Bifurcation and Stability in Nonlinear Discrete Systems [1st ed.] 9789811552113, 9789811552120

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Nonlinear Physical Science

Albert C. J. Luo

Bifurcation and Stability in Nonlinear Discrete Systems

Nonlinear Physical Science Series Editors Albert C. J. Luo , Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL, USA Dimitri Volchenkov , Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX, USA Advisory Editors Eugenio Aulisa , Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX, USA Jan Awrejcewicz , Department of Automation, Biomechanics and Mechatronics, Lodz University of Technology, Lodz, Poland Eugene Benilov , Department of Mathematics, University of Limerick, Limerick, Limerick, Ireland Maurice Courbage, CNRS UMR 7057, Universite Paris Diderot, Paris 7, Paris, France Dmitry V. Kovalevsky , Climate Service Center Germany (GERICS), Helmholtz-Zentrum Geesthacht, Hamburg, Germany Nikolay V. Kuznetsov , Faculty of Mathematics and Mechanics, Saint Petersburg State University, Saint Petersburg, Russia Stefano Lenci , Department of Civil and Building Engineering and Architecture (DICEA), Polytechnic University of Marche, Ancona, Italy Xavier Leoncini, Case 321, Centre de Physique Théorique, MARSEILLE CEDEX 09, France Edson Denis Leonel , Departmamento de Física, Sao Paulo State University, Rio Claro, São Paulo, Brazil Marc Leonetti, Laboratoire Rhéologie et Procédés, Grenoble Cedex 9, Isère, France Shijun Liao, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, China Josep J. Masdemont , Department of Mathematics, Universitat Politècnica de Catalunya, Barcelona, Spain Dmitry E. Pelinovsky , Department of Mathematics and Statistics, McMaster University, Hamilton, ON, Canada Sergey V. Prants , Pacific Oceanological Inst. of the RAS, Laboratory of Nonlinear Dynamical System, Vladivostok, Russia

Laurent Raymond Marseille, France

, Centre de Physique Théorique, Aix-Marseille University,

Victor I. Shrira, School of Computing and Maths, Keele University, Keele, Staffordshire, UK C. Steve Suh , Department of Mechanical Engineering, Texas A&M University, College Station, TX, USA Jian-Qiao Sun, School of Engineering, University of California, Merced, Merced, CA, USA J. A. Tenreiro Machado Porto, Portugal

, ISEP-Institute of Engineering, Polytechnic of Porto,

Simon Villain-Guillot , Laboratoire Ondes et Matière d’Aquitaine, Université de Bordeaux, Talence, France Michael Zaks Germany

, Institute of Physics, Humboldt University of Berlin, Berlin,

Nonlinear Physical Science focuses on recent advances of fundamental theories and principles, analytical and symbolic approaches, as well as computational techniques in nonlinear physical science and nonlinear mathematics with engineering applications. Topics of interest in Nonlinear Physical Science include but are not limited to: • • • • • • • •

New findings and discoveries in nonlinear physics and mathematics: Nonlinearity, complexity and mathematical structures in nonlinear physics: Nonlinear phenomena and observations in nature and engineering: Computational methods and theories in complex systems: Lie group analysis, new theories and principles in mathematical modeling: Stability, bifurcation, chaos and fractals in physical science and engineering: Discontinuity, synchronization and natural complexity in physical sciences: Nonlinear chemical and biological physics

More information about this series at http://www.springer.com/series/8389

Albert C. J. Luo

Bifurcation and Stability in Nonlinear Discrete Systems

With 43 figures

123

Albert C. J. Luo Department of Mechanical and Industrial Engineering Southern Illinois University Edwardsville Edwardsville, IL, USA

ISSN 1867-8440 ISSN 1867-8459 (electronic) Nonlinear Physical Science ISBN 978-981-15-5211-3 ISBN 978-981-15-5212-0 (eBook) https://doi.org/10.1007/978-981-15-5212-0 Jointly published with Higher Education Press The print edition is not for sale in China Mainland. Customers from China Mainland please order the print book from Higher Education Press © Higher Education Press 2020 This work is subject to copyright. All rights are reserved by the Publishers, 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 publishers, 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 publishers nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

This book presents the bifurcation and stability of nonlinear discrete systems. In 2019, the author discussed the bifurcation and stability of nonlinear dynamical systems. In a similar fashion, the monotonic and oscillatory stability and bifurcations of nonlinear discrete systems are discussed. The local monotonic and oscillatory stability and bifurcation of fixed points for a specific eigenvector are presented, and the corresponding higher order singularity of fixed points is discussed. Based on the Yin–Yang theory of discrete systems, the dynamics of forward and backward discrete systems are discussed. The global analysis of monotonic and oscillatory stability of fixed points in one-dimensional discrete systems is presented through polynomial discrete systems. From the local analysis, the normal forms of nonlinear discrete systems are presented, and infinite-fixed-point discrete systems are discussed. This book provides different points of view for nonlinear discrete systems, which can help one better understand nonlinear dynamics of nonlinear discrete systems. This book consists of five chapters. Chapter 1 discusses the local theory of stability and bifurcation of nonlinear discrete systems. In Chap. 2, the monotonic and oscillatory stability and bifurcations of one-dimensional and two-dimensional discrete systems are presented. In Chap. 3, the global stability of one-dimensional, polynomial discrete systems with higher singularity is discussed. Chapter 4 presents the local stability and bifurcation of forward and backward nonlinear discrete systems. Chapter 5 discusses infinite-fixed-point discrete systems, and fixed-point computations and normal forms of nonlinear discrete systems are presented as well. Finally, the author hopes the materials presented herein can last long for science and engineering. Science and technology are good for human being but not for scientists themselves only. The book may have some typos or mistakes there and here. The author hopes readers can point out and make corrections. Herein, the author would like to thank all supporting people during the difficult time period. Edwardsville, IL, USA

Albert C. J. Luo

vii

Contents

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1 1 3 17 17 45 52 53 59 65 72 76

2 Low-Dimensional Discrete Systems . . . . . . . . . . . . . . . . . . . . . . 2.1 One-Dimensional Discrete Systems . . . . . . . . . . . . . . . . . . . 2.2 Monotonic and Oscillatory Stability . . . . . . . . . . . . . . . . . . . 2.3 Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Monotonic Bifurcations . . . . . . . . . . . . . . . . . . . . . . 2.3.2 One-Step Oscillatory Bifurcations . . . . . . . . . . . . . . . 2.3.3 Two-Step Oscillatory Bifurcations . . . . . . . . . . . . . . 2.4 Sample Bifurcation Analysis . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Saddle-Node Appearing Bifurcation . . . . . . . . . . . . . 2.4.2 Saddle-Node Switching Bifurcations . . . . . . . . . . . . . 2.4.3 Third-Order Monotonic Sink Pitchfork Bifurcations . . 2.4.4 Third-Order Monotonic Source Pitchfork Bifurcations 2.5 Two-Dimensional Discrete Systems . . . . . . . . . . . . . . . . . . . 2.5.1 Stability and Singularity . . . . . . . . . . . . . . . . . . . . . . 2.5.2 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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77 77 78 117 117 121 124 127 128 132 140 152 159 159 169 170

1 Local Stability and Bifurcations . . . . . . . . . . . 1.1 Discrete Systems . . . . . . . . . . . . . . . . . . . 1.2 Fixed-Points and Stability . . . . . . . . . . . . . 1.3 Stability Switching . . . . . . . . . . . . . . . . . . 1.3.1 Monotonic and Oscillatory Stability 1.3.2 Spiral Stability . . . . . . . . . . . . . . . . 1.4 Local Bifurcations . . . . . . . . . . . . . . . . . . 1.4.1 Monotonic Bifurcations . . . . . . . . . 1.4.2 One-Step Oscillatory Bifurcations . . 1.4.3 Two-Step Oscillatory Bifurcations . 1.4.4 Neimark Bifurcations . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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ix

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Contents

3 Global Stability of 1-D Discrete Systems . . . . . . . 3.1 Discrete System Classifications . . . . . . . . . . . 3.2 Fixed-Point Global Stability . . . . . . . . . . . . . 3.3 One Fixed-Point Systems . . . . . . . . . . . . . . . 3.3.1 Monotonic Stability . . . . . . . . . . . . . . 3.3.2 Oscillatory Stability . . . . . . . . . . . . . . 3.4 Two-Fixed-Point Systems . . . . . . . . . . . . . . . 3.4.1 Monotonic Two-Fixed-Point Systems . 3.4.2 Oscillatory Two-Fixed-Point Systems . 3.5 Three-Fixed-Point Systems . . . . . . . . . . . . . . 3.5.1 Monotonic Three-Fixed-Point Systems 3.5.2 Oscillatory Three-Fixed-Point Systems References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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171 171 175 181 181 183 185 185 189 194 194 198 206

4 Forward and Backward Discrete Systems . 4.1 Yin-Yang Theory of Discrete Systems . 4.2 Forward Discrete Systems . . . . . . . . . . 4.3 Backward Discrete Systems . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . .

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207 207 218 234 249

5 Infinite-Fixed-Point Discrete Systems . . . . . . . . . . . . 5.1 Fixed-Point Computations . . . . . . . . . . . . . . . . . . 5.2 Normal Forms of Discrete Systems . . . . . . . . . . . 5.3 Infinite-Fixed-Point Systems . . . . . . . . . . . . . . . . 5.3.1 One-Infinite-Fixed-Point Systems . . . . . . . 5.3.2 Two-Infinite-Fixed-Point Systems . . . . . . . 5.3.3 Higher-Order Infinite-Fixed-Point Systems 5.4 Infinite-Fixed-Point-Network Systems . . . . . . . . . 5.4.1 An Infinite-Fixed-Point-Network System . . 5.4.2 A Spiral Infinite-Fixed-Point System . . . . . 5.4.3 A Linear Infinite-Fixed-Point System . . . . 5.4.4 A Hyperbolic Infinite-Fixed-Point System . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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251 251 259 270 271 275 284 289 290 294 298 303 309

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Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

Chapter 1

Local Stability and Bifurcations

In this chapter, basic concepts of nonlinear discrete systems are introduced. The local stability theory of fixed-points in nonlinear discrete systems is presented. The stability switching and bifurcation of fixed-points on the specific eigenvectors of the corresponding linearized systems are discussed. The higher singularity and stability of fixed points in nonlinear discrete systems on the specific eigenvectors are also discussed.

1.1 Discrete Systems Definition 1.1 For α ⊆ R n and  ⊆ R m with α ∈ Z, consider a vector function fα : α ×  → α which is C r (r ≥ 1)-continuous, and there is a discrete (or difference) equation in a form of xk+1 = fα (xk , pα ) for xk , xk+1 ∈ α , k ∈ Z and pα ∈ .

(1.1)

With an initial condition of x0 , the solution of Eq. (1.1) is given by xk = fα (fα (· · · (fα (x0 , pα ))))    k

(1.2)

for xk ∈ α , k ∈ Z and p ∈ . (i) (ii) (iii) (iv)

The difference equation with the initial condition is called a discrete system. The vector function fα (xk , pα ) is called a discrete vector field on α . The solution xk for each k ∈ Z is called a discrete flow of discrete system. The solution xk for all k ∈ Z on domain α is called the discrete trajectory, phase curve or orbit of discrete system, which is defined as  = {xk |xk+1 = fα (xk , pα ) for k ∈ Z and pα ∈ } ⊆ ∪α α .

© Higher Education Press 2020 A. C. J. Luo, Bifurcation and Stability in Nonlinear Discrete Systems, Nonlinear Physical Science, https://doi.org/10.1007/978-981-15-5212-0_1

(1.3) 1

2

1 Local Stability and Bifurcations

(v) The discrete system is called a uniform discrete system if xk+1 = fα (xk , pα ) = f(xk , p) for k ∈ Z and xk ∈ α

(1.4)

Otherwise, this discrete system is called a non-uniform discrete system. Definition 1.2 For the discrete system in Eq. (1.1), the relation between state xk and state xk+1 (k ∈ Z) is called a discrete map if fα

Pα : xk → xk+1 and xk+1 = Pα xk

(1.5)

with the following properties: fα1 ,fα2 ,...,fαn

P(k,n) : xk −−−−−−→ xk+n and xk+n = Pαn ◦ Pαn−1 ◦ · · · ◦ Pα1 xk

(1.6)

where P(k;n) = Pαn ◦ Pαn−1 ◦ · · · ◦ Pα1 .

(1.7)

If Pαn = Pαn−1 = · · · = Pα1 = Pα , then P(α;n) ≡ Pα(n) = Pα ◦ Pα ◦ · · · ◦ Pα

(1.8)

Pα(n) = Pα ◦ Pα(n−1) and Pα(0) = I.

(1.9)

with

The total map with n-different sub-maps is shown in Fig. 1.1. The map Pαk with the relation function fαk (αk ∈ Z) is given by Eq. (1.5). The total map P(k,n) is given in Eq. (1.7). The domains αk (αk ∈ Z) can fully overlap each other or can be completely separated without any intersection. Definition 1.3 For a discrete vector function of fα ∈ R n , fα : R n → R n . The operator norm of fα is defined by

fα k xk

fα1 Pα1

Ωα 2

Ωα k

Pα k

x k +1

Ωα k +1

Ωα n

Ωα1 Fig. 1.1 Maps and discrete vector functions on each sub-domain for a discrete system

1.1 Discrete Systems

3

||fα || =

n  i=1

max | f α(i) (xk , pα )|.

||xk ||≤1,pα

(1.10)

For an n × n matrix fα (xk , pα ) = Aα xk and Aα = (ai j )n×n , the corresponding norm is defined by ||Aα || =

n 

|ai j |.

