorously studied in [59, 62]. We list here the results which are expressed in terms of the delay ration .ν = (δ1 Tw1 )(δ2 Tw2 )−1 . The asymptotic stability of the difference subsystem of (77) is given in the following Theorem 6 The necessary and sufficient condition for the difference system in continuous time w1− (t) = .

2δ1 Rd' 1 − (δ1 − δ2 )Rd' − w (t − 2δ T ) + w+ (t − 2δ2 Tw2 ) 1 w1 1 + (δ1 + δ2 )Rd' 1 1 + (δ1 + δ2 )Rd' 2

1 + (δ1 − δ2 )Rd' + 2δ2 Rd' − w (t − 2δ2 Tw2 ) ' w1 (t − 2δ1 Tw1 ) − 1 + (δ1 + δ2 )Rd 1 + (δ1 + δ2 )Rd' 2 (78) is that the roots of the equation w2+ (t) = −

(s ν − ρ1 )(s + ρ2 ) + (1 − ρ1 )(1 − ρ2 ) = 0

.

(79)

be located inside the unit disk .D1 ⊂ C i.e. have .|s| < 1. This condition is fulfilled iff .ν is rational and its both numerator and denominator are odd numbers. If the numerator is even and the denominator is odd or .ν is irrational, (79) will have at least a single simple root with .|s| = 1 i.e. located on the unit circle .C1 ⊂ C. Here we denoted 1 − (δ1 − δ2 )Rd' 1 + (δ1 − δ2 )Rd' . .ρ1 = ' , ρ2 = 1 + (δ1 + δ2 )Rd 1 + (δ1 + δ2 )Rd' In [61] we called this type of stability—fragile.

On a Class of Applications for Difference Equations in Continuous Time

275

6.2.3 We shall consider now a third case, occurring from another engineering inference adopted in hydraulic engineering: to “lump” the Darcy Weisbach distributed losses together with the local hydraulic losses. In this way the losses through the throttling of the surge tank become negligible and will be indeed neglected in the following, together with the dynamic heads. Consequently we deduce the following model from (56) Twi ∂t qi + ∂ξi h i = 0 δi2 Twi ∂t h i + ∂ξi qi = 0 , i = 1, 2 ; h 1 (0, t) ≡ 1 ) ( λ1 L 1 Tw1 1 R1 + q1 (1, t)|q1 (1, t)| = 1 + z(t) = h 1 (1, t) − . 2 D1 T ( )1 1 λ2 L 2 Tw2 R2 + q2 (0, t)|q2 (0, t)| = h 2 (0, t) + 2 D2 T2 dz = q1 (1, t) − q2 (0, t) ; q2 (1, t) ≡ 0. Ts dt

(80)

Observe that only the boundary conditions have been modified in the sense of their dissipativeness (we use here this notion in the sense of Physics). We compute firstly the steady state given by q¯i ≡ const ; q¯2 (1) = q¯2 (0) = 0 , q¯1 (1) = q¯2 = 0, .

h¯ i (ξi ) ≡ const ; h¯ 1 (0) = 1 = h¯ 1 (1) = h¯ 2 (0) , z¯ = 0.

Introducing the deviations of the piezometric heads .χi (ξi , t) := h i (ξi , t) − 1, the model in deviations is obtained Twi ∂t qi + ∂ξi χi = 0 δi2 Twi ∂t χi + ∂ξi qi = 0 , i = 1, 2 ; χ1 (0, t) ≡ 0 ) ( 1 λ1 L 1 Tw1 χ1 (1, t) − R1 + q1 (1, t)|q1 (1, t)| = z(t) = . 2 D1 T1 ) ( λ2 L 2 Tw2 1 R2 + q2 (0, t)|q2 (0, t)| = χ2 (0, t) + 2 D2 T2 dz = q1 (1, t) − q2 (0, t) ; q2 (1, t) ≡ 0. Ts dt

(81)

We can associate the same Lyapunov functional as in the previous cases i.e. defined as a state functional in (61). Its derivative along the solutions of (81) will be now as follows .

dV ★ d = V (z(t), qi (·, t), χi (·, t)) = −R1' q12 (1, t)|q1 (1, t)| − R2' q22 (0, t)|q2 (0, t)| ≤ 0, dt dt

(82)

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V. R˘asvan

where we denoted .

Ri' :=

1 2

( Ri +

λi L i Di

)

Twi , i = 1, 2. Ti

Again, in order to apply the Barbashin-Krasovskii-LaSalle invariance principle, we shall associate to (81) a system of functional differential with deviated argument, as in the previous cases. Formulae (71)–(72) are valid also. The boundary conditions .χ1 (0, t) ≡ 0, .q2 (1, t) ≡ 0 will give as previously w1+ (t) + w1− (t − δ1 Tw1 ) = 0 , w2− (t) − w2+ (t − δ2 Tw2 ) = 0,

.

thus allowing elimination of .w1+ (t) and .w2− (t) for .t > max{δ1 Tw1 , δ2 Tw2 }. Consequently the following system with deviated argument is obtained, after some additional manipulation - required for giving to the system the form for the construction of the solution by steps dz = −g1 (δ1 z(t) + 2w1− (t − 2δ1 Tw1 )) − g2 (δ2 z(t) + 2w2+ (t − 2δ2 Tw2 )) dt . w − (t) = −w − (t − 2δ T ) + g (δ z(t) + 2w − (t − 2δ T )) (83) 1 w1 1 1 1 w1 1 1 1 Ts

w2+ (t) = w2+ (t − 2δ2 Tw2 ) + g2 (δ2 z(t) + 2w2+ (t − 2δ2 Tw2 )), where the following functions have been introduced g (x) =

. i

1+

√

2x 1 + 4δi Ri' |x|

, i = 1, 2.

(84)

These functions are subject to the following sector conditions 0≤

.

gi (x) ≤ 1. x

(85)

The Lyapunov functional (61), written along the solutions of (83) reads as (65) while its derivative is given by dV ★ = −R1' (w1− (t) + w1− (t − 2δ1 Tw1 ))2 |w1− (t) + w1− (t − 2δ1 Tw1 )|− . dt −R2' (w2+ (t) − w2+ (t − 2δ2 Tw2 ))2 |w2+ (t) − w2+ (t − 2δ2 Tw2 )| ≤ 0.

(86)

On the set where this derivative vanishes we shall have .z(t) ≡ const and .gi (·) = 0, i = 1, 2. Consequently .w1− (·) and .w2+ (·) should be constant, subject to

.

δ z¯ + 2w¯ 1− = δ2 z¯ + 2w¯ 2+ = 0.

. 1

On a Class of Applications for Difference Equations in Continuous Time

277

At the same time the difference subsystem of (83) will give .w¯ 1− = w¯ 2+ = 0 hence .z ¯ = 0. Application of the Barbashin-Krasovskii-LaSalle invariance principle will give again asymptotic stability provided the difference subsystem is asymptotically stable. This subsystem is composed of two independent (for .z(t) ≡ 0) difference equations in continuous time. Consider the first of them which reads w1− (t) = −w1− (t − 2δ1 Tw1 ) + g1 (2w1− (t − 2δ1 Tw1 ))

.

1 Let.ηk+1 (σ ) := −w1− (2(k + 1)δ1 Tw1 + σ ),.−2δ1 Tw1 ≤ σ < 0. Consequently the continuous time difference equation is “sent” in discrete time as

η1

. k+1

= −ηk1 − g1 (ηk1 ).

(87)

Since .g1 is subject to the sector condition (85), we can apply the methods of the absolute stability theory. Consider for instance the frequency domain inequality of Tsypkin, see e.g. [28] 1 + e γ (z) > 0 , ∀|z| = 1, (88) . ϕ¯ where .γ (z) is the transfer function of the linear part of (87) namely η1

. k+1

= −ηk1 + μk , νk = 2ηk1 ⇒ γ (z) =

ν˜ (z) 2 = . μ(z) ˜ z+1

(89)

Having .ϕ¯ = 1 and taking .z = eıθ , .−π ≤ θ < π, the frequency domain condition (88) reads 1+2 e

.

1 + cos θ 1 =1+2 > 0 , ∀θ ∈ [−π, π ) eıθ + 1 (1 + cos θ )2 + sin2 θ

and (87) has global asymptotic stability for all nonlinear functions subject to (85) i.e. not only for .g1 (·) as given by (84). In this way global asymptotic stability is proven for (83) and, making use of the representation formulae (64), for (81). To conclude this section, we have emphasized not only the role of the continuous time difference subsystem in the stability analysis, but also how it is possible to “make” it stable by introducing energy dissipation terms in the basic model.

7 Conclusions and Perspective Research 7.1 Throughout the paper there were tackled stability problems for continuous time difference equations arising from non-standard boundary value problems for .1D hyperbolic partial differential equations. Our basic reference was our own research on engineering applications tackled starting from the pioneering papers of Myshkis

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and his co-workers [1, 48] as well as from those of Cooke and his co-worker [12, 13]. However, the approach appears to be even older, relying in fact on the method of d’Alembert for hyperbolic .1D equations. In [65] there is mentioned another pioneering and seminal paper of Vitt [67]. The aforementioned book itself, together with its references is an invaluable source of both problems and approaches for continuous time difference equations. The reader is sent to its Sect. 4 “Boundary value problems for systems of hyperbolic partial differential equations”. We mention but a few of the studied questions: systems with small parameters, stability in Hausdorff metrics, stability in Skorokhod metrics and asymptotic periodicity, the case of .2D hyperbolic partial differential equations. Another valuable reference of both problems and approaches is [36]. 7.2 An open problem—from the mathematical point of view—is the qualitative behavior and stability in those systems described by neutral functional differential equations with their difference operator (associated to a continuous time difference equation) only marginally stable. We solved this problem via the introduction of energy dissipators—the case of water hammer stability. However, from the methodological point of view the introduction of new terms signifies adopting a new model. This aspect points to a property tackled in [20], pp. 154–163—the dissipative boundary conditions. It is stated there that a hyperbolic system can be reduced to its canonical form in such a way that its boundary conditions become dissipative. Is here any connection to continuous time difference equations which are at least stable if not asymptotically stable? If we remain at the initial model to continue mathematical studies (Hic Rhodus, hic salta!), then we have to consider Lemma 1.7.1 in [31] which we reproduce below Lemma 1 (Lemma 1.7.1 of [31]) Consider the scalar neutral functional differential equation d (y(t) − dy(t − r )) = ay(t) + by(t − r ), . (90) dt with its characteristic equation λ(1 − e−λr ) − a − be−λr = 0.

.

(91)

There exists .α ∈ R such that all roots of (91) are subject to . e(λ) < α. If .d /= 0, all solutions of (91) lie in a vertical strip .β < e(λ) < α in .C. If .d /= 0 and there is a sequence .{λ j } j of roots of (91) such that .|λ j | → ∞ as . j → ∞, then there exists a sequence .{λ'j } j of roots of −λr .1 − de =0 (92) such that .λ j − λ'j → 0 for . j → ∞. Moreover, there exists such a sequence .{λ j } j whenever .d /= 0. This sends to some old papers where marginally stable difference operators were present [21, 22]. Also here there can be useful the work on asymptotic root distri-

On a Class of Applications for Difference Equations in Continuous Time

279

bution of the quasi-polynomials and entire functions [5, 17, 42, 43, 68, 69]. And we restart here our conjecture Conjecture 1 Larger is the modulus of the eigenvalues approaching .ıR (in the line of Lemma 1), smaller is the modulus of the associated mode of the system. In this way the weakly damped oscillations are “filtered” by system’s dynamics. In the same line, dealing with given mathematical models, we must mention the suggestions in [64]. We start from a comment at p. 341. It is mentioned there that the assumption on the asymptotic stability for the difference operator is necessary to obtain pre-compactness of the positive orbits whenever the solution is bounded. It is suggested there to embed the resulting semi-dynamical system in a space wherein the positive orbits are pre-compact. To illustrate this suggestion, the reader is sent to an application in Chap. V, Sect. 4, p. 252. Interesting enough, the application there is a boundary value problem for a hyperbolic partial differential equation. With the one-to-one correspondence between the solutions of the boundary value problem for the hyperbolic partial differential equation and those of the associated system of neutral functional differential equations, the problem becomes one of choosing the state space for the neutral functional differential equations—other than .C —see [29]. To summarize, we consider the applications leading to their associated continuous time difference equations as an interesting, evergreen field generating research problems and stimulating research approaches, both old and new.

References 1. Abolinia, V.E., Myshkis, A.D.: Mixed problem for an almost linear hyperbolic system in the plane (in Russian). Mat. Sbornik 50(92)(4), 423–442 (1960) 2. Amerio, L., Prouse, G.: Almost-Periodic Functions and Functional Equations. The University Series in Higher Mathematics. Van Nostrand Reinhold Company, New York (1971) 3. Aronovich, G.V., Kartvelishvili, N.A., Lyubimtsev, Y.K.: Hydraulic Shock and Surge Tanks. Nauka, Moscow USSR (1968). (in Russian) 4. Bellman, R.E., Cooke, K.L.: Differential Difference Equations. Mathematics in Science and Engineering, vol. 6. Academic, New York (1963) 5. Boas, R.P.: Entire Functions. Pure and Applied Mathematics, vol. 5. Academic, New York (1954) 6. Brayton, R.K.: Bifurcation of periodic solutions in a nonlinear difference-differential equation of neutral type. Quart. Appl. Math. XXIV(3), 215–224 (1966) 7. Brayton, R.K.: Nonlinear oscillations in a distributed network. Quart. Appl. Math. XXIV(4), 289–301 (1967) 8. Brayton, R.K.: Small-signal stability criterion for electrical networks containing lossless transmission lines. IBM J. Res. Develop 12(6), 431–440 (1968) 9. Brayton, R.K., Miranker, W.L.: A stability theory for nonlinear mixed initial boundary value problem. Arch. Rat. Mech. Anal. 17, 358–376 (1964) 10. Burke, V., Duffin, R.J., Hazony, D.: Distortionless wave propagation in inhomogeneous media and transmission lines. Quart. Appl. Math. XXXIV(2), 183–194 (1976) 11. Chaudhry, M.H.: Applied Hydraulic Transients. Springer, New York (2014) 12. Cooke, K.L.: A linear mixed problem with derivative boundary conditions. In: Sweet, D., Yorke, J. (eds.) Seminar on Differential Equations and Dynamical Systems (III), Lecture Notes, vol. 51, pp. 11–17. University of Maryland, College Park (1970)

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13. Cooke, K.L., Krumme, D.W.: Differential-difference equations and nonlinear initial-boundary value problems for linear hyperbolic partial differential equations. J. Math. Anal. Appl. 24, 372–387 (1968) 14. Corduneanu, C.: Almost Periodic Functions. Chelsea Publishing House, New York (1989) 15. Corduneanu, C.: Almost Periodic Oscillations and Waves. Springer (2007) 16. Danciu, D., Popescu, D., R˘asvan, V.: Control of a time delay system arising from linearized conservation laws. IEEE Access 7, 48524–48542 (2019) 17. Dickson, D.G.: Expansions in series of solutions of linear difference- differential and infinite order differential equations with constant coefficients. No. 23 in Mem. Amer. Math. Soc. AMS Publications, Providence RI USA (1957) 18. El’sgol’ts, L.E.: Qualitative Methods in Mathematical Analysis (Russian). Gostekhizdat, Moscow USSR (1955) 19. El’sgol’ts, L.E., Norkin, S.B.: Introduction to the Theory of Differential Equations with Deviated Argument (Russian). Nauka, Moscow USSR (1971). (English version by Academic Press, New York 1973) 20. Godounov, S.K.: Équations de la physique mathématique. Éditions Mir, Moscow USSR (1973) 21. Gromova, P.S.: Stability of solutions of nonlinear equations of neutral type in an asymptotically critical case (in Russian). Matem. zametki 1(6), 715–726 (1967). (English version in Math. Notes of the Acad. Sci. USSR vol. 1, pp. 472–479) 22. Gromova, P.S., Zverkin, A.M.: About the trigonometric series whose sum is a continuous unbounded on the real axis function - solution of an equation with deviated argument (in Russian). Differ. uravnenya 4(10), 1774–1784 (1968) 23. Halanay, A.: Differential Equations. Stability. Oscillations. Time Lags. Mathematics in Science and Engineering, vol. 23. Academic, New York (1966) 24. Halanay, A.: Invariant manifolds for systems with time lag. In: Hale, J.K., LaSalle, J.P. (eds.) Differential and Dynamical Systems, pp. 199–213. Academic, New York (1967) 25. Halanay, A., Popescu, M.: Une propriété arithmétique dans l’analyse du comportement d’un système hydraulique comprenant une chambre d’équilibre avec étranglement. C. R. Acad. Sci. Paris 305, 1227–1230 (1987) 26. Halanay, A., R˘asvan, V.: Periodic and almost periodic solutions for a class of systems described by coupled delay-differential and difference equations. Nonlinear Anal. Theory, Methods Appl. 1, 197–206 (1977) 27. Halanay, A., R˘asvan, V.: Frequency domain conditions for forced oscillations in difference systems. Rev. Roum. Sci. Techn.-Electrot. Energ. 24(1), 141–148 (1979) 28. Halanay, A., R˘asvan, V.: Stability and Stable Oscillations in Discrete Time Systems. Advances in Discrete Mathematics and Applications, vol. 2. CRC Press. Taylor & Francis Group, Boca Raton (2000) 29. Hale, J.K.: Dynamical systems and stability. Journ. Math. Anal. Appl. 26(1), 39–59 (1969) 30. Hale, J.K., Meyer, K.R.: A Class of Functional Equations of Neutral Type. Memoirs of AMS, vol. 76. American Mathematical Society, Providence (1967) 31. Hale, J.K., Verduyn Lunel, S.: Introduction to Functional Differential Equations. Applied Mathematical Sciences, vol. 99. Springer International Edition (1993) 32. Jaeger, C.: Fluid Transients in Hydro-Electric Engineering Practice. Blackie, Glasgow (1977) 33. Kabakov, I.P.: Concerning the control process for the steam pressure (in Russian). Inzh. sbornik 2, 27–60 (1946) 34. Kabakov, I.P., Sokolov, A.A.: Influence of the hydraulic shock on the process of steam turbine speed control (in Russian). Inzh. sbornik 2, 61–76 (1946) 35. Kamenskii, G.A.: On the general theory of the equations with deviated argument (Russian). Dokl. AN SSSR 120(4), 697–700 (1958) 36. Kolesov, A.Y., Mishchenko, E.F., Rozov, N.K.: Asymptotic methods of investigation of periodic solutions of nonlinear hyperbolic equations. Proc. Steklov Inst. Math. 222(3), 3–188 (1998) 37. Kolmanovskii, V.B., Myshkis, A.D.: Applied Theory of Functional Differential Equations. Mathematics and Its Applications (Soviet Series), vo. 85. Springer Science+Business Media (1992)

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38. Kolmanovskii, V.B., Myshkis, A.D.: Introduction to the Theory and Applications of Functional Differential Equations. Mathematics and Its Applications, vol. 463. Kluwer Academic Publishers, Dordrecht (1999) 39. Kolmanovskii, V.B., Nosov, V.R.: Stability of Functional Differential Equations. Mathematics in Science and Engineering, vol. 180. Academic, New York (1986) 40. Krasnosel’skii, M.A., Burd, V.S., Kolesov, Yu.S.: Nonlinear Almost-Periodic Oscillations (in Russian). Nonlinear Analysis and Its Applications. Nauka, Moscow USSR (1970) 41. Kurzweil, J.: Invariant manifolds for flows. In: Hale, J.K., LaSalle, J.P. (eds.) Differential and Dynamical Systems, pp. 431–468. Academic, New York (1967) 42. Langer, R.E.: On the zeros of exponential sums and integrals. Bull. Amer. Math. Soc. 37, 213–239 (1931) 43. Levin, B.Y.: Zeros Distribution for the Entire Functions (in Russian), p. 1964. Gostekhizdat, Moscow USSR: (English version by American Mathematical Society, Providence, 1956) 44. Levitan, B.M., Zhikov, V.V.: Almost-Periodic Functions and Differential Equations. Moscow State University Publishing House, Moscow USSR (1978). (in Russian) 45. Mambretti, S.: Water Hammer Simulations. WIT Press, Southampton (2014) 46. Myshkis, A.D., Filimonov, A.M.: Continuous solutions of quasi-linear hyperbolic systems with two independent variables (in Russian). Differ. Uravn. 17(3), 488–500 (1981) 47. Myshkis, A.D., Filimonov, A.M.: On the global continuous solvability of the mixed problem for one-dimensional hyperbolic systems of quasilinear equations (in Russian). Differ. Uravn. 44(3), 394–407 (2008) 48. Myshkis, A.D., Shlopak, A.S.: Mixed problem for systems of differential-functional equations with partial derivatives and Volterra type operators (in Russian). Mat. Sbornik 41(83)(2), 239– 256 (1957) 49. Nagumo, J., Shimura, M.: Self-oscillation in a transmission line with a tunnel diode. Proc. IRE 49(8), 1281–1291 (1961). https://doi.org/10.1109/JRPROC.1961.287920 50. Popescu, M.: Hydroelectric Plants and Pumping Stations (in Romanian). Editura Universitar˘a, Bucharest (2008) 51. Popov, V.M.: Hyperstability of Control Systems. Grundlehren der mathematischen Wissenschaften, vol. 204. Springer International Edition (1973) 52. R˘asvan, V.: Absolute stability of a class of control systems described by coupled delaydifferential and difference equations. Rev. Roum. Sci. Techn.-Electrot. Energ. 18(2), 329–346 (1973) 53. R˘asvan, V.: Absolute stability of a class of control systems described by functional differential equations of neutral type. In: Jannsens, P., Mawhin, J., Rouche, N. (ed.) Equations différentielles et fonctionnelles non linéaires, pp. 381–396. Editions Hermann, Paris (1973) 54. R˘asvan, V.: Existence, stability and computation of forced oscillations in nonlinear distributed electrical networks. In: Proceedings of the VIIIth International Conference of Nonlinear Oscillations, vol. 1, pp. 589–594. Akademia, Prague Czechoslovakia (1978) 55. R˘asvan, V.: Absolute stability of nonlinear difference systems. Rev. Roum. Sci. Techn.-Electrot. Energ. 24(3), 495–500 (1979) 56. R˘asvan, V.: Functional differential equations associated to propagation. In: Loiseau, J.J., Niculescu, S., Sipahi, R. (eds.) Time Delay Systems, Analysis, Algorithms and Control. Lecture Notes in Control and Information Science, vol. 388, pp. 293–302. Springer (2009) 57. R˘asvan, V.: Augmented validation and a stabilization approach for systems with propagation. In: Miranda, F. (ed.) Systems Theory: Perspectives, Applications and Developments. Systems Science Series, vol. 1, pp. 125–170. Nova Science Publishers, New York (2014) 58. R˘asvan, V.: Models and stabilization for mechanical systems with propagation and linear motion coordinates. In: Witrant, E., Fridman, E., Sename, O., Dugard, L. (eds.) Recent Results on Time-Delay Systems. Advances in Delays and Dynamics, vol. 5, pp. 149–167. Springer (2015) 59. R˘asvan, V.: Critical cases in neutral functional differential equations, arising from hydraulic engineering. Opuscula Math. 42(4), 605–633 (2022) 60. R˘asvan, V.: On functional differential equations connected to Huygens synchronization under propagation. STCC (Syst. Theory, Control Comput.) J. 2(1), 34—43 (2022)

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61. R˘asvan, V.: Propagation, delays and stability (robust versus fragile). Int. J. Robust Nonlinear Control 32(6), 3225–3250 (2022). https://doi.org/10.1002/rnc.5656 62. R˘asvan, V.: Stability results for the functional differential equations associated to water hammer in hydraulics. Electr. J. Qualit. Theory Differ. Equ. 19, 1–19 (2022) 63. R˘asvan, V., Niculescu, S.I.: Oscillations in lossless propagation models - a Liapunov Krasovskii approach. IMA J. Math. Control Inf. 19, 157–172 (2002) 64. Saperstone, S.H.: Semidynamical Systems in Infinite Dimensional Spaces. Applied Mathematical Sciences, vol. 37. Springer, New York (1981) 65. Sharkovskii, A.N., Maystrenko, I.A., Romanenko, E.I.: Difference Equations and Their Applications. Naukova Dumka, Kiev USSR (1986). (in Russian) 66. Sokolov, A.A.: A criterion for stability of linear control systems with distributed parameters and its applications (in Russian). Inzh. sbornik 2, 4–26 (1946) 67. Vitt, A.A.: On the theory of the violin string (Russian). J. Tech. Phys. 6(9), 1450–1479 (1936) 68. Zverkin, A.M.: Series expansion of the solutions of linear differential difference equations i: quasi-polynomials (in Russian). In: El’sgol’ts, L.E., Zverkin, A.M. (eds.) Papers of the Seminar on the Theory of Differential Equations with Deviated Argument, vol. 3, pp. 3–39. University of Peoples’ Friendship, Moscow USSR (1965) 69. Zverkin, A.M.: Series expansion of the solutions of linear differential difference equations ii: series expansions (in Russian). In: El’sgol’ts, L.E., Zverkin, A.M. (eds.) Papers of the Seminar on the Theory of Differential Equations with Deviated Argument, vol. 4, pp. 3–50. University of Peoples’ Friendship, Moscow USSR (1967)

The Interplay Between Dispersal and Allee Effects in a Two-Patch Discrete-Time Model Azmy S. Ackleh and Amy Veprauskas

Abstract We consider a discrete-time two-patch model with dispersal and a nonlinear non-monotone growth rate suggesting that the population in a patch may undergo an Allee effect. We assume that newborns do not disperse while other individuals in the population do. We first analyze the model without dispersal and demonstrate that, under certain conditions, the model exhibits two positive equilibria with both the extinction and larger positive equilibrium being locally asymptotically stable. We also provide conditions where a unique positive equilibrium exists and is globally asymptotically stable. We then study the model with dispersal and show that, if at least one patch undergoes an Allee effect in the absence of dispersal, then introducing dispersal may produce multiple stable positive equilibria. As a result, when dispersal parameters are varied, a discontinuous switching phenomenon between two attractors may be introduced into a patch that otherwise has continuous behavior. While the existence and stability of these equilibria is established theoretically for small dispersal proportions, it is shown through numerical simulation that this phenomena may also occur for large dispersal proportions. Keywords Discrete time model · Dispersal · Allee effect · Multiple stable positive equilibria

1 Introduction Dispersal and Allee effects are two important ecological processes that can significantly affect population dynamics [13, 14]. Dispersal is the movement of individuals from one habitat patch to another, while an Allee effect is a phenomenon where the growth rate of a population decreases at low densities due to difficulties in finding mates or resources, leading to a positive-density dependence [15].

A. S. Ackleh (B) · A. Veprauskas University of Louisiana at Lafayette, Lafayette, LA 70504, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 S. Olaru et al. (eds.), Difference Equations, Discrete Dynamical Systems and Applications, Springer Proceedings in Mathematics & Statistics 444, https://doi.org/10.1007/978-3-031-51049-6_13

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The interplay between dispersal and Allee effects, particularly as it relates to biological invasions, is a topic of current interest in ecological research [1, 4, 5, 8, 19]. Dispersal can have a significant impact on the dynamics of populations experiencing an Allee effect. Specifically, it can affect the ability of the population to persist at low densities and the number and stability of the equilibria. A recent study has shown that dispersal can have a stabilizing effect on populations experiencing an Allee effect and can increase the persistence of the population under certain conditions [9]. In this paper we focus on investigating the interplay between dispersal and Allee effects using a discrete-time two-patch model with dispersal. In this two-patch model, population dynamics are determined by demographic and dispersal parameters that operate at the local and regional levels, respectively. The nonlinearity of the model comes from the assumption that the growth rate of the population is a function of the population size, which includes nonlinear and non-monotone population growth rates. Here, we study the effects of nonlinear non-monotone growth rates on the emergence of multiple positive equilibria, and how the presence of Allee effects can influence these dynamics. Specifically, we examine how dispersal affects the occurrence and persistence of multiple equilibria in the system, and how the strength of an Allee effect influences these outcomes. Our results contribute to a better understanding of the conditions under which multiple equilibria can arise in the presence of dispersal and Allee effects, and how these equilibria can impact the long-term persistence of populations in fragmented landscapes. Previous studies have used discrete-time multi-patch modeling approaches to understand the effect of dispersal on population dynamics (e.g., [9, 10, 16–18]). In [10], the authors study a model that considers multiple patches with a nonlinear growth rate to examine the evolution of dispersal in heterogenous landscapes. The authors show that slower dispersing populations outcompete faster dispersing populations in unconditional dispersal strategies, while conditional dispersers with no dispersal costs can resist invasion attempts through a one parameter family of strategies. In [16] the authors study a discrete-time two-patch dispersal system where the local dynamics are overcompensatory and each patch exhibits an Allee effect. They study the impact of coupling the patches through dispersal and demonstrate that, while the local dynamics produce bistability, the coupling may result in several coexisting attractors. They also provide parameter ranges that protect the spatiallystructured population from essential extinction. In [17, 18] the author studies the impact of synchronous and asychnronous dispersal on the dynamics of discrete-time two-patch models, where the local pre-dispersal patch dynamics are undercompensatory or overcompensatory. Therein, the author shows that whenever the pre-dispersal dynamics of each patch is undercompensatory, then introducing dispersal results in qualitatively equivalent dynamics in the two-patch system. In contrast, when pre-dispersal dynamics are overcompensatory in both patches, then both synchronous and asynchronous dispersal can lead to the creation of multiple attractors but synchronous dispersal results in a larger number of distinct attractors.

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In [9], the authors investigate the impact of insect dispersal intensity on the population dynamics of a two-patch plant-insect model that exhibits Allee like effects, namely, where fluctuations in the plant population may cause local extinct of the insect population when the latter population is sufficiently small. In this model, they assume the two patches are homogenous and that dispersal rates between the two patches are the equal. The results show that when the pre-dispersal patches are permanent, permanence may be maintained for a small dispersal rate while large dispersal and attacking rates can destroy the permanence of the system, leading to the extinction of insects in one patch. Meanwhile, when the pre-dispersal patch dynamics exhibit Allee like effects, small and intermediate dispersal can lead to source-sink dynamics and extinction of the insect population in both patches, respectively. In comparison to these previous studies, our investigation here focuses on a twopatch discrete-time model with a heterogeneous environment that allows for different dispersal rates between the patches and considers a non-monotone growth rate allowing for Allee effects to occur in one or both patches. This differs from the aforementioned studies which assumed either monotone growth rates [10, 17, 18] or uniform habitats [9]. This paper is organized as follows: In Sect. 2 we present the discrete-time twopatch model. In Sect. 3 we summarize the dynamics of the model without dispersal where each patch may undergo an Allee effect. In Sect. 4 we study theoretically and numerically the dynamics of the model with dispersal. Finally, in Sect. 5 we provide concluding remarks.

2 The Two-Patch Model with Allee Effect We study the following discrete-time two-patch model with dispersal: .

x1 (t + 1) = β1 (x1 (t))x1 (t) + s1 ((1 − d1 )x1 (t) + d2 x2 (t)), x2 (t + 1) = β2 (x2 (t))x2 (t) + s2 ((1 − d2 )x2 (t) + d1 x1 (t)),

(1)

where .xi , .i = 1, 2, denotes the population density in patch .i and .si denotes the survival probability of an individual in patch .i. Here the fecundity of an individual in patch .i is given by β (x) =

. i

bi (1 + ai x) , 1 + ci x + ai x 2

where .ai , ci > 0. We assume that newborns do not disperse while surviving individuals do (e.g., tadpoles and frogs where tadpoles remain in the same pond and cannot disperse while frogs can hop from one pond to another). The fraction of individuals that disperse from patch 1 to patch 2 per one time unit is given by .d1 and from patch 2 to patch 1 is given by .d2 , where .0 ≤ di ≤ 1. The system can be written in matrix form

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.

x(t + 1) = P(x(t))x(t),

(2)

where .x := [x1 , x2 ] and [ .

P(x) :=

] β1 (x1 ) + s1 (1 − d1 ) s1 d2 . s2 d1 β2 (x2 ) + s2 (1 − d2 )

3 Summary of Dynamics with No Dispersal . d1 = d2 = 0 For this case the two patches are decoupled and the system takes the following form: .

x1 (t + 1) = β1 (x1 (t))x1 (t) + s1 x1 (t), x2 (t + 1) = β2 (x2 (t))x2 (t) + s2 x2 (t).

(3)

Each patch has two or three equilibria depending on the value of the inherent growth rate of that patch defined by .λi0 := bi + si . Observe that the inherent growth rate .λi0 bi is on the same side of one as the net reproduction number given by . Ri0 = (1−s = i) 2 3 bi (1 + si + si + si + . . . ). The equilibria are given by all nonnegative solutions of the equation .xi = β(xi )xi + si xi , namely .xi = 0 and 1,2 .xi

=

bi ai − ci (1 − si ) ±

√ (bi ai − ci (1 − si ))2 − 4a(1 − si )((1 − si ) − bi )) . 2ai (1 − si )

To this end, we have the following theorem concerning the local dynamics which was obtained in [7]: Theorem 1 ([7]) Consider patch .i, .i = 1, 2, in model (3). (i) Assume .ai ≤ ci . If .λi0 := bi + si < 1, then no positive equilibria exist and the extinction equilibrium, .xi = 0, is locally asymptotically stable. If .λi0 > 1, then there is a unique positive equilibrium, .xi2 , and it is locally asymptotically stable, while the equilibrium .xi = 0 is unstable. √ i (2 1 + ai − ci + ci − 2) + si . If .λi0 < λi , (ii) Assume .ai > ci and let .λi := 1−s ai then no positive equilibria exist and the extinction equilibrium is locally asymptotically stable. If .λi ≤ λi0 < 1, then there are two positive equilibria .xi1,2 and .xi2 is locally asymptotically stable. Furthermore, the extinction equilibrium .xi = 0 is locally asymptotically stable, while the positive equilibrium given by .xi1 is unstable. If .λi0 > 1, then there is one positive equilibrium given by .xi2 and it is locally asymptotically stable. For illustration we present a bifurcation diagram with respect to .λi0 = bi + si for the model without dispersal in the case of .ai > ci in Fig. 1. In this figure we fix

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18 16 14

Equilibria

12 10 8 6 4 2 0 0.2

0.4

0.6

0.8

1

1.2

0 i

Fig. 1 Shown is the bifurcation diagram of the model with.d1 = d2 = 0 for the case.ai > ci . Stable equilibria are given by solid lines and unstable equilibria are given by dashed lines

a = 10, .ci = 2, .si = 0.2 and we vary the parameter .bi between .0 and .1, resulting in λ0 ∈ [0.2, 1.2]. Clearly, the figure demonstrates that a backward bifurcation exists and that each patch has two positive equilibria for .λi0 ∈ (0.68, 1] and one positive equilibrium for .λi0 > 1. Next, in Theorem 2 we establish some results concerning the global dynamics of model (3). In this theorem, by a globally asymptotically stable equilibrium we mean an equilibrium that is locally asymptotically stable and attracts every solution with positive initial condition.

. i

. i

Theorem 2 Consider patch .i, .i = 1, 2, in model (3). (i) Assume .ai ≤ ci . Then the extinction equilibrium, .xi = 0, is globally asymptotically stable if .λi0 < 1. (ii) Assume .ai > ci . Then the extinction equilibrium, .xi = 0, is globally asymptotically stable if .λi0 < λi . (iii) Assume.λi0 > 1. Then the unique positive equilibrium,.xi2 , is globally asymptotically stable if either .ci ≥ 1 or .ci < 1 and .bi ≤ b˜i where .b˜i := si (ai2 + 2ai + 2ai ci + ci2 ) . ai (1 − ci ) Proof (i) Define . f i (xi ) := βi (xi (t)) + si . Since β ' (xi ) := bi

. i

ai − ci − 2ai xi − ai2 xi2 , (1 + ci xi + ai xi )2

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if .ci ≥ ai , then .βi' (xi ) < 0 on the interval .[0, ∞). Therefore, .βi obtains its maximum at .xi = 0. It follows that for any .xi (0) ≥ 0 we have t 0 t 0 ≤ xi (t) = ∏t−1 j=0 f i (x i ( j))x i (0) ≤ f i (0) x i (t) := (λi ) x i (t) → 0 as t → ∞.

.

(ii) If .ai > ci , then . f i obtains its maximum at .xi∗ := follows that for any .xi (0) ≥ 0 we have

1 √ ( ai − ci + 1 − 1). It ai

∗ t 0 ≤ xi (t) = ∏t−1 j=0 f i (x i ( j))x i (0) ≤ f i (x i ) x i (t).

.