(1.11)

i, j=1

Definition 1.4 For α ⊆ R n and  ⊆ R m with α ∈ Z, the vector function fα (xk , pα ) with fα : α ×  → R n is differentiable at xk ∈ α if  ∂fα (xk , pα )  fα (xk + xk , pα ) − fα (xk , pα ) = lim .  xk →0 ∂xk xk (xk ,p)

(1.12)

∂fα /∂xk is called the spatial derivative of fα (xk , pα ) at xk , and the derivative is given by the Jacobian matrix  ∂ f α(i) ∂fα (xk , pα ) = . ∂xk ∂ xk( j) n×n

(1.13)

Definition 1.5 For α ⊆ R n and  ⊆ R m , consider a vector function f(xk , p) with f : α ×  → R n where xk ∈ α and p ∈  with k ∈ Z. The vector function f(xk , p) satisfies the Lipschitz condition ||f(yk , p) − f(xk , p)|| ≤ L||yk − xk ||

(1.14)

with xk , yk ∈ α and L a constant. The constant L is called the Lipschitz constant.

1.2 Fixed-Points and Stability Definition 1.6 Consider a discrete system xk+1 = fα (xk , pα ) in Eq. (1.4). (i) A point xk∗ ∈ α is called a fixed-point or period-1 solution of a discrete nonlinear system xk+1 = fα (xk , pα ) under a map Pα if for xk+1 = xk = xk∗ xk∗ = fα (xk∗ , p)

(1.15)

The linearized system of the nonlinear discrete system xk+1 = fα (xk , pα ) in Eq. (1.4) at the fixed-point xk∗ is given by yk+1 = D Pα (xk∗ , p)yk = Dfα (xk∗ , p)yk

(1.16)

4

1 Local Stability and Bifurcations

where ∗ yk = xk − xk∗ and yk+1 = xk+1 − xk+1 .

(1.17)



(ii) A set of points x∗j ∈ α j α j ∈ Z is called the fixed-point set or period-1 point set of the total map P(k;n) with n-different sub-maps in nonlinear discrete system of Eq. (1.5) if ∗ ∗  xk+ j+1 = fα j  (xk+ j , pα j  ) for j ∈ Z+ and j = mod( j, n) + 1; ∗ ∗ xk+ j = xk+ mod ( j,n) .

(1.18)

Each linearized equation of the total map P(k;n) gives ∗ ∗ yk+ j+1 = D Pα j (xk+ j , pα j )yk+ j = Dfα j (xk+ j , pα j )yk+ j ∗ ∗ with yk+ j+1 = xk+ j+1 − xk+ j+1 and yk+ j = xk+ j − xk+ j for j = 1, 2, 3, · · ·

(1.19)

The resultant equation for the total map is yk+n = D P(k,n) (xk∗ , p)yk

(1.20)

where D P(k,n) (xk∗ , p)

=

1

∗ D Pα j (xk+ j−1 , p)

j=n ∗ ∗ = D Pαn (xk+n−1 , pαn ) · . . . · Pα2 (xk+1 , pα2 ) · D Pα1 (xk∗ , pα1 ) ∗ ∗ = Dfαn (xk+n−1 , pαn ) · . . . · Dfα2 (xk+1 , pα2 ) · Dfα1 (xk∗ , pα1 ). (1.21)

A fixed-point xk∗ lies in the intersected set of two domains k and k+1 , as shown in Fig. 1.2. In the vicinity of the fixed-point xk∗ , the incremental relations in the two domains k and k+1 are different. In other words, setting yk = xk − xk∗ and Fig. 1.2 A fixed-point between domains k and k+1 for a discrete system

Ωα xk

x∗k

fα

x k +1

1.2 Fixed-Points and Stability

5

Ωα j Ωα1 xk

Ωα 2 fα1

xk + j

Ωα j +1

fα j

x k + j +1

Ωα n

Pα j

Pα1 Pα n

xk +n

fα n

Fig. 1.3 Fixed-points with n-maps for discrete system ∗ yk+1 = xk+1 − xk+1 , the corresponding linearization is generated as in Eq. (1.16). Similarly, The fixed-point of the total map with n-different sub-maps requires the intersection set of two domains k and k+n , there is a set of equations for fixedpoints from Eq. (1.18). The other values of fixed-points lie in different domains, i.e., x∗j ∈  j ( j = k + 1, k + 2, . . . , k + n − 1), as shown in Fig. 1.3. The corresponding linearized equations are given in Eq. (1.19). From Eq. (1.20), the local characteristics of the total map can be discussed as a single map. Thus, the dynamical characteristics for the fixed-point of the single map will be discussed, which are applicable to fixed-points of the resultant map. The results can also be extended to any period-m flows with P (m) .

Definition 1.7 Consider a nonlinear discrete system xk+1 = f(xk , p) in Eq. (1.4) with a fixed-point xk∗ . The linearized system of the discrete nonlinear system in the neighborhood of xk∗ is yk+1 = Df(xk∗ , p)yk (yl = xl − xk∗ and l = k, k + 1) in Eq. (1.16). The matrix Df(xk∗ , p) possesses n 1 real eigenvalues |λ j | < 1 ( j ∈ N1 ), n 2 real eigenvalues |λ j | > 1 ( j ∈ N2 ), n 3 real eigenvalues λ j = 1 ( j ∈ N3 ), and n 4 real eigenvalues λ j = −1 ( j ∈ N4 ). N = {1, 2, . . . , n} and Ni = {i 1 , i 2 , . . . , i ni } ∪ ∅ 3 n i = n. Ni ⊆ N ∪ ∅, (i = 1, 2, 3, 4) with i m ∈ N (m = 1, 2, . . . , n i ) and i=1 3 ∪i=1 Ni = N , Ni ∩ N p = ∅ ( p = i).Ni = ∅ if n i = 0. The corresponding eigenvectors for contraction, expansion, invariance and flip oscillation are {v j } ( j ∈ Ni ) (i = 1, 2, 3, 4), respectively. The stable, unstable, invariant and flip subspaces of yk+1 = Df(xk∗ , p)yk in Eq. (1.16) are linear subspace spanned by {v j } ( j ∈ Ni ) (i = 1, 2, 3, 4), respectively, i.e.,

6

1 Local Stability and Bifurcations

E

s

Eu E

i

Ef

   (Df(x∗ , p) − λ I)v = 0, j j  k = span v j  ;  |λ j | < 1, j ∈ N1 ⊆ N ∪ ∅

   (Df(x∗ , p) − λ I)v = 0, j j  k = span v j  ;  |λ j | > 1, j ∈ N2 ⊆ N ∪ ∅

   (Df(x∗ , p) − λ I)v = 0, j j  k = span v j  ;  λ j = 1, j ∈ N3 ⊆ N ∪ ∅

   (Df(x∗ , p) − λ I)v = 0, j j  k = span v j  .  λ j = −1, j ∈ N4 ⊆ N ∪ ∅

(1.22)

where E s = Ems ∪ Eos ∪ Ezs with     (Df(x∗ , p) − λ j I)v j = 0, s k  Em = span v j  ; 0 < λ j < 1, j ∈ N1m ⊆ N ∪ ∅     (Df(x∗ , p) − λ j I)v j = 0, k Eos = span v j  ; −1 < λ j < 0, j ∈ N1o ⊆ N ∪ ∅     (Df(x∗ , p) − λ j I)v j = 0, k Ezs = span v j  ; λ j = 0, j ∈ N1z ⊆ N ∪ ∅

(1.23)

E u = Emu ∪ Eou with     (Df(x∗ , p) − λ j I)v j = 0, u k  Em = span v j  ; λ j > 1, j ∈ N2m ⊆ N ∪ ∅     (Df(x∗ , p) − λ j I)v j = 0, k Eou = span v j  ; λ j < −1, j ∈ N o ⊆ N ∪ ∅

(1.24)

2

where subscripts “m” and “o” represent the monotonic and oscillatory evolutions. Definition 1.8 Consider a nonlinear discrete system xk+1 = f(xk , p) in Eq. (1.4) with a fixed-point xk∗ . The linearized system of the discrete nonlinear system in the neighborhood of xk∗ is yk+1 = Df(xk∗ , p)yk (yl = xl − xk∗ and l = k, k + 1) in Eq. (1.16). The matrix Df(xk∗ , p) has complex eigenvalues α j ±iβ j with eigenvectors u j ± iv j ( j ∈ {1, 2, . . . , n}) and the base of eigenvectors is   B = u1 , v1 , . . . , u j , v j , . . . , un , vn .

(1.25)

The stable, unstable, center subspaces of yk+1 = Dfk (xk∗ , p)yk in Eq. (1.16) are linear subspaces spanned by {u j , v j } ( j ∈ Ni , i = 1, 2, 3), respectively. Set N = {1, 2, . . . , n} plus Ni = {i 1 , i 2 , . . . , i ni }∪∅ ⊆ N ∪∅ with i m ∈ N (m = 1, 2, . . . , n i )

1.2 Fixed-Points and Stability

7

3 3 and i=1 n i = n. ∪i=1 Ni = N with Ni ∩ N p = ∅ ( p = i). Ni = ∅ if n i = 0. The stable, unstable, center subspaces of yk+1 = Df(xk∗ , p)yk in Eq. (1.16) are defined by

  ⎫  r = α 2 + β 2 < 1, ⎪  j ⎬ j j  s E = span (u j , v j ) (Df(x∗ , p) − (α j ± iβ j )I)(u j ± iv j ) = 0, ; k  ⎪ ⎪ ⎩ ⎭  j ∈ N ⊆ {1, 2, . . . , n} ∪ ∅ 1  ⎧  ⎫  r = α 2 + β 2 > 1, ⎪ ⎪  j ⎨ ⎬ j j  E u = span (u j , v j ) (Df(x∗ , p) − (α j ± iβ j )I)(u j ± iv j ) = 0, ; k  ⎪ ⎪ ⎩ ⎭  j ∈ N ⊆ {1, 2, . . . , n} ∪ ∅ 2  ⎧  ⎫  r = α 2 + β 2 = 1, ⎪ ⎪  j ⎨ ⎬ j j  E c = span (u j , v j ) (Df(x∗ , p) − (α j ± iβ j )I)(u j ± iv j ) = 0, . k  ⎪ ⎪ ⎩ ⎭  j ∈ N ⊆ {1, 2, . . . , n} ∪ ∅ 3 ⎧ ⎪ ⎨

(1.26)

Definition 1.9 Consider a nonlinear discrete system xk+1 = f(xk , p) in Eq. (1.4) with a fixed-point xk∗ . The linearized system of the discrete nonlinear system in the neighborhood of xk∗ is yk+1 = Df(xk∗ , p)yk (yl = xl − xk∗ and l = k, k + 1) in Eq. (1.16). The fixed-point or period-1point is hyperbolic if no any eigenvalues of Df(xk∗ , p) are on the unit circle (i.e., |λi | = 1 for i = 1, 2, . . . , n). Theorem 1.1 Consider a nonlinear discrete system xk+1 = f(xk , p) in Eq. (1.4) with a fixed-point xk∗ . The linearized system of the discrete nonlinear system in the neighborhood of xk∗ is yk+1 = Df(xk∗ , p)yk (y j = x j − xk∗ and j = k, k + 1) in Eq. (1.16). The eigenspace of Df(xk∗ , p) (i.e., E ⊆ R n ) in the linearized discrete system is expressed by direct sum of three subspaces E = E s ⊕ E u ⊕ E c.

(1.27)

where E s , E u and E c are the stable, unstable and center subspaces, respectively. Proof The proof can be referred to Luo (2011).



Definition 1.10 Consider a nonlinear discrete system xk+1 = f(xk , p) in Eq. (1.4) with a fixed-point xk∗ . Suppose there is a neighborhood of the fixed-point xk∗ as Uk (xk∗ ) ⊂ k , and in the neighborhood, ||f(xk∗ + yk , p) − Df(xk∗ , p)yk || = 0. ||yk ||→0 ||yk ||

(1.28)

yk+1 = Df(xk∗ , p)yk

(1.29)

lim

and

8

1 Local Stability and Bifurcations

(i) A C r invariant manifold Sloc (xk , xk∗ ) = {xk ∈ U (xk∗ )| lim xk+ j = xk∗ and j→+∞

xk+ j ∈ U (xk∗ ) with j ∈ Z+ }

(1.30)

is called the local stable manifold of xk∗ , and the corresponding global stable manifold is defined as ∗ ( j) ∗ S (xk , xk∗ ) = ∪ j∈Z− f(Uloc (xk+ j , xk+ j )) = ∪ j∈Z− f (Uloc (xk , xk )).

(1.31)

(ii) A C r invariant manifold Uloc (xk , xk∗ ) Uloc (xk , xk∗ ) = {xk ∈ U (xk∗ )| lim xk+ j = xk∗ and j→−∞

xk+ j ∈ U (xk∗ ) with j ∈ Z− }

(1.32)

is called the local unstable manifold of x∗ , and the corresponding global stable manifold is defined as ∗ ( j) ∗ U (xk , xk∗ ) = ∪ j∈Z+ f(Uloc (xk+ j , xk+ j )) = ∪ j∈Z+ f (Uloc (xk , xk )). (1.33)

(iii) A C r −1 invariant manifold Cloc (xk , xk∗ ) is called the canter manifold of xk∗ if Cloc (xk , xk∗ ) possesses the same dimension of E c for xk∗ ∈ S (xk , xk∗ ), and the tangential space of Cloc (xk , xk∗ ) is identical to E c . As in continuous systems, the stable and unstable manifolds are unique, but the center manifold is not unique. If the nonlinear vector field f is C ∞ -continuous, then a C r center manifold can be found for any r < ∞. Theorem 1.2 Consider a nonlinear discrete system xk+1 = f(xk , p) in Eq. (1.4) with a hyperbolic fixed-point xk∗ . The corresponding solution is xk+ j = f(xk+ j−1 , p) with j ∈ Z. Suppose there is a neighborhood of the hyperbolic fixed-point xk∗ (i.e., Uk (xk∗ ) ⊂ α ), and f(xk , p) is C r (r ≥ 1)-continuous in Uk (xk∗ ). The linearized system is yk+ j+1 = Df(xk∗ , p)yk+ j (yk+ j = xk+ j − xk∗ ) in Uk (xk∗ ). If the homeomorphism between the local invariant subspace E(xk∗ ) ⊂ U (xk∗ ) and the eigenspace E of the linearized system exists with the condition in Eq. (1.28), the local invariant subspace is decomposed by E(xk , xk∗ ) = Sloc (xk , xk∗ ) ⊕ Uloc (xk , xk∗ ).

(1.34)

(a) The local stable invariant manifold Sloc (x, x∗ ) possesses the following properties: (i)

for xk∗ ∈ Sloc (xk , xk∗ ), Sloc (xk , xk∗ ) possesses the same dimension of E s and the tangential space of Sloc (xk , xk∗ ) is identical to E s ;

1.2 Fixed-Points and Stability

9

(ii) for xk ∈ Sloc (xk , xk∗ ), xk+ j ∈ Sloc (xk , xk∗ ) and lim xk+ j = xk∗ for all j ∈ Z+ ; j→∞

(iii) for xk ∈ / Sloc (xk , xk∗ ), ||xk+ j − xk∗ || ≥ δ for δ > 0 with j, j1 ∈ Z+ and j ≥ j1 ≥ 0. (b) The local unstable invariant manifold Uloc (xk , xk∗ ) possesses the following properties: for xk∗ ∈ Uloc (xk , xk∗ ), Uloc (xk , xk∗ ) possesses the same dimension of E u and the tangential space of Uloc (xk , xk∗ ) is identical to E u ; (ii) for xk ∈ Uloc (xk , xk∗ ), xk+ j ∈ Uloc (xk , xk∗ ) and lim xk+ j = x∗ for all j ∈ Z− (i)

j→−∞

(iii) for xk ∈ / Uloc (x, x∗ ), ||xk+ j − xk∗ || ≥ δ for δ > 0 with j1 , j ∈ Z− and j ≤ j1 ≤ 0. Proof See Nitecki (1971).



Theorem 1.3 Consider a nonlinear discrete system xk+1 = f(xk , p) in Eq. (1.4) with a fixed-point xk∗ . The corresponding solution is xk+ j = f(xk+ j−1 , p) with j ∈ Z. Suppose there is a neighborhood of the fixed-point xk∗ (i.e., Uk (xk∗ ) ⊂ α ), and f(xk , p) is C r (r ≥ 1)-continuous in Uk (xk∗ ). The linearized system is yk+ j+1 = Df(xk∗ , p)yk+ j (yk+ j = xk+ j − xk∗ ) in Uk (xk∗ ). If the homeomorphism between the local invariant subspace E(xk∗ ) ⊂ U (xk∗ ) and the eigenspace E of the linearized system exists with the condition in Eq. (1.28), in addition to the local stable and unstable invariant manifolds, there is a C r −1 center manifold Cloc (xk , xk∗ ). The center manifold possesses the same dimension of E c for x∗ ∈ Cloc (xk , xk∗ ), and the tangential space of Cloc (x, x∗ ) is identical to E c . Thus, the local invariant subspace is decomposed by E(xk , xk∗ ) = Sloc (xk , xk∗ ) ⊕ Uloc (xk , xk∗ ) ⊕ Cloc (xk , xk∗ ). Proof See Guckenhiemer and Holmes (1990).

(1.35) 

Definition 1.11 Consider a nonlinear discrete system xk+1 = f(xk , p) in Eq. (1.4) on domain α ∈ R n . Suppose there is an metric space (α , ρ), then the map P under the vector function f(xk , p) is called the a contraction map if (1) (2) , xk+1 ) = ρ(f(xk(1) , p), f(xk(2) , p)) ≤ λρ(xk(1) , xk(2) ) ρ(xk+1

(1.36)

for λ ∈ (0, 1) and xk(1) , xk(2) ∈ α with ρ(xk(1) , xk(2) ) = ||xk(1) − xk(2) ||. Theorem 1.4 Consider a nonlinear discrete system xk+1 = f(xk , p) in Eq. (1.4) on domain α ∈ R n . Suppose there is an metric space (α , ρ), if the map P under the vector function f(xk , p) is the a contraction map, then there is an unique fixed-point xk∗ which is globally stable. Proof The proof can be referred to Luo (2011).



10

1 Local Stability and Bifurcations

Definition 1.12 Consider a nonlinear discrete system xk+1 = f(xk , p) in Eq. (1.4) with a fixed-point xk∗ . The linearized system of the discrete nonlinear system in the neighborhood of xk∗ is yk+1 = Df(xk∗ , p)yk (yl = xl − xk∗ and l = k, k + 1) in Eq. (1.16). Suppose U (xk∗ ) ⊂  is a neighborhood of fixed-point xk∗ , and there are n linearly independent vectors v j ( j = 1, 2, . . . , n). The independent eigenvector v j is called a covariant eigenvector. The corresponding contravariant vector of the independent eigenvector v j is defined by v j . For a perturbation of fixed-point yk = xk − xk∗ , let yk = c j v j = c j v j (tensor notation convention). P = (v1 , v2 , . . . , vn )

(1.37)

is called an eigenvector covariant matrix. 

Q = v1 , v2 , . . . , vn

(1.38)

is called an eigenvector contravariant matrix. The eigenvector contravariant vector is defined as v j = (a j1 , a j2 , . . . , a jn )T with Q T = P −1 = (ar s ).

(1.39)

The component c j is called a contravariant component. The corresponding vector T

c¯ = c1 , c2 , . . . , cn

(1.40)

is called a contravariant component vector. The component c j is called a covariant component. The corresponding vector is c = (c1 , c2 , . . . , cn )T

(1.41)

is called a covariant component vector. Remark For the Cartesian coordinate, the covariant and contravariant vectors and components are identical. Since the eigenvectors of the linearized systems of nonlinear systems are not orthogonal, the eigenvector space cannot be used for the Cartesian coordinate. However, the eigenvectors can be used to express the solutions of a discrete system as yk = c j v j = c j v j . The two classes of v j and v j are the covariant and contravariant eigenvectors. Definition 1.13 The metric tensors of the covariant and contravariant eigenvectors are defined as for i, j = 1, 2, . . . , n gi j = (vi )T · v j ,g i j = (vi )T · v j , j

j

gi = (vi )T · v j = δi ;

(1.42)

1.2 Fixed-Points and Stability

11

and the unit vectors of the covariant and contravariant eigenvectors are defined as  √ ei = vi / gii , ei = vi / g ii .

(1.43)

Two transformations based on the covariant and contravariant eigenvectors are defined as yk = c j v j = c j v j

(1.44)

 √ yk = ci gii ei = c j g j j e j

(1.45)

and

with j

(v j )T · yk = (v j )T · ci vi = ci (v j )T · vi = ci δi = c j ; (v j )T · yk = (v j )T · ci vi = ci (v j )T · vi = ci δ ij = c j .

(1.46)

For the orthogonal eigenvectors, vi = vi , g ii = gii = 1, g i j = gi j = 0.

(1.47)

Consider a solution in a two-dimensional system for i, j = 1, 2 as yk = ci vi = c1 v1 + c2 v2 ,

(1.48)

yk = c j v j = c1 v1 + c2 v2 . Based on unit covariant and contravariant eigenvectors, √ √ yk = c1 g11 e1 + c2 g22 e2 ,   yk = c1 g 11 e1 + c2 g 22 e2 .

(1.49)

The projected quantities on the covariant and contravariant eigenvectors are (ei )T · yk = (ei )T · c j v j = √c ii (vi )T · v j = √c ii δ ij = √c ii , (ei )T · yk = (ei )T · c j v j =

j

j

i

g

g

g

c √j gii

(vi )T · v j =

c j √j δ gii i

=

√ci gii

.

(1.50)

Thus, ci = (vi )T · yk =



g ii (ei )T · yk ; √ c j = (v j )T · yk = g j j (e j )T · yk .

(1.51)

12

1 Local Stability and Bifurcations

The covariant and contravariant eigenvectors as the coordinates to express a solution vector of discrete system are presented in Fig. 1.4. The corresponding geometric interpretation of the relations of components is described. A vector yk is depicted by the covariant and contravariant eigenvectors. The unit vectors and corresponding quantities based on the two eigenvectors are presented. Definition 1.14 Consider a nonlinear discrete system xk+1 = f(xk , p) in Eq. (1.4) with a fixed-point xk∗ . The corresponding solution is given by xk+ j = f(xk+ j−1 , p) with j ∈ Z. Suppose there is a neighborhood of the fixed-point xk∗ (i.e., Uk (xk∗ ) ⊂ α ), and f(xk , p) is C r (r ≥ 1)-continuous in Uk (xk∗ ). The linearized system is yk+ j+1 = Df(xk∗ , p)yk+ j (yk+ j = xk+ j − xk∗ ) in Uk (xk∗ ). Consider a real eigenvalue λi of matrix Df(xk∗ , p) (i ∈ N = {1, 2, . . . , n}) and there is a corresponding eigenvector vi . On l vl , thus, the invariant eigenvector vi , consider yk = cki vi and yk+1 = ck+1 e 2 = v 2 / g 22

c2 v 2

e 2 = v 2 / g 22 c2 g 22

c2 / g 22

c1 / g 11

c 2 g 22 c2 v2

yk

c 2 / g 22

c1v1

e1 = v1 / g11

c1 g11 c1 / g11

c1 g 11

c1v1 e1 = v1 / g 11

Fig. 1.4 A solution vector based on the covariant and contravariant eigenvectors in a twodimensional discrete system. The thick black and blue colors are used for covariant and contravariant eigenvectors, respectively

1.2 Fixed-Points and Stability

13

l ck+1 = (vl )T · yk+1 = (vl )T · Df(xk∗ , p)vi cki = λi δil cki = λl ckl .

(1.52)

(i) xk on the direction vl at the fixed-point xk∗ is stable if lim |ckl | = lim |(λl )k | × |c0l | = 0 for |λl | < 1.

k→∞

k→∞

(1.53)

(ii) xk on the direction vl at the fixed-point xk∗ is unstable if lim |ckl | = lim |(λl )k | × |c0l | = ∞ for |λl | > 1.

k→∞

k→∞

(1.54)

(iii) xk on the direction vl at the fixed-point xk∗ is invariant if lim ckl = lim (λl )k c0l = c0l for λl = 1.

k→∞

k→∞

(1.55)

(iv) xk on the direction vl is flipped if lim ckl = lim (λl )2k × c0l = c0l

2k→∞

lim

2k+1→∞

2k→∞

ckl =

⎫ ⎬

lim (λl )2k+1 × c0l = −c0l ⎭

for λl = −1.

(1.56)

2k+1→∞

(v) xk on the direction vl at the fixed-point xk∗ is degenerate if ckl = (λl )k c0l = 0 for λl = 0.

(1.57)

Definition 1.15 Consider a nonlinear discrete system xk+1 = f(xk , p) in Eq. (1.4) with a fixed-point xk∗ . The corresponding solution is given by xk+ j = f(xk+ j−1 , p) with j ∈ Z. Suppose there is a neighborhood of the fixed-point xk∗ (i.e., Uk (xk∗ ) ⊂ ∗ α ), and f(xk , p) is C r (r k ). The linearized system is

≥ 1)-continuous  in Uk (x ∗ ∗ ∗ yk+ j+1 = Df(xk , p)yk+ j yk+ j = xk+ j − xk in Uk (xk ). Consider a pair of complex

√  eigenvalue αi ±iβi of matrix Df(xk∗ , p) i ∈ N = {1, 2, . . . , n}, i = −1 and there is a corresponding eigenvector v2i−1 ± iv2i . On the invariant plane of (v2l−1 , v2l ), ck2l−1 = (v2l−1 )T · yk , ck2l = (v2l−1 )T · yk

(1.58)

Dx f · v2l−1 = αl v2l−1 − βl v2l , Dx f · v2l = βl v2l−1 + αl v2l .

(1.59)

with

Thus, 2l−1 ck+1 = (v2l−1 )T · yk+1 = (v2l−1 )T · Df(xk∗ , p)v j c j = αl c2l−1 + βl c2l , 2l ck+1 = (v2l )T · yk+1 = (v2l )T · Df(xk∗ , p)v j c j = −βl c2l−1 + αl c2l .