Thus, if . f i (xi∗ ) < 1, or equivalently .λi0 < λi , then .xi (t) → 0 as .t → ∞. (iii) Define . Fi (xi ) := f i (xi )xi (t). Then, Fi' (xi ) ai2 si xi4 + 2ai ci si xi3 + (ci2 si + 2ai si − ai bi (1 − ci ))xi2 + 2(ai bi + ci si )xi + bi + si = . (1 + ci xi + ai xi2 )2 ) ( 1 4 3 2 := 2 2 C4 xi + C3 xi + C2 xi + C1 xi + C0 . (1+ci xi +ai xi )

Note that.Ci > 0 for.i = 0, 1, 3, 4. When.ci ≥ 1,.C2 > 0 and, thus. Fi' (xi ) > 0. Meanwhile, the sign of .C2 is not determined when .ci < 1. Note also that .C2 + C1 > 0 and that .C4 + C3 + C2 > 0 if .bi ≤ b˜i . For any .xi ≥ 0, either .xi ∈ [0, 1] or .xi > 1. If .xi ∈ [0, 1] then using the fact that for . P > 1, .xiP ≤ xi , we observe that .

Fi' (xi ) ≥

( ) 1 (C2 + C1 )xi2 + C0 > 0, (1 + ci xi + ai xi2 )2

since .C2 + C1 > 0. On the hand, if .xi > 1 then .xiP > xi . Hence, .

Fi' (xi ) ≥

( ) 1 (C4 + C3 + C2 )xi2 + C0 > 0, 2 2 (1 + ci xi + ai xi )

if .bi ≤ b˜i . Assume either condition in (iii) holds. Then the patch .i dynamics are monotone and we apply Lemma 3 from [2] to obtain global stability. This lemma states that if . Fi is a monotone map with a unique fixed point in the ordered interval .[ p, q] such that . p ≤ Fi ( p) and . Fi (q) ≤ q, then every solution sequence starting in .[ p, q] converges to the unique fixed point. Since the uniqueness of the positive equilibrium was obtained in Theorem 1, it remains to find an ordered interval .[ p, q] satisfying the desired conditions. bi := First note that since .lim xi →∞ βi (xi )xi = bi , we have .lim supt→∞ xi (t) ≤ 1−s i xˆi . Thus every forward patch.i solution of (3) enters and remains in the interval.[0, xˆi ].

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Choose .q := xˆi . Then, . Fi (q) ≤ q. Next, we note that, since .λi0 > 1 and . Fi' (xi ) > 0 for all .xi ≥ 0, there exists an .ϵ > 0 sufficiently small such that .

Fi (ϵ) = λi0 ϵ + o(ϵ) ≥ ϵ.

Thus, for an initial condition.xi (0) ∈ (0, xˆi ] we may choose a sufficiently small.ϵ > 0 such that. p := ϵ ≤ xi (0) and. p ≤ Fi ( p). The result then follows from Lemma 3 from [2] together with the local asymptotic stability of the interior equilibrium obtained in Theorem 1. Remark 1 Consider .b˜i to be a function of .ci and .ai , i.e., .b˜i (ci , ai ). Observe that ∂ b˜i > 0 for .0 < ci < 1, that is, .b˜i is increasing in .ci and that .limci →1 bi = ∞, which . ∂ci agrees with the case .ci ≥ 1. Also, .b˜i (0, ai ) = si (ai + 2) provides a lower bound for the range of global stability for .ci < 1. Clearly, .b˜i (0, ai ) is increasing in .ai with ˜i (0, ai ) = ∞. .lim ai →∞ b

4 Model Dynamics with Dispersal We begin by calculating the growth rate of the two-patch model by looking at the dominant eigenvalue of the zero-density projection matrix: [ .

P(0) =

b1 + s1 (1 − d1 ) s1 d2 s2 d1 b2 + s2 (1 − d2 )

]

which is given by λ0 := .

1 2

( ) b1 + s1 (1 − d1 ) + β20 + s2 (1 − d2 )

√ + 21 [(b1 + s1 (1 − d1 )) − (b2 + s2 (1 − d2 ))]2 + 4s1 s2 d1 d2 .

Since. P(0) is equal to the Jacobian matrix of system (1) evaluated at 0,.λ0 determines the local asymptotic stability of the extinction equilibrium. Observe that for the system without dispersal, that is when .d1 = d2 = 0, the expression reduces to .λ0 = maxi {bi + si }, which agrees with inherent growth rate provided in Sect. 3. Furthermore, in the overall pre-dispersal system, if the inherent growth rate .λ0 = maxi {bi + si } > 1 then the total population .x1 + x2 persists over the entire landscape (even though it may go extinct in one patch). We first show in Theorem 3 that, in the absence of an Allee effect in either predispersal patch, the introduction of dispersal does not change the qualitative dynamics of model (1). Notice that Theorem 3(ii) does not address the local stability of the interior equilibrium. Since finding an explicit formula for the interior equilibrium

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requires solving a high-degree polynomial, this is not possible in general and, thus, we were unable to obtain useful expressions for the eigenvalues of the corresponding Jacobian matrix. Theorem 3 Consider model (1). Assume .a1 ≤ c1 , .a2 ≤ c2 , and .d1 , d2 > 0. (i) If .λ0 < 1, then the extinction equilibrium is globally asymptotically stable. (ii) If .λ0 > 1 and either .ci ≥ 1 or .ci < 1 and .bi ≤ b˜id where si (1 − di )(ai2 + 2ai + 2ai ci + ci2 ) , b˜ d := ai (1 − ci )

. i

for .i = 1, 2, then there is a unique positive equilibrium that globally attracts all solutions of non-negative and non-zero initial conditions. Proof (i) Since the dominant eigenvalue of the inherent project matrix is less than one, .λ0 < 1, the extinction equilibrium is locally asymptotically stable. Moreover, when .ci ≥ ai , the function .βi (xi ) obtains its maximum at .xi = 0. Thus, for any . x(0) ≥ 0 we have t 0 ≤ x(t) = ∏t−1 j=0 P(x( j))x(0) ≤ P(0) x i (t) → 0 as t → ∞,

.

where the limit follows since .λ0 < 1. Hence the extinction equilibrium globally attracts all solutions of non-negative and non-zero initial conditions. (ii) First we prove that when .λ0 > 1, there is a unique positive equilibrium. Suppose there are two positive equilibria .x¯ := (x¯1 , x¯2 ) and . y¯ := ( y¯1 , y¯2 ) such that at ¯ least one of the components in . y¯ is smaller than the corresponding component in .x. Define .α := min{α > 1 : x ≤ αy}. We first show that .x¯ /= α y¯ . If this were the case then we would have 0 = x¯1 − α y¯1 = β1 (x¯1 )x1 − β1 ( y¯1 )αy1 + s1 (1 − d1 )(x¯1 − α y¯1 ) + s1 d2 (x¯2 − α y¯2 )

.

which implies .β1 (x¯1 ) − β1 ( y¯1 ) = 0. However, the only solutions to this equation are a1 − c1 − a1 x¯1 .y ¯1 = x¯1 or . y¯1 = < 0. Therefore, without loss of generality, assume a1 + a12 x¯1 .x ¯1 < α y¯1 and .x¯2 = α y¯2 . Define .G 2 (x1 , x2 ) := β(x2 )x2 + s2 (1 − d2 )x2 + s2 d1 x1 . 2 > 0 and it is straightforward to show (refer to . Fi' (xi ) in the proof of Then . ∂G ∂x1 2 > 0. Moreover, Theorem 2) that if either .c2 ≥ 1 or .c2 < 1 and .b2 ≤ b˜2d then . ∂G ∂x2 ' since .β2 (x2 ) < 0, we have .

x¯2 = G 2 (x¯1 , x¯2 ) < G 2 (α y¯1 , α y¯2 ) = α[β2 (α y¯2 ) y¯2 + s2 (1 − d2 ) y¯2 + s2 d1 y¯1 ] < αG 2 ( y¯1 , y¯2 ) = α y¯2 ,

a contradiction. Given the uniqueness of the positive equilibrium, global attractivity of the positive equilibrium then follows from the same arguments as applied in the proof of

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Theorem 2 where, for the lower bound on the ordered interval, we choose . p := ϵv where .v is the positive eigenvector corresponding to .λ0 (whose existence is guaranteed since . P(0) is irreducible) and .ϵ > 0 is sufficiently small.

4.1 Impact of Dispersal on Species Persistence The value of the inherent growth rate .λ0 determines the stability of the extinction equilibrium and provides insight into population persistence [6] (as is demonstrated in Theorem 3). Here, we consider the impact of dispersal on the inherent growth rate and hence on the population persistence. The following result is useful to understand the impact of dispersal on the inherent growth rate of the system. In particular, it allows for understanding whether dispersal increases or decreases this growth rate. = −A + Lemma 1 Suppose. ∂λ ∂di 0

B C

with. A, C > 0. Then,. ∂λ < 0 if. B ≤ 0 or if. B > ∂di 0

0 and . A2 C 2 − B 2 > 0. Furthermore, if . B > 0 and . A2 C 2 − B 2 < 0, then . ∂λ > 0. ∂di 0

< 0. Proof Clearly, if . B ≤ 0, then . ∂λ ∂di Now, assume that . B > 0 and . A2 C 2 − B 2 > 0. Then, we have 0

.

B 2 < A2 C 2 =⇒ B < AC =⇒

B B < A =⇒ −A + < 0. C C

Thus, .

∂λ0 B = −A + < 0. ∂di C

Similarly, if . A2 C 2 − B 2 < 0, then one can show that .

∂λ0 > 0, ∂di

which establishes the result. We now use Lemma 1 to determine the impact of dispersal on the inherent growth rate of model (1). We find that increasing dispersal from patch .i is deleterious if it has a higher pre-dispersal growth rate than the other patch and is advantageous otherwise. Meanwhile, if dispersal is symmetric, that is.d1 = d2 = d, then increasing dispersal is always deleterious. Corollary 1 Consider model (1). Assume .λ01 /= λ02 . Then (i) if .λ01 > λ02 then . ∂λ < 0 and if .λ01 < λ02 then . ∂λ > 0; ∂d1 ∂d1 0

0

(ii) if .d1 = d2 = d, then . ∂λ < 0. ∂d 0

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Proof (i) Taking the derivative of .λ0 with respect to .d1 we obtain .

∂λ0 1 s1 (b1 − b2 + s1 (1 − d1 ) − s2 (1 − d2 )) − 2 d2 s1 s2 . = − s1 − √ ∂d1 2 2 (b1 − b2 + s1 (1 − d1 ) − s2 (1 − d2 ))2 + 4 d1 d2 s1 s2

Letting .

A=

1 s1 , 2

B = −s1 (b1 − b2 + s1 (1 − d1 ) − s2 (1 − d2 )) + 2 d2 s1 s2

and / C =2

.

(b1 − b2 + s1 (1 − d1 ) − s2 (1 − d2 ))2 + 4 d1 d2 s1 s2 ,

= −A + B/C where . A, C > 0 and we have that . ∂λ ∂d1 0

.

A2 C 2 − B 2 = 16 d2 s1 2 s2 ((b1 + s1 ) − (b2 + s2 )) .

Applying Lemma 1 we have that if .λ01 := b1 + s1 > b2 + s2 := λ02 , then . A2 C 2 − ( 0 B 2 > 0 and hence . ∂λ < 0. On the other hand, if .λ01 < λ02 , then . B = s1 (λ02 − λ01 )+ ∂d1 s1 d1 + s2 d2 ) + 2d2 s1 s2 > 0 and . A2 C 2 − B 2 < 0 in this case. Thus, by Lemma 1 we 0 > 0. This establishes part (i). have that . ∂λ ∂d1 As for part (ii), assume .d1 = d2 = d, then b2 + b1 + (s1 + s2 ) (1 − d) 0 + .λ = 2

√

(b2 − b1 + (s2 − s1 ) (1 − d))2 + 4 d 2 s1 s2 2

and .

s1 + s2 ∂λ0 (s1 − s2 ) (b2 − b1 + (s2 − s1 ) (1 − d)) + 4 d s1 s2 √ =− + . ∂d 2 2 (b2 − b1 + (s2 − s1 ) (1 − d))2 + 4 d 2 s1 s2

Letting, .

A=

s1 + s2 , 2

B = (s1 − s2 ) (b2 − b1 + (s2 − s1 ) (1 − d)) + 4 d s1 s2

and C =2

.

/ (b2 − b1 + (s2 − s1 ) (1 − d))2 + 4 d 2 s1 s2 ,

we get that .

A2 C 2 − B 2 = 16s1 s2 (λ01 − λ02 )2 > 0.

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In this case, it also follows from Lemma 1 that . ∂λ < 0. This completes part (ii). ∂d 0

From part (i) of Corollary 1 we observe that increased dispersal from the patch with the better environment which attains the .λ0 = maxi {bi + si } in the absence of dispersal is deleterious to the system as it reduces the overall inherent growth rate of the system. In this case, it is possible for the population to go extinct when predispersal it would have persisted. However, increased dispersal rates from the patch with a worse environment is advantageous to the entire system as it increases the inherent growth rate.

4.2 Impact of Dispersal on Attractors and Stability Next we consider the impact of dispersal when at least one patch exhibits an Allee effect in isolation. Specifically, with all other parameters held fixed, we consider the existence of positive equilibria .(x1 , x2 ) as a function of .d := (d1 , d2 ). Define .((x 1 , x 2 ), d) to be an equilibrium pair. Theorem 4 Let .xie for .i = 1, 2 denote an equilibrium of patch .i in isolation and assume .λi0 /= 1. Then there exists .d1∗ , d2∗ > 0 such that for .0 < di < di∗ , .i = 1, 2, a branch of equilibria of the form x ≈ xie +

. i

] 1 [ e si xi di − x ej d j , λi0 − 1

j /= i,

bifurcates from the equilibrium pair .((x1e , x2e ), (0, 0)) and retains the same local stability properties as .((x1e , x2e ), (0, 0)). Proof Denote model (1) as .x(t + 1) = F(x(t); d) where . F : R2 × R2 → R2 . Clearly, . F((x1e , x2e ); (0, 0)) = (x1e , x2e ). Moreover, the Jacobian matrix evaluated at e e 0 0 .((x 1 , x 2 ), (0, 0)) is diagonal with eigenvalues .λ1 and .λ2 . It follows by the Implicit Function Theorem (see for example Theorem A4.1 of [3]) that there exists an open neighborhood.U of.R2 containing.(x1e , x2e ), an open neighborhood.V of.R2 containing .(0, 0), and a uniquely determined function .g : V → U whose partial derivatives exist in .V , such that . F(g(d); d) = g(d), for all .d ∈ V with .g(0, 0) = (x1e , x2e ). Hence, we have the existence of an equilibrium pair .((x1 , x2 ), d) for .d ∈ V . Further, since the eigenvalues of the Jacobian matrix depend continuously on .d1 and .d2 , these equilibria retain the local stability properties of .((x1e , x2e ), (0, 0)). Finally, calculation shows that .

−x ej ∂xie ∂xie si x e and for j /= i. = 0 i = 0 ∂di ∂d j λi − 1 λi − 1

Remark 2 By Theorem 4, if .xie = 0, then the branch of equilibria bifurcating from a nontrivial equilibrium pair .(x1e , x2e ) is positive if and only if .λi0 < 1. This means

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that, if at least one patch exhibits an Allee effect, then (up to symmetry) the following cases occur when .d1 and .d2 are sufficiently small: (i) If .λ01 > 1, .a2 > c2 , and .λ2 < λ02 < 1, then model (1) has three positive equilibria. Two of these equilibria are locally asymptotically stable and one is unstable. (ii) If .a1 > c1 , .λ1 < λ01 < 1, .a2 > c2 , and .λ2 < λ02 < 1, then model (1) has eight positive equilibria. Three of these equilibria are locally asymptotically stable and five are unstable. (iii) If .a1 > c1 , .λ01 < λ1 , .a2 > c2 , and .λ2 < λ02 < 1, or if .a1 < c1 , λ01 < 1, a2 < c2 , and .λ2 < λ02 < 1, then model (1) has two positive equilibria. One of these equilibria is locally asymptotically stable and the other is unstable. For all other scenarios, the model either has one positive equilibrium (which occurs if .λi0 > 1 for at least one .i) or none.

4.3 Numerical Studies Next we provide some numerical studies to better understand the impact of dispersal when dispersal is not small. Key insights of these studies are as follows: 1. As established in Theorem 4, introducing dispersal between two patches can produce multiple positive attractors (Figs. 2, 3, and 6). As the dispersal parameters are varied, this may introduce a discontinuous switching phenomenon between two attractors into a patch that would otherwise vary continuously (Fig. 2). 2. Although Theorem 4 only holds for small dispersal values, the range of values over which multiple positive attractors exist may be quite large (Figs. 4 and 6). Moreover, these equilibria retain the same local stability properties as for small dispersal values. 3. It is also possible for multiple positive attractors to be created when dispersal values are increased (Fig. 5). Example 1 Multiple positive attractors for small .di . In this example, we fix the values of the following model parameters: .

b1 = 1, a1 = 1, c1 = 2, s1 = 0.2, x1 (0) = 0.1, b2 = 0.4, a2 = 20, c2 = 2, s2 = 0.2, d2 = 0.3,

while varying .d1 over the interval .(0, 1] and the initial condition .x2 (0) over the interval .[0.01, 0.2]. We simulate the model for a large time .T = 500 and observe that all solutions converge to equilibria. We present .x1 (T ) (left) and .x2 (T ) (right) in Fig. 2. Clearly, for smaller values of .d1 and .x2 (0) we observe that the solution converges to a different equilibrium than for larger values of .d1 and .x2 (0) and that there is a discontinuous switching as .d1 or .x2 (0) are increased. This phenomenon appears because patch 2 without dispersal exhibits an Allee effect and has more

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Fig. 2 Shown is a scenario of discontinuous switching. The graphs give the long-term solutions (equilibria) with respect to varying .x2 (0) and .d1 for patch 1 (left) and patch 2 (right). Notice that for ease of readability, the dispersal axes in these graphs have been transposed

than one positive equilibrium. Furthermore, the extinction equilibrium is locally asymptotically stable. Thus, increasing .x2 (0) or .d1 pushes the population in the patch 2 away from the local attracting region of the equilibrium bifurcating from the extinction equilibrium and into the attracting domain of the positive equilibrium. Hence, the discontinuity observed. In Fig. 3 we further fix the value .x20 = 0.05, where a discontinuity occurs for small .d1 and we vary .d1 ∈ (0, 1]. We present the equilibria bifurcation diagram for patch 1 in the top left panel and equilibria bifurcation diagram for patch 2 in the top right panel. In the bottom panel we present the spectral radius of the Jacobian matrix evaluated at each equilibrium to determine whether the equilibrium is locally stable or unstable. In the bottom panel on the right, we show the solution attractors of .x1 and .x2 for two different values of .x2 (0), small and large. We notice that for the small initial condition the solution undergoes a discontinuous switching for a small value of .d1 while for the large initial condition the attractors are smooth functions of .d1 . Example 2 Multiple positive attractors for all dispersal values .di . While Theorem 4 proves that multiple attractors are possible for small .d1 and .d2 , it does not state what happens for larger .d1 and .d2 . In this example, we show that it is possible to have multiple positive attractors for all values of .d1 or .d2 . We consider the following parameter values and allow .d2 to vary in the interval .(0, 1]: b = 1, a1 = 1, c1 = 3, s1 = 0.2, b2 = 0.4, a2 = 20, c2 = 1.3, s2 = 0.2, d1 = 0.1.

. 1

Observe that at .d2 = 0, since .a1 < c1 and .λ01 > 1, there exist a unique positive equilibrium for patch 1. However, as .d2 becomes positive patch 1 bifurcates into three positive equilibria due to the impact of dispersal from patch 2 which, in isolation, exhibits an Allee effect (see Fig. 4).

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0.4

0.3

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1.25

0.5 d 1

1

x2(0)=0.05

0.4

xi

1.2 1.15 1.1

0.2 0

0

1.05

0.5

1

d1

1 0.95

x2(0)=0.15

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xi

Dominant Eigenvalue of the Jacobian

0.5 d 1

Patch 2

0.4

0.2

0.8 0 0.75

0

0.5

d1

1

0

0.5

1

d1

Fig. 3 The top panel provides bifurcation diagrams for patch 1 (left) and patch 2 (right) with respect to the .d1 . The bottom left panel graphs the spectral radius of the Jacobian matrix evaluated at each equilibrium. Stable equilibria are denoted by circle (blue) curves and unstable ones are denoted by diamond (red) curves. The bottom right graphs present the long-term solution for two initial values of .x2 (0). The solid (black) curves represent .x1 and the dashed (blue) curves represent .x2

The Interplay Between Dispersal and Allee Effects … Patch 1

0.3

297 Patch 2

0.4 0.35

0.25 0.3 0.2

Equilibria

Equilibria

0.25 0.15

0.2 0.15

0.1 0.1 0.05 0.05 0

0

0.5

0

1

0

Spectral Radius of the Jacobian

d2

0.5

1

d2

Local Stability

1.3 1.2 1.1 1 0.9 0.8 0.7

0

0.5

1

d2 Fig. 4 Shown is a scenario where multiple attractors exists for all values of .d2 . The top panel is a bifurcation diagram with respect to .d2 for patch 1 (left) and patch 2 (right). The bottom panel gives the spectral radius of the Jacobian matrix evaluated at each equilibrium. Circles (blue) denote stable equilibria and diamonds (red) denote unstable equilibria

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Fig. 5 Shown is a scenario of discontinuous switching for large, rather than small, values of .d2 . The graphs give the long-term solutions (equilibria) at time .T = 500 with respect to varying .x2 (0) and .d2 for patch 1 (left) and patch 2 (right)

Example 3 Multiple positive attractors for large dispersal. In this example we demonstrate that not only small dispersal can introduce multiple attractors when only one positive attractor exists before dispersal, but that large dispersal can also introduce multiple attractors when for small dispersal only one positive attractor exists. Here we fix the parameters as follows: .

b1 = 1, a1 = 1, c1 = 2, s1 = 0.2, d1 = 0.3, x1 (0) = 0.1, b2 = 0.4, a2 = 20, c2 = 2, s2 = 0.2.

We then vary .d2 over the interval .(0, 1] and .x2 (0) over the interval .[0.01, 0.2] and present the results in Fig. 5. Example 4 A strong Allee effect in both patches. In this example we choose the model parameters as follows so that without dispersal both patches undergo an Allee effect: b = 0.5, a1 = 10, c1 = 2, s1 = 0.2, b2 = 0.4, a2 = 20, c2 = 2, s2 = 0.2.

. 1

We fix .d2 = 0.3 and we vary .d1 over the interval .(0, 1] and present the a bifurcation diagram of the equilibria with respect to the parameter .d1 in Fig. 6. Notice that in this case the model has up to six positive equilibria with two locally asymptotically stable and four unstable. Then, we fix .d1 = 0.1 and we vary .d2 over the interval .(0, 1] and present a bifurcation diagram of the equilibria with respect to the parameter .d2 in Fig. 6. Here we observe that the the model has up to eight positive equilibria with three locally asymptotically stable and five unstable.

The Interplay Between Dispersal and Allee Effects … Patch 1

0.4

Patch 2

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0.3

0.3

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Equilibria

Equilibria

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0.2 0.15

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0 0

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Equilibria

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d1

Patch 1

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0.15 0.1

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0.05

0.05

0

0 0

0.5

d2

1

0

0.5

1

d2

Fig. 6 The top panel provides bifurcation diagrams for patch 1 (left) and patch 2 (right) with respect to the .d1 . The bottom panel provides bifurcation diagrams for patch 1 (left) and patch 2 (right) with respect to the .d2 . Stable equilibria are denoted by circles (blue) and unstable ones are denoted by diamonds (red)

5 Concluding Remarks In this study we considered a discrete-time two-patch model with Allee effects and dispersal. We first established conditions for global stability of the unique positive equilibrium of the system without dispersal. Then we investigated the system with dispersal and provided conditions for the existence of a unique positive equilibrium and its global attractivity. We also studied the impact of dispersal on the inherent

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growth rate of the system to provide insight as to how dispersal affects population persistence. We found that whether increasing dispersal positively or negatively affects the overall system inherent growth rate.λ0 depends on the relation between the pre-dispersal inherent growth rates in each patch. Namely, if one patch has a higher pre-dispersal growth rate, then increasing dispersal out of that patch decreases .λ0 and, thus, is deleterious to population persistence. Conversely, dispersal out of a patch with a lower pre-dispersal inherent growth rate is beneficial as it increases .λ0 . Meanwhile, when dispersal is symmetric (which is equivalent to unconditional in two patches), i.e., .d1 = d2 = d, .λ0 is always a decreasing function of .d. In comparison to this latter result, we note that Theorem 3.1 in [10] (or Theorem 5.2 in [11]) establishes this same monotonicity relationship for a more general .n-patch model. However, the assumption in model (1) that only mature individuals are able to disperse means that this theorem is not applicable to our model. To understand how Allee effects may interact with dispersal to impact system steady states, we applied a perturbation argument to the dispersal parameters .d1 and .d2 . We found that when at least one pre-dispersal patch exhibits an Allee effect (namely, .ai > ci and .λi ≤ λi0 < 1), introducing dispersal between the patches, .d1 , d2 0, increases the number of positive equilibria in each patch with respect to the number of pre-dispersal positive equilibria. Since these equilibria retain the stability properties of the pre-dispersal equilibria, this results in the creation of multiple stable positive equilibria. Numerically, we observe that, as dispersal parameters are varied, discontinuous switching between two attractors may be introduced into a patch that would otherwise vary continuously. In addition, though we only establish the existence of multiple attractors for small dispersal parameters, our numerical studies show that the range of values over which multiple positive attractors exist may be quite large. These theoretical and numerical results complement existing work which has shown that introducing dispersal between patches may increase the number of attractors when local dynamics are overcompensatory [17, 18], or when the interaction between two species creates Allee-like affects [9].

References 1. Ackleh, A.S., Allen, L.J.S., Carter, J.: Establishing a beachhead: a stochastic population model with an Allee effect applied to species invasion. Theor. Popul. Biol. 71(2007), 290–300 (2007) 2. Ackleh, A.S., De Leenheer, P.: Discrete three-stage population model: persistence and global stability results. J. Biol. Dyn. 2(4), 415–427 (2008) 3. Ackleh, A.S., Hossain, M.I., Veprauskas, A., Zhang, A.: Persistence and stability analysis of discrete-time predator-prey models: a study of population and evolutionary dynamics. J. Differ. Equations Appl. 25(11), 1568–1603 (2019) 4. Fernandez, A.A.: Interplay between Allee effects and collective movement in metapopulations. Oikos 121, 813–822 (2012) 5. Berec, L. Angulo, E., Courchamp, F.: Multiple Allee effects and population management. Trends Ecol. Evol. 22, 185–191 (2007) 6. Cushing, J.: An Introduction to Structured Population Dynamics. In: CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM (1998)

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7. Cushing, J.M., Stefanko, K.: A Darwinian dynamic model for the evolution of post-reproduction survival. J. Biol. Syst. 29(02), 433–450 (2021) 8. Etienne, R., Wertheim, B., Hemerik, L., Schneider, P., Powell, J.: The interaction between dispersal, the Allee effect and scramble competition affects population dynamics. Ecol. Model. 148(2), 153–168 (2002) 9. Kang, Y., Armbruster, D.: Dispersal effects on a discrete two-patch model for plant-insect interactions. J. Theor. Biol. 268(1), 84–97 (2021) 10. Kirkland, S., Li, C., Schreiber, S.: On the evolution of dispersal in patchy landscapes. SIAM J. Appl. Math. 66(4), 1366–1382 (2006) 11. Karlin, S.: Classifications of selection-migration structures and conditions for a protected polymorphism. Evol. Biol. 14(61), 204 (1982) 12. Rosenkranz, G.: On global stability of discrete population models. Math. Biosci. 64(2), 227– 231 (1983) 13. Shaw, A.K., Kokko, H.: Dispersal evolution in the presence of Allee effects can speed up or slow down invasions. Am. Nat. 185(5), 631–639 (2015) 14. Stephens, P.A., Sutherland, W.J.: Consequences of the Allee effect for behaviour, ecology and conservation. Trends Ecol. Evol. 14(10), 401–405 (1999) 15. Stephens, P.A., Sutherland, W.J., Freckleton, R.P.: What is the Allee effect? Oikos 87(1), 185– 190 (1999) 16. Vortkamp, I., Schreiber, S.J., Hastings, A. Hilker, F.M.: Multiple attractors and long transients in spatially structured populations with an Allee effect. Bull. Math. Biol. 82, 82 (2020). https:// doi.org/10.1007/s11538-020-00750-x 17. Yakubu, A.A.: Two-patch dispersal-linked compensatory-overcompensatory spatially discrete population models. J. Biol. Dyn. 1(2), 157–182 (2007) 18. Yakubu, A.A.: Asynchronous and synchronous dispersals in spatially discrete population models. SIAM J. Appl. Dyn. Syst. 7(2), 284–310 (2008) 19. Zhou, S., Wang, G.: Allee-like effects in metapopulation dynamics. Math. Biosci. 189, 103–113 (2004)

Krause Mean Processes Generated by Off-Diagonally Uniformly Positive Nonautonomous Stochastic Hyper-Matrices Mansoor Saburov and Khikmat Saburov

Abstract The notion of consensus through repeated averaging was first introduced by DeGroot in the context of structured, time-invariant and synchronous environments. Since then, consensus has emerged as a prevalent phenomenon in multi-agent systems and has been extensively studied in a diverse range of fields, including biology, physics, control engineering, and social science. The Krause mean process is a generalized model of opinion dynamics among many agents that represents opinions as vectors. In this work, we investigate an opinion sharing dynamics in the multi-agent system by means of Krause mean processes which are generated by doubly stochastic hyper-matrices. This is arguably a feasible generalization of the classical DeGroot and Chatterjee-Seneta models from square stochastic matrices to higher-order stochastic hyper-matrices. We then demonstrate how consensus can be achieved in the multi-agent system when doubly stochastic hyper-matrices are off-diagonally uniformly positive. Keywords Multi-agent system · Consensus · Krause mean process · Stochastic hyper-matrix · Polynomial operator

M. Saburov (B) College of Engineering and Technology, American University of the Middle East, Egaila, Kuwait e-mail: [email protected] K. Saburov National University of Uzbekistan, Tashkent, Uzbekistan e-mail: [email protected] YEOJU Technical Institute in Tashkent, Tashkent, Uzbekistan AJOU University in Tashkent, Tashkent, Uzbekistan © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 S. Olaru et al. (eds.), Difference Equations, Discrete Dynamical Systems and Applications, Springer Proceedings in Mathematics & Statistics 444, https://doi.org/10.1007/978-3-031-51049-6_14

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1 Introduction A multi-agent system is a collection of multiple intelligent agents that interact with each other and may possess different information and conflicting interests. These agents can be embodied in the form of robots, individuals, or human teams. Humans, as complex beings, are influenced by various factors including social context, culture, law, and other factors that dictate their behavior. Despite these complexities, human societies exhibit remarkable global patterns that transition from disorder to order. To understand such social behavior, there is a need for a mathematical model to explain the regularities that emerge from the collective interactions among individuals. Opinions, being a fundamental aspect of human behavior, can be viewed as the internal states that drive an individual’s actions. Opinion dynamics refers to the continuous fusion of individual opinions among a group of interacting agents, guided by established rules, leading to a final outcome in the form of consensus, polarization, or fragmentation. The evolution of opinions among a group of interacting individuals has been the subject of various mathematical models. Most of these models adopt a linear approach. In the study of opinion dynamics, a majority of the research has been focused on the problem of reaching consensus. The concept of reaching consensus in a structured time-invariant and synchronous environment was first introduced by DeGroot in [6]. Later, Chatterjee and Seneta [5] extended this model to a structured timevarying and synchronous environment. In these models, the opinion sharing dynamics of a structured time-varying synchronous multi-agent system are represented by the backward product of square stochastic matrices, while a non-homogeneous Markov chain is represented by the forward product of square stochastic matrices. Thus, the consensus in the multi-agent system and the ergodicity of the Markov chain are considered dual problems. Since DeGroot’s introduction of the concept of consensus in structured, time-invariant, and synchronous environments, the phenomenon has become increasingly prevalent in a variety of scientific fields, including biology, physics, control engineering, and social science, as evidenced by a growing body of literature (e.g., [13, 24, 46, 47]). The concept of consensus has emerged as a ubiquitous aspect of multi-agent systems research. Recently, various nonlinear models have been developed to analyze opinion dynamics within social communities (as demonstrated in the references [10–12, 17–20, 23]). A Krause mean process presents a general model for the opinion sharing dynamics, where individual opinions are identified as vectors. For a comprehensive understanding of this model, the reader is advised to consult the monograph [21]. The concept of a quadratic stochastic process was first introduced by Bernstein (see [3]) in 1942 and is considered to be the simplest nonlinear Markov chain (see [16]). Subsequently, the analytic theory of quadratic stochastic processes generated by cubic stochastic matrices was established in the papers [7, 44]. Since then, quadratic stochastic operators have been widely used as a key tool for the study of dynamical properties and modeling in various fields such as biology [14, 22], physics [48], and control systems [35–40]. The properties of quadratic stochastic operators

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defined on the finite-dimensional simplex, including their fixed points and omega limiting sets, were explored in the series of papers [41–43]. Meanwhile, the studies in [28–34] focused on the ergodicity and chaotic dynamics of quadratic stochastic operators defined on the finite-dimensional simplex. Additionally, the correlation between the Krause mean process and quadratic stochastic processes has been established in a series of papers [35–40]. An overview of recent advancements and open problems in the theory of quadratic stochastic operators and processes can be found in the paper [9]. In this paper, we investigate an opinion sharing dynamics in a multi-agent system by means of Krause mean processes which are generated by off-diagonally uniformly positive nonautonomous doubly stochastic hyper-matrices. Our goal is to demonstrate how consensus can be achieved in the multi-agent system. The proposed model (see P ROTOCOL –DSHM) is arguably a feasible generalization of the classical DeGroot and Chatterjee-Seneta models from square stochastic matrices to higher-order stochastic hyper-matrices. The main result (see Theorem 1) of this paper extends, generalizes, and unifies all previous results established in the series of papers [4, 27, 35–40]. The paper’s structure is organized as follows: Sect. 2 presents a concise overview of Krause mean processes, featuring classical examples of both linear and nonlinear models. Subsequently, in Sect. 3, we discuss Markov processes with memory which are dual processes to Krause mean processes. In Sect. 4, we proceed to introduce Krause mean processes generated by doubly stochastic hyper-matrices (see P ROTOCOL –DSHM). This is a plausible extension of the classical DeGroot and Chatterjee-Seneta models from square stochastic matrices to higher-order stochastic hyper-matrices. Section 5 is featured with the main result of this paper. In Sect. 6, we engage in a comprehensive analysis of our results, drawing comparisons with other relevant findings. Finally, concluding remarks will be presented in Sect. 7.

2 The Krause Mean Processes We first overview a general model of the opinion sharing dynamics in a multiagent system, originally introduced in [11], that encompasses all classical models of opinion sharing dynamics studied in the papers [2, 5, 6]. The study focuses on a group of .m individuals, denoted as .Im := {1, · · · , m}, who work together as a team or committee. Each individual holds a subjective distribution for a given task, and it is assumed that they may revise their distribution in response to information received from other members of the group. Let .x(t) = (x1 (t), · · · , xm (t))T be the subjective distributions of the multi-agent system at the time .t where .xi (t) ≥ 0 for all .i ∈ Im . Let . pi j (t, x(t)) denote the weight that the .i th individual assigns to .x j (t) when he/she makes the revision at the time .t + 1. It is assumed that

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M. Saburov and K. Saburov m ∑ .

pi j (t, x(t)) = 1 and pi j (t, x(t)) ≥ 0, ∀ i, j ∈ Im .

j=1

After being informed of the subjective distributions of the other members of the group, the individual numbered .i revises his/her own subjective distribution from . x i (t) to . x i (t + 1) as follows x (t + 1) =

m ∑

. i

pi j (t, x(t))x j (t), ∀ i ∈ Im

j=1

Let .P (t, x(t)) denote an .m × m row-stochastic matrix whose .(i j) element is pi j (t, x(t)). A general model of the structured time-varying synchronous system is defined as follows

.

x(t + 1) = P (t, x(t)) x(t), ∀ t ∈ N.

.

(1)

We say that a consensus is reached in the structured time-varying synchronous multi-agent system (1) if .x(t) converges to .c = (c, · · · , c)T as .t → ∞. Obviously, the consensus .c = c(x(0)) might depend on an initial opinion .x(0). We may then obtain all classical models [2, 5, 6, 11, 12] by choosing suitable matrices .P (t, x(t)). Let us now illustrate some of them here. Example 1 (The DeGroot Model) Let us assume that .P (t, x(t)) = P is a timeinvariant square row-stochastic matrix. We then derive the model in a structured time-invariant and synchronous environment that was first introduced by DeGroot (see [6]). In this case, the opinion sharing dynamics of the DeGroot model is given as follows x(t + 1) = Px(t), ∀ t ∈ N.

.

(2)

Example 2 (The Chatterjee–Seneta Model) Let us assume that .P (t, x(t)) = P(t) is a time-varying square row-stochastic matrix. We then derive the model in a structured time-varying and synchronous environment that was introduced by Chatterjee and Seneta (see [5]). In this case, the opinion sharing dynamics of the Chatterjee–Seneta model is given as follows x(t + 1) = P(t)x(t), ∀ t ∈ N.

.

(3)

Example 3 (The Hegselmann–Krause Model) Roughly speaking, in a bounded confidence model of opinion dynamics, an agent is influenced only by those agents whose opinions are within his/her confidence bound. Formally speaking, we assume that an agent .i engages in interactions and adopts the opinions of only those agents whose opinions deviate from that of agent .i by no more than a specific confidence level .εi . At the step .t, for each agent .i having opinion profile . x i (t), we define a bounded confidence set of agents as follows

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307

CS(i, εi ) := { j ∈ Im : |xi (t) − x j (t)| ≤ εi }. The set.CS(i, εi ) is called the confidence neighborhood of the agent.i at the step.t. We assume that the agent .i assigns an equal weight to all of his/her confidence neighborhoods. Subsequently, we can define a matrix .PCS (t, x(t)) = ( picsj (t, x(t)))i,m j=1 for any .t and for any .x(t) as follows ( picsj (t, x(t))

=

0, 1 , |CS(i,εi )|

if j ∈ / CS(i, εi ) if j ∈ CS(i, εi )

where .|CS(i, εi )| is the number of elements of the finite set .CS(i, εi ). In this case, the opinion sharing dynamics of The Hegselmann–Krause Model is given as follows x(t + 1) = PCS (t, x(t)) x(t), ∀ t ∈ N.

.

(4)

The Krause mean process is a more general representation of opinion sharing dynamics, in which opinions are represented by vectors. For a thorough understanding of these processes, the readers are encouraged to consult the monograph [21] by Krause. Let.S be a non-empty convex subset of.Rd and.Sm be the.m−fold Cartesian product of .S. m Definition 1 (The Krause mean process [21]) A given sequence .{x(t)}∞ t=0 ⊂ S T m where .x(t) = (x1 (t), · · · , xm (t)) is called a Krause mean process on .S if one has

xi (t + 1) ∈ conv{x1 (t), · · · , xm (t)}, ∀ i ∈ Im , t ∈ N, where .conv{·} is a convex hull of a set. m In other words, a given sequence .{x(t)}∞ t=0 ⊂ S is the Krause mean process if one has

conv{x1 (t + 1), · · · , xm (t + 1)} ⊂ conv{x1 (t), · · · , xm (t)}, ∀ t ∈ N Definition 2 (The Krause mean operator [21]) A mapping .T : Sm → Sm is called a t Krause mean operator if its trajectory .{x(t)}∞ t=0 , x(t) = T (x(0)) starting from any m initial point .x(0) ∈ S generates a Krause mean process on .Sm . It is noteworthy to mention that the nonlinear model of opinion sharing dynamics given by (1) is the Krause mean process due to the fact that the action of a stochastic matrix .P = ( pi j )i,m j=1 on a vector .x = (x1 , · · · , xm )T can be viewed as formation of arithmetic means ∑m pi j x j , ∀ i ∈ Im (Px)i = j=1

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with weights . pi j . The various kinds of nonlinear models of mean processes have been studied in the series of papers [11, 12, 17–20].

3 A Markov Chain with Memory In general, the Krause mean process and the Markov chain are dual processes to each other. For the purpose of this discussion, we will briefly review some fundamental definitions in the theory of Markov chains as described in the reference [45]. Recall that a discrete-time Markov chain (or a Markov chain with memory 1) is a stochastic process with a sequence of random variables .{X t , t = 0, 1, 2 . . .} , which takes on values in a discrete finite state space .[m] = {1, . . . , m} for a positive integer .m such that .

pi1 j = Prob (X t+1 = j|X t = i 1 , X t−1 = i 2 , . . . , X 1 = i t , X 0 = i t+1 ) = Prob (X t+1 = j|X t = i 1 )

∑ where .i 1 , · · · , i k , · · · , i t+1 , j ∈ [m] and . mj=1 pi1 j = 1, pi1 j ≥ 0, 1 ≤ i 1 , j ≤ m. In other words, the probability of moving to the next state depends( only )mon the present state and not on the previous states. A stochastic matrix .P = pi1 j i1 , j=1 is called one-step transition matrix of the Markov chain. Let .x j (t) = Prob{X t = j} be a distribution of the state . j at time .t. A distribution T of the ∑m Markov chain at time .t is a stochastic vector .x(t) = (x1 (t), · · · , xm (t)) , i.e., . j=1 x j (t) = 1 and . x j (t) ≥ 0 for any .1 ≤ j ≤ m. The transition of the distributions from .x(t) to .x(t + 1) is then governed by the rule x j (t + 1) =

∑m i 1 =1

pi1 j xi1 (t), 1 ≤ j ≤ m.

{ } ∑ Let .∆m−1 = x ∈ Rm : m be an .(m − 1)− k=1 x k = 1, x k ≥ 0, 1 ≤ k ≤ m m−1 m−1 dimensional simplex. (A linear operator .L : ∆ → ∆ associated with the )m stochastic matrix .P = pi1 j i1 , j=1 by .L(x) = xT P, i.e., .

(L(x)) j =

m ∑

pi1 j xi1 , 1 ≤ j ≤ m

(5)

i 1 =1

is called a linear Markov operator. A discrete-time Markov chain with memory .k (a .k− order Markov chain, see [1, 26]) is a stochastic process with a sequence of random variables .{X t , t = 0, 1, 2 . . .}, which takes on values in a discrete finite state space .Im = {1, . . . , m} for a positive integer .m such that

Krause Mean Processes Generated by Off-Diagonally … .

309

pi1 ···ik j = Prob (X t+1 = j|X t = i 1 , X t−1 = i 2 , . . . , X 1 = i t , X 0 = i t+1 ) = Prob (X t+1 = j|X t = i 1 , · · · , X t−k+1 = i k )

where .i 1 , · · · , i k , · · · , i t+1 , j ∈ Im and m ∑

pi1 ···ik j = 1,

pi1 ···ik j ≥ 0, 1 ≤ i 1 , · · · , i k , j ≤ m.

j=1

In other words, the probability of moving to the next state depends only on the past .k states (see [1, 26]). If.k = 1 then we obtain a Markov chain. If.k > 1 then it is possible to construct a new chain that possesses the Markov property. This can be achieved by considering the state space as the ordered .k−tuples of successive elements. An m,··· ,m,m .(k + 1)−order .m− dimensional stochastic hyper-matrix .P = ( pi 1 ···i k j )i ,··· ,i , j=1 is 1 k called the one-step transition hyper-matrix of the Markov chain with memory .k. Let .x j (t) = Prob{X t = j} be the distribution of the state . j at time .t. The distribution of the Markov chain with ∑m memory .k at time .t is a stochastic vector T .x(t) = (x 1 (t), · · · , x m (t)) , i.e., . j=1 x j (t) = 1 and . x j (t) ≥ 0 for any .1 ≤ j ≤ m. The transition of the distributions from .x(t) to .x(t + 1) is then governed by the following rule ∑

x j (t + 1) =

pi1 i2 ···ik j xi1 (t)xi2 (t − 1) · · · xik (t − k + 1), 1 ≤ j ≤ m.

1≤i 1 i 2 ···i k ≤m

A polynomial stochastic operator .P : ∆m−1 → ∆m−1 associated with the .(k + ,m,m defined 1)−order .m−dimensional stochastic hyper-matrix .P = ( pi1 ···ik j )im,··· 1 ,··· ,i k , j=1 by .

(P(x)) j =

∑

pi1 ···ik j xi1 · · · xik , 1 ≤ j ≤ m

(6)

1≤i 1 ···i k ≤m

is called a nonlinear Markov operator (see [16]). In the case.k = 1, we obtain a linear Markov operator, meanwhile in the case .k = 2, we obtain a quadratic stochastic operator which has an incredible application in population genetics (see [22]).

4 The Krause Mean Process Generated by Doubly Stochastic Hyper-Matrices In this section, we explore the connection between the Krause mean processes and Markov chains with memory .k. To begin with, we will present some definitions and notations that will be utilized throughout this paper.

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Definition 3 (Stochastic Hyper-Matrix) A .(k + 1)−order .m−dimensional hyper,m,m is called stochastic if one has matrix .P = ( pi1 ···ik j )im,··· 1 ,··· ,i k , j=1 m ∑

pi1 ···ik j = 1, pi1 ···ik j ≥ 0,

1 ≤ i 1 , · · · , i k , j ≤ m.

j=1

Definition 4 (Doubly Stochastic Hyper-Matrix) A .(k + 1)−order .m−dimensional ,m,m is called doubly stochastic if one has hyper-matrix .P = ( pi1 ···ik j )im,··· 1 ,··· ,i k , j=1 m ∑

pi1 ···ik j =

i k =1

m ∑

pi1 ···ik j = 1, pi1 ···ik j ≥ 0,

1 ≤ i 1 , · · · , i k , j ≤ m.

j=1

,m,m Let.P = ( pi1 ···ik j )im,··· be the.(k + 1)−order.m−dimensional doubly stochas1 ,··· ,i k j=1 ,m be its .k−order .m−dimensional .l th tic hyper-matrix and .Pl = ( pi1 ···ik l )im,··· 1 ,··· ,i k =1 ,m subhyper-matrix for fixed .l. It is clear that .Pl = ( pi1 ···ik l )im,··· is also stochastic 1 ,··· ,i k =1 hyper-matrix. In the sequel, we write .P = (P1 |P2 | · · · |Pm ) for the .(k + 1)−order .m−dimensional doubly stochastic hyper-matrix. We define a polynomial stochastic operator .P : ∆m−1 → ∆m−1 associated with .(k + 1)−order .m−dimensional doubly stochastic hyper-matrix .P = (P1 | · · · |Pm ) as follows

.

(P(x))l =

m ∑

···

i 1 =1

m ∑

pi1 ···ik l xi1 · · · xik ,

1 ≤ l ≤ m.

(7)

i k =1

We also define a polynomial stochastic operator .Pl : ∆m−1 → ∆m−1 associated ,m as with the .k−order .m−dimensional stochastic hyper-matrix .Pl = ( pi1 ···ik l )im,··· 1 ,··· ,i k =1 follows .

(Pl (x)) j =

m ∑

···

i 1 =1

m ∑

pi1 ···ik−1 jl xi1 · · · xik−1 ,

1≤ j ≤m

(8)

i k−1 =1

for all .l ∈ Im . It follows from (7) and (8) that .

(P(x))l =

m ∑

( ) (Pl (x)) j x j = Pl (x), x , 1 ≤ l ≤ m

j=1

where .(·, ·) stands for the standard inner product of two vectors. Therefore, the polynomial stochastic operator .P : ∆m−1 → ∆m−1 given by (7) can be written as follows

Krause Mean Processes Generated by Off-Diagonally …

P(x) =

.

311

(( ) ( )) T P1 (x), x , · · · , Pm (x), x

where .Pl : ∆m−1 → ∆m−1 is defined by (8) for all .l ∈ Im . We now define an .m × m matrix as follows ) ( ) ( ) ⎞ ⎛( (P1 (x))1 (P1 (x))2 · · · (P1 (x))m ⎜ P2 (x) P2 (x) 2 · · · P2 (x) m ⎟ 1 ⎜ ⎟ .P(x) = ⎜ ⎟. .. .. .. .. ⎝ ⎠ . . . . ( ) ( ) ( ) Pm (x) 1 Pm (x) 2 · · · Pm (x) m

(9)

(10)

We show that .P(x) stochastic matrix for every .x ∈ ∆m−1 . In fact we ( is doubly )m know that .P(x) = pi j (x) i, j=1 where

.

m m ∑ ∑ ( ) pi j (x) = Pi (x) j = ··· pi1 ···ik−1 ji xi1 · · · xik−1 . i 1 =1

(11)

i k−1 =1

Therefore, we have m ∑ .

pi j (x) =

···

i 1 =1

i=1

= m ∑

m ∑

pi j (x) =

j=1

=

m ∑

m ∑ i k−1 =1

···

m ∑

i 1 =1

i k−1 =1

m ∑

m ∑

···

i 1 =1

i k−1 =1

m ∑

m ∑

i 1 =1

···

( m ∑

) pi1 ···ik−1 ji

xi1 · · · xik−1

i=1

xi1 · · · xik−1 = (x1 + · · · + xm )k−1 = 1, ⎛ ⎞ m ∑ ⎝ pi1 ···ik−1 ji ⎠ xi1 · · · xik−1 j=1

xi1 · · · xik−1 = (x1 + · · · + xm )k−1 = 1.

i k−1 =1

Hence, it follows from (9) and (10) that P(x) = P(x)x

.

(12)

and we call the equation (12) a matrix form of the polynomial stochastic operator (7) associated with the .(k + 1)−order .m−dimensional doubly stochastic hyper-matrix. We are now ready to present the main protocol (model) of this paper. This is the Krause mean process generated by doubly stochastic hyper-matrices (for short, we use DSHM). P ROTOCOL –DSHM: Let .{k(n)}∞ of natural numbers such that n=1 be a sequence )m,··· ,m,m ( (n) ∞ .2 ≤ k(n) ≤ k 0 for all .n ∈ N and .{Pn }n=1 , .Pn = pi ···i be a sequence 1 k(n) j i 1 ,··· ,i k(n) , j=1 of.(k(n) + 1)−order.m−dimensional doubly stochastic hyper-matrices. Let.{Pn }∞ n=1 ,

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Pn : ∆m−1 → ∆m−1 be a sequence of polynomial stochastic operators associated with .(k(n) + 1)−order .m−dimensional doubly stochastic hyper-matrices .{Pn }∞ n=1 . Suppose that an opinion sharing dynamics of the multi-agent system is generated by non-autonomous polynomial stochastic operators as follows

.

( ) x(n+1) = Pn x(n) , x(1) ∈ ∆m−1

.

(13)

)T ( where .x(n) = x1(n) , · · · , xm(n) is the subjective distribution after .n revisions. It follows from (12) that the opinion sharing dynamics of the multi-agent system given by P ROTOCOL –DSHM can be written as follows ( ) x(n+1) = Pn x(n) = Pn (x(n) )x(n) , x(1) ∈ ∆m−1

.

(14)

This means that, due to the matrix form (1), the opinion sharing dynamics of the multi-agent system given by P ROTOCOL –DSHM generates the Krause mean process. Consequently, we have the following result. Proposition 1 Let .{k(n)}∞ of natural numbers such that .2 ≤ n=1 ⊂ N be a sequence )m,··· ,m,m ( (n) , . P = p k(n) ≤ k0 for all .n ∈ N and .{Pn }∞ n n=1 i 1 ···i k(n) j i 1 ,··· ,i k(n) , j=1 be a sequence of ∞ .(k(n) + 1)−order .m−dimensional doubly stochastic hyper-matrices. Let .{Pn }n=1 , m−1 m−1 .Pn : ∆ →∆ be a sequence of polynomial stochastic operators associated with .(k(n) + 1)−order .m−dimensional doubly stochastic hyper-matrices .{Pn }∞ n=1 . Then the opinion sharing dynamics of the multi-agent system given by P ROTOCOL – DSHM generates the Krause mean process. Remark 1 Some special cases of P ROTOCOL –DSHM have been considered in the previous studies: the case .k(n) = 2 for all .n ∈ N and .{Pn }∞ n=1 = {P} in [35, 36]; the case .k(n) = 2 for all .n ∈ N with any sequence .{Pn }∞ n=1 in [38, 40]; the case ∞ .k(n) = k ≥ 2 for all .n ∈ N and .{Pn }n=1 = {P} in [37, 39]. In this paper, we unify, extend, and generalize all previous results presented in the papers [35–40]. Remark 2 We now propose the multi-agent system interpretation of P ROTOCOL – DSHM. We assume that each agent has the ability to revise and change his/her own opinion on a given issue or task following the influence exerted by all possible groups which may consist of a minimum of two agents and a maximum of .k0 agents. This obviously creates non-linearity in the proposed model. In order to keep the model homogeneous, we interpret the influence of a single agent as the influence of a group of many identical agents. More precisely, the following assumptions are made: • There is a group of .m agents .Im := {1, · · · , m} acting together as a team or committee; • Each agent can specify his/her own opinion on some given issue or task. An opinion is a generic concept that represents an agent’s belief or behavior or attitude; • An opinion profile at the step .n is a stochastic vector .x(n) = (x1(n) , · · · , xm(n) )T ;

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313

• At the step .n, each agent, say . j, is greatly influenced by a group of ranked agents .{i 1 i 2 · · · i k(n) } with .k(n) members, .2 ≤ k(n) ≤ k0 , in which .i k(n) is the leader or spokesman of the group; • At the step .n, the influence of a group of ranked agents .{i 1 i 2 · · · i k(n) } on an agent . j is . pi(n) ; 1 i 2 ···i k(n) , j • In general, the influences of two groups of the same agents that are ranked in two different ways, have different impacts on the same agent . j. In other words, it is possible to have /= pi(n) ; pi(n) 1 i 2 ···i k(n) , j π(1) i π(2) ···i π(k(n)) , j • The influence profile of the group of ranked agents .{i 1 step .n is a stochastic vector

i2

···

i k(n) } at the

)T ( (n) (n) (n) pi(n) := p , p , · · · , p ; i i i i ···i • i ···i ,1 i ···i ,2 i ···i ,m 1 2 k(n) 1 2 k(n) 1 2 k(n) 1 2 k(n) • The collective influence of all possible groups of agents with .k(n) members on an agent . j at the step .n is a stochastic hyper-matrix (n) m,··· ,m P (n) j = ( pi 1 ···i k(n) , j )i 1 ,··· ,i k(n) =1 ;

• An agent . j believes that a group of ranked agents .{i 1 i 2 · · · i k(n) } is trusted/influential at the step .n if the influence . pi(n) of the group on the 1 i 2 ···i k(n) , j agent . j is high; (n) (n) • The trust . p (n) ji (x ) of an agent . j towards an agent .i with the opinion profile .x at the step .n is the total joint influences exerted on the agent . j by all possible groups of .k(n) agents, in which the agent .i always serves as a spokesman, while maintaining the opinion profile .x(n) , i.e., (n) p (n) ji (x ) =

m ∑

···

i 1 =1

m ∑

pi(n) xi(n) · · · xi(n) ; 1 ···i k(n)−1 i, j 1 k(n)−1

i k(n)−1 =1

• The trust matrix with) the opinion profile .x(n) is a square stochastic matrix ( (n) (n) m .Pn (x ) = p (n) (x ) j,i=1 ; ji • The opinion profile at the next step .n + 1 is then revised as follows ( ) x(n+1) := Pn x(n) x(n) . It is obvious that for (any nonzero vector .x ≥ 0, i.e, .xk ≥(0 for)all .1 ≤ k ≤ m, ) x x x k ∈ ∆m−1 . Con∈ ∆m−1 and .P ||x|| we have .P(x) = ||x||1 P ||x||1 in which . ||x|| 1 1

sequently, if .||x(0) ||1 > 1 then . lim ||x(n) ||1 = lim ||x(0) ||k1 = ∞ and if .||x(0) ||1 < 1 n

n→∞

then . lim ||x(n) ||1 = lim ||x(0) ||k1 = 0. n

n→∞

n→∞

n→∞

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{ } Hence, unlike the linear case, the simplex .∆m−1 = x ∈ Rm + : ||x||1 = 1 is only invariant set under the action of the polynomial stochastic operator, i.e., ) ( an m−1 ⊂ ∆m−1 . In other words, for any initial opinion .x(0) ∈ ∆m−1 the opin.P ∆ ion sharing dynamics of the multi-agent system given by P ROTOCOL –DSHM (n) ∈ ∆m−1 for all .n ∈ N. Simigenerates a vector sequence .{x(n) }∞ n=0 for which .x lar to the linear case, we say that a consensus is reached in the structured timevarying synchronous multi-agent system if a vector sequence .{x(n) }∞ n=0 converges to T m−1 .c = (c, · · · , c) ∈ ∆ as .n → ∞. However, it is obvious that the only stochastic vector of the simplex with equal distributions is the center of the simplex, i.e., 1 1 T .c = ( , · · · , ) . Consequently, unlike the linear case, the consensus in the opinm m ion sharing dynamics given by P ROTOCOL –DSHM does not depend on the initial opinion .x(0) ∈ ∆m−1 and it can be stated as follows. Definition 5 (Consensus) We say that the opinion sharing dynamics of the multiagent system given by P ROTOCOL –DSHM eventually reaches a consensus if an opinion sharing dynamics converges to the center .c = ( m1 , · · · , m1 )T of the simplex m−1 .∆ .

5 The Main Result Let us first introduce a notion of off-diagonally uniformly positive stochastic hypermatrices. Definition 6 (Off-diagonally uniformly positive hyper-matrices) of natural numbers such that .2 ≤ k(n) ≤ k0 Let .{k(n)}∞ n=1 ⊂ N be a sequence )m,··· ,m,m ( (n) for all .n ∈ N and .{Pn }∞ , . P = p n n=1 i 1 ···i k(n) j i 1 ,··· ,i k(n) , j=1 be a sequence of .(k(n) + 1)−order .m−dimensional doubly stochastic hyper-matrices. We say that a sequence of stochastic hyper-matrices .{Pn }∞ n=1 is off-diagonally uniformly positive if there exists .δ > 0 such that . pi(n) > δ for any .n ∈ N and for any .i 1 , · · · i k(n) , j ∈ Im 1 ···i k(n) j with at least two different indices .i 1 , · · · i k(n) . We are now ready to state the main result of this paper. Theorem 1 (Consensus of doubly stochastic hyper-matrices) Let .{k(n)}∞ n=1 ⊂ N be a sequence of natural numbers such that.2 ≤ k(n) ≤ k0 for all.n ∈ N and.{Pn }∞ n=1 , )m,··· ,m,m ( (n) .Pn = pi ···i be a sequence of . (k(n) + 1)−order . m−dimensional 1 k(n) j i 1 ,··· ,i k(n) , j=1 m−1 doubly stochastic hyper-matrices. Let .{Pn }∞ → ∆m−1 be a sequence n=1 , .Pn : ∆ of polynomial stochastic operators associated with .(k(n) + 1)−order .m− T dimensional doubly stochastic hyper-matrices .{Pn }∞ n=1 . Let .e1 = (1, 0, 0, · · · , 0) , T T m−1 .e2 = (0, 1, 0, · · · , 0) , .em = (0, 0, 0, · · · , 1) be vertices of the simplex .∆ and ( ) (n+1) .ek := Pn ek(n) , where .ek(0) := ek for all .k ∈ Im and .n ∈ N. If a sequence of (n k ) stochastic hyper-matrices .{Pn }∞ ∈ n=1 is off-diagonally uniformly positive and .ek int∆m−1 for each .k ∈ Im with some .n k ∈ N then the opinion sharing dynamics of the

Krause Mean Processes Generated by Off-Diagonally …

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multi-agent system given by P ROTOCOL –DSHM eventually reaches a consensus for any initial opinion. T T Proof Let .e1 = (1, 0, 0, · · · , 0)T , .e2 = (0, 0, m = (0, 0, 0, · · · , 1) { 1, } · · · , 0) , .e(n+1) m−1 (n) ∞ (n) be vertices of the simplex .∆ and let . x n=1 where .x = Pn (x ) be a m−1 m−1 trajectory of the polynomial stochastic operators .Pn : ∆ → ∆ starting from { }

an initial point .x(1) ∈ ∆m−1 . Particularly, let . ek(n)

∞

n=1

be a trajectory of the polyno-

→∆ starting from a vertex .ek of the simplex mial stochastic operator .Pn : ∆ Sm−1 for all .k ∈ Im . According to the definition, the multi-agent system eventually 1 T 1 reaches a consensus if .{x(n) }∞ n=1 converges to the center .c = ( m , · · · , m ) of the simplex .∆m−1 for any initial point .x(1) ∈ ∆m−1 . We accomplish it under two hypotheses: m−1

m−1

.

(i) For each .k ∈ Im one has .ek(n k ) ∈ int∆m−1 for some .n k ∈ N; (ii) A sequence of stochastic hyper-matrices .{Pn }∞ n=1 is off-diagonally uniformly positive (see Definition 6). Step-1 We first show that .Pn (int∆m−1 ) ⊂ int∆m−1 for any .n ∈ (N. Indeed, )m let .x ∈ (x) is a int∆m−1 . This means that .xi > 0 for all .i ∈ Im . Since .Pn (x) = pk(n) j k, j=1 square doubly stochastic matrix and .Pn (x) = Pn (x)x, we derive that 0 < min x j ≤ j∈Im

m ∑

pk(n) j (x)x j = (Pn (x))k ,

∀ k ∈ Im .

j=1

This means that .Pn (x) ∈ int∆m−1 . Step-2 We now show that there exists .n 0 ∈ N such that for any initial point .x(1) ∈ ∆m−1 one has .x(n 0 ) ∈ int∆m−1 . It has been noted that .n 0 does not depend on an initial point .x(1) ∈ ∆m−1 . Since for each .k ∈ Im one has .ek(n k ) ∈ intSm−1 for some .n k ∈ N, it then follows from the previous step that for each .k ∈ Im one has (n) .ek ∈ int∆m−1 for any .n > n k . Let .n 0 := max n k . Then .ek(n 0 ) ∈ int∆m−1 for all .k ∈ Im . We now show that k∈Im

x(n 0 +1) = Pn 0 (· · · (P1 (x(1) )) · · · ) ∈ int∆m−1 for any initial point .x(1) ∈ ∆m−1 . In order to prove it, we first prove the following inequality for any initial point (1) .x ∈ ∆m−1 .

x(n+1) ≥ x1K(n) e1(n) + x2K(n) e2(n) + · · · + xmK(n) em(n) , n ∈ N

.

(15)

where .K(n) = k(1) · k(2) · . . . · k(n) for .n ∈ N. Let us first introduce some necessary notations. Let .MPn : (Rm )×k(n) → Rm be a multi-linear operator asso+ 1)−order .m−dimensional stochastic hyper-matrix .Pn = ciated )m,···.(k(n) ( (n) with ,m,m pi1 ···ik(n) j i1 ,··· ,ik(n) , j=1 as follows m m ∑ ( ) ∑ MPn y(1) , y(2) , · · · , y(k(n)) = ··· yi(1) yi(2) · · · yi(k(n)) pi1 ···ik(n) • 1 2 k(n)

.

i 1 =1

i k(n) =1

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where .pi1 ···ik(n) • = ( pi1 ···ik(n) 1 , · · · , pi1 ···ik(n) m ) ∈ ∆m−1 for any .i 1 , · · · , i k(n) ∈ Im . It is clear that .Pn (x) = MPn (x, x, · · · , x) for any .x ∈ ∆m−1 . Moreover, if m−1 .x = λ1 v1 + · · · + λq vq ∈ ∆ with .v1 , · · · , vq ∈ ∆m−1 , λ1 + · · · + λq = 1, and .λ1 , · · · , λq ≥ 0 then Pn (x) =

q ∑

.

i 1 =1

···

q ∑

( ) λi1 · · · λik(n) MPn vi1 , · · · , vik(n)

i k(n) =1 k(n) = λk(n) 1 Pn (v1 ) + · · · + λq Pn (vq )+ ∑ ( ) + λi1 · · · λik(n) MPn vi1 , · · · , vik(n)

(16)

at least for two i μ ,i ν : i μ /=i ν

Hence, it follows form (16) that .x

(2)

K(1) (1) e1 K(2) (2) x 1 e1

+ x2

(1) + · · · + xmK(1) em + remaining parts,

+

K(1) (1) e2 K(2) (2) x 2 e2

(2) + · · · + xmK(2) em + remaining parts,

K(n) (n) e1

+ x2

K(n) (n) e2

(n) + · · · + xmK(n) em + remaining parts.

= P1 (x(1) ) = x1

x(3) = P2 (x(2) ) = .. .

x(n+1) = Pn (x(n) ) = x1

Consequently, the last equality yields the inequality (15). Moreover, it follows from the inequality (15) and .ei(n) > 0 for any .n > n 0 , i ∈ Im (see Step-2) that .x(n+1) > 0 for any .n > n 0 and for any .x(1) ∈ ∆m−1 . This shows that .x(n 0 +1) ∈ int∆m−1 . ({ }) Step-3 We now show that for any .x(1) ∈ ∆m−1 the omega limit set .ω x(n) (n) ∞ of the sequence a subset of the interior .int∆m−1 of the simplex ({ .{x}) }n=1 is m−1 m−1 .∆ i.e., .ω x(n) int∆ . This indeed follows from the previous step that .Pn 0 +1 (· · · (P1 (∆m−1 )) · · · ) int∆m−1 . Since the image of simplex under the polynomial stochastic operator is a compact set, there exists .α > 0 such that Pn 0 +1 (· · · (P1 (∆m−1 )) · · · ) ≥ αe := (α, α, · · · , α)T

∀ x ∈ ∆m−1 .

On the other hand, since the interior .int∆m−1 of the simplex .∆m−1 is an invariant set (see Step-1) and Pn (· · · (P1 (∆m−1 )) · · · ) ⊂ Pn 0 +1 (· · · (P1 (∆m−1 )) · · · ) ⊂ ∆α (n) for any.n > n 0 , we have that.{x(n) }∞ ≥ αe for any.n > n 0 where n=n 0 ⊂ ∆α , i.e.,.x

∆α := {x ∈ ∆m−1 : x ≥ αe}. ({ (n) }) of the sequence .{x(n) }∞ Consequently, the omega n=1 is a subset }) set .ω x m−1 ({ (n)limit ⊂ ∆α ⊂ int∆ for any .x(1) ∈ ∆m−1 . of the set .∆α , i.e., .ω x

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Step-4 As we showed in the previous step .Pn (· · · (P1 (∆m−1 )) · · · ) ⊂ ∆α for any .n > n 0 . It is therefore enough to study the dynamics of the polynomial stochastic operator over the set .∆α which is an invariant set. Let .x(1) ∈ ∆α . Then .x(n) ∈ ∆α , i.e., .x(n) ≥ αe for any .n ∈ N. It follows from the matrix form (12) of the polynomial stochastic operator that ( ) ( ) ( ) ( ) x(n+1) = Pn (x(n) ) = Pn x(n) x(n) = Pn x(n) · · · P2 x(2) P1 x(1) x(1)

.

where .Pn (x) is the square doubly stochastic matrix defined by (10). Let us set for any two integer numbers .n > r ( ) ( ) ( ) ( ) (n) (r ) P[x ,x ] := Pn x(n) Pn−1 x(n−1) · · · Pr +1 x(r +1) Pr x(r ) . Then for any .n ≥ r ≥ 0, we obtain (n) (1) (n) (r ) x(n+1) = P[x ,x ] x(1) = P[x ,x ] x(r ) .

.

We set .ρ := inf

n∈N

min

i 1 ,··· ,i k(n) , j∈Im at least two il1 /=il2

pi(n) > 0. 1 ···i k(n) j

( ) (n) m Then from (11), for a stochastic matrix .Pn (x(n) ) = pi(n) j (x ) i, j=1 we have

.

(n) pi(n) j (x ) =

m ∑ i 1 =1

m ∑

···

i k(n)−1 =1

≥

pi(n) xi(n) · · · xi(n) 1 ···i k(n)−1 ji 1 k(n)−1 ∑

pi(n) xi(n) · · · xi(n) ≥ ρα k0 > 0 1 ···i k(n)−1 ji 1 k(n)−1

i 1 ,··· ,i k(n)−1 ∈Im at least two il1 /=il2

for any .i, j (∈ Im and).n ∈ N where .2 ≤ k(n) ≤ k0 for any .n ∈ N. Consequently, m Pn (x(n) ) = pi j (x(n) ) i, j=1 is a uniformly positive square stochastic matrix for any (1) .x ∈ ∆α and for any .n ∈ N. m ∑ | pi1 j − pi2 j | be Dobrushin’s ergodicity coefficient Step-5 Let .δ(P) = 21 max .

i 1 ,i 2 j=1

(see [45]) of a square stochastic matrix .P = ( pi j )i,m j=1 . Since .Pn (x(n) ) = ( )m pi j (x(n) ) i, j=1 is positive and its entries are uniformly bounded away from zero for any .x(1) ∈ ∆α and for any .n ∈ N, we obtain ( ) δ Pn (x(n) ) ≤ λ < 1, ∀ n ∈ N.

.

Hence, we have ( (n) (1) ) δ P[x ,x ] ≤ λn+1 ,

( (n) (1) ) lim δ P[x ,x ] = 0.

n→∞

(17)

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Therefore, the backward product of doubly stochastic matrices.{Px(n) }∞ n=1 is weakly (n) (1) ergodic (see [45]) which is also strongly ergodic (see [45]), i.e., . lim P[x ,x ] = n→∞

mc T c and

lim x(n+1) = lim P[x

n→∞

(n)

,x(1) ] (1)

n→∞

x

= c, ∀ x(1) ∈ ∆α ,

where .c = ( m1 , · · · , m1 )T . This completes the proof.

6 Discussions We now discuss the main result of this paper and compare it with the previous results [4, 27, 35–40] on the consensus problem. Definition 7 (Triply Stochastic Hyper-Matrix) A .(k + 1)−order .m− dimensional ,m,m is called triply stochastic if for any .i 1 , · · · , i k , hyper-matrix .P = ( pi1 ···ik j )im,··· 1 ,··· ,i k , j=1 j ∈ Im m ∑

pi1 ···ik−1 ik j =

i k−1 =1

m ∑ i k =1

pi1 ···ik−1 ik j =

m ∑

pi1 ···ik−1 ik j = 1, pi1 ···ik j ≥ 0.

j=1

,m,m It is obvious that if .P = ( pi1 ···ik j )im,··· is a triply stochastic hyper-matrix then 1 ,··· ,i k , j=1 it is also a doubly stochastic hyper-matrix. Consequently, a polynomial stochastic operator associated with a triply stochastic hyper-matrix also generates a Krause mean process. Also, the following P ROTOCOL –TSHM was studied in the series of papers [4, 27, 35–40]. P ROTOCOL –TSHM ( SEE [4, 27, 35–40]): Let .{k(n)}∞ n=1 be a sequence of natural numbers such that .2 ≤ k(n) ≤ k0 for all .n ∈ N and .{Pn }∞ n=1 , .Pn = ( (n) )m,··· ,m,m pi1 ···ik(n) j i1 ,··· ,ik(n) , j=1 be a sequence of .(k(n) + 1)−order .m−dimensional triply m−1 stochastic hyper-matrices. Let .{Pn }∞ → ∆m−1 be a sequence of polyn=1 , .Pn : ∆ nomial stochastic operators associated with.(k(n) + 1)−order.m−dimensional triply stochastic hyper-matrices .{Pn }∞ n=1 . Suppose that an opinion sharing dynamics of the multi-agent system is generated by non-autonomous polynomial stochastic operators

( ) x(n+1) = Pn x(n) , x(1) ∈ ∆m−1

.

(18)

( )T where .x(n) = x1(n) , · · · , xm(n) is the subjective distribution after .n revisions (see Remark 2). Remark 3 We would like to emphasize that, in this paper, we do not require the ,m,m , i.e., symmetricity of hyper-matrix .P = ( pi1 ···ik j )im,··· 1 ,··· ,i k , j=1 pi1 ···ik−1 ik j = piπ(1) ···iπ(k−1) iπ(k) j

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for any .i 1 , · · · , i k−1 , i k , j ∈ Im and any permutation .π of the set .Ik . As a result, in general, the property of double stochasticity for hyper-matrices does not necessarily imply triple stochasticity, leading to the class of all doubly stochastic hyper-matrices being larger than the class of triply stochastic hyper-matrices. Therefore, the main result (Theorem 1) of this paper extends, generalizes, and unifies all results of the papers [4, 27, 35–40]. To gain a complete understanding of the problem at hand, it is necessary to outline some of the results established in the papers [4, 27, 35–40]. Theorem 2 (Consensus of triply stochastic hyper-matrix [35–40]) Let .P = (P∗∗1 | · · · |P∗∗m ) be a cubic triply stochastic matrix and let .Q : Sm−1 → Sm−1 be the associated quadratic stochastic operator. If .P > 0 then the opinion sharing dynamics of the multi-agent system given by P ROTOCOL –TSM eventually reaches a consensus. Let us now compare Theorems 1 and 2 in the following examples. Example 4 We consider the cubic doubly stochastic matrix .P = (P1∗∗ |P2∗∗ |P3∗∗ ) where .P1∗∗ , .P2∗∗ , and .P3∗∗ are square doubly stochastic matrices given as follows ⎛

P1∗∗

.