(1.60)

14

1 Local Stability and Bifurcations

With ckl = (ck2l−1 , ck2l )T , l ck+1 = El ckl = rl Rl ckl

(1.61)

where  El =

αl βl −βl αl

cos θl =

αl rl



 and Rl =

and sinθl =

cos θl sin θl , − sin θl cos θl

βl , rl rl

=

(1.62)

αl2 + βl2 ;

and 

Elk

αl βl = −βl αl

k

cos kθl sin kθl . = − sin kθl cos kθl 

and

Rlk

(1.63)

(i) xk on the plane of (v2l−1 , v2l ) at the fixed-point xk∗ is spirally stable if lim ||ckl || = lim rlk ||Rlk || × ||c0l || = 0 for rl = |λl | < 1.

k→∞

k→∞

(1.64)

(ii) xk on the plane of (v2l−1 , v2l ) at the fixed-point xk∗ is spirally unstable if lim ||ckl || = lim rlk ||Rlk || × ||c0l || = ∞ for rl = |λl | > 1.

k→∞

k→∞

(1.65)

(iii) xk on the plane of (v2l−1 , v2l ) at the fixed-point xk∗ is on the invariant circles if, lim ||ckl || = lim rlk ||Rlk || × ||c0l || = ||c0l || for rl = |λl | = 1.

k→∞

k→∞

(1.66)

(iv) xk on the plane of (v2l−1 , v2l ) at the fixed-point xk∗ is degenerate in the direction of v2l−1 if βl = 0. Definition 1.16 Consider a nonlinear discrete system xk+1 = f(xk , p) in Eq. (1.4) with a fixed-point xk∗ . The corresponding solution is given by xk+ j = f(xk+ j−1 , p) with j ∈ Z. Suppose there is a neighborhood of the fixed-point xk∗ (i.e., Uk (xk∗ ) ⊂ ∗ α ), and f(xk , p) is C r (r ≥ 1)-continuous

in Uk (xk ) with∗ Eq. (1.28).∗ The linearized ∗ system is yk+ j+1 = Df(xk , p)yk+ j yk+ j = xk+ j − xk in Uk (xk ). The matrix Df(xk∗ , p) possesses n real eigenvalues λi (i = 1, 2, . . . , n). (i) (ii) (iii) (iv)

The fixed-point xk∗ is called a hyperbolic point if |λi | = 1 (i = 1, 2, . . . , n). The fixed-point xk∗ is called a sink if |λi | < 1 (i = 1, 2, . . . , n). The fixed-point xk∗ is called a source if |λi | > 1 (i = 1, 2, . . . , n). The fixed-point xk∗ is called a center if |λi | = 1 (i = 1, 2, . . . n) with distinct eigenvalues.

Definition 1.17 Consider a nonlinear discrete system xk+1 = f(xk , p) in Eq. (1.4) with a fixed-point xk∗ . The corresponding solution is given by xk+ j = f(xk+ j−1 , p)

1.2 Fixed-Points and Stability

15

with j ∈ Z. Suppose there is a neighborhood of the fixed-point xk∗ (i.e., Uk (xk∗ ) ⊂ α ), and f(xk , p) is C r (r ≥ 1)-continuous in Uk (xk∗ ) with Eq. (1.28). The linearized system is yk+ j+1 = Df(xk∗ , p)yk+ j (yk+ j = xk+ j − xk∗ ) in Uk (xk∗ ). The matrix Df(xk∗ , p) possesses n eigenvalues λi (i = 1, 2, . . . , n). (i) The fixed-point xk∗ is called a stable node if |λi | < 1 (i = 1, 2, . . . , n). (ii) The fixed-point xk∗ is called an unstable node if |λi | > 1 (i = 1, 2, . . . , n). (iii) The fixed-point xk∗ is called an (l1 : l2 )-saddle if at least one |λi | > 1 (i ∈ L 1 ⊂ {1, 2, . . . n}) and the other |λ j | < 1 ( j ∈ L 2 ⊂ {1, 2, . . . n}) with L 1 ∪ L 2 = {1, 2, . . . , n} and L 1 ∩ L 2 = ∅. (iv) The fixed-point xk∗ is called an lth-order degenerate case if |λi | = 1(i ∈ L ⊆ {1, 2, . . . n}) . Definition 1.18 Consider a 2n-dimnesional nonlinear discrete system xk+1 = f(xk , p) in Eq. (1.4) with a fixed-point xk∗ . The corresponding solution is given by xk+ j = f(xk+ j−1 , p) with j ∈ Z. Suppose there is a neighborhood of the fixedpoint xk∗ (i.e., Uk (xk∗ ) ⊂ α ), and f(xk , p) is C r (r ≥ 1)-continuous in Uk (xk∗ ) with Eq. (1.28). The linearized system is yk+ j+1 = Df(xk∗ , p)yk+ j (yk+ j = xk+ j − xk∗ )in Uk (xk∗ ). The matrix Df(xk∗ , p) possesses n-pairs of complex eigenvalues λi (i = 1, 2, . . . , n). The fixed-point xk∗ is called a spiral sink if |λi | < 1 (i = 1, 2, . . . , n) and Imλ j = 0 ( j ∈ {1, 2, . . . , n}). (ii) fixed-point xk∗ is called a spiral source if |λi | > 1 (i = 1, 2, . . . , n) with Imλ j = 0 ( j ∈ {1, 2, . . . , n}). (iii) fixed-point xk∗ is called a center if |λi | = 1 with distinct Imλi = 0 (i = 1, 2, . . . , n). (i)

The generalized stability and bifurcation of flows in linearized, nonlinear discrete systems in Eq. (1.4) will be discussed as follows. Definition 1.19 Consider a nonlinear discrete system xk+1 = f(xk , p) in Eq. (1.4) with a fixed-point xk∗ . The corresponding solution is given by xk+s = f(xk+s−1 , p) with s ∈ Z. Suppose there is a neighborhood of the fixed-point xk∗ (i.e., Uk (xk∗ ) ⊂ α ), and f(xk , p) is C r (r ≥ 1)-continuous in Uk (xk∗ ) with Eq. (4.28). The linearized system is yk+s+1 = Df(xk∗ , p)yk+s (yk+s = xk+s − xk∗ ) in Uk (xk∗ ). The matrix Df(xk∗ , p) possesses n eigenvalues yk = xk −xk∗ (i = 1, 2, . . . n). Set N = {1, 2, . . . , m, m + 1, . . . , (n + m)/2}, N j = {l1 , l2 , . . . , ln j } ∪ ∅ with l p ∈ N ( p = 1, 2, . . . , n j ; j = 1, 2, . . . , 7), 4j=1 n j = m and Σ 7j=5 n j = (n − m)/2. ∪7j=1 N j = N and N j ∩ Nl = ∅ (l = j). N j = ∅ if n j = 0. Nα = Nαm ∪ Nαo (α = 1, 2) and o Nαm ∩ Nαo = ∅ with n m α + n α = n α where superscripts “m” and “o” represent monotonic and oscillatory evolutions. The matrix Df(xk∗ , p) possesses n 1 -stable, n 2 -unstable, n 3 -invariant, and n 4 -flip real eigenvectors plus n 5 -stable, n 6 -unstable and n 7 -center pairs of complex eigenvectors. Without repeated complex eigenvalues of |λi | = 1 (i ∈ N3 ∪ N4 ∪ N7 ), an iterative response of xk+1 = f(xk , p) is an o m o ([n m 1 , n 1 ] : [n 2 , n 2 ] : [n 3 : κ3 ] : [n 4 : κ4 ]|n 5 : n 5 : n 7 ) flow in the neighborhood of ∗ the fixed-point xk . With repeated complex eigenvalues of |λi | = 1 (i ∈ N3 ∪ N4 ∪ N7 ),

16

1 Local Stability and Bifurcations

o m o an iterative response of xk+1 = f(xk , p) is an ([n m 1 , n 1 ] : [n 2 , n 2 ] : [n 3 : κ3 ] : [n 4 : κ4 ]|n 5 : n 5 : [n 7 , l; κ7 ]) flow in the neighborhood of the fixed-point xk∗ , where κ p ∈ {∅, m p } ( p = 3, 4), (m i )th with (m i )th (s = 1, 2, . . . , l). The meanings of notations in the aforementioned structures are defined as follows:  m 0 o (i) n 1 , n 1 represents n 1 -sinks with n m 1 -monotonic convergence and n 1 oscillatory convergence among n 1 -directions of vi (i ∈ N1 ) if |λi | < 1 (i ∈ N1 and 1 ≤ n 1 ≤ m) with distinct or repeated eigenvalues. o m o (ii) [n m 2 , n 2 ] represents n 2 -sources with n 2 -monotonic divergence and n 2 oscillatory divergence among n 2 -directions of vi (i ∈ N2 ) if |λi | > 1 (i ∈ N2 and 1 ≤ n 2 ≤ m) with distinct or repeated eigenvalues. (iii) n 3 = 1 represents an invariant center on 1-direction of vi (i ∈ N3 ) if λi = 1 (i ∈ N3 and n 3 = 1). (iv) n 4 = 1 represents an flip center on 1-direction of vi (i ∈ N4 ) if λi = −1 (i ∈ N4 and n 4 = 1). (v) n 5 represents n 5 -spiral sinks on n 5 -pairs of (ui , vi ) (i ∈ N5 ) if |λi | < 1 and Im λi = 0 (i ∈ N5 and 1 ≤ n 5 ≤ (n − m)/2) with distinct or repeated eigenvalues. (vi) n 6 represents n 6 -spiral sources on n 6 -pairs of (ui , vi ) (i ∈ N6 ) if |λi | > 1 and Imλi = 0 (i ∈ N6 and 1 ≤ n 6 ≤ (n − m)/2) with distinct or repeated eigenvalues. (vii) n 7 represents that n 7 -invariant centers on n 7 -pairs of (ui , vi ) (i ∈ N7 ) if |λi | = 1 and Imλi = 0 (k ∈ N7 and 1 ≤ n 7 ≤ (n − m)/2) with distinct eigenvalues. (viii) ∅ represents none if n j = 0 ( j ∈ {1, 2, . . . , 7}). (ix) [n 3 ; κ3 ] represents (n 3 − κ3 ) invariant centers on (n 3 − κ3 ) directions of vi3 (i 3 ∈ N3 ) and κ3 -sources in κ3 -directions of v j3 ( j3 ∈ N3 and j3 = i 3 ) if λi = 1 (i ∈ N3 and n 3 ≤ m) with the (κ3 + 1)th -order nilpotent matrix N3κ3 +1 = 0 (0 < κ3 ≤ n 3 − 1). (x) [n 3 ; ∅] represents n 3 invariant centers on n 3 -directions of vi (i ∈ N3 ) if λi = 1 (i ∈ N3 and 1 < n 3 ≤ m) with a nilpotent matrix N3 = 0. (xi) [n 4 ; κ4 ] represents (n 4 − κ4 ) flip oscillatory centers on (n 4 − κ4 ) directions of vi4 (i 4 ∈ N4 ) and κ4 -sources in κ4 -directions of v j4 ( j4 ∈ N4 and j4 = i 4 ) if λi = −1 (i ∈ N4 and n 4 ≤ m) with the (κ4 + 1)th -order nilpotent matrix N4κ4 +1 = 0 (0 < κ4 ≤ n 4 − 1). (xii) [n 4 ; ∅] represents n 4 flip oscillatory centers on n 4 -directions of vi ((i ∈ N4 ) if λi = −1 (i ∈ N4 and 1 < n 4 ≤ m) with a nilpotent matrix N4 = 0. l l κ7s ) invariant centers on (n 7 − s=1 (xiii) [n 7 , l; κ7 ] represents (n 7 − s=1  κ7s ) pairs l of (ui7 , vi7 ) (i 7 ∈ N7 ) and s=1 κ7s sources on κ7 pairs of u j7 , v j7 ( j7 ∈ N7 and j7 = i 7 ) if |λi | = 1 and Imλi = 0 (i ∈ N7 and n 7 ≤ (n − m)/2) for l κ7s -pairs of repeated eigenvalues with each (κ7s + 1)th -order nilpotent s=1 matrix N7κ7s +1 = 0 (0 < κ7s ≤ l, s = 1, 2, . . . , l). (xiv) [n 7 , l; ∅] represents n 7 -invariant centers on n 7 -pairs of (ui , vi ) (i ∈ N7 ) if l κ7s -pairs |λi | = 1 and Imλi = 0 (i ∈ N7 and 1 ≤ n 7 ≤ (n − m)/2) for s=1

1.2 Fixed-Points and Stability

17

(0 < κ7s ≤ l, s = 1, 2, . . . , l) of repeated eigenvalues with a nilpotent matrix N7 = 0.

1.3 Stability Switching To extend the idea of Definitions 1.14 and 1.15, a new function will be defined to determine the stability and the stability state switching.

1.3.1 Monotonic and Oscillatory Stability Definition 1.20 Consider a nonlinear discrete system xk+1 = f(xk , p) ∈ R n in Eq. (1.4) with a fixed-point xk∗ . The corresponding solution is given by xk+ j = f(xk+ j−1 , p) with j ∈ Z. Suppose there is a neighborhood of the fixed-point xk∗ (i.e., Uk (xk∗ ) ⊂ ), and f(xk , p) is C r (r ≥ 1)-continuous in Uk (xk∗ ) with Eq. (1.28). The linearized system is yk+ j+1 = Df(xk∗ , p)yk+ j (yk+ j = xk+ j − xk∗ ) in Uk (xk∗ ) and there are linearly independent vectors vi (i = 1, 2, . . . , n). For a perturbation of i vi , fixed-point yk = xk − xk∗ , let yk = cki vi and yk+1 = ck+1 cki = (vi )T · yk = (vi )T · (xk − xk∗ ).