⎛ ⎛ ⎞ ⎞ ⎞ p111 p112 p113 p211 p212 p213 p311 p312 p313 = ⎝ p121 p122 p123 ⎠ P2∗∗ = ⎝ p221 p222 p223 ⎠ P3∗∗ = ⎝ p321 p322 p323 ⎠ p131 p132 p133 p231 p232 p233 p331 p332 p333

The following quadratic stochastic operator.Q : S2 → S2 presents Protocol–DSHM ( T ) ( T ) ( T ) Q(x) = x1 P1∗∗ x + x2 P2∗∗ x + x3 P3∗∗ x = P (x) x

.

(19)

T T T where .P (x) = x1 P1∗∗ + x2 P2∗∗ + x3 P3∗∗ is a square doubly stochastic matrix. It has been demonstrated in previous works [35–40] that the consensus in the system governed by the Protocol–TSHM is established when the square doubly stochastic matrices .P1∗∗ > 0, .P2∗∗ > 0, and .P3∗∗ > 0 are positive and

⎛

P1∗∗ + P2∗∗ + P3∗∗

.

⎞ 111 = ⎝1 1 1⎠. 111

(20)

This result was extended in the work [4], where consensus was reached even without the constraint (20), albeit still requiring positivity of square doubly stochastic matrices .P1∗∗ > 0, .P2∗∗ > 0, .P3∗∗ > 0. Subsequently, these results were further improved in the paper [27]. Namely, without positivity of the square doubly stochastic matrices .P1∗∗ , P2∗∗ , P3∗∗ and without the constraint (20), the consensus is still established in the system described by Protocol–DSHM if the cubic doubly stochastic matrix .P is (only) diagonally primitive, i.e., for some .s ∈ N we have

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[

diag(P)

]s

⎛

⎞s p111 p112 p113 = ⎝ p221 p222 p223 ⎠ > 0. p331 p332 p333

However, there is still room for further advancement in these results. Indeed, Theorem 1 is further extension and generalization of the results published in the papers [4, 27, 35–40]. To illustrate it, let us now consider another example. Example 5 We consider the following square doubly stochastic matrices ⎛1 ⎜ P1∗∗ = ⎝

.

2 1 4 1 4

1 2 1 4 1 4

0 1 2 1 2

⎛1

⎞

⎜ ⎟ ⎠ P2∗∗ = ⎝

4 1 2 1 4

1 4 1 2 1 4

1 2

⎛1

⎞

⎜ 0⎟ ⎠ P3∗∗ = ⎝ 1 2

4 1 4 1 2

1 4 1 4 1 2

1 2 1 2

⎞ ⎟ ⎠

0

We then obtain the following quadratic stochastic operator .Q : S2 → S2 Q(x) = (x12 + x22 + x32 )

.

e1 + e2 e1 + e2 + 2e3 + (x1 x2 + x1 x3 + x2 x3 ) 2 2

where.e1 = (1, 0, 0)T ,.e2 = (0, 1, 0)T , and.e3 = (0, 0, 1)T are vertices of the simplex S2 . On the one hand, since for any .s ∈ N

.

⎛1 [

diag(P)

]s

⎜ = diag(P) = ⎝

2 1 2 1 2

1 2 1 2 1 2

0

⎞

0⎟ ⎠, 0

the cubic doubly stochastic matrix .P = (P1∗∗ |P2∗∗ |P3∗∗ ) is not diagonally primitive. But, since.pi j∗ = ( 41 , 41 , 21 ) > 0 for all.i, j ∈ I3 with.i /= j, i.e.,.P = (P1∗∗ |P2∗∗ |P3∗∗ ) is off-diagonally positive, due to Theorem 1, the opinion sharing dynamics of the multi-agent system given by P ROTOCOL –DSHM eventually reaches a consensus for any initial opinion .x ∈ S2 . In this sense, Theorem 1 extends, generalizes, and unifies all results of the papers [4, 27, 35–40]. Finally, we would like to mention that, in general, the Krause mean process eventually reaches a consensus if it has an eventually shrinking property (see [18–21]). In this paper, we have improved Krause’s result (see [18, 19]) in the special case when the Krause mean processes are generated by doubly stochastic hyper-matrices. Let us give some examples to support our claim. Example 6 We consider the following square doubly stochastic matrices ⎛

P1••

.

⎞ 100 = ⎝0 21 21 ⎠ , 0 21 21

⎛1

P2••

⎞ 0 21 = ⎝0 1 0⎠ , 1 0 21 2 2

⎛1 P3•• =

⎞ 0 0⎠ . 001

1 2 2 ⎝1 1 2 2

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321

We then obtain the following quadratic stochastic operator .Q : S2 → S2 Q(x) = x12 e1 + x22 e2 + x32 e3 + x1 x2

.

3c + e3 3c + e2 3c + e1 + x1 x3 + x2 x3 , 2 2 2

where.e1 = (1, 0, 0)T ,.e2 = (0, 1, 0)T , and.e3 = (0, 0, 1)T are vertices of the simplex S2 and .c = (1/3, 1/3, 1/3)T is a center of the simplex .S2 . We also obtain .Q(e1 ) = e1 , .Q(e2 ) = e2 , and .Q(e3 ) = e3 . This means that the Krause mean process generated by cubic doubly stochastic matrix does not have an eventually shrinking property. On the other hand, the Krause mean process generated by cubic doubly stochastic matrix eventually reaches a consensus for any initial opinion expect vertices of the simplex. Indeed, it is easy to check for any .x ∈ S2 that

.

min xk ≤ min (Q(x))k ≤ min (Q2 (x))k ≤ · · · ≤ max(Q2 (x))k ≤ max(Q(x))k ≤ max xk

k∈I3

k∈I3

k∈I3

k∈I3

k∈I3

k∈I3

Since .Q(x) ∈ intS2 for any .x ∈ S2 \ {e1 , e2 , e3 } and the center .c of the simplex is the only fixed point in the interior of the simplex, a trajectory of the quadratic stochastic operator .Q : S2 → S2 starting from any initial point except the vertices of the simplex always converges to the center of the simplex.

7 Conclusion Historically, an idea of reaching consensus for a structured, time-invariant, and synchronous environment was introduced by DeGroot [6]. Later, Chatterjee and Seneta [5] generalized DeGroot’s model to incorporate structured, time-varying, and synchronous environments. Since then, the consensus, which is a widely observed phenomenon in multi-agent systems, has garnered interest in numerous scientific fields, including biology, physics, control engineering and social science. To some extent, the Krause mean process is a general model for opinion sharing dynamics, in which opinions are represented by vectors, and serves as a general framework for understanding consensus. In the present study, we have introduced a model for opinion sharing dynamics employing Krause mean processes generated by doubly stochastic hyper-matrices. The proposed P ROTOCOL –DSHM is arguably a feasible generalization of the classical DeGroot and Chatterjee-Seneta models from square stochastic matrices to higher-order stochastic hyper-matrices. Specifically, in contrast to the classical case, we have assumed that each agent is capable of revising his/her opinion on certain issues following the influence of all possible groups of many (at least two) agents. To maintain homogeneity, the influence of a single agent was treated as that of a group of identical agents. We have demonstrated that if every agent holds an

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eventually positive opinion and the influence of each group of agents with varying members on any other agent is positive, then the multi-agent system will ultimately reach consensus. The main result (Theorem 1) of this paper extends, generalizes, and unifies all previous results established in the series of papers [4, 27, 35–40]. Acknowledgements The author is greatly indebted to anonymous reviewers for several useful suggestions and comments which improved the presentation of the paper.

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20. Krause, U.: Opinion dynamics – local and global. In: Liz, E., Manosa, V. (eds.) Proceedings of the Workshop Future Directions in Difference Equations, pp. 113–119. Universidade de Vigo, Vigo (2011) 21. Krause, U.: Positive Dynamical Systems in Discrete Time: Theory, Models, and Applications. Walter de Gruyter (2015) 22. Lyubich, Y.I.: Mathematical Structures in Population Genetics. Springer (1992) 23. Malinowska, A.B., Schmeidel, E., Zdanowicz, M.: Discrete leader-following consensus. Math. Meth. Appl. Sci. 40(18), 7307–7315 (2017) 24. Moreau, L.: Stability of multiagent systems with time-dependent communication links. IEEE Trans. Autom. Control 50(2), 169–182 (2005) 25. Pulka, M.: On the mixing property and the ergodic principle for non-homogeneous Markov chains. Linear Algebra App. 434, 1475–1488 (2011) 26. Raftery, A.: A model of high-order Markov chains. J. Roy. Stat. Soc. 47, 528–539 (1985) 27. Saburov, K.: Krause mean processes generated by cubic stochastic matrices I: diagonally primitive cubic stochastic matrices, Math Notes (Accepted) (2023) 28. Saburov, M.: Ergodicity of nonlinear Markov operators on the finite dimensional space. Non. Anal. Theo. Met. Appl. 143, 105–119 (2016) 29. Saburov, M.: Quadratic stochastic Sarymsakov operators. J. Phys: Conf. Ser. 697, 012015 (2016) 30. Saburov, M.: On regularity of diagonally positive quadratic doubly stochastic operators. Results Math. 72, 1907–1918 (2017) 31. Saburov, M.: On regularity of positive quadratic doubly stochastic operators. Math. Notes 103(2), 328–333 (2018) 32. Saburov, M.: Ergodicity of .p−majorizing quadratic stochastic operators. Markov Processes Relat. Fields 24(1), 131–150 (2018) 33. Saburov, M.: Ergodicity of.p−majorizing nonlinear Markov operators on the finite dimensional space. Linear Algebra Appl. 578, 53–74 (2019) 34. Saburov, M.: The discrete-time Kolmogorov systems with historic behavior. Math. Meth. Appl. Sci. 44(1), 813–819 (2021) 35. Saburov, M., Saburov, K.: Reaching a consensus in multi-agent systems: a time invariant nonlinear rule. J. Educ. Vocat. Res. 4(5), 130–133 (2013) 36. Saburov, M., Saburov, K.: Mathematical models of nonlinear uniform consensus. Sci. Asia 40(4), 306–312 (2014) 37. Saburov, M., Saburov, K.: Reaching a nonlinear consensus: polynomial stochastic operators. Inter. J. Cont. Auto. Syst. 12(6), 1276–1282 (2014) 38. Saburov, M., Saburov, K.: Reaching a nonlinear consensus: a discrete nonlinear time-varying case. Inter. J. Syst. Sci. 47(10), 2449–2457 (2016) 39. Saburov, M., Saburov, K.: Reaching consensus via polynomial stochastic operators: a general study. In: Springer Proceedings in Mathematics and Statistics, vol. 212, pp. 219–230 (2017) 40. Saburov, M., Saburov, K.: Mathematical models of nonlinear uniformly consensus II. J. Appl. Nonlinear Dyn. 7(1), 95–104 (2018) 41. Saburov, M., Yusof, N.A.: Counterexamples to the conjecture on stationary probability vectors of the second-order Markov chains. Linear Algebra Appl. 507, 153–157 (2016) 42. Saburov, M., Yusof, N.: The structure of the fixed point set of quadratic operators on the simplex. Fixed Point Theory 19(1), 383–396 (2018) 43. Saburov, M., Yusof, N.: On uniqueness of fixed points of quadratic stochastic operators on a 2D simplex. Methods Funct. Anal. Topol. 24(3), 255–264 (2018) 44. Sarymsakov, T., Ganikhodjaev, N.: Analytic methods in the theory of quadratic stochastic processes. J. Theoret. Prob. 3, 51–70 (1990) 45. Seneta, E.: Nonnegative Matrices and Markov Chains. Springer (1981) 46. Touri, B., Nedi´c, A.: Product of random stochastic matrices. IEEE Trans. Autom. Control 59(2), 437–448 (2014) 47. Tsitsiklis, J., Bertsekas, D., Athans, M.: Distributed asynchronous deterministic and stochastic gradient optimization algorithms. IEEE Trans. Autom. Control 31(9), 803–812 (1986) 48. Ulam, S.: A Collection of Mathematical Problems. New York (1960)

Control Design Techniques and Numerical Methods in Relationship with Discrete-Time Models

Passivity Techniques and Hamiltonian Structures in Discrete Time Dorothée Normand-Cyrot, Salvatore Monaco, Mattia Mattioni, and Alessio Moreschini

Abstract The object of this paper is to show the impact of representing discrete-time dynamics as two coupled difference/differential equations in establishing passivity properties and describing port-Hamiltonian structures as well as the related energybased control strategies. Keywords Discrete-time systems · Port-Hamiltonian structures · Passivity based control · Energy-based control

1 Introduction Energy-based modeling and control are fundamental evergreen concepts that are extensively investigated because of their sustained impact towards the newest methodologies and technologies (see the textbooks or survey-oriented contributions [1, 3, 45, 49, 51, 52, 58]). The underlying idea consists in deducing a system representation explicitly catching the energy features of the dynamics through dissipative or conservative components. Then, energy based control strategies can be designed. D. Normand-Cyrot (B) Laboratoire des Signaux et Systèmes (CNRS, Université ParisSaclay, CentraleSupelec), 91190 Gif-sur-Yvette, France e-mail: [email protected] S. Monaco · M. Mattioni Dipartimento di Ingegneria Informatica, Automatica e Gestionale A. Ruberti Sapienza University of Rome, 00185 Rome, Italy e-mail: [email protected] M. Mattioni e-mail: [email protected] A. Moreschini Department of Electrical and Electronic Engineering, Imperial College London, London SW7 2AZ, UK e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 S. Olaru et al. (eds.), Difference Equations, Discrete Dynamical Systems and Applications, Springer Proceedings in Mathematics & Statistics 444, https://doi.org/10.1007/978-3-031-51049-6_15

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In this framework, the class of port-Hamiltonian systems is paradigmatic [9, 47, 50]. From a theoretical point of view, most of the literature is devoted to the continuoustime setting in spite of a pervasive interest in computer-oriented applications and thus in digital structures at large. The general obstacles to perform an equivalent analysis in a discrete-time framework are well-known and related to the difficulty to describe the geometric structure behind the state and output discrete evolutions [32]. In this sense, the definition of dissipation itself is an open question [4, 5, 33, 44]. As a consequence, a shared definition of discrete-time port-Hamiltonian structures has been missing in spite of a variety of approaches often inspired by discretization schemes aimed to preserve the energy properties and/or power-balance exchanges (to cite a few see [2, 7, 8, 11, 13, 15–17, 53–56, 56, 57, 60]). The present work presents recent advances in the definition of dissipative notions for discrete-time systems when adopting a Differential Difference Representation (DDR) and their impact for characterizing port-Hamiltonian structures. The idea behind such a representation consists in separating discrete-time dynamics into two components: a difference equation that describes the control-free evolution and a differential equation that models the variation of the dynamics with respect to the control variable. This hybrid structure is adapted to cope with the intrinsic nonlinearity (that becomes more and more complex through iterative composition along successive time steps) in both the state and input variables when representing discretetime dynamics in the map form. Splitting the free evolution from the controlled part leads naturally to a description of a discrete-time dynamics through two coupled difference and differential equations. An exponential representation of the discretetime ow results through the integration of the differential controlled part. This is useful for further composing the state evolutions along successive time steps; the composition of exponential ows replace the composition of nonlinear functions. Analogously, when considering an output function, one gets an exponential form representation of the Volterra series expansions characterizing the input-to-output evolutions. Recalling, if necessary, that typical numerical burdens have to be faced when considering both purely discrete systems or systems issued from sampling due to their intrinsic nonlinearities, these exponential forms provide efficient computational tools. Accordingly, the combinatorial properties of the series expansions that describe the solutions can be exploited to qualify suitable polynomial approximations of increasing order. Section 2 recalls these developments as useful prolegomena to the paper while details can be found in [27, 28, 30, 35]. In Sect. 3, the aforementioned representation is used to characterize passivity. This notion, and more generally the one of dissipativity, relies upon energy exchanges and the way the system interacts with the environment. Roughly speaking, a system is passive when the internally stored energy does not exceed the one externally supplied [58, 59]. This property is caught by characterizing the variation of a particular function along the system’s trajectories [45, 49, 52]. Such a function, referred to as the storage function, generally represents the energy and is strictly linked to Lyapunov and/or Hamiltonian functions. With this in mind and for a fixed storage quantity, the DDR form we propose immediately allows us to characterize average passivity [33]; a novel notion of passivity in discrete time. This is done by isolating the control

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dependent part and thus defining in a very natural way the corresponding average passive output that is, in turn, a conjugate quantity whose product with the control variable is a power unit. The notion of average dissipation can be then exploited for control purposes by extending the usual concept of negative output damping to cope with stabilization at the origin. This approach can be further generalized to describe Passivity-Based Control (PBC) at large for nonlinear discrete-time systems towards the so-called second generation of passivity-based control, aimed at managing the system energy to satisfy control specifications. Those techniques endow Energy Balancing (EB) and Interconnection and Damping Assignment (IDA) to deal with more complex systems including networked or cascade dynamics. The foundations of average passivity-based techniques are recalled in Sect. 3, while more detailed studies are in [34, 36]. As already mentioned, port-Hamiltonian dynamics are of pervasive interest due to their mathematical structure and their foundations in physics (to cite a few [9, 18–20, 46, 48, 50]). Hereinafter a novel state-space representation of discrete-time port-Hamiltonian structures that represents a breakthrough in the literature is naturally deduced from the DDR form. The so defined forms are endowed with average passivity properties that validate the proposed choice of conjugate output. Further on, the fundamental characteristics of port-Hamiltonian structures as the qualifying closeness property under power-preserving interconnection [38, 40], are validated. As a consequence, these forms are efficient for the design of average passivity based control strategies for complex and networked discrete-time dynamics as illustrated in [22, 23, 38, 41]. Port-Hamiltonian structures are discussed in Sect. 4 while dedicated studies are in [38, 39, 41].

1.1 Notations Throughout the paper all the functions and vector fields defining the dynamics are assumed smooth and complete over the respective definition spaces. The sets .R, .N and .Z denote the set of real, natural numbers including .0 and integers respectively. For any vector .v ∈ Rn , .|v| and .v T define the norm and transpose of .v respectively. . Id denotes the identity function on the definition space while . I denotes the identity operator and the identity matrix when related to a linear operator. The symbols .“.> 0” and .“.< 0” denote positive and negative definite functions (or matrices), respectively. The symbol .◦ denotes the composition of two functions or operators, depending on the context. Given a real-valued function .V : Rn → R assumed differentiable, setting .∇ = col{ ∂∂xi }i=1,n , .∇V (x) represents the gradient column-vector. Given a vector-valued function . F(x) = col(F1 (x), . . . , Fn (x)), the operator Jx [F](x) =

∂ F(x) = ∂x

(

∂ Fi (x) ∂x j

) i, j=1,n

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denotes the Jacobian of the function . F evaluated at .x. Given a smoothi vector field ∑ L over .Rn , .e f denotes the exponential Lie operator .e f := I + i≥1 i!f with .L f = ∑n ∂ i i=1 f i (x) ∂ xi and indicating by .L f the operator at power .i with respect to the usual composition of vector fields (for a linear vector field, the exponential Lie operator recovers the exponential of the matrix representing the operator). For any smooth function .h : Rn → R, one verifies .e f h(x) = h(e f (x)) = e f h|x where .|x denotes the evaluation of the function at .x. Given two vector fields . f, g, their Lie bracket is again a vector field denoted by .ad f g = [ f, g] = (L f ◦ Lg Id − Lg ◦ L f )Id and, iteratively ◦ ad f g with .ad 1f g = ad f g. A function . R(x, u) is said in for .i ≥ 1, .ad if g = ad i−1 f p p−1 ˜ . O(u ) for. p ≥ 1 if, whenever it is defined, it can be written as. R(x, u) = u R(x, u) ˜ and there exist a function .θ ∈ K∞ and .u ⋆ > 0 such that .∀u ≤ u ⋆ , .| R(x, u)| ≤ θ (u).

2 Basics on Discrete-Time Dynamics 2.1 Differential/Difference Representation A single-input nonlinear discrete-time dynamics over .Rn is usually represented by a function . F : Rn × R → Rn , smooth in both the state and input variables x

. k+1

= F(xk , u k ) = xk + F0 (xk ) + g(xk , u k )u k .

(1)

For any pair of state and input variables .(xk , u k ) fixed at time instant .k ∈ N, .xk+1 denotes the state reached at time .k + 1 from .xk , under the action of the control .u k . For convenience that will be clear later on and without loss of generality, the free evolution is decomposed as . F(x, 0) = x + F0 (x) while .g(x, u)u (with .g(x, 0) /= 0) represents the control dependent part of the dynamics. In [30], we proposed to represent discrete dynamics as two coupled difference and differential equations (DDR). More in detail, for any time step .k and under mild conditions (e.g., submersivity of . F(x, u)), (1) can be represented as .

x + (0) = x + F0 (x)

(2a)

+

.

d x (u) = G(x + (u), u) du

(2b)

when denoting .(x, u) = (xk , u k ) ∈ Rn × R, any pair of state and input variables at generic time instant .k, .x + (u) = x + (u k ) = xk+1 , the state reached at time .k + 1 starting from .xk under the action of .u k and .x + (0), the state reached along free evolution .u k = 0. In doing so, we underline that .x + (u) is viewed as a curve in .Rn , parameterized by the control variable .u ∈ R. .G(·, u) is a properly defined vector field on .Rn , parameterized by .u, satisfying .G(F(x, u), u) = ∂∂uF (x, u). With this in mind, the difference equation (2a) describes the jump of the state evolving in free

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evolution while the differential equation (2b) models the rate of change of the state dynamics with respect to control variation. Remark 1 Provided . F(x, 0) be invertible, the vector field .G(x, u) can be defined for .u sufficiently small as | ∂ F(x, u) || . G(x, u) := , ∂u |x=F −1 (x,u)

(3)

and expanded, around .u = 0, as .

G(x, u) = G 1 (x) +

∑ ui i≥1

i!

G i+1 (x); G 1 (x) = G(x, 0).

(4)

The vector fields .G i in the power expansion (4) define a family of canonical vector ' fields associated to the dynamics (2). The .(G i ) s have been proved useful in the study of analysis and design problems linked to the geometry of the evolutions in the state space (e.g.[6, 27, 28, 30, 31]). Remark 2 The DDR form can be generalized to multi-input dynamics by modeling the rate of change of the dynamics under the action of each control. Setting .u = (u 1 , . . . , u m )T , one replaces (2b) by the set of partial derivative equations .

∂ x + (u) = G j (x + (u), u) for j = (1, . . . , m) with x + (0) = F(x, 0) ∂u j

and .G j (x, u) satisfying the condition .G j (F(x, u), u) :=

(5)

∂ F(x,u) . ∂u j

2.2 Some Useful Manipulations To better grasp the DDR forms, some useful concepts are specified below. Particular classes can be discussed besides the linear one that corresponds to set in (2), . F0 (x) = Ax with constant control vector . G(x, u) = B and matrices .(A, B) of suitable dimensions. A peculiar class is represented by the input-affine one. Proposition 1 (Input-affine DDR) Assuming .G(x, u) = G 1 (x) in (2), the dynamics in map form (1) preserves a nonlinear dependence on u which admits the exponential representation .

| | F(x, u) = euG 1 (x)|

x+F0 (x)

(6)

where .euG 1 denotes the flow associated with the solution to the differential equation (2b) when .G(x, u) = G 1 (x).

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This result follows from the fact that integrating (2b) from .0 and .u with initial condition .x + (0) = x + F0 (x), one recovers a dynamics in map form (1); i.e. .

x + (u) = x + F0 (x) +

ʃ

u

G(x + (s), s)ds = x + F0 (x) + g(x, u)u.

(7)

0

When assuming .G(x, u) = G 1 (x), this integral form admits the simple exponential representation described in Proposition 1 that is generalized in the sequel to a control dependent vector field .G(·, u). Remark 3 We note for completeness that given an input-affine dynamics in map form, .g(x, u) = g(x) in (1), the associated control vector field .G(x, u) in (4) is no more linear in .u and satisfies the algebraic constraint .G(F(x, u), u) = g(x). Easy manipulations show how DDR forms are transformed under coordinates change and feedback. Lemma 1 Consider the coordinates change .z = T (x) defined by the diffeomorphism .T : Rn → Rn , then the DDR dynamics (2) is transformed into .

z + (0) = z + F¯0 (z)

dz + (u) ¯ + (u), u) = G(z du

(8)

with .

| | z + F¯0 (z) = T (x + F0 (x))| ¯ G(z, u) = AdT G(z, u)

x=T −1 (z)

where . AdT G(·, u) indicates the transport of the vector field .G(·, u) along .T ; i.e. .

AdT G(z, u) =

[ ∂T ∂x

G(·, u)

] x=T −1 (z)

| = LG(·,u) T (x)|x=T −1 (z) .

(9)

Lemma 2 Consider the state feedback.u(x, v) = α(x) + v with.α : Rn → R smooth and external control .v ∈ R, then the DDR dynamics (2) is transformed into x + (0) = x + Fα (x)

. α

d xα+ (v) = G α (xα+ (v), v) dv where

(10)

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333

x + (v) = F(x, α(x) + v) = x + (α(x) + v)

. α

xα+ (0) = F(x, α(x)) = x + Fα (x) = x + (α(x)) G α (x, v) = G(x, α(x) + v). The proof works out by showing that ∂ F(x, u) || d xα+ (v) ∂(α(x) + v) = = G(xα+ (v), α(x) + v) = G α (xα+ (v), v). | u=α(x)+v dv ∂u ∂v

2.3 Input-to-State and Input-to-Output Trajectories In our context, a discrete-time system .∑d (h) is given by a state dynamics in the DDR form (2) and a smooth output function .h : Rn → R. The generalization of Proposition 1 to a non input-affine form is reported and further extended to describe input-output evolutions over one or several time-steps. Theorem 1 Given .∑d (h) and a pair of state and input variables .(xk , u k ) at generic time instant k, one gets at time .k + 1 x

. k+1

= x + (u k ) = eu k G(·,u k ) Id |xk +F0 (xk ) = F(xk , u k )

h(xk+1 ) = h(x + (u k )) = eu k G(·,u k ) h|xk +F0 (xk ) = h(F(xk , u k )) where the series exponent is a Lie series .G(·, u) ∈ Lie{G 1 , . . . , G p , . . . } that can be described through its expansion in powers of .u. For the first term one reports G(·, u) = G 1 +

.

u u2 G 2 + (G 3 + [G 1 , G 2 ]) + O(u 3 ). 2 3!

The proof, developed in [27, 35], follows from the Lie properties endowed by the ow characterizing the solutions to nonlinear .u-dependent ordinary differential equations of the form (2b), when expressed through a chronological exponential. A major property of these exponential representations of discrete ows is to be easily composed along successive time steps. For, it is instrumental to define, according to (9), the transport of any vector field .G i along the free evolution .(I d + F0 ) so j ' getting for . j ≥ 0, the family of transported vector fields .(G i ) s with j+1 .G i (x)

j+1

and iteratively .G i

=

j Ad Id +F0 G i (x)

| | = LG j (x + F0 (x))|| i

(Id +F0 )−1 (x)

j+1

(x) = Ad Id +F0 G i (x) with .G i0 = G i .

The following result holds when denoting by .G j (·, u), j ≥ 0, the series exponent

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G j (·, u) ∈ Lie{G 1 , . . . , G pj , . . . }

.

'

j

defined as in Theorem 1 with respect to the transported vector fields .(G i ) s. Theorem 2 Given .∑d (h), initial state value .x0 ∈ Rn and input sequences .{u 0 , . . . , u k }, .k ∈ N, then at generic time instant .k + 1 > 0, one gets = x + (u k , . . . , u 0 ) = eu 0 G

k

(·,u 0 )

◦ · · · ◦ eu k G

0

(·,u k )

|(Id +F0 )k+1 (x0 )

yk+1 = y + (u k , . . . , u 0 ) = eu 0 G

k

(·,u 0 )

◦ · · · ◦ eu k G

0

(·,u k )

h|(Id +F0 )k+1 (x0 ) .

x

. k+1

'

According to these results it becomes clear that the vector fields .(G i )s and their transport along the drift term enter in a differential geometric characterization of the structure of the accessible states. In fact, accessibility can be reported to the properties of the orbit of the associated Lie groups (see [10, 14] for further details). Further on, j j invariance can be characterized in terms of the Lie algebra . Lie{G 1 , . . . , G p , . . . } as discussed in [25]. The same holds regarding the properties of controlled invariance, feeback linearization [31] or decoupling [6], up to characterize the corresponding control solutions.

2.4 Concluding Comments In this section, an alternative description to the usual map form is proposed for discrete-time dynamics. The free evolution that generates a purely discrete evolution defines the initial condition of the differential equation modeling the control action. This suggests a modeling approach that would separately identify the free evolution as a map . F0 and the variation of the dynamics with respect to .u as a vector field . G(·, u), that may depend on .u. In doing so, a family of control vector fields, the ' .(G i ) s, that enters in the characterization of the structural and control properties of the dynamics, is defined. Accordingly, it becomes possible to combine a more visual geometric approach with a more computational algebraic one to provide intriguing relationships between the continuous-time and discrete-time settings that should be further investigated towards a unified vision. Further on, being the solution to an ordinary differential equation, the state dynamics can be rewritten in terms of the exponential operator characterizing the associated ow. It follows that the whole differential geometry apparatus behind this operator form can be used to describe the input-to-state and input-to-output behaviours along successive time steps. In particular, the composition of nonlinear maps is replaced with the composition of exponential forms that is more tractable in practice. This is well illustrated with the characterization of the Volterra series and the computation of its kernels in terms j ' of the vector fields .(G i ) s and their Lie brackets. We refer to [26, 29] for further discussion along these lines in relation with realization problems.

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3 Passivity Techniques The notion of average passivity has been introduced in [34] to weaken the necessary requirement of direct throughput when adopting the standard notion in a discretetime framework. This definition is in fact directly inspired by the splitting of the state dynamics into free and control parts.

3.1 Average Passivity Denoting by .∑d (h) a discrete-time system described by the dynamics (1) (equivalently (2)) with output .h : Rn → R, the definition below is recalled from [34]. Definition 1 (Average passivity) .∑d (h) is said average passive if there exists a positive semi-definite function . S : Rn → R≥0 (the storage function) such that for all n .(x, u) ∈ R × R the following passivity inequality holds .

ʃ

+

S(x (u)) − S(x) ≤

u

h(x + (s))ds := uh av (x, u)

(11)

0

with respect to the average output defined as .

h av (x, u) :=

1 u

ʃ

u

h(x + (s))ds.

(12)

0

This definition is directly inspired by the DDR form of the dynamics that yields to rewrite the rate of change of the storage function . S, between two successive states, in an integral form as .

S(x + (u)) − S(x) = S(x + (0)) − S(x) +

ʃ

u

L G(·,s) S(x + (s))ds.

0

The so-defined average map .h av (·, u) in (12), introduces a direct input-output link in such a way that average passivity with respect to .h is in fact equivalent to usual passivity with respect to .h av (·, u). As an immediate consequence of average passivity, setting .u = 0 in (11), one verifies . S(x + (0)) − S(x) ≤ 0, so concluding stability of any equilibrium .xe ∈ Rn when . S qualifies as a Lyapunov function at .xe (. S(xe ) = 0 and . S(x) > 0 for .x /= xe ) and asymptotic stability provided . S(x + (0)) − S(x) < 0. More in general, average dissipativity can be defined making reference to the average output and a supply rate function .s : R × R → R, so requiring that the dissipation inequality below be verified for all .(x, u) ∈ Rn × R .

S(x + (u)) − S(x) ≤ s(u, h av (x, u)).

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The notion of average passivity can be extended to average passivity from some nominal non-zero constant value .u¯ /= 0, [21, 38, 40]. It is instrumental when discussing the action of a feedback law over passivity. Definition 2 (Average passivity from .u) ¯ .∑d (h) is said average passive from a given u¯ with .u¯ ∈ U ⊆ R, if there exists a positive semi-definite function . S : Rn → R≥0 (the storage function) such that, for all .(x, u) ∈ Rn × R

.

av S(xu+ ¯ (u)) − S(x) ≤ uh u¯ (x, u)

.

(13)

with .u-average output from .u¯ defined as h av (x, u) :=

. u¯

1 u

ʃ

u

h(x + (u¯ + s))ds.

0

av When .u¯ = 0, one recovers average passivity since .h av 0 (x, u) = h (x, u).

3.2 Feedback Stabilization and Interconnection On these bases, Passivity Based Control (PBC) techniques can be developed as discussed below. Two basic ingredients of control strategies exploiting passivity are revisited hereafter making reference to the average notion. Firstly, stabilization through negative output feedback and additional damping is specified. Secondly, closeness under power preserving input-output interconnection is discussed so enlarging the control design to interconnected dynamics. The following definition is instrumental. Definition 3 (Zero-state detectability) Given .∑d (h) let .xe ∈ Rn be an equilibrium and.Z ∈ Rn be the largest invariant set contained in the set.{x ∈ Rn s.t. h(x + (0), 0) = 0}. .∑d (h) is said zero-state detectable (ZSD) if .x = xe is an asymptotically stable equilibrium conditionally to .Z. The following theorem characterizes the negative output damping feedback. Theorem 3 (Negative average output feedback) Given .∑d (h) with equilibrium .xe ∈ Rn , .(F0 (xe ) = 0), assumed average passive with storage function . S > 0, then the feedback .u = α(x) solving the algebraic equality u = −κh av (x, u) with κ > 0

.

(14)

ensures asymptotic stability provided that .∑d (h) is Zero State Detectability (ZSD). Moreover, setting .u(x, v) = α(x) + v with external control .v ∈ R, then the closed loop dynamics is average passive again with respect to .h .

S(xα+ (v)) − S(x) ≤ vh av α (x, v).

(15)

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To conclude average passivity of the closed-loop dynamics it is sufficient to note that under the feedback .u(x, v) = α(x) + v, the average output associated with .h along the closed-loop dynamics described in Lemma 2, recovers the average output from .u ¯ = α(x) defined in Definition 2. In fact, one has h av (x, v) :=

. α

1 v

ʃ

v 0

h(xα+ (s))ds.

Then, by definition of .α(x), one gets ʃ .

α(x)+v

h(x + (s))ds =

0

ʃ ʃ

α(x)

h(x + (s))ds +

0 α(x)

=

h(x + (s))ds +

ʃ

α(x)+v

α(x) ʃ v

h(x + (α(x) + s))ds

ʃ

0

h(x + (s))ds

0

= −κ(h av (x, α(x)))2 + 0

v

h(xα+ (s))ds

≤ vh av α (x, v) so concluding that average passivity from .α(x) coincides with average passivity under preliminary feedback .α(x) with respect to the same output map .h. The negative average output feedback solving (14) is the first step towards stabilizing strategy through additional damping or average PBC feedback. Remark 4 Computational aspects are beyond the scope of the paper. However, we underline that the control solution being expressed as the solution to the algebraic equality (14), its computation may be a difficult. In practice, it can be performed according to suitable approximation methods as discussed in [21, 22]. The second fundamental property to verify regards the interconnection of two average passive systems through their respective input and output variables when setting i av i .u = φ(h (x, u)). Given two average passive systems .∑d (h ) with storage function i . S for .i = 1, 2, a power-preserving input-output interconnection making reference to the average outputs can be naturally defined as in [40, 43]. Definition 4 (Power preserving interconnection) The input-output interconnection between.∑d (h 1 ) and.∑d (h 2 ) is said power preserving if it satisfies the integral equality ʃ

u1

.

+

ʃ

u2

h 1 (x 1 (s))ds +

0

+

h 2 (x 2 (s))ds = 0

(16)

0

equivalently rewritten as av

av

u 1 h 1 (x 1 , u 1 ) + u 2 h 2 (x 2 , u 2 ) = 0.

.

(17)

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We easily note that the simplest way to solve (17) is to set ( 1) ( 1 av 1 1 ) ( ) ( 1 av 1 1 ) 0 −1 u h (x , u ) h (x , u ) = φ 2 av 2 2 = . av u2 1 0 h (x , u ) h 2 (x 2 , u 2 )

(18)

so recovering the classical power preserving interconnection expressed with respect to the average outputs. The solution to the implicit equality (17) defines a preliminary power preserving state-feedback that we denote .α(x) = (α 1 (x), α 2 (x))T with .x = (x 1 , x 2 )T . The following Theorem can be stated. Theorem 4 (Average passivity under power preserving interconnection) Let, for.i = 1, 2, the systems .∑d (h i ) be average passive with respective storage functions . S i . Let .α(x) be the power-preserving interconnection satisfying (18) and set .u = α(x) + v with external control .v = (v 1 , v 2 )T . Then, the interconnected system 1 1 1 xα1+ 1 (v ) =Fα 1 (x, v )

(19a)

2+ 2 . x 2 (v ) α

(19b)

.

=Fα22 (x, v 2 )

with output.h = (h 1 , h 2 )T is average passive with storage function. S(x) := S 1 (x 1 ) + S 2 (x 2 ). Namely, the dissipation inequality holds; i.e. .

1 1 1 2 2+ 2 2 2 T av S 1 (xα1+ 1 (v )) − S (x ) + S (x α 2 (v )) − S (x ) ≤ v h α (x, v)

(20)

with the average output of the closed-loop dynamics defined as ( av .h α (x, v)

=

1 v1 1 v2

) ʃ v1 1 1+ h (x (s))ds 1 . ʃ0v2 2 α2+ 0 h (x α 2 (s))ds

(21)

The proof, detailed in [40], works out in two steps. First, one concludes that under power-preserving input-output interconnection, the interconnected system is average passive from .α(x) according to the Definition 2. Then, because of the feedback structure, average passivity from .α recovers average passivity of the dynamics under preliminary feedback .α(x), so concluding the claim.

3.3 Passivating Output Map Theorem 3 shows how stabilization under feedback can be achieved by exploiting average passivity. However, stabilization to some target equilibria that are local extrema of suitably shaped energy functions can be requested. For, the following proposition is instrumental, it specifies a dummy output function that preserves average passivity and is zero at the local minima of the storage function. It is reminiscent of the continuous-time context (see [34, 38] for details).

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Proposition 2 (Average passivating output) Let.∑d (h) be average passive with storage function . S, then it is also average passive with respect to the dummy output function .

.

Y (x, u) = LG(·,u) S(x).

(22)

Y (·, u), computed as the Lie derivative of . S along .G(·, u), is referred to as an average passivating output because it satisfies the Energy Balance Equality (EBE)

.

S(x + (u)) − S(x) = S(x + (0)) − S(x) + S(x + (u)) − S(x + (0)) . stored energy

dissipated energy

supplied energy

Average passivity with respect to the output (22) holds since by definition ( )av S(x + (u)) − S(x + (0)) = u LG(·,u) S (x, u) = uY av (x, u) with . S(x + (0)) − S(x) ≤ 0 from the average passivity assumption of .∑d (h). Moreover, .Y (x, u) is zero at local extrema of . S since .Y (x, u) = ∂∂ xS G(x, u). Remark 5 Proposition 2 generalizes to assuming .∑d (h) average passive from a ¯ In that case, average passivity from .u¯ with respect to .Y (·, u) follows given .u. .

S(x + (u¯ + u)) − S(x) ≤ S(x + (u¯ + u)) − S(x + (u)) ¯ = uYu¯av (x, u)

because by assumption . S(x + (u)) ¯ − S(x) ≤ 0 and by definition ʃ ( )av 1 u LG u¯ (·,u) S (x, u) = LG(·,u+s) S(x + (u¯ + s))ds = Yu¯av (x, u) ¯ u 0

.

with .G u¯ (·, s) = G(·, u¯ + s). Remark 6 Specifying the result in Theorem 3 on such output function (22), the stabilizing feedback .u = α(x) results to be the solution to the algebraic equality ( )av u = −κ LG(·,u) S (x);

.

κ>0

that can be solved in first approximation around .u = 0 so getting | )| G 1 (x) || ∂x x=x+F0 (x)

( ∂ S(x)

α (x) = −κλ(x)

. ap

with a suitable gain .λ(x) > 0 as discussed in [21]. Setting .u = α(x) + v, the closed loop dynamics with output .LG α (·,v) S(x) = Yα (x, v) remains average passive.

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In the present paper, oriented to characterize Hamiltonian dynamics in discrete time, it is instrumental to relate the passivating output defined in (22) to a certain discrete gradient function. The definition below is recalled from [24, 38]. Definition 5 (Discrete gradient function) Given a smooth real-valued function . S : Rn → R, its discrete gradient is a function of two variables .∇¯ S|zx : Rn × Rn → Rn satisfying for all .x, z ∈ Rn (z − x)T ∇¯ S|zx = S(z) − S(x) with

.

lim ∇¯ S|zx = ∇ S(x).

(23)

z→x

Definition 5 properly states that the discrete gradient function satisfying (23) describes the rate of change of this function between two states. It is not uniquely defined and different methods to solve the equality can be worked out [11, 12, 24]. Through component-wise integration, one gets the computable expression below ]T [ ∇¯ S|zx = ∇¯ 1 S|zx11 . . . ∇¯ n S|zxnn

.

with some simplified notations ¯ i S|zxi = .∇ i

1 z i − xi

ʃ

zi xi

∂ S(x1 , ..., xi−1 , s, z i+1 , ..., z n ) ds. ∂s

Remark 7 When . S(x) = 21 x T P x with . P ∈ Rn×n , the discrete gradient is uniquely expressed as 1 ∇¯ S|zx = P(x + z). 2

(24)

.

By definition of the discrete-gradient function, the EBE in Proposition 2 rewrites .

( ) ( ) S(x + (u)) − S(x) = S(x + (0)) − S(x) + S(x + (u)) − S(x + (0)) ʃ u + = S(x (0)) − S(x) + LG(·,s) S(x + (s))ds . =F0T (x)∇¯ S|xx

+ (0)

0 +

=ug T (x,u)∇¯ S|xx + (u) (0)

It is instrumental to describe the passivating output defined in (22) in terms of the discrete gradient function. One gets. Lemma 3 (Average passivating output in discrete gradient form) Given the dynamics (1) and a real-valued smooth function map . S : Rn → R, the following equalities hold .

S(x + (0)) − S(x) = S(x + F0 (x)) − S(x) = F0T (x)∇¯ S|xx

+

(0) +

S(x + (u)) − S(x + (0)) = S(F(x, u)) − S(x + F0 (x)) = ug T (x, u)∇¯ S|xx + (u) (0) .

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with the relation +

ug T (x, u)∇¯ S|xx + (u) (0) =

ʃ

.

u

( )av LG(·,s) S(x + (s))ds = u LG(·,u) S (x, u).

(25)

0

Example 1 The discrete integrator .

x + (u) = x + u;

y = h(x) = x

is the simplest storage element. Setting . S(x) = 21 x 2 , as storage function, the system is average passive as .

1 S(x + (u)) − S(x) = xu + u 2 = 2

ʃ 0

u

x + (v)dv =

ʃ

u

(x + s)ds = uh av (x, u)

0

with .h av (x, u) = x + 21 u. Accordingly, one conclude passivity of the input-stateoutput system .

1 x + (u) = x + u; h av (x, u) = x + u. 2

The associated negative average output feedback satisfies the algebraic equality u = −κ(x + 21 u) so computing .u = − 1+κ κ x; κ > 0 that recovers negative output 2 feedback with suitably shaped gain.

.

3.4 Concluding Comments In this section, the notion of average passivity it introduced and is shown to be qualifying for the design of discrete-time average passivity based stabilizing strategies through damping (Theorem 3) or interconnection (Theorem 4). A passivating output and its related negative average output feedback are described in Proposition 2, so enlarging the control objectives to stabilization to target equilibria. This method of passivation can be further exploited to describe cascade stabilizing procedures for triangular state dynamics through backstepping or feedforward strategies in a discrete-time context, as in [21]. Further on, the second generation of average passivity based control including an energy-shaping component to shape the energy of the system and fulfil required control specifications can be developed. Such extensions include Interconnection and Damping Assignment (IDA) techniques that modify the internal port-Hamiltonian structure to assign a new equilibrium, or Control by Interconnection techniques that manage energy exchanges through an interconnection pattern. Preliminary works in this direction are in [22, 39–41]. Finally, Lemma 3, that rewrites the average passivating output in its discrete gradient form, directly inspires the novel port-control Hamiltonian structure we propose in the next section.

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4 Port-Hamiltonian Structures in Discrete Time Port-Hamiltonian structures have a pervasive impact in numerous applied domains enlarging the more traditional mechanical one. These structures are essentially described in the continuous-time domain while in discrete time, a consensus on a specific structure is not reached in spite of a rich literature. In this section, a novel description of port-Hamiltonian structures is proposed exploiting the DDR form and the average passivating output map introduced in Proposition 2.

4.1 Control-Free Port-Hamiltonian Dynamics From [38, 39, 43], we first recall the definition of a control-free port-Hamiltonian structure. A unified definition that mimics the continuous-time structure exists in the discrete-time framework, just replacing the gradient function with the discrete gradient function. Let . H : Rn → R≥0 , be a smooth real-valued function that denotes the Hamiltonian function. Definition 6 A control-free discrete-time port-Hamiltonian dynamics over .Rn can be described by the first-order difference equation .

x + − x = (J (x) − R(x))∇¯ H |xx

+

(26)

where . J (x) = −J T (x), . R(x) = R T (x) 0, are matrices of functions representing the interconnection and resistive parts respectively. By construction, one immediately verifies that: • any local extremum of . H (x) (.∇¯ H |xxee = ∇ H (xe ) = 0), is an equilibrium; .• the rate of change of the Hamiltonian along the dynamics satisfies the equality .

.

+ + + + H (x + ) − H (x) = ∇¯ H T |xx J (x)∇¯ H |xx −∇¯ H T |xx R(x)∇¯ H |xx

=0

≤0

by skew symmetry of . J (x) and semi-positiveness of . R(x). Taking the sum of these increments, energy dissipation from time .0 to time .k is described by the equality

.

H (xk ) − H (x0 ) = −

k−1 ∑

+

+

x x ∇¯ H T |xii R(xi )∇¯ H |xii .

i=0 dissipated energy≤0

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343

• When . J (x) = 0, the dynamics is dissipative. The simplest example is the gradient dynamics defined with . R(x) = I and . J (x) = 0 that is

.

.

x + − x = −∇¯ H |xx

+

satisfying .

+ + + H (x + ) − H (x) = −∇¯ H  |xx ∇¯ H |xx = −||∇¯ H |xx |2 ≤ 0.

• When . R(x) = 0, the dynamics is conservative

.

H (x + ) = H (x) so concluding that the Hamiltonian is a constant of motion for (26).

4.2 Canonical Discrete Hamiltonian Dynamics Let us illustrate the proposed definition on a peculiar class. Canonical Hamiltonian dynamics are defined over .R2n , when setting as skew symmetric interconnection matrix

 0 Id . Jc = −Id 0 1+ 2+ + Setting .x = (x 1 , x 2 ) , .x i ∈ Rn for .i = 1, 2 and .∇¯ H |xx = col(∇¯ 1 H |xx 1 , ∇¯ 2 H |xx 2 ), the canonical discrete Hamiltonian vector field associated to a given . H , denoted by ¯ H , satisfies .X

.

x 1+ − x 1 x 2+ − x 2



= Jc ∇¯ H |xx = +



2+ ∇¯ 2 H |xx 2 1+ −∇¯ 1 H |x 1



= X¯ H

(27)

x

For completeness, we note that for a given Hamiltonian function over . R 2n , the canonical Hamiltonian dynamics is solution for all .v of the condition   +

(x + − x, v) = ∇¯ H |xx , v

.

where . (u, v) =< u, Jc v > denotes the usual symplectic form. In fact, easy computations show that this equality rewritten as   < (x + − x), Jc v >= ∇¯ H (x), v is solved by .x + − x = X¯ H defined in (27).

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Further on, for any given real-valued smooth function .C : R2n → R, its rate of change along the Hamiltonian dynamics . X¯ H is given by C(x + ) − C(x) = {C, H } D

.

where .{C, H } D indicates the discrete Poisson bracket interestingly defined as the usual Poisson bracket but with respect to the discrete gradient; i.e. {C, H } D :=

n ∑

.

1i+ 2i+ 2i+ 1i+ ∇¯ 1i C|xx 1i ∇¯ 2i H |xx 2i − ∇¯ 2i C|xx 2i ∇¯ 1i H |xx 1i .

i=1

Canonical discrete Hamilton’s equations can be alternately written as C(x + ) − C(x) = {C, H } D , ∀C : R2n → R.

.

In our formalism, any function .C satisfying .{C, H } D = 0 is referred to as a discrete integral or constant of the motion with respect to the discrete dynamics generated by ¯ H. .X

4.3 Port-Controlled Hamiltonian Structures A novel description of port-controlled Hamiltonian structures exploiting the DDR form and the passivating average output map defined in Proposition 2 can now be proposed. This form is further validated by the Energy Balance Equation, its relation with feedback strategies, and the power-preserving interconnection. The definition below is recalled from [38, 39]. Definition 7 (Port-controlled Hamiltonian system (pH)) Given a smooth real-valued function . H : Rn → R≥0 , a discrete-time port-Hamiltonian system .∑dH over .Rn can be described according to the input-state-output form below .

x + (0) = x + (J (x) − R(x))∇¯ H |xx

+

(0)

(28a)

+

.

d x (u) = G(x + (u), u) du .Y (x, u) = LG(·,u) H (x)

(28b) (28c)

J (x) = −J T (x), . R(x) = R T (x) 0, are matrices of functions representing the interconnection and the resistive parts. .

Accordingly, the following result holds. Theorem 5 Given a discrete-time port-Hamiltonian system of the form (28), then the following holds:

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345

• any local extremum of . H (x) is an equilibrium; • the rate of change of the Hamiltonian along the dynamics satisfies

. .

.

H (x + (u)) − H (x) = −(∇¯ H |xx

+

) R(x)∇¯ H |xx

(0) T

+

(0)

ʃ

u

+

LG(·,s) H (x + (s)) ds

0

≤0

=uY av (x,u)

(29) so concluding average passivity with respect to the output map (28c). Some comments are in order. We note that the so defined output (28c), that corresponds to the passivating output introduced in Proposition 2 when substituting the storage function . S with the Hamiltonian function . H , qualifies as conjugate output: the product .uY av (x, u) describes the energy brought to the system through the external input and output variables between two successive time steps. Taking the sum of each increment (29) from time .0 to .k, one gets the Energy Balance Equality in a form that perfectly splits into the total stored energy, the internally dissipated energy from the one supplied by the input/output variables:

.

H (xk ) − H (x0 ) = − stored energy

k−1 ∑

+

x ∇¯ H T |xii

(0)

x + (0)

R(xi )∇¯ H |xii

+

k−1 ∑

u i Y av (xi , u i ) .

i=0

i=0 dissipated energy

supplied energy

As an alternative to the DDR form of .∑dH in (28), integration with respect to .u transforms the port-Hamiltonian system into its map form. Adopting the discrete gradient form representation of the average output described in Lemma 3, one gets equivalently to (28) the port-Hamiltonian structure in map form. The following proposition specifies the equivalence between these two forms. Proposition 3 (Port-Hamiltonian systems in map form) The pH structure (28) can be equivalently rewritten in map form as . .

x + (u) = x + (J (x) − R(x))∇¯ H |xx

y(x, u) =

+

(0)

+ ug(x, u)

+ g T (x, u)∇¯ H |xx + (u) (0)

(30a) (30b)

where by definition and from Lemma 3 .ug(x, u)

:=

ʃ u 0

G(x + (s), s)ds;

y(x, u) = Y av (x, u) =

H (x + (u)) − H (x + (0)) . u

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Remark 8 The average representation of the conjugate output is instrumental to describe its series expansion in power of .u that gives for the first terms .

y(·, u) = Y av (x, u) = LG 1 Hx (0) +

) u( 2 LG 1 + LG 2 H |x (0) + O(u 2 ) 2

(31)

where. O(u 2 ) contains all the remaining terms of higher order in the control variable.u. Further details regarding the complete series expansion and its iterative computation are in [34, 38]. It is interesting to highlight a matrix representation of the port-Hamiltonian structure we propose as a preamble to describe the associated Dirac structure. Easy computations show that the Eq. (28) satisfy Lemma 4 Easy computations show that the Eq. (28) satisfies ⎞ ⎞⎛ ⎞ ⎛ x + (0) J (x) − R(x) 0 0 x + (0) − x ∇¯ H |x ⎜ ⎟ 0 0 G(x + (u), u)⎠⎝∇ H (x + (u))⎠ d x + (u) ⎠ = ⎝ .⎝ + T + −Y (x (u), u) 0 −G (x (u), u) 0 du ⎛

so revealing the hybrid representation that couples a one step ahead difference free evolution with a differential form with respect to the control .u.

4.4 Average PBC Strategies for Port-Hamiltonian Systems Port-controlled Hamiltonian systems represent a common class of average passive systems over which PBC strategies can be applied. Specifying the result in Theorem 3, one gets. Theorem 6 Let the port-Hamiltonian system .∑dH described in (28) (equivalently in (30)) be ZSD with . H having a minimum in .x⋆ . Then, the feedback .α(x), solution of the implicit damping equality κ .α(x) = −κY (x, α(x)) = − α(x)

ʃ

α(x)

av

LG(·,s) H (x + (s))ds

(32)

0

equivalently rewritten in terms of the discrete gradient function as +

α(x) = −κgαT (x)∇¯ H |xx + (α(x)) .

(33)

.

with .gα (x) := g(x, α(x)) achieves asymptotic stabilization of the equilibrium .x⋆ . Accordingly the closed loop dynamics rewrites +

.

+ x + (α(x)) = x + (J (x) − R(x))∇¯ H |xx − κgα (x)gαT (x)∇¯ H |xx + (α(x)) .

(34)

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Setting now .u(x, v) = α(x) + v and .xα+ (v) = x + (α(x) + v), the closed loop portHamiltonian structure can be represented in matrix form over .R3n+1 as ⎛

⎞ x+ − x ⎜x + (α(x)) − x + ⎟ ⎟ .⎜ ⎝ ⎠ dxα+ (v) −Yα (xα+ (v), v) ⎞⎛ ¯ x + (0) ⎞ ∇ H |x J (x) − R(x) 0 0 0 T ⎟⎜ ∇¯ H |x ++ (α(x)) ⎟ ⎜ (x)g (x) 0 0 0 −κg α α ⎟⎜ ⎟ x (0) . = ⎜ ⎝ 0 0 0 G α (xα+ (v), v)⎠⎝∇ H (xα+ (v))⎠ + 0 0 0 −G T dv α (x α (v), v) ⎛

with symmetric matrix ( .

R(x) 0 0 κgα (x)gαT (x)

) 0

expressing the modified closed-loop dissipation matrix. The conjugate output Y av (x, v), computed as the average from .α(x) of the output .Y (x, u) = LG(·,u) H (x) can be rewritten in discrete gradient form as

. α

Y av (x, v) =

. α

1 v

ʃ

v 0

+

x (v) LG α (·,s) H (xα+ (s))ds = g˜ T (x, v)∇¯ H |xαα+ .

with ʃ

v

v g(x, ˜ v) =

.

ʃ

+

G(x (α(x) + s), α(x) + s)ds =

0

0

v

G α (xα+ (s), s)ds

= α(x)g(x, α(x) + v) − α(x)g(x, α(x)) + vg(x, α(x) + v). Theorem 6 is the first step towards a variety of stabilizing techniques. Specifying the result in Theorem 4 to port-Hamiltonian structures, one gets preservation of the portHamiltonian structure under power-preserving interconnection; a qualifying property to discuss energy management based control schemes. Examples in this direction are developed in [22, 23, 38, 41–43]. Remark 9 For completeness, we report the port-controlled Hamiltonian structure usually proposed in the literature [2, 16, 60]. One sets .

x + (u) = x + (J (x) − R(x))∇¯ H |xx

h (x, u) =

. lit

+ T glit (x, u)∇¯ H |xx (u)

+

(u)

+ uglit (x, u)

(35a) (35b)

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where, with respect to the form we propose, the discrete gradient of . H from .x to .x + (0) is substituted with the discrete gradient of . H from .x to .x + (u) plus an additive controlled part .uglit (x, u). It results that the rate of change of . H between two successive time instants rewrites as .

H (x + (u)) − H (x) = −(∇¯ H |xx 

+

) R(x)∇¯ H |xx 

(u) 

+

≤0

(u)



 +uglit (x, u)∇¯ H |xx

+

(u)

(36)

so naturally concluding passivity with respect to the output map .h lit (x, u). However, the fact that the resistive part depends on the input variable in an unpredictable + way through the term .∇¯ H |xx (u) may represent an obstacle to managing damping or energy exchanges under feedback. As a consequence in (36), the inner product  ¯ H |xx + (u) , does not contain the total power supplied to the system because .uglit (x, u)∇ the resistive part has a control dependent element too. Example 2 Specifying the state equations (28) (equivalently (30)) to a linear dynamics with quadratic Hamiltonian function . H (x) = 21 x  P x and symmetric positive matrix . P, a linear Port-Hamiltonian structure can be be described as .

x + (0) = x + (J − R)

P (x + x + (0)); 2

d x + (u) = B; Y (x) = B  P x du

(37)

or equivalently in map form as .

x + (u) = x + (J − R)

P P (x + x + (0)) + Bu; Y av (x, u) = B  (x + (0) + x + (u)) 2 2 (38)

with matrices of appropriate dimensions and constant elements. Because the discrete gradient function can be explicitly expressed as a function of .x and .x + (0), the state equations (37) (equivalently (38)) can be rewritten in their explicit form so getting a linear dynamics with an output map admitting a feed-through term as it is required to encompass a passivity property. One gets .

with

x + (u) = A H x + Bu;  AH = I −

y(x, u) = B  P A H x +  (J −R)P −1 2



I+

(J −R)P 2



B P B u 2

(39)

.

Easy computations show that the output . y(x, u) is exactly the average output associated with .Y (x) = B  P x since by definition

Passivity Techniques and Hamiltonian Structures in Discrete Time

( .

BT P x

)av

349

ʃ 1 u T + B P x (s)ds u 0 ʃ BT P B 1 u T u. B P(A H x + Bs)ds = B T P A H x + = u 0 2

(x, u) =

It is also possible to raise the question: when and how a passive system satisfies a pH structure? In the linear case, the question can be answered. Proposition 4 Consider the average passive linear system x + (u) = Ax + Bu;

Y (x) = B T P x

with positive definite storage. 21 x T P x, then it can be rewritten in the port-Hamiltonian form (37) (equivalently (38)) according to the decomposition in skew-symmetric and symmetric part as follows 2(A − I )(I + A)−1 P −1 = J − R; J = −J T ; R = R T ≥ 0.

.

5 Concluding Comments In the proposed differential algebraic framework, discrete-time Port-controlled Hamiltonian structures that validate the usually required energy balance properties are described in the proposed differential algebraic framework. Accordingly, the basic stabilizing techniques behind energy-based control strategies are revisited to confirm the open perspectives regarding control design through energy management along the lines developed in [22, 23, 38, 41]. All the material discussed in this paper regards a purely discrete time setting but it can be specified to the sampled-data context. How port-Hamiltonian structures are transformed under sampling when assuming both the measurements and control variables available at discrete time instants is thus a natural and challenging question addressed in [30, 35, 37]. In that digital framework, it comes out that the sampled-data dynamics are necessarily parameterized by the sampling period as well as the control solutions that are described around the continuous-time ones by infinite series expansions. As a result, the solutions can be computed through an iterative procedure and approximated at any order so rendering the approach constructive in a digital environment. Finally, we stress that in our opinion the proposed approach sets an unifying framework to investigate controlled finite-dimensional dynamical systems in discrete-time as well as under sampling. Acknowledgements Dorothée Normand-Cyrot expresses her gratitude to the Committee Members for the invitation to give a plenary lecture at the ICDEA 2022.

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Explicit MPC Solution Using Hasse Diagrams: Construction, Storage and Retrieval Stefan ¸ S. Mihai, Florin Stoican and Bogdan D. Ciubotaru

Abstract This chapter provides new methods for the construction, storage and retrieval of the explicit MPC solution in the case with quadratic cost and linear constraints. By exploiting the geometric interpretation of the MPC problem, we: (i) construct the explicit solution (i.e., enumerate the critical regions and associated affine laws) in an efficient manner; (ii) store it as a partially ordered set; and (iii) provide a modified graph traversal algorithm for efficient point location (i.e., identifying the currently active critical region and its associated control law). Keywords Explicit MPC · Hasse diagram · Point location problem · Multi-parametric quadratic programming

1 Introduction Model Predictive Control (MPC) is a popular control strategy which has time and time again proved its mettle in both theoretical and industrial process control applications [5]. Essentially, MPC is a feedback control problem that uses an accurate model of the process to predict the evolution of its state vector and optimize a sequence of constrained control actions over a finite time horizon. A major impetus for the popularity of MPC is the ease of imposing constraints on the state, input and the output vectors, respectively. While a mature technique, there is still active interest and ongoing work in the field, with many variants and extensions proposed in recent S. ¸ S. Mihai .· F. Stoican .(B) .· B. D. Ciubotaru Department of Automatic Control and Systems Engineering, National University of Science and Technology Politehnica Bucharest, 313 Splaiul Independen¸tei, Sector 6, 060042 Bucharest, Romania e-mail: [email protected] S. ¸ S. Mihai e-mail: [email protected] B. D. Ciubotaru e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 S. Olaru et al. (eds.), Difference Equations, Discrete Dynamical Systems and Applications, Springer Proceedings in Mathematics & Statistics 444, https://doi.org/10.1007/978-3-031-51049-6_16

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years. One of these variants is the explicit MPC formulation, described at length for the case with linear constraints and quadratic cost in [2, 4], exploits the multiparametric nature of the optimization program and which will be discussed at length in the remainder of this chapter. Not least, the explicit MPC is an attractive choice in embedded applications because of its ease of implementation. More often than not, embedded hardware lacks the computational capabilities to solve online an optimization problem whilst under strict time constraints. The explicit MPC is a good alternative to the classical MPC because all the control gains are computed offline, after which may then be stored as look-up tables. Under the aforementioned assumptions, the optimal control law is a piecewise affine function with polyhedral support that partitions the statespace into critical regions. Consequently, the online effort amounts to identifying which control gains to use at any moment in time by locating the currently active critical region. The main issue of this approach is that the number of regions increases exponentially with the prediction horizon and the search of all critical regions becomes tedious (the so-called point location problem). Significant research has been directed towards implementations which provide the solution in a reasonable amount of time, many of them based on the enumeration of candidate sets of active constraints [1, 15, 19]. Others have followed a combinatorial interpretation by recasting the problem as a ‘multiparametric linear complementarity’ one [9, 10, 17]. Not least, there are papers that explore a geometric interpretation [20, 21], similar to our approach in the previous work [13]. The improvements discussed in this chapter cover: (i) an extension of [13] by constructing a directed graph whose nodes represent the solution of a multi-parametric quadratic program, giving the critical regions and their associated affine laws characterizing the control action; (ii) a graph constructed by imposing stopping criteria in the face-lattice algorithm of [11], thus significantly reducing the number of enumerated nodes (and, consequently, the computation time); (iii) a graph traversal algorithm for the identification of the active critical region (i.e., the one in which the current state-vector lies) via a small number of inclusion checks (tests of neighboring regions). The paper is organized as follows. Section 2 discusses the essential notions about polyhedral sets, later used to describe the constraints of the optimization problem, and the multi-parametric quadratic problem which represents the scaffolding for the explicit MPC. This section also discusses the construction of a polyhedral set’s face lattice and its representation using its Hasse diagram, and provides an algorithm that is later modified to compute the solution of the optimization problem. Section 3 gives details about the computation of the solution and introduces a novel algorithm that gives the solution in the form of a directed graph whose elements are partially ordered by the set inclusion operation. We also discuss the identification and retrieval of the solution in the online mode, and we propose a graph traversal algorithm that explores the particular structure of the solution that we mentioned. Lastly, in Sect. 4

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we validate the C++ implementation of our algorithm and compare the solution with a state-of-the-art toolbox. Moreover, we test the solution retrieval algorithm and discuss the results. Notation: The matrix . X T denotes the transpose of matrix . X ; the matrix . X A , with .A = {1, . . . , n}, is the sub-matrix of . X , composed by selecting rows of indices .i ∈ A; the comparison operators (e.g., ‘.≤’) are applied component-wise; .all(·) returns logical false if at least one element of the input vector is not logical true. represents the For polyhedral sets, ‘conv’ { } i.e., ∑ ∑convex hull (of the input vertices), .conv({v1 , . . . , vn }) = x : x = λ v , λ = 1, λ ≥ 0, ∀i = {1, . . . , n} and i i i i i { ∑ ‘cone’ is the nonnegative hull, .cone({r1 , . . . , rn }) = x : x = i λi ri , λ ≥ 0, ∀i = {1, . . . , n}}.

2 Preliminaries 2.1 Polyhedral Tools In [8], Fukuda has an in-depth discussion about convex polyhedra in arbitrary dimensions and the common computational problems related to them. This section highlights the relevant notions, required for the remainder of the chapter. Definition 1 (Convex polyhedron) Let .x ∈ Rd be a vector and .(A, b) be a matrixvector pair with . A ∈ Rn×d , .b ∈ Rn . The set .

P(A, b) = {x ∈ Rd : AiT x ≤ b}

(1)

is the half-space representation of a polyhedron, defined as the intersection of.d linear inequalities. Here, . AiT represents the .i th line of . A. If the polyhedral set in (1) is bounded, we call . P a polytope. Remark 1 The polyhedral set in (1) has the equivalent vertex-ray representation .

P = conv({v1 , . . . , vdV }) + cone({r1 , . . . , rd R }),

(2)

where .vi , ∀i = {1, . . . , dV }, and .ri , ∀i = {1, . . . , d R }, are the vertices and rays of . P, respectively. If . P is a polytope, then .cone({r1 , . . . , rd R }) = ∅. Definition 2 (Face of a polytope) Let . P(A, b) be a polytope. Then a subset . F of . P is a face of . P iff d . F = {x ∈ R : A A x = bA , A I x ≤ bI }. (3) Here, .A ⊆ {1, . . . , n} is a set of indices corresponding to the active inequalities, and the set .I is constructed as .{1, . . . , n} \ A, which corresponds to the inactive inequalities.

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Remark 2 A face’s dimension may go up to the dimension of the space in which the polytope is embedded. Particularly, the .0-faces of a fully-dimensional polytope in .Rd are called vertices, the .1-faces represent its edges, and the .(d − 1)-faces are the facets. Definition 3 (Face lattice) Let .F(P) be the finite set of all .k-faces of . P, .∀k ∈ {0, . . . , d}, ordered by set inclusion. Then .F(P) is the face lattice of . P and it is a partially ordered set, i.e., a poset. Definition 4 (Polar polytope) Let . P ◦ be a polytope with the face lattice .F(P ◦ ). We say that . P ◦ is the polar, or dual, polytope of . P if .F(P ◦ ) is anti-isomorphic to .F(P). Remark 3 We can easily construct . P ◦ using the matrix .V of all vertices of . P as (by the polar polytope’s definition, we have that .∀x ∈ P, ∃y ∈ P ◦ s.t. .x T y ≤ 1) .

P ◦ = {x ∈ Rn : V T x ≤ 1}.

(4)

2.2 The Multi-parametric Quadratic Problem Essentially, the multi-parametric quadratic program (mpQP) is a convex constrained optimization problem where the solution is formulated in terms of a multi-variate parameter, hence the name of “multi-parametric”. Such problems are common in engineering applications, the object of study in this chapter (the MPC problem) being a famous exponent of this class [12]. Definition 5 (mpQP) Let .b, .u ∈ Rm , .x ∈ Rn be vectors, and . Q, . H , . A and . E be matrices with appropriate dimensions. Then 1 u ★ (x) = arg min u T Qu + x T H u, 2 .s.t. Au ≤ b + E x,

.

(5a) (5b)

is a multi-parametric quadratic program with the solution .u ★ (x) a piecewise affine function in the parameter .x, having polyhedral support, composed from the so-called critical regions. The first step in solving (5) is to put the optimization problem in its dual form and give the necessary (and sufficient in the case of quadratic cost) Karush-Kuhn-Tucker (KKT) system of equations .

Qu ★ + H T x + AT λ★ = 0, . Au − E x ≤ b,

(6a) (6b)

λ★ ≥ 0, (Au − E x − b) = 0.

(6c) (6d)

.

λ

.

★

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In (6d), the . operator is the element-wise product. This equation is known as the complementarity condition and induces a separation of constraints into the active and inactive sets .A = {i ∈ N : Ai u − E i x − bi = 0} and .I = {1, . . . } \ A, respectively. It follows that both the primal and dual solution .u ★ (x), and, respectively, .λ★ (x) can be derived by rewriting (6) in the matrix form [ .

Q AT A AA 0

] [ ★] [ ] −H T x u = . λ★ bA + E A x

(7)

Lemma 1 (Existence condition for KKT system (6)) Let .A be a set of active indices and [ ] Q AT A .KA = , AA 0 then .A is a feasible candidate set if . AA Q −1 AT A has full column rank. Proof Assume that the cost matrix . Q is nonsingular. Then the inverse of .KA is −1 KA =

.

[

] −1 −1 −1 −Q −1 AT Q −1 + Q −1 AT A (KA /Q) AA Q A (KA /Q) , −(KA /Q)−1 AA Q −1 (KA /Q)−1

where .KA /Q = −AA Q −1 AT A is the Schur complement of .KA . It stands to reason ◻ that .KA admits an inverse whenever .KA /Q has full column rank. The challenge is to find the collection of all feasible sets of active indices .A to compute .u ★ (x) with (7). Each such set of active constraints uniquely describes a particular affine control law .u ★ (x).

2.3 Hasse Diagram Construction The Hasse diagram is an intuitive mathematical tool that allows us to represent the order between elements of a set. For polytopes, the diagram illustrates the elements of its face lattice and the relations between them, as depicted in Fig. 1. Regarding the practical implementation, we can represent the Hasse diagram using a directed graph, which is introduced by Kaibel in his work [11]. In Sect. 3, we will modify this algorithm to obtain the solution of mpQP (5). Remark 4 Kaibel’s algorithm is computationally expensive for polytopes embedded in high dimensional spaces, as further detailed in [11]. Moreover, the algorithm requires the incidence matrix of the polytope which can be obtained after determining its vertices with a vertex enumeration algorithm (e.g., see [3]). This demand may lead to an unpredictable increase in the computation time of the graph.

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Fig. 1 Face lattice representation using the Hasse diagram

Remark 5 Recalling the definition of the face lattice and of the polar polytope, we note that the same Hasse diagram describes both a polytope and its polar (with a different interpretation for the nodes content and a switch in the inclusion order). This will prove helpful in reducing the computation time later on.

3 The Solution of the Multi-parametric Problem The solution of the mpQP, .u ★ (x), is a piecewise affine function with polyhedral support which partitions the state-space into critical regions. Each critical region can be constructed from a set of active indices .A introduced in (7). The challenge is to find the collection of feasible .A that leads to a nonempty critical region. In a previous work [13], we introduced the lifted feasible domain as .

P

([

] ) A −E , b =

([ ] [ ] ) [ ] u u ∈ Rm+n : A −E ≤b , x x

(8)

which, through simple algebraic manipulations, is shown to be the feasible domain of the mpQP problem from (5), put in the extended form

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Fig. 2 Illustrative example: the solution corresponds to the projection (blue) on the lifted feasible domain (light blue) of the unconstrained optimum (red) [14]

[ ] [ ][ ] 1 u T Q HT u , H 2H T Q −1 H x u 2 x [ ] [ ] u A −E ≤ b. .s.t. x

u ★ (x) = arg min

.

(9a) (9b)

Remark 6 Representation (9) is a fixed-shape problem in the lifted space .Rm+n , unlike the original form (5) which is in the lower-dimensional space .Rm but which changes with each value of the parameter .x ∈ Rn . The critical regions ([ mentioned ] ) in Definition 5 can be interpreted as the mapping of some faces of . P A −E , b on the unconstrained optimum subspace . Q −1 H T x, as illustrated in Fig. 2. The analytical solution of the mpQP is derived from the KKT system (6) as u ★ (x) = L A x0 +

.

A,

where the locally optimal gain matrices . L A ,

x ∈ CRA , A,

(10)

and the critical region .CRA are

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CRA =

.

.

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

LA = Q A

[ x ∈ Rn :

−1

(

] [ b − AI AI L A − E I x≤ I −SA sA zA

ZA −1 T −1 AT AA ) (E A A (AA Q

+ AA Q

−1

A

⎫ ⎪ ⎪ ]⎪ ⎬ ⎪ ⎪ ⎪ ⎭

,

(11a)

) H T) − H T ,

−1 T −1 = Q −1 AT AA ) bA , A (AA Q

−1 −1 T SA = −(AA Q −1 AT H ), A ) (E A + AA Q

sA = −( AA Q

−1

(11b)

−1 AT A ) bA .

In the following subsection, we show how to identify the critical regions using the face lattice of the lifted feasible domain.

3.1 Critical Regions Identification Using the Hasse Diagram So far we mentioned the link between the lifted feasible domain and the critical regions that define the solution. Moreover, we have stated that a critical region can be constructed from a set of active indices .A using (11a). It is obvious that one may attempt to identify the critical regions (thus the solution of the mpQP) by iterating through combinations of indices to create.A and test if the resulting.CRA is nonempty. Conversely, this approach may never converge in a feasible time for problems with a large number of inequalities. A geometrical insight is that the critical regions are linked to the faces of the feasible domain (see Remark 6). It is natural then to reduce the search space of .A’s by looking only at those combinations of indices that encode the lifted feasible domain. The proposed in [13] is to construct the entire face lattice of ([ ] approach ) A −E , b . While this leads to a reduced number of candidate sets of active .P indices, the generation of the face lattice can take a large amount of time depending on the shape of . P. In this chapter, we propose to modify Kaibel’s algorithm [11] to enumerate potential solutions of the mpQP by exploiting the duality of the lifted feasible domain1 . Noteworthy, we partially compute the face lattice by imposing stopping criteria which leads to the pruned graph .L' that contains only the solutions of the mpQP whilst preserving the poset property. This approach is detailed in Algorithm 1. We later argue that the conservation of relations between the set of active indices facilitates the identification of the critical region in the online scenario. To sum up briefly, the algorithm computes the partial face lattice of the polar lifted feasible domain. It uses a queue to store nodes (i.e., a collection of vertices) that are evaluated. Because of the duality of polytopes, the selection of vertices which define 1

Recalling the properties of the dual polytope, we know that its vertices correspond to the active inequalities of the original.