(1.67)

Define the following functions i = G i (xk , p) = (vi )T · yk+1 = (vi )T · (xk+1 − xk∗ ) ck+1

= (vi )T · [f(xk , p) − xk∗ ] ∞  1 (ri ) jr j j G i( j1 j2 ··· jr ) (xk∗ , p)ck1 ck2 · · · ck i = i r ! r =1 i

(1.68)

i

and (1) ∗ i T ∗ 1 2 n G i( j1 ) (xk , p) = (v ) · Dc j1 f(xk (ck , ck , . . . , ck ), p) k

= (vi )T · Dc( j1 ) f(xk∗ (¯ck ), p) k

= (vi )T · Dxk f(xk∗ , p)∂c( j1 ) xk k

= (vi )T · Dxk f(xk∗ , p)v j1

(1.69)

18

1 Local Stability and Bifurcations (m) (m) ∗ i T G i( j1 j2 ... jm ) (xk , p) = (v ) · D j1 j2

j

f(xk∗ (ck1 , ck2 , . . . , ckn ), p)

= (vi )T · D (m) j1 j2

j

f(xk∗ (¯ck ), p)

ck ck ...ckm ck ck ...ckm

= (v ) · i T

Dx(m) f(xk∗ , p)(v j1 v j2 k

(1.70)

. . . v jm )

(m = 1, 2, . . . ; j1 , j2 , . . . , jm = 1, 2, . . . , n), with G i(0) (xk∗ , p) = G i (xk∗ , p) if m = 0, and Dx(m) f(xk∗ (c1 , . . . , cn ), p) is the (m)th k ∗ order derivative with respect to xk at xk . Definition 1.21 Consider a nonlinear discrete system xk+1 = f(xk , p) ∈ R n in Eq. (1.4) with a fixed-point xk∗ . The corresponding solution is given by xk+ j = f(xk+ j−1 , p) with j ∈ Z. Suppose there is a neighborhood of the fixed-point xk∗ (i.e., Uk (xk∗ ) ⊂ ), and f(xk , p) is C r (r ≥ 1)-continuous in Uk (xk∗ ) with Eq. (1.28). The linearized system is yk+ j+1 = Df(xk∗ , p)yk+ j (yk+ j = xk+ j − xk∗ ) in Uk (xk∗ ) and there are linearly independent vectors vi (i = 1, 2, . . . , n). For a perturbation of (i) (i) = ck+1 vi . fixed-point yk = xk − xk∗ , let yk(i) = ck(i) vi and yk+1 (i) xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is monotonically stable if 0 < (vi )T · (xk+1 − xk∗ ) < (vi )T · (xk − xk∗ ) for (vi )T · (xk − xk∗ ) > 0, 0 > (vi )T · (xk+1 − xk∗ ) > (vi )T · (xk − xk∗ ) for (vi )T · (xk − xk∗ ) < 0,

(1.71)

for xk ∈ U (xk∗ ) ⊂ α . The fixed-point xk∗ is called a monotonic sink (or a monotonically stable node) on the direction vi . (ii) xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is monotonically unstable if (vi )T · (xk+1 − xk∗ ) > (vi )T · (xk − xk∗ ) > 0 for (vi )T · (xk − xk∗ ) > 0, (vi )T · (xk+1 − xk∗ ) < (vi )T · (xk − xk∗ ) < 0 for (vi )T · (xk − xk∗ ) < 0,

(1.72)

for xk ∈ U (xk∗ ) ⊂ α . The fixed-point xk∗ is called a monotonic source (or a monotonically unstable node) on the direction vi . (iii) xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is invariant if

1.3 Stability Switching

19

(vi )T · (xk+1 − xk∗ ) = (vi )T · (xk − xk∗ ) for (vi )T · (xk − xk∗ ) > 0, (vi )T · (xk+1 − xk∗ ) = (vi )T · (xk − xk∗ ) for (vi )T · (xk − xk∗ ) < 0,

(1.73)

for xk ∈ U (xk∗ ) ⊂ α . The fixed-point xk∗ is called an invariant degenerate case on the direction vi . (iv) xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is zerostable if (vi )T · (xk+1 − xk∗ ) = 0 for (vi )T · (xk − xk∗ ) > 0, (vi )T · (xk+1 − xk∗ ) = 0 for (vi )T · (xk − xk∗ ) < 0,

(1.74)

for xk ∈ U (xk∗ ) ⊂ α . The fixed-point xk∗ is called a zero-sink on the direction vi . (v) xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is oscillatorilly stable if 0 > (vi )T · (xk+1 − xk∗ ) > −(vi )T · (xk − xk∗ ) for (vi )T · (xk − xk∗ ) > 0, 0 < (vi )T · (xk+1 − xk∗ ) < −(vi )T · (xk − xk∗ ) for (vi )T · (xk − xk∗ ) < 0,

(1.75)

for xk ∈ U (xk∗ ) ⊂ α . The fixed-point xk∗ is called an oscillatory sink (or an oscillatorilly stable node) on the direction vi . (vi) xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is oscillatorilly unstable if (vi )T · (xk+1 − xk∗ ) < −(vi )T · (xk − xk∗ ) < 0 for (vi )T · (xk − xk∗ ) > 0, (vi )T · (xk+1 − xk∗ ) > −(vi )T · (xk − xk∗ ) > 0 for (vi )T · (xk − xk∗ ) < 0,

(1.76)

for xk ∈ U (xk∗ ) ⊂ α . The fixed-point xk∗ is called an oscillatory source (or an oscillatorilly unstable node) on the direction vi . (i) ∗ (vii) xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk on the direction vi is symmetrically flipped if

(vi )T · (xk+1 − xk∗ ) = −(vi )T · (xk − xk∗ )

(1.77)

20

1 Local Stability and Bifurcations

for xk ∈ U (xk∗ ) ⊂ α . The fixed-point x∗ is called a flipped degenerate case on the direction vi . (viii) xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is monotonically unstable of the second-order lower-saddle if 0 < (vi )T · (xk+1 − xk∗ ) < (vi )T · (xk − xk∗ ) for (vi )T · (xk − xk∗ ) > 0, (vi )T · (xk+1 − xk∗ ) < (vi )T · (xk − xk∗ ) < 0 for (vi )T · (xk − xk∗ ) < 0,

(1.78)

for xk ∈ U (xk∗ ) ⊂ α . The fixed-point xk∗ is called a monotonic lower-saddle of the second kind on the direction vi . (ix) xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is monotonically unstable of the second order upper-saddle if (vi )T · (xk+1 − xk∗ ) > (vi )T · (xk − xk∗ ) > 0 for (vi )T · (xk − xk∗ ) > 0, 0 > (vi )T · (xk+1 − xk∗ ) > (vi )T · (xk − xk∗ ) for (vi )T · (xk − xk∗ ) < 0,

(1.79)

for xk ∈ U (xk∗ ) ⊂ α . The fixed-point xk∗ is called a monotonic upper-saddle of the second on the direction vi . (x) xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is oscillatorilly unstable of the second-order lower-saddle if (vi )T · (xk+1 − xk∗ ) < −(vi )T · (xk − xk∗ ) < 0 for (vi )T · (xk − xk∗ ) > 0, 0 < (vi )T · (xk+1 − xk∗ ) < −(vi )T · (xk − xk∗ ) for (vi )T · (xk − xk∗ ) < 0,

(1.80)

for xk ∈ U (xk∗ ) ⊂ α . The fixed-point xk∗ is called an oscillatory lower-saddle of the second-order on the direction vi . (xi) xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is oscillatorilly unstable of the second-order upper-saddle if 0 > (vi )T · (xk+1 − xk∗ ) > −(vi )T · (xk − xk∗ ) for (vi )T · (xk − xk∗ ) > 0, (vi )T · (xk+1 − xk∗ ) > −(vi )T · (xk − xk∗ ) > 0 for (vi )T · (xk − xk∗ ) < 0,

(1.81)

for xk ∈ U (xk∗ ) ⊂ α . The fixed-point xk∗ is called an oscillatory upper-saddle of second order on the direction vi .

1.3 Stability Switching

21

The stability of fixed-points for a specific eigenvector based on the linearization is presented in Fig. 1.5. The solid curve is (vi )T · xk+1 = (vi )T · f(xk , p). The circular symbol is fixed-point. The shaded regions are stable. The horizontal solid line is for a degenerate case. The vertical solid line is for a line with infinite slope. The monotonically stable node (a monotonic sink) is presented in Fig. 1.5i. The i = cki ) and dashed and dotted lines are for (vi )T · yk+1 = (vi )T · yk+1 (or ck+1 i i i T i T (v ) · yk+1 = −(v ) · yk (or ck+1 = −ck ), respectively. The iterative responses approach the fixed-point. However, the monotonically unstable point (a monotonic source) is presented in Fig. 1.5ii. The iterative responses go away from the fixedpoint. The special cases are the invariant node and zero-sink, as shown in Fig. 1.5iii and iv, respectively. For an invariant node, any selected points are invariant. The zero-invariant sink will bring any points to the fixed point. Similarly, the oscillatory stable node (or an oscillatory sink) is presented in Fig. 1.5v. The oscillatory unstable node (or an oscillatory source) is presented in Fig. 1.5vi. In Fig. 1.5vii and viii, the symmetrically flipped nodes for the fixed-points are invariant. Thus, one called such a case is period-doubling. The symmetrically flipped nodes to the fixed-point via the i i = −cki and ck+1 = cki are illustrated. lines of ck+1 The lower- and upper-saddle of fixed-points for a specific eigenvector is presented in Fig. 1.6. The semi-sink and semi-source can form a saddle. Shaded areas are i ∗ ). ck+1 = (vi )T · yk+1 and stable zones. (yk = xk − xk∗ and yk+1 = xk+1 − xk+1 i i T ck = (v ) · yk . For a monotonic lower-saddle, the upper part is a monotonic sink in the stable zone (shaded) and the lower part is a monotonic source in the unstable zone. For a monotonic upper-saddle, the lower part is a monotonic sink in the stable zone (shaded) and the upper part is a monotonic source in the unstable zone. The i = cki are presented monotonic lower- and upper-saddle based on the line of ck+1 in Fig. 1.6i and ii. The oscillatory lower- and upper-saddle based on the line of i = −cki are presented in Fig. 1.6iii and iv. For an oscillatory lower-saddle, the ck+1 i = −cki is a sink in the stable zone (shaded) and upper part based on the line of ck+1 the lower part is a source in the unstable zone. Thus, such a lower-saddle is called an oscillatory lower-saddle. Similarly, for an oscillatory upper-saddle, the lower part i = −cki is a sink in the stable zone (shaded) and the upper based on the line of ck+1 part is a source in the unstable zone. Theorem 1.5 Consider a nonlinear discrete system xk+1 = f(xk , p) ∈ R n in Eq. (1.4) with a fixed-point xk∗ . The corresponding solution is given by xk+ j = f(xk+ j−1 , p) with j ∈ Z. Suppose there is a neighborhood of the fixed-point xk∗ (i.e., Uk (xk∗ ) ⊂ ), and f(xk , p) is C r (r ≥ 1)-continuous in Uk (xk∗ ) with Eq. (1.28). The linearized system is yk+ j+1 = Df(xk∗ , p)yk+ j (yk+ j = xk+ j − xk∗ ) in Uk (xk∗ ) and there are linearly independent vectors vi (i = 1, 2, . . . , n). For a perturbation of (i) (i) = ck+1 vi . fixed-point yk = xk − xk∗ , let yk(i) = ck(i) vi and yk+1 (i) xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is monotonically stable if and only if (1) (xk∗ , p) = λi ∈ (0, 1) G i(i)

(1.82)

22

1 Local Stability and Bifurcations

( v i )T ⋅ x k +1

( v i )T ⋅ x k +1

cki +1 = cki

( v i )T ⋅ x k

( v i )T ⋅ x k ( v i )T ⋅ x∗k i T

(v ) ⋅ x

∗ k

cki +1 = cki

(i) ( v i )T ⋅ x k +1

(ii) ( v i )T ⋅ x k +1

cki +1 = cki

( v i )T ⋅ x k

cki +1 = cki

( v i )T ⋅ x k

( v i )T ⋅ x∗k

( v i )T ⋅ x∗k

(iii)

(iv) i k +1

c

i k

= −c

cki +1 = cki

cki +1 = cki

( v i )T ⋅ x∗k

( v i )T ⋅ x∗k

( v i )T ⋅ x k +1

( v i )T ⋅ x k +1 cki +1 = −cki i T

( v ) ⋅ xk

(v)

( v i )T ⋅ x k

(vi)

Fig. 1.5 Stability of fixed-points after linearization: (i) a monotonically stable node (a monotonic sink), (ii) an monotonically unstable node (a monotonic source), (iii) an invariant node, (iv) a zero-sink, (v) an oscillatorilly stable node (an oscillatory sink) and (vi) an oscillatorilly unstable i node (an oscillatory source) (vii) a symmetrically flipped via one time of ck+1 = −cki and (viii) i i symmetrically flipped via two times of ck+1 = ck . Shaded areas are stable zones. (yk = xk − xk∗ i ∗ ). ci i T i T and yk+1 = xk+1 − xk+1 k+1 = (v ) · yk+1 and ck = (v ) · yk

1.3 Stability Switching

23 cki +1 = cki

( v i )T ⋅ x∗k

cki +1 = cki

( v i )T ⋅ x∗k

( v i )T ⋅ x k +1

( v i )T ⋅ x k +1 cki +1 = −cki i T

cki +1 = −cki i T

( v ) ⋅ xk

( v ) ⋅ xk

(vii)

(viii)

Fig. 1.5 (continued)

for xk ∈ U (xk∗ ) ⊂ α . (ii) xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is monotonically unstable if and only if (1) (xk∗ , p) = λi ∈ (1, ∞) G i(i)

(1.83)

for xk ∈ U (xk∗ ) ⊂ α . (iii) xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is invariant if and only if (1) (m i ) ∗ (xk∗ , p) = λi = 1 and G i( G i(i) j1 j2 ... jm ) (xk , p) = 0 i

for m i = 2, 3, . . . ; j1 , j2 , . . . , jm i = 1, 2, . . . , n

(1.84)

for xk ∈ U (xk∗ ) ⊂ α . (iv) xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is zerostable if and only if (1) (m i ) ∗ G i(i) (xk∗ , p) = λi = 0 and G i( j1 j2 ··· jm ) (xk , p) = 0 i

for m i = 2, 3, . . . ; j1 , j2 , . . . , jm i = 1, 2, . . . , n

(1.85)

for xk ∈ U (xk∗ ) ⊂ α . (v) xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is oscillatorilly stable if and only if (1) (xk∗ , p) = λi ∈ (−1, 0) G i(i)

(1.86)

24

1 Local Stability and Bifurcations

( v i )T ⋅ x k +1

( v i )T ⋅ x k +1

cki +1 = cki

( v i )T ⋅ x k

v iT ⋅ x k

( v i )T ⋅ x∗k

( v i )T ⋅ x∗k

cki +1 = cki

(i) cki +1 = −cki

(ii) cki +1 = cki

cki +1 = −cki

( v i )T ⋅ x∗k

cki +1 = cki

( v i )T ⋅ x∗k

i T

( v ) ⋅ x k +1

( v i )T ⋅ x k +1 ( v i )T ⋅ x k

( v i )T ⋅ x k

(iii)

(iv)

Fig. 1.6 The lower- and upper-saddles fixed-points: (i) a monotonic lower-saddle of the secondorder and (ii) a monotonic upper-saddle of the second-order, (iii) an oscillatory lower-saddle of the second-order and (iv) an oscillatory upper-saddle of the second-order. Shaded areas are stable i ∗ ). ci i T i T zones. (yk = xk − xk∗ and yk+1 = xk+1 − xk+1 k+1 = (v ) · yk+1 and ck = (v ) · yk

for xk ∈ U (xk∗ ) ⊂ α . (vi)

xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is oscillatorilly unstable if and only if (1) (xk∗ , p) = λi ∈ (−∞, −1) G i(i)

(1.87)

for xk ∈ U (xk∗ ) ⊂ α . (i) ∗ (vii) xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk on the direction vi is symmetrically flipped if and only if (1) (m i ) ∗ (xk∗ , p) = λi = −1 and G i( G i(i) j1 j2 ··· jm ) (xk , p) = 0 i

for m i = 2, 3, . . . ; j1 , j2 , . . . , jm i = 1, 2, . . . , n.