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a node in the Hasse diagram of the dual polytope correspond the active inequalities that define a face of the original polytope. We only add a node to the queue after it passes two eliminatory tests (steps 14 and 17). Firstly, we compute the matrix.KA in step 12. Recalling Lemma 1, a solution may only exist if .KA is full rank (step 14). The complexity of this step is related to the choice of algorithm for computing the rank. Secondly, even if .A respects the rank condition, the resulting critical region must be nonempty. To tackle this problem, we used two different approaches in step 17: i) the cdd library [7] and ii) an algebraic method [23]. An example is discussed in Sect. 4. Algorithm 1 mpQP solution.

[ ] Require: the lifted feasible domain H-representation ( A −E , b) and the mpQP cost matrices Q and H Ensure: the poset L' with the solution of the mpQP [ ] 1: compute the vertex matrix of the polar lifted feasible domain V ◦ = A −E /b 2: compute the H-representation ( A◦ , b◦ ) of the polar polytope from V ◦ (see [3]) and the vertexfacet incidence matrix 3: initialize L' with ∅ (a node containing ∅) 4: initialize a queue Q by ∅ 5: while the size of Q > 0 do 6: extract H from Q (the ‘pop’ operation for a queue) 7: compute the collection H of all H (v), v ∈ V \ H (as detailed in [11]) 8: compute the set G of minimal sets in H (as detailed in [11]) 9: for each A ∈ G do 10: find or create the node A corresponding to A in L' 11: if A was not found in L' then 12: compute KA = AA Q −1 AA T 13: if KA is not full rank then 14: continue 15: end if 16: compute the H-representation (Z A , z A ) of the critical region [12] 17: if the region (Z A , z A ) is empty then 18: continue 19: end if 20: add A to Q 21: create the edge ( H , A ) in L' 22: end if 23: end for 24: end while

Remark 7 We used the LU decomposition ) method which gives, in our case, a com( plexity of approximately .O (m + |A|)3 . Noteworthy, the complexity of [11] is .O(min{d V , d H } · α · ϕ), where .α is the number of facet-vertex incidences and .ϕ is [ ] the total number of faces of . P( A −E , b). However, it is difficult to estimate the complexity of the face enumeration problem in step 2.

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3.2 Storage and Retrieval by Exploiting the Graph Structure As stated, the solution of the mpQP (5) is a piecewise affine function in the parameter .x. When the optimization program is related to an explicit MPC problem, the solution .u ★ (x) must later be implemented on the specific hardware to dictate the appropriate control laws in the online mode. Classically, either the critical regions’ half-spaces or the sets of active indices are stored in look-up tables through which the micro-controller iterates to indentify the critical region where the current state vector.x lies. This solution search is equivalent with the point location problem which aims to identify the active piece, i.e., the one containing the point, of a piecewise affine function. There is a plethora of methods aiming to reduce either the storage demands, the search times, or both. Most of the approaches propose variations of a binary search tree implementation [16, 22] but also lattice expressions [24] or some geometrical insight as in [6]. The common drawback of these methods is that they require pre-processing steps which may be impractical for large problems. Granted, these steps are performed offline and should not hinder the execution of the point location problem. We have shown in Sect. 3 that the sets of active indices corresponding to a solution come together in a directed graph that represents a partially ordered set. Our goal is to exploit this intrinsic structure of the solution to reduce the number of inclusion checks of the state vector .x. To do this, we have designed Algorithm 2, which is a modified breadth-first search (BFS) algorithm. As illustrated in Fig. 3, we argue that the next iteration .xk+1 of the current state .xk will not, in most cases, jump far in the solution space, but rather that it should visit a local neighbourhood of .xk . It is well-known [1] that, disregarding degenerate cases, neighboring critical regions differ through a single index, either added or removed from the set. Since our geometric insight makes it clear that neighboring critical regions correspond to neighboring faces of the lifted feasible domain, it is reasonable to search for the regions containing .xk+1 near to the one containing .xk .

Fig. 3 Illustrative example: the neighbourhood of .A1 where .xk is located

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To simplify the notations in Algorithm 2, we will use parentheses to index elements. The map .V declared in step 7 should be interpreted as an associative container with key-value pairs, where the keys are unique. The solution graph .L' is described through its adjacency list and it has the following interpretation: the head of the list .L' (node) points to a collection of nodes which we call children; we abuse the notation .nodes = L' (node) in step 18 to indicate that we retrieve the collection of children from the parent node .L' (node). Our algorithm works by processing unvisited nodes in .L' that are stored in a queue. Initially, the .r oot node is added to the queue and it is checked whether .xk is contained in that region. Then we iterate through the parents in .L' and we store in the queue all the parents which contain .node as a child. After this step, we add to the queue all the children of the .node. Algorithm 2 Point location algorithm. Require: xk ∈ Rn , the root node from where the search begins, the finite collection of critical regions S = {i ∈ N, Ai : Z Ai , z Ai } and the solution graph L' Ensure: Z A , z A which describe the critical region in which xk is contained 1: if A corresponding to xk−1 is known then 2: r oot = A 3: else 4: r oot = ∅ 5: end if 6: initialize a queue Q with r oot 7: declare a map V for the visited nodes and let V (r oot) = true 8: while Q is not empty do 9: extract node and remove it from Q 10: locate node in S and retrieve Z node , z node 11: if all(Z node xk ≤ z node ) is true then 12: return Z A , z A , A = node 13: end if 14: for each par ent in L' do 15: if V ( par ent) is true then 16: continue 17: end if 18: let childr en = L' ( par ent) 19: if node is not contained among childr en then 20: add par ent to Q and make V ( par ent) true 21: end if 22: end for 23: for each child in L' (node) do 24: if V (child) is false then 25: add child to Q and make V (child) true 26: end if 27: end for 28: end while

Remark 8 The similarity between our algorithm and the standard BFS is that we use a queue, i.e., a “first in, first out” mechanism, to store the visited nodes.

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4 Numerical Example and Discussions In this section, we introduce an arbitrary MPC problem that is translated to the explicit MPC formulation equivalent to the mpQP (5). For performance considerations, we implemented Algorithm 1 in C++ using the Eigen library, which is a C++ template specialized for linear algebra computations. Additionally, we use the cdd library [7] for polyhedral computations, i.e., vertex enumeration and emptyness test from the step 17 of Algorithm 1. Furthermore, we have also implemented an algebraic test to ascertain the emptyness of a polyhedron [23], which is useful for large problems where cdd might be slow. To illustrate the performance of our algorithms described in Sect. 3, consider a linear discrete state-space system with two inputs ⎡

⎤ ⎡ ⎤ −0.3574 0.1518 −0.7103 0.0411 −0.1733 . x k+1 = ⎣−0.6596 −0.2792 −0.1984⎦ x k + ⎣−0.3084 −0.0484⎦ u k , 0.3042 0.6718 −0.3859 0.2922 0.1498 [ ] [ ] . yk = −0.1847 −0.4079 −0.3729 x k + −0.6224 −0.0909 u k .

(12a) (12b)

Validation is done with the Parametric Optimization Toolbox (POP) [18], which is installed in MATLAB. We use the same toolbox to convert (12) to the mpQP form whilst adding the following box constraints for input and output magnitude bounds, and input variation bounds, that were randomly generated [ ]T |xk | ≤ 1.0263 2.0014 0.4006 , [ ]T . |u k | ≤ 1.1371 0.1528 , [ ]T . |u k − u k−1 | ≤ 0.1137 0.0153 , . |yk | ≤ 1. .

(13a) (13b) (13c) (13d)

The cost matrices associated to the MPC problem are similarly generated .

[ ] Q x = diag( 1.7543 0.0958 2.1694 ), [ ] . Ru = diag( 1.0946 1.4240 ), .

Px = I3 .

(14a) (14b) (14c)

After choosing the prediction horizon . N = 3, the MATLAB POP toolbox solved this problem in .31 seconds, while our C++ implementation of Algorithm 1 took .0.5 seconds, giving the same solution illustrated in Fig. 4. Both of these algorithms were run on a machine with Windows 11 OS and an AMD Ryzen 5 4000 series processor. While it is difficult to compare two programs that run in different environments, it is clear that the C++ implementation is significantly faster. The resulted .215 critical regions are illustrated in Fig. 4. The associated graph (the restriction of the Hasse .k-skeleton associated to the lifted domain, computed as in Algorithm 1) is shown in Appendix.

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Fig. 4 The critical regions associated to the solution of the numerical example

After the computation of critical regions and gain matrices . L A , . A , the explicit MPC controller can be implemented using look-up tables. The only challenging part in the online scenario is to find in which critical region, described by .A, lies the current state vector .xk . To test Algorithm 2 described in Sect. 3, we have to generate a trajectory that crosses the critical regions illustrated in Fig. 4 without leaving the boundary of the feasible domain. Moreover, each point of the trajectory should not be situated randomly in the feasible region, but rather in the vicinity of the previous one, i.e., .||xk+1 − xk || ≤ δ, ∀k ∈ N, δ > 0 finite. Alternatively, we can generate the test trajectory by measuring the closed-loop state .xk when using the explicit MPC controller; however, .xk quickly converges to the origin, thus making it less than ideal for testing the point location performance. The performance metric that we chose is the total number of inclusion checks of a state vector .xk in a critical region .CRA , until its corresponding region is identified. The critical regions are generated with (11a), so we only look at sets of active indices ' .A ∈ L that are searched using the point location procedure detailed in Algorithm 2. For comparison, we added the same elements .A in a stack and we applied the same metric on a linear search, i.e., a simple iteration through the active sets until the correct region is identified. Our results are illustrated in Fig. 5 where the normalized distribution of total inclusion checks and the output is as expected. The point location problem particularized for the graph traversal Algorithm 2, in gray, reveals that over .40% of inclusion checks are below .20 checks, which represents .9.3% of the total possible .215 checks.

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Fig. 5 The distribution of inclusion checks . Z A x0 ≤ z A , when searching the solution graph (gray) and the solution stack (blue) Table 1 Performance indicators for the critical region identification algorithms Median (checks) Mean (checks) Std. dev. (checks) Graph traversal Linear search

.28

.43.4

.47.6

.86

.81.8

.63.5

By contrast, the linear search, in blue, has more of a uniform distribution. Extensive results are illustrated in Table 1, such as the median, mean and the standard deviation of inclusion checks. It is important to mention that Algorithm 2 is not optimized for time and has some overhead from the data structures used, i.e., the queue and the associative container. Conversely, the graph traversal outperforms a classical linear search by the chosen metric. Noteworthy, by increasing the prediction horizon . N , the step 2 in Algorithm 1 becomes a challenge because the number of inequalities of the polar polytope quickly increases. Table 2 summarizes this undesired effect.

Explicit MPC Solution Using Hasse Diagrams: Construction, … Table 2 The effect on the computation time of increasing the prediction horizon ' .N Inequalities in . A◦ .L computation time (s) Face enumeration computation time (s)

367

Regions

.3

.976

.0.285

.0.020

.215

.4

.6384

.4.284

.1.356

.333

.5

.42352

.13.537

.42.424

.281

.6

.168662

.59.225

.417.707

.269

5 Conclusion In this chapter, we have introduced a geometric interpretation of the multi–parametric quadratic program by reformulating the problem to emphasize the lifted feasible domain, as in (9). Next, we discussed the particularization of Kaibel’s face-lattice algorithm [11] in Algorithm 1, to generate a directed graph .L' that gives the mpQP solution as a poset. This property was exploited in Sect. 3.2 to identify in a small number of inclusion checks which critical region contains the current state vector . x k ; as mentioned, this is a critical component of the explicit MPC strategy, where it is required to quickly locate the state vector in a critical region to retrieve the locally optimal affine laws and close the loop with a state feedback action. Lastly, in Sect. 4 we tested the computation of the explicit MPC’s solution with a C++ implementation of Algorithm 1, which was compared with the state-of-the-art POP toolbox [18], obtaining an identical solution, but with significant reduction in computation time. Additionally, we tested the point location Algorithm 2 and compared it with a classical linear search approach, outperforming it with .53.31% in the mean number of inclusion checks. Open problems still remain on this subject. In future work, we plan to investigate how to update the solution graph .L' when the prediction horizon of the problem is increased, without recomputing the entire face lattice. We are also interested in understanding how the Hasse diagram of particular polyhedral shapes (e.g., zonotopes) modify when intersecting these shapes; this is justified by the nature of the constraint set, which is in fact the intersection of simpler sets (those corresponding to the stage constraints). Acknowledgements This work was supported by a grant of the Ministry of Research, Innovation and Digitization, CCCDI—UEFISCDI, project number PN-III-P2-2.1-PED-2021-1626, within PNCDI III.

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Appendix: Restricted Hasse Diagram of Example (12)–(14)

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References 1. Ahmadi-Moshkenani, P., Johansen, T.A., Olaru, S.: Combinatorial approach toward multiparametric quadratic programming based on characterizing adjacent critical regions. IEEE Trans. Autom. Control 63(10), 3221–3231 (2018) 2. Alessio, A., Bemporad, A.: A survey on explicit model predictive control. In: Nonlinear Model Predictive Control, pp. 345–369. Springer (2009) 3. Avis, D., Fukuda, K.: Reverse search for enumeration. Discrete Appl. Math. 65(1–3), 21–46 (1996) 4. Bemporad, A., Morari, M., Dua, V., Pistikopoulos, E.N.: The explicit linear quadratic regulator for constrained systems. Automatica 38(1), 3–20 (2002) 5. Darby, M.L., Nikolaou, M.: MPC: current practice and challenges. Control Eng. Practice 20(4), 328–342 (2012) 6. Fuchs, A.N., Jones, C., Morari, M.: Optimized decision trees for point location in polytopic data sets-application to explicit MPC. In: Proceedings of the American Control Conference, pp. 5507–5512. IEEE (2010) 7. Fukuda, K.: Cddlib reference manual. Report version 093a, McGill University, Montréal, Quebec, Canada (2003) 8. Fukuda, K.: Polyhedral computation (2020). https://doi.org/10.3929/ethz-b-000426218 9. Jones, C.N., Bari´c, M., Morari, M.: Multiparametric linear programming with applications to control. Europ. J. Control 13(2–3), 152–170 (2007) 10. Jones, C.N., Morrari, M.: Multiparametric linear complementarity problems. In: Proceedings of the 45th IEEE Conference on Decision and Control, pp. 5687–5692. IEEE (2006) 11. Kaibel, V., Pfetsch, M.E.: Computing the face lattice of a polytope from its vertex-facet incidences. Comput. Geom. 23(3), 281–290 (2002) 12. Kvasnica, M., Jones, C.N., Pejcic, I., Holaza, J., Korda, M., Bakaráˇc, P.: Real-time implementation of explicit model predictive control. In: Handbook of Model Predictive Control, pp. 387–412. Springer (2019) 13. Mihai, S.S., ¸ Stoican, F., Ciubotaru, B.D.: On the link between explicit MPC and the face lattice of the lifted feasible domain. IFAC-PapersOnLine 55(16), 308–313 (2022) 14. Mihai, S.S., ¸ Stoican, F., Ciubotaru, B.D.: Computing the explicit MPC solution using the Hasse diagram of the lifted feasible domain (2023). Submitted to ECC’23, preprint available on researchgate 15. Mönnigmann, M.: On the structure of the set of active sets in constrained linear quadratic regulation. Automatica 106, 61–69 (2019) 16. Mönnigmann, M., Kastsian, M.: Fast explicit MPC with multiway trees. IFAC Proc. Vol. 44(1), 1356–1361 (2011) 17. Morari, M., Jones, C., Zeilinger, M.N., Baric, M.: Multiparametric linear programming for control. In: 2008 27th Chinese Control Conference, pp. 2–4. IEEE (2008) 18. Oberdieck, R., Diangelakis, N.A., Papathanasiou, M.M., Nascu, I., Pistikopoulos, E.N.: Popparametric optimization toolbox. Ind. & Eng. Chem. Res. 55(33), 8979–8991 (2016) 19. Oberdieck, R., Diangelakis, N.A., Papathanasiou, M.M., Nascu, I., Pistikopoulos, E.N.: Popparametric optimization toolbox. Ind. & Eng. Chem. Res. 55(33), 8979–8991 (2016). ACS Publications 20. Olaru, S.B., Dumur, D.: A parameterized polyhedra approach for explicit constrained predictive control. In: 43rd IEEE Conference on Decision and Control (CDC), pp. 1580–1585. IEEE (2004) 21. Seron, M.M., De Dona, J.A., Goodwin, G.C.: Global analytical model predictive control with input constraints. In: Proceedings of the 39th IEEE Conference on Decision and Control, pp. 154–159 (2000) 22. Tøndel, P., Johansen, T.A., Bemporad, A.: Evaluation of piecewise affine control via binary search tree. Automatica 39(5), 945–950 (2003) 23. Truffet, L.: Some ideas to test if a polyhedron is empty (2020). arXiv:2004.12818 24. Xu, J., Lou, Y., De Schutter, B., Xiong, Z.: Error-free approximation of explicit linear MPC through lattice piecewise affine expression (2021). arXiv:2110.00201

Tube Model Predictive Control for Flexible Satellite Dynamics Sabin Diaconescu, Florin Stoican, and Bogdan D. Ciubotaru

Abstract This chapter presents a 3-axis robust model predictive control algorithm for a flexible spacecraft during pointing maneuvers. The spacecraft’s linear system dynamics are affected by parametric model uncertainties and exogenous disturbances. In order to mitigate the effect of pointing jitters on image-capturing quality, the proposed controller creates a tube as a sequence of invariant polytopic sets, with the aim of ensuring pre-imposed bounds on the state trajectories of the system. Keywords Robust MPC · Robust positive invariance · Satellite dynamics

1 Introduction Model predictive control (MPC) is among the most popular control techniques due to the high degree of flexibility it provides, for both controller design and constraint handling, as well as for providing an effective platform for negotiating trade-offs between computational complexity and cost-function (sub)optimality [6, 27]. Since most mathematical models employed for MPC design are based upon approximations of the real system’s dynamics, they rarely describe the underlying dynamics of the system to the latter’s full extent. This issue has been well-known for some time [19] and has given rise to specialized fields, such as robust and stochastic MPC [16]. S. Diaconescu · F. Stoican (B) · B. D. Ciubotaru Department of Automatic Control and Systems Engineering, National University of Science and Technology Politehnica Bucharest, 313 Splaiul Independen¸tei, Sector 6, 060042 Bucharest, Romania e-mail: [email protected] S. Diaconescu e-mail: [email protected] B. D. Ciubotaru e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 S. Olaru et al. (eds.), Difference Equations, Discrete Dynamical Systems and Applications, Springer Proceedings in Mathematics & Statistics 444, https://doi.org/10.1007/978-3-031-51049-6_17

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The ultimate goal of the robust MPC technique is to produce a controller which ensures closed-loop stability, constraint fulfilment and satisfactory performance in situations when the target plant is affected by both model uncertainties and exogenous disturbances. In contrast to the well-known classical MPC, which has been widely exploited in the fields of robotics, process and automotive control [8, 11], robust MPC has received less attention with respect to real-world applications [13], owing primarily to its theoretical and numerical complexity. Quite notably, tube-based MPC approaches tackle model uncertainties and exogenous disturbances by employing a sequence of sets, constructed as functions of online optimization variables. This sequence ensures that all future trajectories of the system will be contained within the aforementioned set [5]. However, in order to obtain a numerically tractable optimization problem, a parameterized version of the sets (used to define the tube) must be considered. To this end, there exist several approaches for the tube’s parameterization. The simplest one is called rigid tube MPC, and it employs fixed size set translations to construct the tube [18]. The translations used in rigid tube MPC can be combined with a set of scalings to produce the so-called homothetic tube MPC approaches [26]. Yet another approach is to employ a fixed number of hyperplanes, in order to generate polytopic sets. This method has the advantage that the constructed tube can be made to have any desired shape [10, 25]. An alternative to these polytopic invariant sets consists in the use of ellipsoidal sets, thus generating a problem whose size scales linearly with respect to the order of the system [22]. Midway between these two set families lie the zonotopic sets, which possess most of the flexibility associated with polyhedra, all the while being similarly easy to manipulate as the ellipsoidal sets [3]. Nevertheless, all these tube-MPC flavors mask the complexity of bounding the uncertainty via the computation of the (invariant, in the case of a rigid tube [18]) sets which describe the uncertainty’s profile. While conceptually simple, this operation may become challenging for high-dimensional problems or iterative procedures [5].

Contributions and Chapter Structure The goal of the chapter is to show how robust MPC techniques can be used for precision in satellite pointing maneuvers. The focus is on disturbance rejection, in the presence of parametric uncertainty and flexible modes in the model. Specifically, the chapter’s contributions are: (i) the application of standard MPC for the uncertain dynamics of a satellite with 4 reaction wheels and a flexible solar array; (ii) the adaptation of the MPC scheme to the robust case - rigid tube formulation; (iii) the particularization of the set invariance construction procedures for the case at hand. The rest of the chapter is organized as follows. Section 2 describes the flexible satellite dynamics and states the control problem. Section 3 reiterates upon and

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adapts the tube-MPC approach, with an emphasis on set invariance computation. The details pertaining to MPC implementation are given in Sect. 4. Simulations are then presented in Sect. 5, while conclusions and directions for further research are stated in Sect. 6.

2 Problem Statement 2.1 Flexible Satellite Modelling The model used to obtain the MPC controller has been derived via the Satellite Dynamics Toolbox (see [1, 2, 7, 29, 30]). The impact of considering the structural dynamics of the solar panels, along with the development of the full nonlinear model, is also given in [33]. The following assumptions were made for the derivation of the linear parameter varying (LPV) models: the hub is considered rigid, the nonlinear terms are neglected (under the small angular rate assumption) and the only external forces and torques acting on the appendage are the ones being applied by the hub at the anchorage point [1, 29]. The required precision for high stability pointing missions has the order of magnitude of microradians. Thus, disregarding the nonlinear terms is in no way restrictive. Considering a rigid appendage .A cantilevered to a satellite hub .B, the derivation of the attitude dynamics model follows from the classical Newton-Euler equations under the previously defined assumptions as [ .

] [ ] Fext − FB/A B aP , = DP Text,P − TB/A,P ω˙

where . Fext and .Text,P denote the external forces and torques, . FB/A and .TB/A,P the reaction force and the reaction torque due to the interaction with the appendage at point . P [12], . D B P the model of the body .B about the point . P, .a P the inertial acceleration of the main body and .ω the angular speed of the main body with respect to the inertial frame. With these considerations, the model of the body .B about the point . P is given by T DB P = τB P .

[

] m B I3 O3 τ , O3 I BB B P

(1)

DB B

where .m B is the hub’s mass, . I BB the .3 × 3 inertia matrix of the hub at point . B, .τ P B the kinematic model between the points . P and . B, and . D B B the hub’s model at point . B. Considering that the appendage is flexible, the expressions of . FB/A and .TB/A,P are described by the hybrid-cantilever model [17] as

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] [ ] FB/A aP + L TP η¨ , = DA P TB/A,P ω˙ [ ] . ( ) a η¨ + diag (2ζi ωi ) η˙ + diag ωi2 η = −L P P , ω˙ [

(2)

where . D A P is the model of the appendage .A expressed at point . P, obtained by replacing . B with . A and .B with .A in (1), . L P the . N f lex × 6 matrix of modal participation factors, .η the array of the modal coordinates, and .ωi and .ζi (i = 1, · · · , N f lex ) are the frequencies and the damping ratios of the . N f lex flexible modes, respectively. Further, the solar array is driven by a Solar Array Driving Mechanism (SADM) located inside the revolute joint, which connects the base and the appendage. The augmented model is only recalled here from [1] ⎡

⎤ 0 ] ⎤ ⎢ 0 ⎥ ⎡[ ] ⎤ ⎡[ ⎢ ⎥ ] [ FB/A ⎢ 0 ⎥ aP I6 A ⎢ ⎥ ⎣ ω˙ ⎦= ⎦, [M . ⎣ TB/A,P ] P Ra ⎢ I x ⎥ Ra Ra 0 0 0 xa ya z a ⎢ 6 a⎥ ¨ Cm θ ⎣ ya ⎦ za

(3)

where .Ra is the appendage reference frame, .[xa ya z a ]T Ra the unit vector along the revolute joint axis, .Cm the torque applied by the actuator along .(P, z a ), .θ¨ the angular acceleration inside the revolute joint and . M PA the hybrid-cantilever model (2), described previously. More details about the modeling, the impact of the harmonic disturbances produced by the SADM, along with a general framework to generate linearized models of satellites with large flexible appendages are given in [12, 31]. In order to express the dynamics of the entire assembly in the hub’s body frame, .Rb , the kinematic model, .τ P B , and the direction cosine matrix, . Pa/b , are used ⎛ [ .

Fext Text,B

] Rb

⎞

⎜ ⎟[ ] [ ] [ ]T ⎜ B ⎟ aB Pa/b O3 Pa/b O3 T A ⎟ D [M , =⎜ + τ ] τ Ra PB⎟ PB P ⎜ B O3 Pa/b O3 Pa/b ⎝ ⎠ ω˙ Rb M BA

⎡

cos θ − sin θ where Pa/b = ⎣ sin θ cos θ 0 0

.

⎤

0 0⎦ . 1

(4) (5)

Finally, for control law synthesis purposes, the inverse dynamics model of (4) must be also computed by using [1, 12] ( .

A DB B + MB

)−1

( )−1 ]−1 ( )−1 [ I6 + M BA D B = DB , B B

(6)

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such that a classical state-space representation is obtained. This operation is not further detailed here as it involves rather standard matrix manipulations. If the appendage is an embedded angular momentum, the development of the model is similar to the solar array and fully detailed in [1, 29]. The following assumptions are considered for the derivation of a simple reaction wheel model: the angular speed and the spin axis of the spinning top are considered constant in the appendage reference frame, the spinning top is balanced, and the spin axis is along the .z axis of the appendage body frame.

2.2 Design Specifications and Uncertainties The satellite has a typical configuration consisting of a flexible solar array, cantilevered to a rigid base, and four reaction wheels in a pyramidal configuration, which ensures the vanishing of the gyroscopic effect [1, Chap. 10]. The obtained model can be treated as a quasi-LPV model (see (3)), parameterized according to the geometric configuration, .θ, of the solar array with respect to the main body. The model exhibits .22 state variables coming from the three flexible modes of the solar array (6 states), one mode for SADM (2 states), four pairs of integrators in the reaction wheels (8 states) and 6 integrators on the outputs (6 states). The inputs of the system are the 3 torques (.Tx , .Ty , .Tz ) generated by the reaction wheels, while the outputs are the three angles (.Φ, .Θ, .𝚿) to be controlled. The Bode magnitude plot of the transfer between the torque, .Ty , applied by the reaction wheels, at point . B, the center of mass of the spacecraft, and its pitch angle, .Θ, is given in Fig. 1. The other transfers exhibit a similar behavior. The uncertain parameters considered are: the mass, .m B , the diagonal terms of the inertia matrix, . IxB , . I yB , . IzB , and the frequencies of the bending modes, .ω1 , .ω2 and .ω3 . The uncertainty level considered is .±20% from the respective nominal value of each parameter. The full development along with the associated numerical values are given in Chaps. 2 and 10 of [1].

2.3 Control Problem Formulation The next generation of Earth observation and space science missions involves tight pointing requirements and lower line-of-sight jitter, all while employing lighter and more flexible structures [32]. If the mass of the solar array is not negligible, its rotation has a significant impact on the pointing precision. In this regard, the evolution of the flexible modes during the solar array’s rotation is also showcased in Fig. 1. The usage of high-performance instruments comes with severe pointing requirements and constraints on the attitude control system. The pointing requirement to be satisfied, in spite of low frequency orbital disturbances, is given by the Absolute

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amplitude [dB]

50 0 −50 −100 −150 −200 −250 −2 10

10−1

100 frequency [rad/sec]

101

102

Fig. 1 Bode diagram from input torque to pitch angle

Performance Error (APE) metric, which is defined as the instantaneous value of the performance error at any given time [20, 21].

3 MPC-Based Control Scheme 3.1 State-Space Model The state-space representation of a discrete-time system, to be employed in the sequel, is defined as x

= A(δk , θk )xk + B(δk , θk )u k + νk , . yk = C(δk , θk )x k + D(δk , θk )u k + μk ,

. k+1

(7a) (7b)

where .xk ∈ Rn , .u k ∈ Rm and . yk ∈ R p denote the state, input and output vectors, respectively; the matrices . A, B, C, D have appropriate dimensions to enable the operations in (7). The internal uncertainties and the scheduling parameter (performing a full turn over one orbit) characterizing the system are denoted by .δk ∈ Rdδ and .θk ∈ R, respectively. The former are unknown but bounded, either deterministically or stochastically, while the latter are known a-priori (the scheduling parameter is measured on-board).

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The variables .νk ∈ Rdν and .μk ∈ Rdμ stand for the disturbance and the measurement noise, respectively, that affect the dynamics. Both the input and the output must remain bounded throughout the evolution of the system, i.e., they must be kept within predefined sets: .u k ∈ U ⊂ Rm and p . yk ∈ Y ⊂ R , respectively. The sets considered further are C-sets [15], i.e., they are convex, compact sets containing the origin in their non-empty interior. We now proceed to define the nominal counterpart of (7), where all unknown uncertainties, disturbances and noises vanish, thus producing x

. k+1

= A(θk )x k + B(θk )u k + ν k ,

y = C(θk )x k + D(θk )u k .

. k

(8a) (8b)

We point out the shorthand notation . X (θk ) = X (δk = 0, θk ), the lack of measurement noise and the presence of only the known (if available) part of the disturbance, denoted .ν k ; specifically, .νk = ν k + νk , with .νk denoting the unknown part. Note also that the nominal input, state and output are denoted by .u k , .x k , . y k , respectively. Lastly, we define the state-based tracking error .xk = xk − x k and the input-based one .u k = u k − u k , which, used in (7) and (8), produce the tracking error dynamics x

. k+1

= A(θk )xk + B(θk )u k + νk + ∆A(δk , θk )xk + ∆B(δk , θk )u k ,

(9a)

y = C(θk )xk + D(θk )u k

. k

+ ∆C(δk , θk )xk + ∆D(δk , θk )u k + μk ,

(9b)

having denoted . X (δk , θk ) = X (θk ) + ∆X (δk , θk ).

3.2 MPC Control Problem The discrete-time dynamics (8) are steered through the control action .u k , which is obtained by repeatedly solving a model predictive control (MPC) problem. In its standard form [24], this reduces to solving the problem min

.

u˘ k ,...,u˘ k+N −1

N −1 ∑

(x˘k+i+1 , u˘ k+i , ν k+i )

(10a)

i=0

s.t. x˘k+i+1 = f (x˘k+i , u˘ k+i , ν k+i ), x˘k = x k ,

.

(10b)

. k+i

y˘

= g(x˘k+i , u˘ k+i ),

(10c)

u˘

∈ U , y˘k+i ∈ Y , ∀i = 0, . . . , N − 1,

(10d)

. k+i

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where . f : Rn+m+dν |→ Rn , .g : Rn+m |→ R p are merely shorthand notations for the right-hand terms from (8a) and (8b), respectively. Also,.U ⊂ Rm and.Y ⊂ R p are sets which bound the nominal input and output, respectively; they constitute restrictions of the original ones, .U, Y , as the constrained optimization problem (10) predicts the behavior of the nominal dynamics (8). This is done via a sequence of states .{x˘k+1 , . . . , x˘k+N }, updated through (10b)– (10c) via a suitable sequence of inputs .{u˘ k , . . . , u˘ k+N −1 }. The aim of the latter sequence is to minimize the stage cost from (10a) and to guarantee the validation of the input/output constraints from (10d). The feedback loop is closed by: (i) initializing the prediction with the current plant state value, as per (10b), and (ii) applying to the nominal dynamics (8) the first element of the predicted input sequence .u k = u˘ k . We point out that the MPC control law applies only to the nominal dynamics (8) and gives the nominal input .u k . Thus, we consider a tube-based approach, in order to describe the uncertain dynamics behavior (7), via the tracking error dynamics (9). More specifically, by taking u = u k + K (θk ) · (xk − x k ) ,

. k

(11)

we arrive at the closed-loop form of (9), namely x

. k+1

] [ = A(θk ) + B(θk )K (θk ) xk + νk

+ ∆A(δk , θk )xk + ∆B(δk , θk )u k , ] [ . yk = C(θk ) + D(θk )K (θk ) x k + ∆C(δk , θk )xk + ∆D(δk , θk )u k + μk .

(12a) (12b)

The dynamics (12) will be employed in the next section to characterize the tube centered around the nominal state .x k , in which the uncertain states .xk are guaranteed to remain. This will provide an explicit link between the restricted sets .U , Y and their counterparts .U, Y , while providing a constructive result for the tube profile. Remark 1 Finding . K (θk ) is not always a trivial matter. The static gain matrix has to not only stabilize the dynamics for a given .θk , but to also ensure stability for any admissible variation of it. In particular, this may lead to “dwell-time” considerations where a minimum time between successive switches has to be enforced. The issue is mitigated here by the slow, linear and smooth variation between .−π and .π of the scheduling parameter .θk . Remark 2 There is no impediment to considering other control laws in (11), e.g., dynamic ones. The increase in performance may, on the other hand, be negated by the fact that this approach makes the associated invariant set computation more challenging.

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3.3 Set-Based Descriptions For two sets . P, Q ⊂ Rn , . P ⊕ Q = { p + q : ∀ p ∈ P, ∀q ∈ Q} denotes their Minkowski sum and . P Q = { p ∈ P : { p} ⊕ Q ⊆ P} denotes their Pontryagin difference. To implement the rigid tube-based MPC approach, the following high-level procedure must be executed: (i) define the robust positive invariant (RPI, see, e.g., [5]) set . S associated with dynamics (12a), such that x ∈ S =⇒ xk ' ∈ S, ∀k ' ≥ k;

. k

(13)

(ii) restrict .U in such a way so to guarantee .u k ∈ U as U = {u k : u k ∈ U, ∀xk ∈ S} ∐ =U K (θk ) S;

.

(14)

k

(iii) restrict .Y in such a way so to guarantee . yk ∈ Y as .

Y = {y k : yk ∈ Y, ∀xk ∈ S} ([ ] ) =Y C(θk ) + D(θk )K (θk ) S ⊕ ∆Y ,

(15)

where .∆Y is shorthand for the set which bounds the term .∆C(δk , θk )xk + ∆D(δk , θk )u k + μk from (12b). If the theoretic considerations given in steps (i)–(iii) of the procedure hold, then we have that (13) holds under the admissibility conditions .u k ∈ U =⇒ u k ∈ U and . y k ∈ Y =⇒ yk ∈ Y . This, in turn, leads to x − x k ∈ S ⇔ xk ∈ {x k } ⊕ S, ∀k,

. k

thus producing the tube which contains the uncertain state .xk , that is characterized by its time-varying center .x k resulted from the repeated resolution of the MPC problem (10), and by its fixed profile . S, obtained by a procedure which will be described in the sequel.