(1.88)

1.3 Stability Switching

25

for xk ∈ U (xk∗ ) ⊂ α . (viii) xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is monotonically unstable of the second-order lower-saddle if and only if (1) (xk∗ , p) = λi = 1 G i(i) j1 j2 (2) ∗ G i( j1 j2 ) (xk , p)z k z k < 0

(ix)

for xk ∈ U (xk∗ ) ⊂ α . xk+ j ( j ∈ Z) at fixed-point xk∗ on the direction vi is monotonically unstable of the second-order upper-saddle if and only if (1) (xk∗ , p) = λi = 1, G i(i) j j (2) G i( j1 j2 ) (xk∗ , p)z k1 z k2 > 0

(x)

(1.90)

xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is oscillatorilly unstable of the second-order lower-saddle if and only if (1) G i(i) (xk∗ , p) = λi = −1, j j (2) G i( j1 j2 ) (xk∗ , p)z k1 z k2 < 0

(xi)

(1.89)

(1.91)

for xk ∈ U (xk∗ ) ⊂ α . xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is oscillatorilly unstable of the second-order upper-saddle if and only if (1) (xk∗ , p) = λi = −1, G i(i) j1 j2 (2) ∗ G i( j1 j2 ) (xk , p)z k z k > 0

for xk ∈ U (xk∗ ) ⊂ α . Proof Because i = (vi )T · (xk+1 − xk∗ ) ck+1 (1) ∗ 1 = (vi )T · xk∗ + G i( ck ||) − (vi )T · xk∗ j1 ) (xk , p)ck + o(||¯ j

(1) ∗ 1 = G i( ck ||) j1 ) (xk , p)ck + o(||¯ j

due to any selection of cki as an infinitesimal, we have (1) i ∗ 1 ck+1 = G i( j1 ) (xk , p)ck , j

(1) ∗ i T ∗ G i( j1 ) (xk , p) = (v ) · Dx f(xk , p)v j1

= (vi )T · λ j1 v j1 = λ j1 δ ij1 .

(1.92)

26

1 Local Stability and Bifurcations

(i) From definition in Eq. (1.71), we have 0 < (vi )T · (xk+1 − xk∗ ) < (vi )T · (xk − xk∗ ) for (vi )T · (xk − xk∗ ) > 0, 0 > (vi )T · (xk+1 − xk∗ ) > (vi )T · (xk − xk∗ ) for (vi )T · (xk − xk∗ ) < 0 which gives (1) i 0 < ck+1 = G i(i) (xk∗ , p)cki = λi cki < cki for cki > 0, (1) i 0 > ck+1 = G i(i) (xk∗ , p)cki = λi cki > cki for cki < 0.

Thus, if (1) (xk∗ , p) = λi ∈ (0, 1), G i(i)

xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is monotonically stable, vice versa. (ii) From definition in Eq. (1.72), we have (vi )T · (xk+1 − xk∗ ) > (vi )T · (xk − xk∗ ) for (vi )T · (xk − xk∗ ) > 0, (vi )T · (xk+1 − xk∗ ) < (vi )T · (xk − xk∗ ) for (vi )T · (xk − xk∗ ) < 0 which gives (1) i = G i(i) (xk∗ , p)cki = λi cki > cki for cki > 0, ck+1 (1) i = G i(i) (xk∗ , p)cki = λi cki < cki for cki < 0. ck+1

Thus, if (1) (xk∗ , p) = λi ∈ (1, ∞), G i(i)

xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is monotonically unstable, vice versa. (iii) Because (i) = (vi )T · (xk+1 − xk∗ ) ck+1 (1) ∗ 1 = (vi )T · xk∗ + G i( j1 ) (xk , p)ck j

+

∞  1 (m i ) jm j j G i( j1 j2 ··· jm ) (xk∗ , p)(ck1 ck2 · · · ck i ) − (vi )T · xk∗ i m ! i m =2 i

(1) ∗ 1 = G i( j1 ) (xk , p)ck + j

∞  1 (m i ) jm j j G i( j1 j2 ··· jm ) (xk∗ , p)(ck1 ck2 · · · ck i ) i mi ! m =2 i

1.3 Stability Switching

27

from definition in Eq. (1.73) i = cki . (vi )T · (xk+1 − xk∗ ) = (vi )T · (xk − xk∗ ) ⇒ ck+1 i as infinitesimals, we have Due to any selection of cki and ck+1 (1) (m i ) ∗ (xk∗ , p) = λi = 1 and G i( G i(i) j1 j2 ··· jm ) (xk , p) = 0 i

for m i = 2, 3, . . . ; j1 , j2 , . . . , jm i = 1, 2, . . . , n. Therefore, xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is invariant, vice versa. (iv) From definition in Eq. (1.74) i = 0. (vi )T · (xk+1 − xk∗ ) = 0 ⇒ ck+1 i as an infinitesimal, we have Due to any selection of cki and ck+1 (1) (m i ) ∗ (xk∗ , p) = λi = 0 and G i( G i(i) j1 j2 ··· jm ) (xk , p) = 0 i

for m i = 2, 3, . . . ; j1 , j2 , . . . , jm i = 1, 2, . . . , n. Therefore, xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is zero-invariant, vice versa. (v) From definition in Eq. (1.75), we have 0 > (vi )T · (xk+1 − xk∗ ) > −(vi )T · (xk − xk∗ ) for (vi )T · (xk − xk∗ ) > 0, 0 < (vi )T · (xk+1 − xk∗ ) < −(vi )T · (xk − xk∗ ) for (vi )T · (xk − xk∗ ) < 0 which gives (1) i = G i(i) (xk∗ , p)cki = λi cki > −cki for cki > 0, 0 > ck+1 (1) i = G i(i) (xk∗ , p)cki = λi cki < −cki for cki < 0. 0 < ck+1

Thus, if (1) (xk∗ , p) = λi ∈ (−1, 0), G i(i)

xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is oscillatorilly stable, vice versa. (vi) From definition in Eq. (1.76), we have (vi )T · (xk+1 − xk∗ ) < −(vi )T · (xk − xk∗ ) < 0 for (vi )T · (xk − xk∗ ) > 0, (vi )T · (xk+1 − xk∗ ) > −(vi )T · (xk − xk∗ ) > 0 for (vi )T · (xk − xk∗ ) < 0

28

1 Local Stability and Bifurcations

which gives (1) i = G i(i) (xk∗ , p)cki = λi cki < −cki < 0 for cki > 0, ck+1 (1) i ck+1 = G i(i) (xk∗ , p)cki = λi cki > −cki > 0 for cki < 0.

Thus, if (1) (xk∗ , p) = λi ∈ (−∞, −1), G i(i)

xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is oscillatorilly unstable, vice versa. (vii) From definition in Eq. (1.77) i = −cki . (vi )T · (xk+1 − xk∗ ) = −(vi )T · (xk − xk∗ ) ⇒ ck+1 i as an infinitesimal, we have Due to any selection of cki and ck+1 (1) (m i ) ∗ (xk∗ , p) = λi = −1 and G i( G i(i) j1 j2 ··· jm ) (xk , p) = 0 i

for m i = 2, 3, . . . ; j1 , j2 , . . . , jm i = 1, 2, . . . , n. Therefore, xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is flipped, vice versa. (viii) From definition in Eq. (1.78), we have 0 < (vi )T · (xk+1 − xk∗ ) < (vi )T · (xk − xk∗ ) for (vi )T · (xk − xk∗ ) > 0, (vi )T · (xk+1 − xk∗ ) < (vi )T · (xk − xk∗ ) < 0 for (vi )T · (xk − xk∗ ) < 0 which, with the first-order approximation, gives (1) i 0 < ck+1 = G i(i) (xk∗ , p)cki = λi cki < cki for cki > 0, (1) i = G i(i) (xk∗ , p)cki = λi cki < cki < 0 for cki < 0. ck+1 (1) (xk∗ , p) = λi ∈ (0, 1) ∩ (1, ∞), which implies Df(xk , p) does not Thus, G i(i) exist at point xk∗ . However, because the discrete vector function of f(xk , p) is C r (r ≥ 1)-continuous at point xk∗ , Df(xk , p) exists and is continuous at point xk∗ . Therefore, (1) (xk∗ , p) = λi = 1. G i(i)

and the first-order approximation of Eq. (1.78) is not adequate. The higheri can be approximated order approximation should be considered. Further, ck+1 by

1.3 Stability Switching

29

(2) 1 2 i i i ∗ 0 < ck+1 = cki + 2!1 G i( j1 j2 ) (xk , p)ck ck < ck for ck > 0, j j (2) i = cki + 2!1 G i( j1 j2 ) (xk∗ , p)ck1 ck2 < cki < 0 for cki < 0. ck+1 j

j

Therefore, (2) ∗ 1 2 G i( j1 j2 ) (xk , p)ck ck < 0. j

j

In summary, if (1) (xk∗ , p) = λi = 1, G i(i) j j (2) G i( j1 j2 ) (xk∗ , p)ck1 ck2 < 0,

xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is monotonically unstable of the second-order lower-saddle, vice versa. (ix) From definition in Eq. (1.79), we have (vi )T · (xk+1 − xk∗ ) > (vi )T · (xk − xk∗ ) > 0 for (vi )T · (xk − xk∗ ) > 0, 0 > (vi )T · (xk+1 − xk∗ ) > (vi )T · (xk − xk∗ ) for (vi )T · (xk − xk∗ ) < 0 which, with the first-order approximation, gives (1) i = G i(i) (xk∗ , p)cki = λi cki > cki > 0 for cki > 0, ck+1 (1) i 0 > ck+1 = G i(i) (xk∗ , p)cki = λi cki > cki for cki < 0 (1) Thus, G i(i) (xk∗ , p) = λi ∈ (0, 1) ∩ (1, ∞), which implies Df(xk , p) does not exist at point xk∗ . However, because the discrete vector function of f(xk , p) is C r (r ≥ 1)-continuous at point xk∗ , Df(xk , p) exists and is continuous at point xk∗ . Therefore, (1) G i(i) (xk∗ , p) = λi = 1.

and the first-order approximation of Eq. (1.79) is not adequate. The higher-order approximation of the discrete vector function should be considered. Further, i can be approximated by ck+1 1 (2) j j G (x∗ , p)ck1 ck2 > cki for cki > 0, 2! i( j1 j2 ) k 1 (2) j1 j2 ∗ i i = cki + G i( j1 j2 ) (xk , p)ck ck > ck for ck < 0. 2!

i ck+1 = cki + i ck+1

Therefore, (2) ∗ 1 2 G i( j1 j2 ) (xk , p)ck ck > 0. j

j

30

1 Local Stability and Bifurcations

In summary, if (1) G i(i) (xk∗ , p) = λi = 1, j1 j2 (2) ∗ G i( j1 j2 ) (xk , p)ck ck > 0,

xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is monotonically unstable of the second-order upper-saddle, vice versa. (x) From definition in Eq. (1.80), we have (vi )T · (xk+1 − xk∗ ) < −(vi )T · (xk − xk∗ ) < 0 for (vi )T · (xk − xk∗ ) > 0, 0 < (vi )T · (xk+1 − xk∗ ) < −(vi )T · (xk − xk∗ ) for (vi )T · (xk − xk∗ ) < 0. which, with the first-order approximation, gives (1) i = G i(i) (xk∗ , p)cki = λi cki < −cki < 0 for cki > 0, ck+1 (1) i = G i(i) (xk∗ , p)cki = λi cki < −cki for cki < 0. 0 < ck+1 (1) Thus, G i(i) (xk∗ , p) = λi ∈ (−∞, −1) ∩ (−1, 0), which implies Df(xk , p) does not exist at point xk∗ . However, because the discrete vector function of f(xk , p) is C r (r ≥ 1)-continuous at point xk∗ , Df(xk , p) exists and is continuous at point xk∗ . Therefore, (1) (xk∗ , p) = λi = −1. G i(i)

and the first-order approximation of Eq. (1.80) is not adequate. The higher-order i approximation of the discrete vector function should be considered. Further, ck+1 can be approximated by (2) 1 2 i i i ∗ ck+1 = −cki + 2!1 G i( j1 j2 ) (xk , p)ck ck < −ck < 0 for ck > 0, j j (2) 1 2 1 i = −cki + 2! G i( j1 j2 ) (xk∗ , p)ck ck < −cki for cki < 0. 0 < ck+1 j

j

Therefore, (2) ∗ 1 2 G i( j1 j2 ) (xk , p)ck ck < 0. j

j

In summary, if (1) (xk∗ , p) = λi = −1, G i(i) j j (2) G i( j1 j2 ) (xk∗ , p)ck1 ck2 < 0,

xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is oscillatorilly unstable of the second-order lower-saddle, vice versa.