4 Controller Implementation The main difficulty in computing (13) is that the result has to be invariant under dynamics (12), which are both nonlinear and suffer from state-dependent noise. Finding such a set is possible with tools available in literature, yet the latter often

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prove, in practice, to be rather cumbersome and conservative in their approach. Hence, we restrict ourselves to the particularities of the model, as given in Sect. 2. The three main assumptions are: (A1) the output equation is not affected by uncertainties and there is no direct input action, i.e., (12b) reduces to . yk = C xk , and .∆Y = {0}. Hence, .Y = Y C S; (A2) the scheduling parameter, .θk , is known, periodic and slowly varying, with a period of .5000 sec; (A3) while the gain matrix . K (θk ) may change at each simulation step .k, it is reasonable to implement a gain scheduling procedure; i.e., taking . M samples .{θ1 , . . . , θ M } to uniformly partition the interval .[−π, π], we get matrices j j j j j . A := A(θ ), . B := B(θ ) used to determine the gains . K afterwards. Remark 3 The assumptions above are not restrictive as (A1) simply notes that uncertainties usually exhibit in the internal and/or actuation dynamics. The scheduling parameter, .θk , mentioned in (A2) is measured on-board. Lastly, the approach proposed in (A3) is the usual one when the scheduling parameter varies slowly and predictably. The numerical data extracted from the Satellite Dynamics Toolbox is used in order to construct an LPV envelope around the uncertain dynamics. The dynamics from (12) describe a linear parameter-varying (LPV) system [1], where the scheduling parameter is the rotation angle of the solar array, .θk , performing a full turn over one orbit (i.e., it varies slowly, linearly and smoothly between .−π and .π). It affects the dynamics through its own input channel and hence its effect on the state space representation cannot be considered as a simple offset added to input/state. With the aim of obtaining an explicit dependence of this parameter (for the employed state-space representations), the chosen approach involves sampling for different values of .θk . For each of these values, an uncertain state-space model is obtained, with the uncertainties being owed to the terms described in Sect. 2. A sufficiently dense sampling, at each degree from .−π to .π, highlights the fact that the nominal state and input matrices depend on the harmonics .θk /2, .θk and .2θk in a truncated Fourier trigonometric series, given by A(θ . k ) = A0 +

1 ∑

c

s

c

s

Ai cos 2i θk + Ai sin 2i θk ,

(16a)

i=−1

B(θ . k ) = B0 +

1 ∑

B i cos 2i θk + B i sin 2i θk ,

(16b)

i=−1

with the constant matrices being obtained through a nonlinear fitting procedure (carried out via MATLAB). This parameterization is next used for the computation of the various prerequisites for the set invariant computations. j j Solving a discrete-time algebraic Riccati equation for each pair (. A , B ) from Assumption 3 gives the static gain . K j stabilizing the system and, due to the convex sum properties and with Assumption 3, we may reformulate (12) as

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x

. k+1

] [ = A(θk ) + B(θk )K j xk + (νk + ∆A(δk , θk )xk + ∆B(δk , θk )u k ) ,

y = C xk ,

. k

381

(17a) (17b)

defined over a domain of validity .θk ∈ [θ j,− , θ j,+ ]. The benefits of this approach are that, instead of computing . S as the RPI set for dynamics (12), we may compute . M RPI sets . S j , for each of the systems indexed by . j in (17). That is, the RPI sets are computed for dynamics whose j j nominal state matrix is of the form . A + B K j to which are appended various sources of disturbance, as detailed next. Specifically, we have to deal with [ ] j the state- and input-dependent “disturbance” terms . A(θk ) − A + ∆A(δk , θk ) xk [ ] j and . B(θk ) − B K j + ∆B(δk , θk ) u k , also affected by the model uncertainties. In order to tackle this, we first replace .xk = x k + xk , .u k = u k + u k , then we bound . x k ∈ X, u k ∈ U , while providing upper bounds (by taking the worst-case values for a dense sampling of the uncertainties) for .∆A(δk , θk ), ∆B(δk , θk ). Given that part of the disturbance is state-dependent, we may employ appropriate techniques available in literature (see, for example, [14]) which enable the computation of the robust positive invariant sets . S j . Note that at every switch . j |→ j + 1 transitional sets map the evolution . S j |→ S j+1 . In order to deal with the associated computational burden, we make use of the state-dependent “ultimate bounds” construction from [14], which provides a skewed hypercube and iterates it for several steps in order to achieve a reasonable shape (i.e., one where the contraction between consecutive steps is small enough). Lastly, to simplify the online part of the procedure, we take . S as the bounding box of sets . S j (i.e., we take the worst-case bounds over the entire range of .θk ). Remark 4 Note that switching between models may lead to a transitional behavior where transitional sets “jump” from the invariant set associated to the current dynamics to the one associated to the next dynamics. In general, this is a significant issue which requires either the construction of a common Lyapunov function or to consider ‘dwell-time’ arguments. Since the scheduling parameter varies slowly, linearly and smoothly between .−π and .π and each dynamics are stable in closed-loop by construction, here it is reasonable to ignore the transitional artifacts (at the switch between .θ j → θ j+1 ) and it is not required to share a common Lyapunov function. Remark 5 The novelty of our approach comes from the nature of the application and from the technicalities involved in computing an invariant set for dynamics of type (12). However, caution must be taken in tackling . S. Relations (14) and (15) clearly show that the original sets .U, Y are restricted by the Pontryagin difference. If . S is too large, the result may easily be the empty set.

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5 Simulation and Validation The simulation is performed during a period of time corresponding to a complete orbit, which has an orbital period of .Tor b = 5000 sec, while the solar array’s geometrical configuration varies in the interval .[−π, π]. The optimal control problem is solved via CasADi [4], for a prediction horizon . N pr ed = 20. The cost is a standard quadratic function with . Q = diag(I3 , O11 ) and . R = 0.01 · I3 . The maximum allowed values for the control torques generated by the reaction wheels are .±50 mNm. The sampling time used is .Ts = 1/8 sec (equivalently stated, the sampling frequency is .8 Hz), as imposed by the star tracker (ST) dynamics. Moreover, the application presented below is a benchmark, and the pointing and control requirements are aligned with the industrial standards [23, 28]. The pointing requirement considered is given by [APEx APE y APEz ] = [±175 ± 175 ± 900] μrad,

.

which describes the bounds for the output tracking error (note that the reference signal is zero). The orbital perturbations, acting as additive disturbances at the plant’s input, are given by (

) 2π , Tor b ) ( π 2π , . Ty [N m] = 0.01 · sin + Tor b 3 ) ( 2π 2π . Tz [N m] = 0.02 · sin + Tor b 3 Tx [N m] = 0.03 · sin

The first step is to consider the LPV description from (17) and the prescribed bounds for the disturbance. With these, we arrive at the RPI sets . S j for each of the . M = 12 systems sampled at .{θ j }1,...,M . The bounding box of the convex hull covering these sets is used to extract the worst-case values for both input and output bounds (to be used as tightening factors, in (14) and (15), respectively). The discrete-time algebraic Riccati equation which gives the static[ feedback matrices is tuned with ] the state weighting matrix . Q K [= 2.5]· 106 · diag( 175 175 45 ) · C T C and the input weighting matrix . R K = diag( 3 1 1 ). Note that the result of these operations is still not easily amenable to illustration. What we depict in Fig. 2 are the 2D projections and 3D representations of the sets .C S j and . K j S j which allow to obtain the tightening bounds for the original input and output constraints, respectively. Specifically, Figs. 2a, b, c, d, e and f depict the 2D projections onto the planes (x,y), (x,z) and (y,z) of each of the sets .Y j and, respectively, .U j . The convex hulls of these sets (.Y and .U ) are depicted in Figs. 2g and h. Recall that the tightened constraint sets are those obtained by subtracting .U , Y , via the Pontryagin difference, from .U, Y , the original ones. While the result is still a convex set (as long as both operands are), the increase in complexity

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20

15

15

10

10 5

[µrad]

[µrad]

5 0 −5

0 −5

−10 −10

−15 −20 −30 −25 −20 −15 −10 −5

0

5

10

15

20

25

−15 −20

30

−15

−10

−5

[µrad]

8

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0 −2

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[mNm]

20

−15

5

10

15

20

(c) (y, z) projection for Y

(b) (x, z) projection for Y

[mNm]

[mNm]

(a) (x, y) projection for Y

−20 −20

0

[µrad]

−8 −20

−4 −6 −15

−10

−5

0

5

10

15

[mNm]

[mNm]

(d) (x, y) projection for U

Y

20

−8 −20

−15

−10

−5

0

5

10

15

20

[mNm]

(e) (x, z) projection for U

(g) 3D representation of

0 −2

(f) ( y, z) projection for U

(h) 3D representation of

U

Fig. 2 Sets used in the tube-MPC implementation

for the result directed us[to over-approximate ] .U , Y by their bounding boxes, deter. 19.68 19.21 7.14 mNm and, respectively, output bounds by input bounds mined [ ] . 29.58 18.29 11.70 .μrad. The resulting, tightened, constraints are now used in the resolution of the nominal MPC problem. The APEs and their associated torques, along with the imposed constraints, are shown in Figs. 3 and 4, with the presented results being guaranteed for any solar array configuration, i.e., for any .θk ∈ [−π, π]. As shown, the absolute performance error and the control signals fulfill the constraints in spite of the orbital perturbations, . T , thus validating the proposed design procedure. We observe the expected behavior: the nominal inputs and outputs lie in their restricted bounds (more precisely, in their projection along each axis). Furthermore, the gap between nominal (solid red) and original (dashed black) lines denotes the

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Fig. 3 The APEs of the nominal model with tightened and original constraints

margin of safety imposed by the construction: under uncertainty, the real trajectories will remain within the original bounds.

6 Conclusion A tube-based robust MPC design procedure has been proposed and validated for image jitter reduction, in the case of a flexible satellite. Due to the constraint-based formulation of MPC, pre-imposed limitations on command actuation and specific tracking error signals (APE) are easily and naturally handled. For future study, the real-time implementation [34] of robust MPC algorithms must be further considered along with the usage of homotethic [26] or elastic tubes [25]. In order to improve the validation procedure, a computation of the metrics complementary to the APE (as defined by the European Cooperation for Space Standardization [9]) has to be performed, following the steps presented in [20, 21]. Moreover, the validation procedure can be further improved by running an extensive Monte Carlo simulation for the uncertain system.

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Fig. 4 The control torques of the nominal model with tightened and original constraints

Acknowledgements The authors are grateful to Dr. Daniel Alazard and his team at ISAESUPAERO for making their Satellite Dynamics Toolbox [1] available. This work was supported by a grant of the Ministry of Research, Innovation and Digitization, CCCDI—UEFISCDI, project number PN-III-P2-2.1-PED-2021-1626, within PNCDI III.

References 1. Alazard, D.: Satellite Dynamics Toolbox Library (SDTlib) (2021) 2. Alazard, D., Sanfedino, F.: Satellite dynamics toolbox for preliminary design phase. In: 43rd AAS Guidance and Control Conference, Breckenridge (2020) 3. Althoff, M., Krogh, B.H.: Zonotope bundles for the efficient computation of reachable sets. In: 50th IEEE Conference on Decision and Control and European Control Conference, pp. 6814–6821. IEEE (2011) 4. Andersson, J.A.E., Gillis, J., Horn, G., Rawlings, J.B., Diehl, M.: CasADi: a software framework for nonlinear optimization and optimal control. Math. Program. Comput. 11(1), 1–36 (2019) 5. Blanchini, F., Miani, S.: Set-Theoretic Methods in Control. Springer (2008)

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6. Borrelli, F., Bemporad, A., Morari, M.: Predictive Control for Linear and Hybrid Systems. Cambridge University Press (2017) 7. Chebbi, J., Dubanchet, V., Perez Gonzalez, J.A., Alazard, D.: Linear dynamics of flexible multibody systems: a system-based approach. Multibody Syst. Dyn. 41(1), 75–100 (2017) 8. Darby, M.L., Harmse, M., Nikolaou, M.: MPC: current practice and challenges. IFAC Proc. Vol. 42(11), 86–98 (2009). 7th IFAC Symposium on Advanced Control of Chemical Processes 9. European Cooperation for Space Standardization: Pointing performance standard ECSS-E-ST60-10C. In: ESA-ESTEC Requirements and Standards Division (2008) 10. Fleming, J., Kouvaritakis, B., Cannon, M.: Robust tube MPC for linear systems with multiplicative uncertainty. IEEE Trans. Autom. Control. 60(4), 1087–1092 (2015) 11. Forbes, M.G., Patwardhan, R.S., Hamadah, H., Gopaluni, R.B.: Model predictive control in industry: challenges and opportunities. IFAC-PapersOnLine 48(8), 531–538 (2015). 9th IFAC Symposium on Advanced Control of Chemical Processes 12. Guy, N., Alazard, D., Cumer, C., Charbonnel, C.: Dynamic modeling and analysis of spacecraft with variable tilt of flexible appendages. J. Dyn. Syst. Meas. Control. 136(2) (2014) 13. Houska, B., Villanueva, M.: Robust optimization for MPC. In: Rakovi´c, S.V., Levine, W.S. (eds.) Handbook of Model Predictive Control, pp. 413-443. Springer International Publishing (2019) 14. Kofman, E., Haimovich, H., Seron, M.M.: A systematic method to obtain ultimate bounds for perturbed systems. Int. J. Control. 80(2), 167–178 (2007) 15. Kouramas, K., Rakovic, S., Kerrigan, E., Allwright, J., Mayne, D.: On the minimal robust positively invariant set for linear difference inclusions. In: Proceedings of the 44th IEEE Conference on Decision and Control, pp. 2296–2301 (2005) 16. Kouvaritakis, B., Cannon, M.: Model Predictive Control: Classical, Robust and Stochastic. Springer International Publishing (2016) 17. Manceaux-Cumer, C., Chretien, J.P.: Minimal LFT form of a spacecraft built up from two bodies. In: AIAA Guidance, Navigation, and Control Conference and Exhibit, Montreal (2001) 18. Mayne, D., Seron, M., Rakovi´c, S.: Robust model predictive control of constrained linear systems with bounded disturbances. Automatica 41(2), 219–224 (2005) 19. Morari, M., Lee, J.H.: Model predictive control: past, present and future. Comput. & Chem. Eng. 23(4), 667–682 (1999) 20. Navarro-Tapia, D., Marcos, A., Veenman, J.: Enhanced AOCS verification techniques for Euclid’s high-pointing performance. IFAC-PapersOnLine 55(25), 91–96 (2022). 10th IFAC Symposium on Robust Control Design 21. Ott, T., Benoit, A., van den Braembussche, W., Fichter, W.: ESA pointing error engineering handbook. In: 8th International ESA Conference on Guidance, Navigation & Control Systems (2011) 22. Parsi, A., Iannelli, A., Smith, R.S.: Scalable tube model predictive control of uncertain linear systems using ellipsoidal sets (2022). arXiv:2204.02134 23. Ponche, A., Marcos, A., Ott, T., Geshnizjani, R., Loehr, J.: Advanced guidance approach for multi-body/multi-actuator spacecraft repointing under attitude constraints. In: 9th European Conference for Aeronautics and Aerospace Sciences (2022) 24. Rakovi´c, S.V., Levine, W.S.: Handbook of Model Predictive Control. Springer (2018) 25. Rakovi´c, S.V., Levine, W.S., Açikmese, B.: Elastic tube model predictive control. In: 2016 American Control Conference, pp. 3594–3599 (2016) 26. Rakovi´c, S.V., Kouvaritakis, B., Findeisen, R., Cannon, M.: Homothetic tube model predictive control. Automatica 48(8), 1631–1638 (2012) 27. Rawlings, J., Mayne, D., Diehl, M.: Model Predictive Control: Theory, Computation, and Design. Nob Hill Publishing (2017) 28. Rodrigues, R., Preda, V., Sanfedino, F., Alazard, D.: Modeling, robust control synthesis and worst-case analysis for an on-orbit servicing mission with large flexible spacecraft. Aerosp. Sci. Technol. 129, 107865 (2022) 29. Sanfedino, F.: Experimental validation of a high accuracy pointing system. Ph.D. thesis, ISAESUPAERO, Toulouse, France (2019)

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30. Sanfedino, F., Alazard, D., Pommier-Budinger, V., Falcoz, A., Boquet, F.: Finite element based N-Port model for preliminary design of multibody systems. J. Sound Vib. 415, 128–146 (2018) 31. Sanfedino, F., Alazard, D., Preda, V., Oddenino, D.: Integrated modeling of microvibrations induced by solar array drive mechanism for worst-case end-to-end analysis and robust disturbance estimation. Mech. Syst. Signal Process. 163, 108168 (2022) 32. Sanfedino, F., Thiébaud, G., Alazard, D., Guercio, N., Deslaef, N.: Advances in fine line-ofsight control for large space flexible structures. Aerosp. Sci. Technol. 130, 107961 (2022) 33. Sidi, M.J.: Spacecraft Dynamics and Control: a Practical Engineering Approach. Cambridge University Press (1997) 34. Zeilinger, M.N., Raimondo, D.M., Domahidi, A., Morari, M., Jones, C.N.: On real-time robust model predictive control. Automatica 50(3), 683–694 (2014)

Numerical Modeling and Some Optimal Control Problems of Dynamic Systems Describing Contact Problems with Friction in Elasticity Nicolae Pop, Tudor Sireteanu, Luige Vladareanu, Mihaiela Iliescu, Ana-Maria Mitu, and Vicentiu Marius Maxim Abstract The purpose of the article is to make use of difference equations to solve the dynamic visco-elastic contact problems with friction. First of all, we propose the discretization in space and time of the equilibrium equations and the use of regularization methods to approximate the non-differentiable terms that model the contact friction, Coulomb type. For the discretization in space of the problem of dynamic elastic contact with friction, the finite element method is used and a system of ordinary differential equations of the second order will result, which will be solved using the finite difference method, the Newmark method and the NewtonRaphson iterative method of solving the linearized dynamic system. The model predictive control (MPC) and the linear quadratic regulator (LQR), described by the linear discrete-time difference equations derived from the system of nonlinear differential equations, have been used to solve some optimal control problems. This problem is of great importance in engineering applications, for example for the control of deformations, velocities and accelerations in areas of interest, and vibration mitigation. Areas of interest could be: the tool tip of a machine tool or the end efector of a robotic arm with links and joints etc. Keywords Finite differences · Discrete-time difference equations · Nonlinear system · Newton-Raphson iterative method · Model predictive control · Linear quadratic regulator · Constrained control · Dynamic systems · Unilateral dry contact problem · Finite element method · Coulomb law

N. Pop (B) · T. Sireteanu · L. Vladareanu · M. Iliescu · A.-M. Mitu · V. M. Maxim Institute of Solid Mechanics of Romanian Academy, Constantin Mille, 15, 10141 Bucharest, Romania e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 S. Olaru et al. (eds.), Difference Equations, Discrete Dynamical Systems and Applications, Springer Proceedings in Mathematics & Statistics 444, https://doi.org/10.1007/978-3-031-51049-6_18

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1 Introduction For numerical modeling and solving of dynamic systems describing contact problems with friction in elasticity, it is necessary to specify the conditions of existence and uniqueness of the solution. The existence of the solution for dynamic frictional contact problems is guaranteed only for the case when the material is viscoelastic with elastic behavior. To define the law of friction, a regularity of the displacement velocity on the contact boundary is necessary. The regularity of the velocity can only be defined under the assumption that the bilinear form (Hooke’s law) also includes the value of the displacement velocity, as in the viscoelastic case. The uniqueness of the solution for dynamic frictional contact problems is ensured, as in the case of static frictional contact problems, assuming that the coefficient of friction is sufficiently small, see [1, 2]. For an approximate solution of the dynamic contact problem, we will make use of the finite element method, which gives a nonlinear system of ordinary differential equations. Finite element methods, together with numerical schemes of finite differences and Newmark method, Newton-Raphson iterative method for solving the linearized dynamic system, are able to model stick-slip motion, sliding, friction damping and related phenomena in a significant range of practical problems. To solve some optimal control problems, such as the control of deformations, velocities or accelerations of deformations, shocks, forces in areas of interest and vibration attenuation, we propose using the model predictive control (MPC) and the linear quadratic regulator (LQR), described by the linearized equations with the discretetime difference derived from the system of nonlinear differential equations, see [3–5]. The novelty consists in the numerical approximation and the approach of the optimisation by using the MPC. The outline of the article is as follows: the equilibrium equations of the dynamic frictional contact problem in classical and variational form and regularization of dynamic contact problems are described in Sect. 2. Section 3 contains the discretization, in space (with the finite element method) and in time (with finite difference, Newmark method and Newton-Raphson iterative method), of the dynamic contact problem with friction, and in Sect. 4, MPC and LQR are presented in the case of a linearized dynamic system, in order to control the deformations, the deformation velocities (the variables in the state space) and the control of the output variables: forces, shocks in areas where they are of interest and where sensors and actuators can be applied. We made use of MPC and LQR from MATLAB/Simulink evironment.

2 The Dynamic Contact Problem with Friction. Classical and Variational Formulation The studies concerning the dynamic contact problem with friction are limited to the contact problems with normal tension on the contact area, to those with nonlocal friction law or which describe the contact using some contact function, which

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connects the contact zone with the normal tension on the area. The solution existence for these problems is proved only for viscoelastic materials with an elastic behavior, the similar useful results can be found in [22, 23]. In order to define the friction law it is necessary to have a certain regularization of velocity on the (Hooke’s law) also contains a value of velocity as in the case of viscosity. Let .Ω ⊂ R d be a domain of dimension .d = 2 or .d = 3 with its boundary .∂Ω = Γ U ∪ Γ N ∪ Γ C , where .ΓU , .Γ N , .ΓC are open and Lipschitz continuous. The three parts, subsets of the boundary, correspond to the three types of conditions: the boundary condition in displacements: .ΓU , the boundary condition in tractions (forces): .Γ N and respectively the boundary condition for the contact with friction: .ΓC . We assume that the domain .Ω satisfies following the hypotheses: it is a bordered, compact domain with a Lipschitz boundary. These hypotheses guarantee the ellipticity of the bilinear forms. The domain .Ω contains viscoelastic and linear materials therefore the relation between tension and deformation (Hooke’s law) follows in the form: .σi j = σi j (u, ˙ u) = Ci(1) ˙ + Ci(2) (1) jkl ekl (u) jkl ekl (u). The tensor .σi j is the tension tensor of the second order, and the coefficients .Ci(1) jkl and (2) .C i jkl are bounded and are tensors of the 4th order, from Hooke’s law, according to the theory of elasticity and represent the material constants. |Ci(id) jkl | ≤ C 0 (id), id = 1, 2; symmetric

(id) (id) Ci(id) jkl = C jikl = C kli j

as well as elliptic (id) Ci(id) jkl (x)ξi j ξkl ≥ C 0

d ∑

ξi2j

i, j=1

for each symmetric tensor .ξi j = ξ ji . Furthermore we suppose that at the initial moment .t = 0 the body is motionless. After that on the body interact the following forces: volume forces . f on .Ω, surface forces on .Γ N , the displacements .U on .ΓU and finally on the boundary .ΓC the body may be in contact with the foundation. This friction is modeled by Coulomb’s law. The contact problem in a time interval . IT = [0, T ], .T > 0 can be described as follows .u ¨ − σ i j, j (u, ˙ u) = f i on Q T := I T × Ω (2) u = U on SUT = IT × ΓU

.

(3)

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σ (n) (u, ˙ u) = h on S NT = IT × Γ N

(4)

⎫ u˙ N ≤ 0, σ N ≤ 0, u˙ N · σ N = 0 ⎬ u˙ T = 0 ⇒ |σT | ≤ F(0)|σ N | on SCT := IT × ΓC u˙ T ⎭ u˙ T /= 0 ⇒ σT = −F(u˙ T )|σ N | |u˙ T |

(5)

u(0, x) = u(0, ˙ x) = 0 on Ω,

(6)

.

.

and .

where .n represents the exterior normal on the boundary bodies, .σi(n) (u, ˙ u) = .σi j (u, ˙ u)n j , the components on the tension vector on the boundary with .σ N = σi(n) n i its normal component and .σT = σ (n) − σ N n its tangential component. Furthermore the corresponding displacements on the boundary are denoted by .u N and .u T , respectively. .F is the friction coefficient which depends or not on the velocity of displacement. The weak form of the problem (2)–(6) is in fact a variational equation. Let . I ⊂ R be an interval and .W be a Banach space, then we define . B0 (I, W ) as the set of bounded functions defined on . I with values on .W , with the norm given by ʃ ||u||2L 2 (I,W )

:=

||u(t)||2W dt. I

The set of admissible functions is { } K = v ∈ L 2 (IT , H 1 (Ω, R d )) | v = u on SUT and v N ≤ 0 on SCT , where we denoted with . L 2 , the Hilbert space of integrable quadratic functions, and with . H 1 the Hilbert space of continuous functions with continuous first-order derivative and the variational equation takes the following form: ˙ x) = 0 such that for all Find .u˙ ∈ K ∩ B0 (IT , L 2 (Ω, R d )) and .u(0, x) = u(0, .v ∈ K , ʃ .

{

⟨u, ¨ v − u⟩ ˙ Ω + a (1) (u, ˙ v − u) ˙ + a (2) (u, v − u) ˙

IT

} + ⟨F (u˙ T )|σ N (u, ˙ u)|, |vT | − |u˙ T |⟩ΓC dt =

ʃ ⟨L , v − u⟩dt ˙ IT

with the bilinear forms

(7)

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a (1) (u, ˙ v) =

ʃ

.

393

Ci(1) ˙ kl (v)d x, jkl ei j (u)e

Ω

ʃ

(2)

a (u, v) =

Ci(2) jkl ei j (u)ekl (v)d x,

Ω

the linear functional ⟨L , u⟩ = ⟨ f, v⟩Ω + ⟨h, v⟩Γ N , and the frictional functional ʃ jT : V × V → R, jT (u, v) =

F(u˙ T )|σ N (u, ˙ u)||vT |ds. ΓC

We noted .⟨., .⟩ the scalar product, defined as: .⟨u, v⟩ X =

ʃ

u(x)v(x)d x.

X

The conditions upon the coefficients .Ci(id) jkl , .id = 1, 2 and the hypotheses over .Ω ensure the continuity and the ellipticity of the bilinear forms .a (id) (· , ·), .id = 1, 2. Therefore (id) ≥ a0(id) ||u||1,Ω , id = 1, 2, a (id) (u, v) ≤ A(id) 0 ||u||1,Ω ||v||1,Ω and a

for all .u, v ∈ W := {w ∈ H 1 (Ω, Rd )|v = 0 on ΓU } with the constants .0 < a0(id) ≤ A(id) 0 . The problems (2)–(6) and (7) are equivalent in the following sense: Proposition 1 .(a) Each solution.u ∈ C 2 (Q T , Rd ) of the problem (2)–(6) solves also the problem (7). 2 d .(b) Each solution of the problem (7) contained in .C (Q T , R ) solves also the initial problem (2)–(6).

2.1 Regularization of Dynamic Contact Problems The first step in obtaining the numerical algorithm is the regularization of the nondifferentiable friction functional . jT . For this purpose, we will estimate the functional . jT by a family of regularized functional . jTε , which are convex and differentiable in the second argument and the contact normal tension is approximated by a power type law, also called normal compliancies of the following type .σn (u) = −cn (u n − g)m n : ʃ n cn (u n − g)m + ψε (vT )ds,

jTε : V × V → R, jT ε (u, v) = ΓC

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where .cn and .m n are the material parameters of the contact interface and the function ψε : (L q (ΓC ))d → L q (ΓC ) represents an approximation of the modulus function, q d q .| · | : (L (ΓC )) → L (ΓC ) and it can be defined in many other ways, for example, q for .ε > 0, .ξ ∈ (L (ΓC ))d and .x ∈ ΓC .

⎧ | | ( | |) | ξ |2 ⎪ 1 |ξ| ⎪ ⎪ ε| | 1 − | | , if |ξ(x)| ≤ ε ⎪ ⎪ ⎨ | ε| 3 | ε| ψε (ξ ) = ) (| | ⎪ ⎪ |ξ| 1 ⎪ ⎪ε | | − ⎪ if |ξ(x)| > ε. ⎩ | ε| 3 , The most frequent example for a regularization function is: ψε (ξ ) =

√

||ξ ||2 + ε, or ψε (ξ ) =

√ ||ξ ||2 + ε2 − ε.

The regularized version of the variational equation is: Find .u ε , such that .

< u¨ ε (t), v > +a(u ε (t), v)+ jn (u ε , v)+ jT ε (u ε (t), v) =< F(t), v >, ∀ v ∈ V (8)

with the initial conditions .u ε (x, 0) = u 0 , .u˙ ε (x, 0) = u 1 . Equation (8) is necessary, because this is a new, regularized equation, where discontinuities do not appear in the first-order derivatives, as they appear in the equation without regularization, because of the modulus function .|.| in the Coulomb friction law. In order to solve the dynamic problem, it is necessary to prove the existence of an equilibrium point for the case of stationary sliding, then we also need to study the dynamic stability of sliding for that equilibrium position ([7] ch. 13). For an approximate solving of the dynamic contact problem, the regularized variational Eq. (8) will be approximated in space by the finite element method, achieving a nonlinear system of ordinary differential equations, which will be solved with specific approximation methods.

3 Approximation with Finite Element Methods of Regularized Dynamic Contact Problems and Newmark Algorithm for Dynamic Contact Problems We propose an approximation of this mathematical model with a sequence of variational inequalities that model the contact condition with the penalty method and the undifferentiable friction functional with a convex function, see [7]. Finite element methods, together with numerical schemes of finite differences for solving associated systems of nonlinear ordinary differential equations, are capa-

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ble of modeling stick-slip motion, dynamic sliding, friction damping and related phenomena in a significant range of practical problems. Using standard finite element procedures, the approximation of the variational Eq. (8) and noting the space of admissible functions with: | V = {v ∈ L 2 (I, H 1 (Ω d ))|v = U on ΓU × (0, T )}. can be constructed in finite-dimensional subspaces.Vh (⊂ V ⊂ V ' ). For certain.(h) the approximations of displacements, velocities and accelerations at each time.t ∈ (0, T ) are elements of .Vh , .u h (t, x), .v h (t, x), .a h (t, x) ∈ Vh . Within each element .Ωhe (e = 1, . . . , Nh ) the components of the displacements, velocities and accelerations are expressed in the form: Ne ∑ u dI (t)N I (x), u dh (t, x) = 1

h .vd (t, x)

=

Ne ∑

vdI (t)N I (x)

1

adh (t, x) =

Ne ∑

adI (t)N I (x)

(9)

1

where .d = 2 or 3, . Ne =the element’s number of node, .u dI (t), .vdI (t), .adI (t) are the nodal values of the displacements, velocities and accelerations, at the time .t and . N I is the element shape function associated with the nodal point . I . Here we have a discretization/approximation with finite elements, but we consider .x as a bi- or tri-dimensional vector, depending on the size of the domain where we look for the solution. If . NdΩ is the number of the nodes of the finite element mesh from .Ω, then this problem is equivalent to the following matrix problem: Ω

Problem . Pε h . Find the function .u : [0, T ] → Rd×Nd , s.t. .

Ma(t) + K u(t) − P(u(t)) + J (u(t), v(t)) = F(t)

(10)

with the initial conditions u(0) = u 0 , v(0) = u 1 .

.

(11)

Here we have introduced the following matrix notations: .u(t), .v(t), .a(t): the column vectors of nodal displacements, velocities and accelerations, respectively; . M: standard mass matrix; . K : standard stiffness matrix; . F(t): consistent nodal exterior forces vector; . P(u(t)): vector of consistent nodal forces on .ΓC ; . J (u(t),v(t)): vector of consistent nodal friction forces on .ΓC ; .u 0 , .u 1 : initial nodal displacement, velocity.

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The components of the element vector .(e) P have the form: (e)

ʃ P=−

σ N · n · N I ds

(e) Γ C

and the components of the element vector .(e) J have the form: (e)

ʃ J =−

σT · n · N I ds.

(e)ΓC

In order to obtain the components of the element vector . P and . J it is used a contact finite element.

3.1 Newton-Raphson Algorithm The algorithms that we shall use for solving the discrete dynamical system involve variants of standard schemes used in nonlinear structural dynamics calculations: the Newmark-type algorithm or the central-difference scheme. Let us consider a partition N [ ] U tk−1 , tk with .0 = t0 < t1 < . . . < t N = T , we denote of the time interval . I = k=1 [ ] .∆t = tk − tk−1 for the length of the sub-interval . tk−1 , tk . In the dynamic case, the inertial terms are restored and issues associated with temporal accuracy and stability come to the fore (must analysis). If we denote in the ordinary differential Eq. (10) . K N ≡ K − P + J (where the matrices P and J contain nonlinearities), with the size of the matrices equal to .d × NdΩ , and (10) becomes: .

Ma(t) + K N (u(t)) = F(t)

(12)

In dynamic case inertial terms cannot be neglected and the variable .t in this case does have the interpretation of real time. We have to find approximations .u k+1 , .vk+1 , .ak+1 at time .tk+1 from Eq. (12), with given displacement vector .u k , velocity .vk and acceleration .ak at time .tk Mak+1 + K N (u k+α ) = F (tk+α ) u k+α = αu k+1 + (1 − α)u k ] ∆t 2 [ u k+1 = u k + ∆tvk + (1 − 2β) ak + 2βak+1 2 ] [ vk+1 = vk + ∆t (1 − γ ) ak + γ ak+1

.

(13)

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where .α, .β and .γ are algorithmic parameters that define the stability and accuracy characteristics of the method. In particular, when .α = 1, the algorithm reduces to the classical Newmark algorithm. A wide range of algorithms exists corresponding to the different available choices of these parameters, we illustrate the implicit methods. To introduce the concept of an implicit method we examine the trapezoidal rule, which is simply a member of the Newmark family obtained by setting.α = 1,.β = 1/4 and .γ = 1/2. The substitution of these values into (13) yields: ⎧ ⎨ Mak+1 + K N (u k+1 ) = F[(tk+1 ) , ] ∆t 2 . = u k + ∆tv u [ k + 4 a]k + ak+1 , ⎩ k+1 ∆t vk+1 = vk + 2 ak + ak+1 .

(14)

Finally, in each node of the structure partition (.Ω domain), we obtain: deformations, velocities and accelerations on each subinterval .[tk−1 , tk ]. This method is the most expensive procedure involved in updating the solution from .tk to .tk+1 . First equation is not only fully coupled, but is also highly nonlinear, in general, due to interval force vector. We could write the first equation of (14) in terms of a dynamic incremental residual . R (u k+1 ) via .

4 R (u k+1 ) : = F (tk+1 ) − K N (u k+1 ) − Mu k+1 2 ∆t ( ) 4 + M ak + ∆tvk + uk = 0 ∆t 2

(15)

The implicit and explicit methods are valid only for linear or linearized problems. In this section we give a general framework for solving the nonlinear discrete equations associated with computation of an unknown state at step .tk+1 , in either context of a dynamic contact problem formulation as in (15). In either case, the equation to be solved takes the form . R (u k+1 ) = 0 (16) with . R, a nonlinear function of the solution vector .u k+1 , is considered. The general concept of a Newton-Raphson iterative solution technique for (16) (identical with (14) and with (15)) is defined in iteration . j by .

( ) [∂ R ] j j R u k+1 + ∆u k = 0 j ∂u u k+1

following by the update u

j+1

. k+1

j

j

= u k+1 + ∆u k

(17)

(18)

|| || || j || Iteration on . j typically continue until the Euclidian norm .||∆u k || is smaller than some tolerance.

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The residual at iteration . j, from (15) is of the form .

( ) ( ) 4 j j j R u k+1 ≡F (tk+1 ) − K N u k+1 − Mu k+1 2 ∆t ( ) 4 j u =0 + M ak + ∆tvk + ∆t 2 k

(19)

with (17), Eq. (19) to take the form [ .

( ( )] ) 4 j j M + K L u k+1 ∆u kj = R u k+1 ∆t

(20)

( ) j where the stiffness matrix . K L u k+1 is given as

.

( ) (∂K ) N j K L u k+1 = j ∂u u k+1

(21)

We note that a variety of iterative procedures exist as alternatives to the NewtonRaphson nonlinear solution procedure (quasi-Newton, secant methods etc.).

3.2 Finite Contact Element for Two-dimensional Bodies In order to analyze this issue we shall start with the perturbed functional of the Lagrangean, which contains both the penalty method and the method of Lagrange multipliers, in a single equation ) 1 ( L u h , ∑n , ∑t = a(u h , u h ) − L(u h ) + ∑nT G n + ∑tT G t 2 1 1 ∑ T ∑t , ∑ T ∑n − − 2ωt t 2ωn n

.

(22)

where .u h ∈ V h is the nodal displacements vector, .V h represents the finite element space, .∑n , ∑t are the vectors of the nodal forces, the normal vector .G n , and .G t , the tangential one, respectively on the contact area, and finally .ωn , .ωt are the penalization parameters. The geometry of the finite element of bidimensional contact consists of two nodes called master nodes, which belong to a body, and of a slave node, which belongs to the second body in contact [8, 9]. Where, .x1 is the vector that defines the coordinates of this master node, .x2 is the vector that defines the coordinates of the second master node, and .xs is the vector that defines the coordinates / of the slave node. Thus if we have .x1 (x11 , y11 ) and .x2 (x22 , y22 ), results |x2 − x1 | =

.

(x22 − x11 )2 + (y22 − y11 )2 ). The gap .g between the two bodies in con-

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399

Fig. 1 The finite contact element in .2D case contact

tact, which corresponds to a finite element contact, is described by the slave node s. This gap .g represents the projection of the vector .xs − x1 on the normal of the segments .x1 x2 where .x1 and .x2 are the master nodes of the finite contact element, meaning s .gn = (x s − x 1 )n.

.