1.3 Stability Switching

31

(xi) From definition in Eq. (1.81), we have 0 > (vi )T · (xk+1 − xk∗ ) > −(vi )T · (xk − xk∗ ) for (vi )T · (xk − xk∗ ) > 0, (vi )T · (xk+1 − xk∗ ) > −(vi )T · (xk − xk∗ ) > 0 for (vi )T · (xk − xk∗ ) < 0 which, with the first-order approximation, gives (1) i 0 > ck+1 = G i(i) (xk∗ , p)cki = λi cki > −cki for cki > 0, (1) i ∗ ck+1 = G i(i) (xk , p)cki = λi cki > −cki > 0 for cki < 0. (1) (xk∗ , p) = λi ∈ (−∞, −1) ∩ (−1, 0), which implies Df(xk , p) does Thus, G i(i) not exist at point xk∗ . However, because the discrete vector function of f(xk , p) is C r (r ≥ 1)-continuous at point xk∗ , Df(xk , p) exists and is continuous at point xk∗ . Therefore, (1) (xk∗ , p) = λi = −1. G i(i)

and the first-order approximation of Eq. (1.81) is not adequate. The higher-order approximation of the discrete vector function should be considered. Further, i can be approximated by ck+1 1 (2) j j G (x∗ , p)ck1 ck2 > −cki for cki > 0, 2! i( j1 j2 ) k 1 (2) j1 j2 ∗ i i = −cki + G i( j1 j2 ) (xk , p)ck ck > −ck for ck < 0. 2!

i ck+1 = −cki + i ck+1

Therefore, (2) ∗ 1 2 G i( j1 j2 ) (xk , p)ck ck > 0. j

j

In summary, if (1) (xk∗ , p) = λi = −1, G i(i) j j (2) G i( j1 j2 ) (xk∗ , p)ck1 ck2 > 0

xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is oscillatorilly unstable of the second-order upper-saddle, vice versa. The theorem is proved.



Definition 1.26 Consider a nonlinear discrete system xk+1 = f(xk , p) in Eq. (1.4) with a fixed-point xk∗ . The corresponding solution is given by xk+ j = f(xk+ j−1 , p) with j ∈ Z. Suppose there is a neighborhood of the fixed-point xk∗ (i.e., Uk (xk∗ ) ⊂ ),

32

1 Local Stability and Bifurcations

and f(xk , p) is C r (r ≥ 1)-continuous in Uk (xk∗ ) with Eq. (1.28). The linearized system is yk+ j+1 = Df(xk∗ , p)yk+ j (yk+ j = xk+ j − xk∗ ) in Uk (xk∗ ) and there are linearly independent vectors vi (i = 1, 2, . . . , n). For a perturbation of fixed-point i i yk = xk − xk∗ , let yk = cki vi and yk+1 = ck+1 vi . (i) xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is monotonically stable of the (m i )th-order sink if (1) (ri ) ∗ G i(i) (xk∗ , p) = λi = 1 and G i( j1 j2 ··· jr ) (xk , p) = 0 i

for j1 , j2 , . . . , jri = 1, 2, . . . , n; ri = 2, 3, . . . , m i − 1; (m i ) ∗ G i( j1 j2 ··· jm ) (xk , p)  = 0,

i  0 < (vi )T · (xk+1 − xk∗ ) < (vi )T · (xk − xk∗ ) for (vi )T · (xk − xk∗ ) > 0, 0 > (vi )T · (xk+1 − xk∗ ) > (vi )T · (xk − xk∗ ) for (vi )T · (xk − xk∗ ) < 0 (1.93)

for xk ∈ U (xk∗ ) ⊂ α . The fixed-point xk∗ is called a monotonic sink (or a monotonic stable node) of the (m i )th-order on the direction vi . (ii) xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is monotonically unstable of the (m i )th-order source if (1) (ri ) ∗ (xk∗ , p) = λi = 1 and G i( G i(i) j1 j2 ··· jr ) (xk , p) = 0 i

for j1 , j2 , . . . , jri = 1, 2, . . . , n; ri = 2, 3, . . . , m i − 1; (m i ) ∗ G i( j1 j2 ··· jm ) (xk , p)  = 0,

i  (vi )T · (xk+1 − xk∗ ) > (vi )T · (xk − xk∗ ) > 0 for (vi )T · (xk − xk∗ ) > 0, (vi )T · (xk+1 − xk∗ ) < (vi )T · (xk − xk∗ ) < 0 for (vi )T · (xk − xk∗ ) < 0 (1.94)

for xk ∈ U (xk∗ ) ⊂ α . The fixed-point xk∗ is called a monotonic source (or a monotonically unstable node) of the (m i )th-order on the direction vi . (iii) xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is monotonically unstable of the (m i )th-order, lower-saddle if (1) (ri ) ∗ G i(i) (xk∗ , p) = λi = 1 and G i( j1 j2 ··· jr ) (xk , p) = 0 i

for j1 , j2 , . . . , jri = 1, 2, . . . , n; ri = 2, 3, . . . , m i − 1; (m i ) ∗ G i( j1 j2 ··· jm ) (xk , p)  = 0,

i  0 < (vi )T · (xk+1 − xk∗ ) < (vi )T · (xk − xk∗ ) for (vi )T · (xk − xk∗ ) > 0, (vi )T · (xk+1 − xk∗ ) < (vi )T · (xk − xk∗ ) < 0 for (vi )T · (xk − xk∗ ) < 0 (1.95)

for xk ∈ U (xk∗ ) ⊂ α . The fixed-point xk∗ is called a monotonic lower-saddle of the (m i )th-order on the direction vi .

1.3 Stability Switching

33

(iv) xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is monotonically unstable of the (m i )th-order, upper-saddle if (1) (ri ) ∗ (xk∗ , p) = λi = 1 and G i( G i(i) j1 j2 ··· jr ) (xk , p) = 0 i

for j1 , j2 , . . . , jri = 1, 2, . . . , n; ri = 2, 3, . . . , m i − 1; (m i ) ∗ G i( j1 j2 ··· jm ) (xk , p)  = 0,

i  (vi )T · (xk+1 − xk∗ ) > (vi )T · (xk − xk∗ ) > 0 for (vi )T · (xk − xk∗ ) > 0, 0 > (vi )T · (xk+1 − xk∗ ) > (vi )T · (xk − xk∗ ) for (vi )T · (xk − xk∗ ) < 0 (1.96)

for xk ∈ U (xk∗ ) ⊂ α . The fixed-point xk∗ is called the monotonic upper-saddle of the (m i )th-order on the direction vi . (v) xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is oscillatorilly stable of the (m i )th-order if (1) (ri ) ∗ G i(i) (xk∗ , p) = λi = −1 and G i( j1 j2 ··· jr ) (xk , p) = 0 i

for j1 , j2 , . . . , jri = 1, 2, . . . , n; ri = 2, 3, . . . , m i − 1; (m i ) ∗ G i( j1 j2 ··· jm ) (xk , p)  = 0,

i  0 > (vi )T · (xk+1 − xk∗ ) > −(vi )T · (xk − xk∗ ) for (vi )T · (xk − xk∗ ) > 0, 0 < (vi )T · (xk+1 − xk∗ ) < −(vi )T · (xk − xk∗ ) for (vi )T · (xk − xk∗ ) < 0 (1.97)

for xk ∈ U (xk∗ ) ⊂ Ωα . The fixed-point xk∗ is called an oscillatory sink (or an oscillatorilly stable node) of the (m i )th-order on the direction vi . (vi) xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is oscillatorilly unstable of the (m i )th-order if (1) (ri ) ∗ G i(i) (xk∗ , p) = λi = −1 and G i( j1 j2 ··· jr ) (xk , p) = 0 i

for j1 , j2 , . . . , jri = 1, 2, . . . , n; ri = 2, 3, . . . , m i − 1; (m i ) ∗ G i( j1 j2 ··· jm ) (xk , p)  = 0,

i  (vi )T · (xk+1 − xk∗ ) < −(vi )T · (xk − xk∗ ) < 0 for (vi )T · (xk − xk∗ ) > 0, (vi )T · (xk+1 − xk∗ ) > −(vi )T · (xk − xk∗ ) > 0 for (vi )T · (xk − xk∗ ) < 0 (1.98)

for xk ∈ U (xk∗ ) ⊂ α . The fixed-point xk∗ is called an oscillatory source (or an oscillatorilly unstable node) of the (m i )th-order on the direction vi . (vii) xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is oscillatorilly unstable of the (2m i )th-order, lower-saddle if

34

1 Local Stability and Bifurcations (1) (ri ) ∗ G i(i) (xk∗ , p) = λi = −1 and G i( j1 j2 ··· jr ) (xk , p) = 0 i

for j1 , j2 , . . . , jri = 1, 2, . . . , n; ri = 2, 3, . . . , m i − 1; (m i ) ∗ G i( j1 j2 ··· jm ) (xk , p)  = 0,

i  (vi )T · (xk+1 − xk∗ ) < −(vi )T · (xk − xk∗ ) < 0 for (vi )T · (xk − xk∗ ) > 0, 0 < (vi )T · (xk+1 − xk∗ ) < −(vi )T · (xk − xk∗ ) for (vi )T · (xk − xk∗ ) < 0 (1.99)

for xk ∈ U (xk∗ ) ⊂ α . The fixed-point xk∗ is called the oscillatory lower-saddle of the (m i )th-order on the direction vi . (viii) xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is oscillatorilly unstable of the (2m i )th-order, upper-saddle if (1) (ri ) ∗ G i(i) (xk∗ , p) = λi = −1 and G i( j1 j2 ··· jr ) (xk , p) = 0 i

for j1 , j2 , . . . , jri = 1, 2, . . . , n; ri = 2, 3, . . . , m i − 1; (m i ) ∗ G i( j1 j2 ... jm ) (xk , p)  = 0,

i  0 > (vi )T · (xk+1 − xk∗ ) > −(vi )T · (xk − xk∗ ) for (vi )T · (xk − xk∗ ) > 0, (vi )T · (xk+1 − xk∗ ) > −(vi )T · (xk − xk∗ ) > 0 for (vi )T · (xk − xk∗ ) < 0 (1.100)

for xk ∈ U (xk∗ ) ⊂ α . The fixed-point xk∗ is called the oscillatory, uppersaddle of the (m i )th-order on the direction vi . Theorem 1.6 Consider a nonlinear discrete system xk+1 = f(xk , p) ∈ R n in Eq. (1.4) with a fixed-point xk∗ . The corresponding solution is given by xk+ j = f(xk+ j−1 , p) with j ∈ Z. Suppose there is a neighborhood of the fixed-point xk∗ (i.e., Uk (xk∗ ) ⊂ ), and f(xk , p) is C r (r ≥ 1)-continuous in Uk (xk∗ ) with Eq. (1.28). The linearized system is yk+ j+1 = Df(xk∗ , p)yk+ j (yk+ j = xk+ j − xk∗ ) in Uk (xk∗ ) and there are linearly independent vectors vi (i = 1, 2, . . . , n). For a perturbation of i i = ck+1 vi . fixed-point yk = xk − xk∗ , let yk = cki vi and yk+1 (i) xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is monotonically stable of the (m i )th-order sink if and only if (1) G i(i) (xk∗ , p) = λi = 1, (ri ) G i( j1 j1 ··· jr ) (xk∗ , p) = 0 for ri = 2, 3, . . . , m i − 1, i

(1.101)

(m i ) 1 2 i ∗ cki × G i( j1 j1 ... jm ) (xk , p)(ck ck · · · ck ) < 0 j

j

jm

i

for xk ∈ U (xk∗ ) ⊂ α . (ii) xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is monotonically unstable of the (m i )th-order source if and only if

1.3 Stability Switching

35

(1) G i(i) (xk∗ , p) = λi = 1, (ri ) G i( j1 j1 ··· jr ) (xk∗ , p) = 0 for ri = 2, 3, . . . , m i − 1, i

cki

×

j1 j2 (m i ) ∗ G i( j1 j1 ··· jm i ) (xk , p)(ck ck

···

jm ck i )

(1.102)

>0

for xk ∈ U (xk∗ ) ⊂ α . (iii) xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is monotonically unstable of the (m i )th-order, lower-saddle if and only if (1) (xk∗ , p) = λi = 1, G i(i) (ri ) G i( j1 j1 ··· jr ) (xk∗ , p) = 0 for ri = 2, 3, . . . , m i − 1, i

j1 j2 (m i ) ∗ G i( j1 j1 ··· jm i ) (xk , p)(ck ck

jm ck i )

···

(1.103)

0

for xk ∈ U (xk∗ ) ⊂ α . (v) xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is oscillatorilly stable of the (m i )th-order sink if and only if (1) G i(i) (xk∗ , p) = λi = −1, (ri ) ∗ G i( j1 j1 ··· jr ) (xk , p) = 0 for ri = 2, 3, . . . , m i − 1, i

cki

×

j1 j2 (m i ) ∗ G i( j1 j1 ··· jm i ) (xk , p)(ck ck

···

jm ck i )

(1.105)

>0

for xk ∈ U (xk∗ ) ⊂ α . (vi) xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is oscillatorilly unstable of the (m i )th-order source if and only if (1) G i(i) (xk∗ , p) = λi = −1, (ri ) G i( j1 j1 ··· jr ) (xk∗ , p) = 0 for ri = 2, 3, . . . , m i − 1, i

cki

×

j1 j2 (m i ) ∗ G i( j1 j1 ··· jm i ) (xk , p)(ck ck

for xk ∈ U (xk∗ ) ⊂ α .