We denote by x = X s + u s ; x1 = X 1 + u 1 ; x2 = X 2 + u 2

. s

the current positions of the nodes of the finite element. X s , X 1 and. X 2 are the reference coordinates, and .u 1 , u 2 and .u s are the displacements of the finite contact element’s nodes, denoted by 1, 2, and .s respectively (Fig. 1). The tangent and the normal versors which follow the direction given by the master nodes are given by the relation below t=

x2 − x1 , n = e3 × t, |x2 − x1 |

where .e3 is the normal versor at the plane defined by these three nodes 1, 2 and .s). One defines the contact parameter, .a, through which we write the finite contact element in normal coordinates, in order to use the finite isoparametric element who will change the value at each incremental step of the linearization process a=

.

xs − x1 t. |x2 − x1 |

The projection of the gap .gts on the versors .t direction is given by gts = (xs − x1 )t − a 0 (x2 − x1 ), where .a 0 is the contact parameter given at the previous step. We introduce the following notations d = |x1 − x2 | =

√ y2 − y1 (x2 − x1 )2 + (y2 − y1 )2 , s = sin θ = , c = cos θ, d

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N. Pop et al.

hence, it results a=

.

1 [(xs − x1 )c + (ys − y1 )s] , d

gns = −(xs − x1 )s + (ys − y1 )c, gts = (xs − x1 )c + (ys − y1 )s − a 0 d. The maximum of the functional (22) is obtained by canceling its derivatives with respect to .u h , .∑n and .∑t , i.e. ⎧ ∂u h π(u h ) + ∑nT ∂u h · G n + ∑tT ∂u h · G t = 0 ⎪ ⎪ ⎪ ( ) ⎪ ⎪ ⎪ ⎪ ⎨ ∂∑ T − 1 ∑n + G n = 0 n ωn . ⎪ ( ) ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎩ ∂∑tT − ω ∑t + G t = 0,

(23)

t

1 where .π(u h ) = a(u h , u h ) − L(u h ) is the total energy of the contact bodies. 2 In order to obtain the explicit expressions from (23), we shall calculate the variations of .g, which depends on the slave node .s and on the master ones 1 and 2, respectively. δg s =

] d [ s g (xs + αηs , x1 + αη1 , x2 + αη2 ) α = 0, dα

where .η1 , .η2 , .η3 are the increments of the displacements .u 1 , u 2 and .u s , meaning that η(η1 , η2 , ηs ) = ∂u h (∂u 1h , ∂u 2h , ∂u sh ). Similarly, we have δt =

] d [ t(x1 + αη1 , x2 + αη2 ) α = 0. dα

The variation of the versor .n is obtained from the following relation n = e3 × t.

.

The components .gns and .gts are g s = [ηs − (1 − a)η1 − aη2 ]

. n

gts = [ηs − (1 − a 0 )η1 − a 0 η2 ]t.

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The linearization of the Eq. (23) leads to a system of equations in the incremental displacements,.∆u h and incremental multipliers,.∆∑n and.∆∑t , which can be solved applying Newton-Raphson’s method. In order to facilitate the computer implementation, we shall use the matrix formulation. To this end .

[ ]T [ ]T Ns0 = −s, c, (1 − a 0 )s, −(1 − a 0 )c, a 0 s, −a 0 c = n, −(1 − a 0 )n, −a 0 n , Cn = Ns = [−s, c, (1 − a)s, −(1 − a)c, as, −ac]T = [n, −(1 − a)n, −an]T , Ts = [c, s, −(1 − a)c, −(1 − a)s, −ac, −as]T = [t, −(1 − a)t, −at]T , T = [0, 0, −c, −s, c, s]T = [0, −t, t]T ,

N = [0, 0, s, −c, −s, c]T = [0, −n, n]T , ]T [ ∆u h = ∆u sh , ∆u 1h , ∆u 2h , [ gn gn gn ]T gn s, −(1 − a 0 )s − c, a 0 c − s, −a 0 s + c Ct = c, s, −(1 − a 0 )c + d d d d [ gn gn ]T n, −a 0 t + n . = t, −(1 − a 0 )t − d d The tangent matrix of the contact finite element, for a fixed contact, will be (

.

] gn [ gn N TsT + Ts N T + NTT d d ) [ ] gn gn 0 T n s N + N NsT − (N T T + T N T ) , + d d

K c = ω [Ns NsT + Ct CtT ] −

while the free term vector of the contact element is .

Rc = − [ω(gn · Cn + gt · Ct )] ,

where .ωn = ωt = ω. After assembling the matrices . K ck and . Rck , of the finite contact element .k, to both the global stiffness matrix . K and the global loading vector . R results KT = K +

S ∑

( K ck ,

RT = − R +

s=1

S ∑

) Rck

,

k=1

where . S is the total number of slave nodes (equal to the total number of finite contact elements). For a sliding contact, we will modify only, g = μ sgn(gt )gn ,

. t

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where .μ is the friction coefficient. In this case we obtain the nonsymmetric matrix . K ck . Now, if we denote the linearized algebraic system with U = (∆u h , ∆∑n , ∆∑t )T ,

.

we can rewrite in the matrix form .

K T U = RT .

3.3 Finite Contact Element for Three-dimensional Bodies To generalize the results from [8] and [9] for the two-dimensional case, we obtained a finite contact element for the three-dimensional case, which has four nodes out of which three are master and belong to the same body and the fourth, called slave, belongs to the second body in contact, see [10]. In all numeric applications we shall use formulations of the perturbed Lagrangean, for the case of a fixed contact and also for a sliding one. For the first case, of a fixed contact with friction, the perturbed Lagrangean has the following form 1 a(u h , u h ) − L(u h ) + ∑nT G n + ∑tT G t + ∑τT G τ 2 1 1 1 ∑ T ∑n − ∑ T ∑t − ∑ T ∑τ , − 2ωn n 2ωt t 2ωτ τ

L(u h , ∑n , ∑t , ∑τ ) =

.

(24)

where .u h ∈ V h is the nodal displacements vector, .V h is the finite element space, .∑n , ∑t , ∑τ are the normal and tangential, nodal force vectors at the contact surface, respectively .G n , G t , G τ are the normal and tangential, nodal interstitial vectors respectively and finally .ωn , ωt , ωτ represent the normal and tangential, penalization parameters, respectively. The normal vector to the plane defined by the nodes 1, 2 and 3, from the contact zone, and the vectors that define the directions given by the nodes 1-2 and 1-3, respectively as Fig. 2 depicts, will be n=

.

(x2 − x1 )(x3 − x1 ) x2 − x1 x3 − x1 , t= , τ= , |(x2 − x1 )(x3 − x1 )| |x2 − x1 | |x3 − x1 |

(25)

where .x1 = X 1 + u 1h , .x2 = X 2 + u 2h , .x3 = X 3 + u 3h represents the current position of the master nodes. . X 1 , X 2 , X 3 are the reference coordinates and .u 1h , .u 2h , .u 3h are the current nodal displacements of the nodes 1, 2 and 3 respectively.

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τ

τ

τ t

t

t t

Fig. 2 Finite contact element in .3D case contact

Furthermore we will define the contact surface parameters as a =

. t

xs − x1 x S − x1 t, aτ = τ, |x2 − x1 | |x3 − x1 |

(26)

where .xs = X s + u sh defines the current position of the slave node. The normal and the tangential interstitia, respectively .gn , gt , gτ are given by g = (xs − x1 )n, gt = (at − at0 )|x2 − x1 |, gτ = (aτ − aτ0 )|x3 − x1 |,

. n

(27)

where .at0 and .aτ0 are the surface parameters, known from the last incremental step. Thus, it is obvious that the interstitium .g depend on the nodes 1, 2, 3 and 5. So the variation of .g is given by the following expression δg =

.

d g(xs + αηs , x1 + αη1 , x2 + αη2 , x3 + αη3 ), dα

(28)

where η(η1 , η2 , η3 , ηs ) ≡ ∂u h (∂u 1h , ∂u 2h , ∂u 3h , ∂u sh ).

.

(29)

The implementation of finite element and the expression of the matrices from the perturbed Lagrangean functional will be obtained in the following ∂ ∏(u h ) + ∑nT ∂u h G n + ∑tT ∂u h G t + ∑τT ∂u h G τ = 0;

. uh

∂

. ∑T n

( ) 1 − ∑n + G n = 0; ωn

(30) (31)

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( ∂

−

. ∑T t

1 ∑t + G t ωt

( ∂

.

∑τt

)

1 − ∑τ + G τ ωτ

= 0;

(32)

= 0,

(33)

)

1 where .∏(u h ) = a(u h , u h ) − L(u h ) is the potential energy of the bodies in contact, 2 and )T )T ( ( ∂ h G n = ∂uh gn1 , ∂uh gn2 , . . . , ∂uh gns , ∂uh G t = ∂uh gt1 , ∂uh gt2 , . . . , ∂uh gts , )t ( ∂uh G τ = ∂uh gτ1 , ∂uh gτ2 , . . . , ∂uh gτs .

. u

S represents the total number of the contact slave nodes, namely .s = 1, 2, . . . , S analogous for .∂∑n , .∂∑t , .∂∑τ . The variation of the normal component of the nodal interstitium .gn ∈ G n has the following form 3 3 3 ∑ ∂gn j ∑ ∑ ∂gn j η + η δgn = j s j i j=1 ∂u s i=1 j=1 ∂u i

.

If we use the notation (29) and ( cn =

∂gn ∂gn ∂gn ∂gn ∂gn , , , ,..., 3 ∂u 1s ∂u 2s ∂u 3s ∂u 11 ∂u 3

) , η = (ηs1 , ηs2 , ηs3 , η11 , . . . , η33 ),

we obtain .δgn = η T cn . The variation of the tangential component of the nodal interstitium .gt ∈ G t , .gτ ∈ G τ can be obtained similarly, δgt = η T ct , δgτ = η T cτ . Furthermore, the residual vector . R and the tangential stiffness matrix . K results from the total potential energy of the bodies in contact, thus ∂

∏

. uh

(u h ) = η T R and ∂u h R = η T K .

Using the conventions .(u h1 , . . . , u h12 ) = (u sh1 , u sh2 , u sh3 , u 1h1 , . . . , u 3h3 ), the Eq. (29) becomes [ η

.

T

] S ∑ s s s s s s (σn cn + σt ct + στ cτ ) = 0, R+ s=1

and similarly for the relations (31)-(33), where .σn ∈ ∑n , .σt ∈ ∑t , .στ ∈ ∑τ .

(34)

Numerical Modeling and Some Optimal Control Problems of Dynamic …

405

Applying the iterative algorithm of Newton-Raphson, one obtains a linearization of the relations (34) and (31)-(33) in .(u h , ∑u h , ∑t , ∑τ ), hence ⎧ ⎫⎫ ⎫ ⎧ ⎧ R1 ⎪⎪ ⎡ ⎤ ⎪ ∆u h ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ R2 ⎪ ⎬ ⎬ ⎬⎪ ⎨ A1 A2 A3 A4 ⎨ ∆∑ n T T T T T ⎣ ⎦ A2 B2 O O =− [η , ∂∑n , ∂∑t , ∂∑τ ] ⎪ ⎪⎪ ⎪ ⎪ ∆∑t ⎪ ⎪ ⎪ ⎪ ⎩ R3 ⎪ ⎭ ⎭ ⎭⎪ ⎩ A3T O C3 O A4T O O D4 ⎪ ⎩ ∆∑τ R4 where . A1

=K+

S ∑

(kns + kts + kτs ), A2 =

s=1

B2 = − R2 = − (kns ) ji =

S ∑

cns , A3 =

s=1

S ∑

cts , A4 =

s=1

1 1 1 I, C3 = − I, D4 = − I, R1 = R + ωn ωt ωτ

S ∑

cτs

s=1 S ∑

(σns cns + σts cts + στs cτs )

s=1

1 1 1 ∑n + G n , R3 = − ∑t + G t , R4 = − ∑τ + G τ , ωn ωt ωτ

∂cnsi ∂u hj

∂ 2 gns

=

∂u ih ∂u hj

, (kis ) ji =

∂ctsi ∂u hj

=

∂ 2 gts

, (kτs ) ji =

∂u ih ∂u hj

∂cτsi ∂u hj

=

∂ 2 gτs ∂u ih ∂u hj

.

Finally, after discretization with finite elements, using the standard assembly procedure and adding the contribution of each contact element to the global tangent stiffness matrix one obtains . K T U = RT , where . K T = K +

S ∑

( K Cs , . RT

=− R+

s=1

S ∑

) RCs

. In the above relations . K and

s=1

R represent the global stiffness matrix and respectively the loading vector without contribution over the contact zone and . K Cs and . RCs are the contribution of the contact ( )T element corresponding to the node .s, and finally .U = ∆u h , ∆∑n , ∆∑t , ∆∑τ and . S denotes the total number of slave nodes. Now, for .ωn = ωt = ω and .σn = ωgn , .σt = ωgt , .στ = ωgτ , it results .

.

KC =

S ∑

) ( ω gns kns + gts kts + gτs kτs + cnsT cns + ctsT cts + cτsT cτs

(35)

s=1

.

RC =

S ∑

) ( ω gnsT cns + gtsT cts + gτsT cτs .

(36)

s=1

In the case of a sliding contact, we shall use the relation .|∑tan | = μ|∑n |, where .μ denotes the Coulomb friction coefficient and .∑tan represents the result of forces .∑t

406

N. Pop et al.

and .∑τ in the tangent contact area determined by the principal nodes 1 and 2. If .β denotes the/angle between the segments .x2 −x1 and .x3 −x1 we obtain .cos β = tτ and |λtan | = μ gt2 + gτ2 + 2ε|gt | |gτ | cos β, where .ε = sgn (gt gτ ). As a direct / consequence of Coulomb’s friction law, it results that .μω|gn | = ωr

.

where .r =

gt2 gτ2 + 2ε|gt | |gτ | cos β, because

λ1 = λtan

gt gt |gt | |gτ | ωgn = −μ sgn (gt ) ωgn = −μ ωgn , λτ = −μ ωgn . r r r r

|gt | |gτ | ∂dt ∂dτ , .dτ = , .bt = h , .bτ = h from the kinetic linr r ∂u ∂u S ∑ (S L s1 + earization (neglecting the nonlinear terms .kt and .kτ ), we obtain . K c = If we denote by .dt =

s=1

S L s2 ) with .

) ( S L s1 = ω gns kns − μgn dts kts − μgns dτs kτs + cnsT cns − μdts cnst cts ) ( S L s2 = ω −μdts cnsT cτs − μgn btsT cts − μgn bτsT cτs ,

and Rc =

S ∑

( ) ω μgn dtsT cts + μgn dτsT cτs − gnsT cns ,

s=1

where we denoted by .(·)T the transposed vector.

4 Optimal Control Algorithm Using MPC and LQR MPC is an open loop optimization method used to minimize the differences between the predicted behavior of a dynamic system and the desired (the reference) behavior of that system. MPC is applied on a large scale in robotics, due to its ability to manage the input and output restrictions in optimal control problems. An advantage of the method is that the optimal problem restrictions are included into the difference equations of the MPC algorithm. What differentiate MPC from other optimization algorithms is that MPC provides an online optimization of a optimal control problem on a finite horizon. The basic idea of MPC is to use a model to predict the result of a process on a future finite time horizon by obtaining a control sequence which minimizes an objective function. Assuming that, in the dynamic analysis, we also consider depreciation, the Eq. (12) becomes: . Ma(t) + Dv(t) + K N (u(t)) = F(t), (37)

Numerical Modeling and Some Optimal Control Problems of Dynamic …

407

where . D is the damping matrix, which approximates the fundamental damping properties of the material. In this case, we will consider the Rayleigh type damping, keeping from the Caughey series, from [11], only the first two terms: . D = a1 K N + a2 M (an example could be:.a1 = 10−5 , a2 = 10−12 ). To write the MPC algorithm, in order to obtain the cost function of the form “output tracking”, which determines how the system input affects the state change and we will start from the dynamic system written in the form: q(t) ¨ + M −1 D q(t) ˙ + M −1 K N q(t) = M −1 F(t)u c (t),

.

(38)

where .u c (t) is the control input vector, .q(t) = u(t), .q(t) ˙ = v(t) and .q(t) ¨ = a(t). Next, we will consider a discrete-time model of the system (16), written in an equivalent form, i.e. instead of a system of .n ordinary differential equations of the second order, a system of .2n differential equations of the first order. The formulated optimization problem will be solved based on an objective function (cost), where the number of control inputs is . Nu the number of outputs is .2n and the number of states is .2n. We introduce the notation, usually called the state vector: .

T x(t) = [q(t), q(t)] ˙ .

(39)

It is known that MPC can also be applied to non-linear systems, after their linearization. For this purpose, we will use the Taylor series development around the reference point and approximate the formula keeping only the terms up to the first order derivatives. The general form of the state equation corresponding to the dynamic system ˙ = f (x, u c ). The same equation is verified by the reference state variables is: .x(t) and the control reference variables:.(xd , u d )), that is : .x˙d (t) = f (xd , u d ). Performing a Taylor expansion of the equation .x(t) ˙ = f (x, u c ), in the reference ˙ = A∆x(t) + B∆u(t), wehere .∆x = point, we get the following state equation: .x(t) x − xd , .∆u = u c − u d . The last equation is the continuous state equation, but MPC is a discrete time control method, so that the discretized equation is: . .

x(k + 1) = A(k)∆x(k) + B(k)∆u(k),

(40)

y(k) = C x(k) x(0) = x0 ,

(41)

where . y(k) is output vector, . A(k) = I + Ts A is the state matrix, obtained as the first partial derivative of . f with respect to .x and . B(k) = Ts B the input matrix, obtained as the first partial derivative of . f with respect to .u and .C is the distribution matrix. We noted with .Ts the sampling time, and . I the unit matrix. Using the transformation (39) in the Eq. (38), we get: ( A=

0

−M −1 K N

) ( ) ( ) 0 I , B= , C = Cq Cv , −M −1 D M −1 F

408

N. Pop et al.

where . I is .n-dimensional identity matrix, .Cq is the distribution matrix of the deformations and .Cv is the distribution matrix of the velocities. The objective function to be minimized can be chosen as a quadratic function of the states and control inputs: . L(k)

=

N ∑

(∆x T (k + j|k)Q∆x(k + j|k) + ∆u kT (k + j − 1|k)R∆u k (k + j − 1|k)),

j=1

(42) where . N is the length of the prediction horizon, the symmetric matrices . Q and . R are used to penalize the state error and the control error, respectively, with . Q ≥ 0 . R > 0. The linear feedback matrix .∆u k is defined using the matrix responsible for defining optimality in the linear quadratic regulator (LQR), which is obtained by solving the Riccati equation, using MATLAB. The basic difference between MPC and LQR is that in the case of predictive control the optimization the problem is solved using a moving time horizon window, while in the case of LQR the same problem is solved in a fixed time window. The advantage of using a moving time window is the ability to perform real-time optimization with constraints on certain system variables. Therefore, the optimization problem can be written as: ∆u ∗ =

.

min

∆u,eq. (10) and (11)

L(k)

(43)

The problem of minimizing (42) is solved at each time step .k, yelding a sequence of optimal control .∆u ∗ (k|k), ..., ∆u ∗ (k + N − 1|k) and the optimal cost function ∗ . L (k). The minimization is with restrictions, the restrictions are given by satisfying Eqs. (10) and (11). If we choose a cost function, in which the second term is of the form: .∆u k = x¨k − x¨k−1 , we will obtain constant nodal accelerations and implicitly the shocks will be zero. In the same way, if we choose .∆u k = f k − f dk we will be able to control the forces acting on one side of the boundary of the studied body, where .( f dk ) are the (desired) reference forces, see [12, 13]. The validity of the proposed methods is demonstrated by the following simple but representative example. We analyze the following boundary value problem with Neumann boundary conditions (Eq. (4), for the general case), with the following specifications: the domain is the square unit .Ω = [0, 1] × [0, 1], the cost functional that is to be minimized and with inequality constraints on the control and states. The boundary of the domain is: ∂Ω ≡ Γ = {(x, 0) ∪ (x, 1)|0 ≤ x ≤ 1} ∪ {(0, y) ∪ (1, y)|0 ≤ y ≤ 1}.

.

Consider the partial differential equation with Neumann boundary condi-tions: −∆w(x, y) − ex p(w(x, y)) = u c (x, y) on Ω, ∂w(x,y) . + w(x, y) = 0 on Γ, the Neumann boundary condition, ∂n with inequality constraints: . w(x, y) ≤ 0.364, −7 ≤ u c (x, y) ≤ 8, (x, y) ∈ Ω, then the desired (reference) state variable and the desired (reference) control variable will be: .

Numerical Modeling and Some Optimal Control Problems of Dynamic … Table 1 The information obtained in solving the present example .N .J .w(0.5, 0.5) 0.0753 0.0762

50 100

–0.0112 –0.0081

409

.u c (0.5, 0.5)

–1.6774 –1.6312

wd (x, y) = sin(2π x)sin(2π y), on Ω, u d (x, y) = 0 on Γ for α = 0.002 :

.

1 . J = min [w,u] 2

ʃ

α (w(x, y) − wd (x, y)) dΩ + 2 Ω

ʃ

2

Γ

(u c (x, y) − u d (x, y))2 dΓ .

The numerical solution that achieves the minimum cost functional was obtained for two discretizations of the domain, coarser (. N =50) and finer (. N =100). The minimum values of the cost functional, . J , the value of the state variable .w, and of the control input variable .u c in the center of the square, for these two cases, are presented in Table 1. Another example which validates the proposed methods is the case of a rehabilitation robot leg with 3 degrees of freedom, taken as a dynamic system. We used VIPRO platform [24] for simulation and validation, by applying the virtual projection method, known as the Vladareanu-Munteanu method [25]. This work is in progress and will be the subject of a future manuscript. The main goal of the optimization is to obtain the best approximation, more precisely the minimization of the error between the desired deformations and the deformations calculated from the model and the cancellation of shocks (imposing the restrictions that the time derivatives of the nodal accelerations to be zero), in a future time interval, from time horizon chosen using MPC. From a mathematical point of view, it means choosing the inputs that minimize a cost function (objective) that leads to the fulfillment of the objective. In the case of our problem, the use of MPC allows many possibilities in choosing the cost function. The cost function the form of “output tracking” aims to minimize (in real time) the errors between the state variables calculated at step .k (the deformations calculated at step .k), and the desired (reference) state variables and the errors between the control variables at step .k (for example incremental forces, acting on the specified boundary .Γ N ) and the reference control variables. This is possible only if the sensors transmit the measured values in a timely manner, and the actuators actuate the controls in the desired sense. If we want to act on an eigenmode, to suppress or minimize it, we propose to transform the system of dynamic Eq. (38), using a transformation like (39), from generalized coordinates into a coordinate system given by the directions of the eigenvectors. We denote by .Φ the n-dimensional modal matrix, which is an orthonormalized Meigenmodes matrix. In this case, we obtain the equivalent system of discrete-time difference equations. ) ( ) ( 0 I 0 , A= , B= −Ω − Φ −1 F

410

N. Pop et al.

where .Ω = diag(ωi2 ), . = diag(2ξi ωi ), i = 1...n, .ωi and .ξi being the natural frequency an the damping ratio respectively, of the .i-th mode, see [19–25]. The advantage of this approach is that by changing the last term of the cost function we can control the eigenmodes. After obtaining the optimal solution, the minimum, we return to the initial coordinate system, through the inverse transformation.

5 Conclusions In this paper, we investigated some methods for solving and optimizing the dynamic equations of elastic and viscoelastic bodies in the case of elastic contact with friction. We made use in this respect of discrete-time difference equations to solve the dynamic visco-elastic contact problems with friction. In our approach we regularized the terms that describe the laws of friction and then we space-discretized the equations, with the finite element method, thus obtaining a system of second-order, non-linear ordinary differential equations. Then we did the successive linearization of the nonlinear dynamic system and then we did the time-discretization, using the finite difference method, resulting in discrete-time difference equations, linear and discrete in time. In the last stage, to solve the optimization problem with constraints, we applied the predictive control model (MPC) and the linear quadratic regulator (LQR), to control the deformations and to cancel the shocks from the vibrations of the analyzed mechanical structure, as well as to minimize the control effort of the actuators. The cost function is a pondered sum between the penalty on the magnitude of the output errors (the difference between the current solution of the state system and the desired solution) and the penalty on the control effort performed by the traction force applied on the boundary. For a future research project, a comparison of the mentioned optimal control performances of four types of controllers is proposed: MPC, LQR and robust controllers, . H∞ loop shaping and .μ-synthesis.

References 1. Stammers, C.W., Sireteanu, T.: Vibration: control of machines by use of semi-active dry friction. J. Sound Vib. 209(4), 671–684 (1998) 2. Sebe¸san, I., Mitu, A.M., Sireteanu, T.: On the stick-slip phenomena in traction railway vehicles. Proc. Rom. Acad. Ser. A 16(2), 209–216 (2015) 3. Pop, N., Ungureanu, M., Pop, A.I.: An approximation of solutions for the problem with Quasistatic contact in the case of dry friction. Math. 9, 904 (2021). https://doi.org/10.3390/ math9080904 4. Pop, A.I., Lung, C., Sabou, S., Tarc˘a, R., Pop, N.: Experimental evaluation of LoRa for remote vehicle tracking and control in urban areas. In: 2022 IEEE 28th International Symposium for Design and Technology in Electronic Packaging (SIITME), Bucharest, Romania, pp. 128–131 (2022). https://doi.org/10.1109/SIITME56728.2022.9987752

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5. Pop, N., Vladareanu, L., Migdalovici, M., Pop, A.I., Radulescu. M.: Trajectory optimization for mobile robots using model predictive control. Period. Eng. Nat. Sci. 7(1), 242–248 (2019). ISSN 2303-4521. Available online at: http://pen.ius.edu.ba 6. Kikuchi, N., Oden, J.T.: Contact Problems in Elasticity: a Study of Variational Inequalities and Finite Element Methods. SIAM, Philadelphia (1988) 7. Pop, N.: Analysis of an evolutionary variational inequality arising in elasticity quasi-static contact problems. Adv. Stud. Pure Math. 53, 213–223 (2009) 8. Ju, J.W., Taylor, R.L.: A perturbed lagrangean formulation for the finite element solution of nonlinear frictional contact problems. Journal de Mecanique Teorique et Appliquée, Spec. issue, suppl. 7(1), 1–14 (1988) 9. Wriggers, P., Simo, J.C.: A note on tangent stiffness for fully nonlinear contact problems. Commun. Appl. Numer. Methods 1, 199–203 (1985) 10. Pop, N.: A finite element solution for a three.−dimensional quasistatic frictional contact problem. Rev. Roumaine des Sciences Techn. serie Mec. Appliq, Editions de l’Academie Roumaine, tom. 42, (1997) 11. Midha, A., Erdman, A.G., Frohrib, D.A.: Finite element approach to mathematical modeling of high-speed elastic linkages. Mech. Mach. Theory 13(6), 603–618 (1978). ISSN 0094-114X. https://doi.org/10.1016/0094-114X(78)90028-9 12. Tak’acs, G., Zometa, P., Findeisen, R. and Rohal’-Ilkiv, B.: Efficiency and performance of embedded model predictive control for active vibration attenuation. In: Proceedings of the European Control Conference, ECC 2016, Aalborg, Denmark, pp. 1334–1340 (2016) 13. Li, J., Nguyen, Q.: Force-and-moment-based model predictive control for achieving highly dynamic locomotion on bipedal robots. In: Proceedings of the 2021 60th IEEE Conference on Decision and Control (CDC), Austin, TX, USA, 14–17, December 2021, pp. 1024–1030 14. Vladareanu, L., Melinte, O., Bruja, A., Hongbo, W., Wang, X., Cang, S., Yu, H., Hou, Z.-G., Xie, X.-L.: Haptic interfaces for the rescue walking robots motion in the disaster areas. In: 2014, UKACC International Conference on Control (CONTROL), pp. 498–503. IEEE. https:// doi.org/10.1109/CONTROL.2014.6915190 15. Vladareanu, L., Capitanu, L.: Hybrid force-position systems with vibration control for improvement of hip implant stability. J. Biomech. (45), S279. ISSN 0021-9290. https://doi.org/10.1016/ S0021-9290(12)70280-4 16. Noje, D., Dzitac, I., Pop, N., Tarca, R.: IoT devices signals processing based on Shepard local approximation operators defined in Riesz MV-algebras. Informatica 31(1), 131–142 (2020) 17. Stephens, B.J., Atkeson, C.G.: Dynamic balance force control for compliant humanoid robots. In: 2010 IEEE/RSJ International Conference on Intelligent Robots and Systems, Taipei, Taiwan, pp. 1248–1255 (2010). https://doi.org/10.1109/IROS.2010.5648837 18. Sentis, L., Park. J. and Khatib, O.: Compliant Control of Multicontact and Center-of-Mass Behaviors in Humanoid Robots. IEEE Trans. Robot. 26(3), 483–501 (2010). https://doi.org/ 10.1109/TRO.2010.2043757 19. Varalakshmi, K.V., Srinivas, J.: Dynamic analysis of flexible-link planar parallel manipulator with platform rigidity considerations. Am. J. Mech. Ind. Eng. 2(4), 174–188 (2017). https:// doi.org/10.11648/j.ajmie.20170204.13 20. Shao, M., Huang, Y., Silberschmidt, V.V.: Intelligent manipulator with flexible link and joint: modeling and vibration control. Shock. Vib. 2020, 1–15 (2020). Article ID 4671358. https:// doi.org/10.1155/2020/4671358 21. Scibilia, F., Olaru, S., Hovd, M.: On feasible sets for MPC and their approximations. Automatica 47(1), 133–139 (2010). https://doi.org/10.1016/j.automatica.2010.10.022 22. Matei, A., Micu, S.: Boundary optimal control for a frictional contact problem with normal compliance. Appl. Math. Optim. 78, 379–401 (2018) 23. Cristescu N.D., Craciun E.-M., Soós, E.: Mechanics of Elastic Composites. CRC Press (2003)

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24. Vladareanu, L., Vladareanu, V., Pop, N., Migdalovici, M., Bo¸scoianu, M., Pop, S., Ciocîrlan, A.: Robot extenics control developed by versatile, intelligent and portable robot Vipro platform applied on firefighting robots. International Association of Online Engineering. Retrieved from https://www.learntechlib.org/p/218015/, 28 May 2023 25. Vladareanu, L., Munteanu, R. I., Curaj, A., Cononovici, S., Sireteanu, T.: Real time control method and device for robots in virtual projection, patent EPO-09464001, 18.05. 2009, EP2105263. Patent OSIM, 123527(30.04), (2013)

A Particular Solution for Higher Order Non-homogeneous Discrete Cauchy-Euler Equations Miloud Assal and Skander Belhaj

Abstract In this paper, we introduce a nonstandard’s method to find a particular solution for non-homogeneous discrete Cauchy-Euler equations of higher order. The proposed method uses a new concept called atoms on discrete sets. This method provides an explicit particular solution for non-homogeneous discrete Cauchy-Euler equations whose characteristic equations have distinct roots. Keywords Difference equations · Discrete Cauchy-Euler equations · Particular solution · Numerical solution AMS classification: 39A06 · 39A05

1 Introduction Problems related to discrete equations, which appears in a wide range of applications, such as population dynamics, economics, physics and among others, are classical problems in mathematics [1]. One of the most important class of difference equation is the discrete Cauchy-Euler equation of higher order which is stated as N ∑ .

a p Ppn ∆ p f = g,

(1)

p=0

M. Assal (B) University of Jeddah, Jeddah, Saudi Arabia, University of Tunis Carthage, IPEIN, Tunis, Nabeul, Tunisia e-mail: [email protected]; [email protected] S. Belhaj University of Tunis El Manar, ENIT, LAMSIN, 1002 Tunis, Tunisia University of Manouba, ISAMM, Campus Universitaire de la Manouba, 2010 Tunis, Tunisia © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 S. Olaru et al. (eds.), Difference Equations, Discrete Dynamical Systems and Applications, Springer Proceedings in Mathematics & Statistics 444, https://doi.org/10.1007/978-3-031-51049-6_19

413

414

M. Assal and S. Belhaj p

where . Pn = n(n − 1)(n − 2) · · · (n − p + 1) with Pn0 = 1, and .∆ f (n) = f (n) − f (n − 1) is the backward operator. .a p , . p = 1, . . . , N are constant real numbers, while .g is a suitable function defined for .n ≥ n 0 for some integer .n 0 . We mention here that this equation is obtained by analogy from the standard one where the classical derivation is replaced by the backward operator while the monomial .x k is replaced by the discrete monomial . Pnk . Various approaches have been suggested to solve the 2nd-order discrete Cauchy-Euler equation, where we refer the reader to [3, 4] and to the references therein. For higher order, it seems that the discrete Cauchy-Euler Eq. (1) has ) been studied before. ∑ pnot(yet Using the fact that .∆ p f (n) = k=0 kp (−1)k f (n − k), p ∈ N, Eq. (1) can be expressed as p N ∑ ∑ .

a p Ppn

p=0 k=0

( ) p (−1)k f (n − k) = g(n). k

(2)

Thanks to Eq. (2) one can show easily that Eq. (1) is linear and nonhomogenous for .g /= 0. In this work we introduce a new concept of atoms over finite sets of real numbers and we prove new combinator properties that we apply further to give an explicit particular solution of Eq. (1). This paper is organized as follows; First we present new tools to solve Eq. (1) where we consider the concept of atoms on finite sets by means of [2] that we present in the second section. The third section of this manuscript covers the main result together with its proof which is based on atoms properties. Section 4 is devoted to present some applications that illustrate our approach. Finally, in Sect. 5 we summarize our work with some comments.

2 Atoms of Discrete Sets Definition 1 Let . X = {r1 , r2 , . . . , r N }, . N ≥ 1 be any finite set of distinct real numbers. For a valued real function . A we define its moment . M(A, s) of order .s to be

.

M(A, s) :=

N ∑

ris A(ri ).

i=1 .

A is called an . X -atom if it satisfies the following two conditions

(i) . M(A, s) = 0 for .0 ≤ s ≤ N − 2, (ii) . M(A, N − 1) = 1 and we mention that if . X = {r0 } be a singleton then . A(r0 ) = 1.

A Particular Solution for Higher Order Non-homogeneous …

415

Remark 1 We mention that the concept of atoms given here is inspired from definition of atoms given in [2], where, (i) is called the cancellation moment condition, while (ii) the size condition of an atom. In what follows we introduce an underlying kind of atoms that will be the main tool to give an explicit particular solution for the non-homogeneous discrete Cauchy-Euler Eq. (1). Theorem 1 Let . X = {r1 , r2 , . . . , r N } be any discrete set of distinct real numbers and let . A : X −→ R be the function defined by A(ri ) =

N ∏

(ri − r j )−1 .

j=1 j/=i

Then . A is an . X -atom. Proof For all .s ∈ N we have N N N ∏ ∑ ∏ M(A, s) = r1s (r1 − r j )−1 + ris (ri − r j )−1 j=2

.

=

N ∏

i=2

r1s (r1 − r j )−1 +

j=2

=

N ∏

N ∑

r sj

j=2

r1s (r1 − r j )−1 +

j=2

N ∑

j=1 i/= j N ∏

(r j − ri )−1

i=1 j/=i

r sj (r j − r1 )−1

j=2

N ∏

(r j − ri )−1 .

i=2 j/=i

Taking into account that the first product can be expressed for all .0 ≤ s ≤ N − 2 as N ∏ .

r1s (r1 − r j )−1 =

j=2

N ∑

(r1 − r j )−1 r sj

j=2

N ∏

(r j − ri )−1 .

(3)

i=2 j/=i

So we obtain for .0 ≤ s ≤ N − 2 N N N ∏ ∑ ∏ ∑ M(A, s) = Nj=2 (r1 − r j )−1r sj (r j − ri )−1 + r sj (r j − r1 )−1 (r j − ri )−1 i=2 j/=i

.

=

j=2

∑N

s −1 + (r j − r1 )−1r sj ] j=2 [r j (r 1 − r j )

N ∏

i=2 j/=i

(r j − ri )−1 = 0.

i=2 j/=i

=0

Now for .s = N − 1, we proceed as above, so we obtain N N N ∏ ∑ ∏ N −1 N −1 −1 −1 M(A, N − 1) = r (r − r ) + r (r − r ) (r j − ri )−1 , 1 j j 1 1 j . j=2

j=2

i=2 j/=i

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and Eq. (3) becomes N ∏ .

r1N −1 (r1

− rj)

−1

j=2

=1+

N ∑

(r1 − r j )−1r jN −1

j=2

N ∏

(r j − ri )−1 .

(4)

i=2 j/=i

Making use of Eq. (4) we obtain N ∏ ∑ M(A, N − 1) = 1 + Nj=2 [r jN −1 (r1 − r j )−1 + (r j − r1 )−1r jN −1 ] (r j − ri )−1 .

i=2 j/=i

=0

=1 which gives the desired result.

Definition 2 Let . X = {r1 , r2 , . . . , r N }, . N ≥ 1 be any finite set of distinct real numbers. We define the mixed moment of order .s of an . X -atom . A to be

.

Mm (A, s) :=

N ∑

σs,i A(ri ), where σs,i =

i=1

∑ ri1 ri2 ...ris , 1 ≤ i, s ≤ N .

i 1