···

jm ck i )

0, 0 > (vi )T · (xk+1 − xk∗ ) > (vi )T · (xk − xk∗ ) for (vi )T · (xk − xk∗ ) < 0



(1) (xk∗ , p) = 1, we have For G i(i) i |ck+1 |=|1 +

(vi )T · (xk − xk∗ ) > 0 for (vi )T · (xk − xk∗ ) > 0, (vi )T · (xk+1 − xk∗ ) < (vi )T · (xk − xk∗ ) < 0 for (vi )T · (xk − xk∗ ) < 0 we have    jm i  j1 j2 ∗ 1 + 1 1 G (m i ) (x , p)(ck ck · · · ck )>1.  cki m i ! i( j1 j1 ··· jmi ) k



38

1 Local Stability and Bifurcations

For the arbitrarily infinitesimal cki , the foregoing equation requires 1 1 (m i ) jm j j G (x∗ , p)(ck1 ck2 · · · ck i ) > 0 cki m i ! i( j1 j1 ··· jmi ) k equivalent to (m i ) ∗ 1 2 i cki × G i( j1 j1 ··· jm ) (xk , p)(ck ck · · · ck ) > 0. j

jm

j

i

Therefore, xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is monotonically unstable of the (m i )th-order source, vice versa. (m i ) ∗ (xk , p) = λi = 1 (iii) From the Tayler series, G i(i) i ck+1 =cki +

1 (m i ) jm j j G (x∗ , p)(ck1 ck2 · · · ck i ). m i ! i( j1 j1 ··· jmi ) k

If 0 < (vi )T · (xk+1 − xk∗ ) < (vi )T · (xk − xk∗ ) for (vi )T · (xk − xk∗ ) > 0, (vi )T · (xk+1 − xk∗ ) < (vi )T · (xk − xk∗ ) < 0 for (vi )T · (xk − xk∗ ) < 0



then we have cki +

1 (m i ) ( jm ) (j ) (j ) G i( j1 j1 ··· jm ) (xk∗ , p)(ck 1 ck 2 · · · ck i ) < cki i mi !

Therefore, (m i ) ∗ 1 2 i G i( j1 j1 ··· jm ) (xk , p)(ck ck · · · ck ) < 0 j

j

jm

i

Thus, xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is monotonically unstable of the (m i )th-order, lower-saddle, vice versa. (iv) From the Tayler series, i =cki + ck+1

1 (m i ) jm j j G i( j1 j1 ··· jm ) (xk∗ , p)(ck1 ck2 · · · ck i ). i mi !

If (vi )T · (xk+1 − xk∗ ) > (vi )T · (xk − xk∗ ) > 0 for (vi )T · (xk − xk∗ ) > 0, 0 > (vi )T · (xk+1 − xk∗ ) > (vi )T · (xk − xk∗ ) for (vi )T · (xk − xk∗ ) < 0 then we have



1.3 Stability Switching

39

cki +

1 (m i ) ( jm ) (j ) (j ) G (x∗ , p)(ck 1 ck 2 · · · ck i ) > cki m i ! i( j1 j1 ··· jmi ) k

Therefore, (m i ) ∗ 1 2 i G i( j1 j1 ··· jm ) (xk , p)(ck ck · · · ck ) > 0. j

jm

j

i

Thus, xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is monotonically unstable of the (m i )th-order, upper-saddle, vice versa. (v) Similar to case (i), consider 0 > (vi )T · (xk+1 − xk∗ ) > −(vi )T · (xk − xk∗ ) for (vi )T · (xk − xk∗ ) > 0, 0 < (vi )T · (xk+1 − xk∗ ) < −(vi )T · (xk − xk∗ ) for (vi )T · (xk − xk∗ ) < 0



(1) (xk∗ , p) = −1, we have For G i(i)

|−1+

1 1 (m i ) jm j j G (x∗ , p)(ck1 ck2 · · · ck i )|< 1. i m ! i( j1 j2 ··· jm i ) k ck i j

jm i

j

For infinitesimals of ck1 , ck2 , . . . , ck equation gives

are arbitrarily selected, the foregoing

(m i ) ∗ 1 2 i cki × G i( j1 j2 ··· jm ) (xk , p)(ck ck · · · ck ) > 0. j

j

jm

i

Therefore, xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is oscillatorilly stable of the (m i )th-order sink, vice versa. (vi) Similar to case (ii), consider (vi )T · (xk+1 − xk∗ ) < −(vi )T · (xk − xk∗ ) < 0 for (vi )T · (xk − xk∗ ) > 0, (vi )T · (xk+1 − xk∗ ) > −(vi )T · (xk − xk∗ ) > 0 for (vi )T · (xk − xk∗ ) < 0



(1) (xk∗ , p) = −1, we have For G i(i)

|−1+

1 1 (m i ) jm j j G (x∗ , p)(ck1 ck2 · · · ck i )|> 1. i m ! i( j1 j2 ··· jm i ) k ck i

Due to an arbitrary infinitesimal cki , the foregoing equation gives (m i ) ∗ 1 2 i cki × G i( j1 j2 ··· jm ) (xk , p)(ck ck · · · ck ) < 0. j

j

jm

i

Therefore, xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is oscillatorilly unstable of the (m i )th-order source, vice versa.

40

1 Local Stability and Bifurcations

(vii) Similar to (iii), consider (vi )T · (xk+1 − xk∗ ) < −(vi )T · (xk − xk∗ ) < 0 for (vi )T · (xk − xk∗ ) > 0, 0 < (vi )T · (xk+1 − xk∗ ) < −(vi )T · (xk − xk∗ ) for (vi )T · (xk − xk∗ ) < 0



(1) For G i(i) (xk∗ , p) = −1, we have

−cki +

1 (m i ) jm j j G (x∗ , p)(ck1 ck2 · · · ck i ) < −cki . m i ! i( j1 j2 ··· jmi ) k j

jm i

j

For infinitesimals of ck1 , ck2 , · · · , ck equation gives

are arbitrarily selected, the foregoing

(m i ) ∗ 1 2 i G i( j1 j2 ··· jm ) (xk , p)(ck ck · · · ck ) < 0. j

j

jm

i

Thus, xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is oscillatorilly unstable of the (m i )th order, lower-saddle, vice versa. (viii) Similar to (vii), consider 0 > (vi )T · (xk+1 − xk∗ ) > −(vi )T · (xk − xk∗ ) for (vi )T · (xk − xk∗ ) > 0, (vi )T · (xk+1 − xk∗ ) > −(vi )T · (xk − xk∗ ) > 0 for (vi )T · (xk − xk∗ ) < 0



(1) (xk∗ , p) = −1, we have For G i(i)

−cki +

1 (m i ) jm j j G i( j1 j2 ··· jm ) (xk∗ , p)(ck1 ck2 · · · ck i ) > −cki . i mi ! j

jm i

j

For infinitesimals of ck1 , ck2 , . . . , ck equation gives

are arbitrarily selected, the foregoing

(m i ) ∗ 1 2 i G i( j1 j2 ··· jm ) (xk , p)(ck ck · · · ck ) > 0. j

j

jm

i

Thus, xk+ j ( j ∈ Z) in a neighborhood of fixed-point xk∗ on the direction vi is oscillatorilly unstable of the (m i )th-order, upper saddle, vice versa. This them is proved.  The stability of fixed-points for a specific eigenvector based on the linearization is presented in Fig. 1.7. The solid curve is (vi )T · xk+1 = (vi )T · f(xk , p). The circular symbol is a fixed-point. The shaded regions are stable. The horizontal solid line is for a degenerate case. The vertical solid line is for a line with infinite slope. A monotonically stable node (a monotonic sink) is presented in Fig. 1.7i. The dashed i = cki ) and (vi )T · yk+1 = and dotted lines are for (vi )T · yk+1 = (vi )T · yk (or ck+1 i i i T −(v ) · yk (or ck+1 = −ck ), respectively. The iterative responses approach the

1.3 Stability Switching

41

( v i )T ⋅ x k +1

( v i )T ⋅ x k +1

cki +1 = cki

( v i )T ⋅ x k

( v i )T ⋅ x k ( v i )T ⋅ x∗k

( v i )T ⋅ x∗k cki +1 = cki

cki +1 = −cki

cki +1 = −cki

(i)

(ii) i k +1

c

cki +1 = −cki

i k

=c

cki +1 = cki

( v i )T ⋅ x∗k

( v i )T ⋅ x∗k

( v i )T ⋅ x k +1

( v i )T ⋅ x k +1 i k +1

c

i k

= −c

( v i )T ⋅ x k

i T

( v ) ⋅ xk

(iii)

(iv)

Fig. 1.7 Stability of fixed-points based on a linearization: (i) monotonically stable node (sink), (ii) monotonically unstable node (source), (iii) oscillatorilly stable node (sink) and (iv) oscillatorilly ∗ ). unstable node (sink). Shaded areas are stable zones. (yk = xk − xk∗ and yk+1 = xk+1 − xk+1 i i i T i T ck+1 = (v ) · yk+1 and ck = (v ) · yk

fixed-point. However, a monotonically unstable (a monotonic source) is presented in Fig. 1.7ii. The iterative responses go away from the fixed-point. Similarly, an oscillatorilly stable node (an oscillatory sink) is presented in Fig. 1.7iii). The dashed i = −cki ) and (vi )T · yk+1 = and dotted lines are for (vi )T · yk+1 = −(vi )T · yk (ck+1 i i i T (v ) · yk (ck+1 = ck ), respectively. An oscillatorilly unstable node (an oscillatory source) is presented in Fig. 1.7iv. The monotonic stability of fixed-points with higher-order singularity for a specific eigenvector is presented in Fig. 1.8. The solid curve is viT · xk+1 = viT · f(xk , p). The circular symbol is fixed-pointed. The shaded regions are stable. The horizontal solid line is also for the degenerate case. The vertical solid line is for a line with

42

1 Local Stability and Bifurcations

( v i )T ⋅ x k +1

( v i )T ⋅ x k +1

cki +1 = cki

( v i )T ⋅ x k

( v i )T ⋅ x k

( v i )T ⋅ x∗k i T

(v ) ⋅ x

∗ k

i k +1

c

i k

= −c

cki +1 = cki cki +1 = −cki

(i) ( v i )T ⋅ x k +1

(ii) i k +1

c

i k

=c

( v i )T ⋅ x k

( v i )T ⋅ x k +1

( v i )T ⋅ x k

( v i )T ⋅ x∗k

( v i )T ⋅ x∗k

cki +1 = −cki

(iii)

cki +1 = −cki

cki +1 = cki

(iv)

Fig. 1.8 Monotonic stability of fixed-points with higher-order singularity: (i) a monotonically stable node (a monotonic sink) of the (m i )th-order, (ii) a monotonically unstable node (a monotonic source) of the (m i )th-order, (iii) a monotonic lower-saddle of (m i )th-order and (iv) a monotonic upper∗ ). saddle of the (m i )th-order. Shaded areas are stable zones. (yk = xk − xk∗ and yk+1 = xk+1 − xk+1 i i i T i T ck+1 = (v ) · yk+1 and ck = (v ) · yk

infinite slope. The monotonically stable node (sink) of the (m i )th order is sketched i = cki ) in Fig. 1.8i. The dashed and dotted lines are for viT · xk = viT · xk+1 (ck+1 i i T T and vi · xk+1 = −vi · xk (ck+1 = −ck ), respectively. The nonlinear curve lies in the stable zone, and the iterative responses approach the fixed-point. However, the monotonically unstable (source) of the (m i )th order is presented in Fig. 1.8ii. The nonlinear curve lies in the unstable zone, and the iterative responses go away from the fixed-point. The monotonically lower-saddle of the (m i )th order is presented in i = cki with the (m i )th Fig. 1.8iii. The nonlinear curve is tangential to the line of ck+1 order, and the one branch is in the stable zone and another branch is in the unstable

1.3 Stability Switching

43

zone. Similarly, the monotonically upper saddle of the (m i )th order is presented in Fig. 1.8iv. As similar to the monotonic stability of fixed-points, the oscillatory stability of fixed-points with higher order singularity for a specific eigenvector is presented in Fig. 1.9. An oscillatorilly stable node (an oscillatory sink) of the (m i )th order is sketched in Fig. 1.9i. The dashed and dotted lines are for viT · yk+1 = −viT · yk i i (ck+1 = −cki ) and viT · yk+1 = viT · yk (ck+1 = cki ), respectively. The nonlinear curve lies in the stable zone, and the iterative responses approach the fixed-point. However, an oscillatorilly unstable (an oscillatory source) of the (m i )th order is presented in Fig. 1.9ii. The nonlinear curve lies in the unstable zone, and the iterative responses go away from the fixed-point. An oscillatory lower-saddle of the (m i )th order is cki +1 = cki

cki +1 = cki

cki +1 = −cki

( v i )T ⋅ x∗k

( v i )T ⋅ x∗k

( v i )T ⋅ x k +1 ( v i )T ⋅ x k +1

cki +1 = −cki

( v i )T ⋅ x k

( v i )T ⋅ x k

(i) cki +1 = −cki

(ii) cki +1 = cki

cki +1 = cki

cki +1 = −cki

( v i )T ⋅ x∗k

( v i )T ⋅ x∗k

i T

( v ) ⋅ x k +1

( v i )T ⋅ x k +1 ( v i )T ⋅ x k

( v i )T ⋅ x k

(iii)

(iv)

Fig. 1.9 Oscillatory stability of fixed-points with higher-order singularity: (i) an oscillatorilly stable node (an oscillatory sink) of the (m i )th-order, (ii) an oscillatorilly unstable node (an oscillatory source) of the (m i )th-order, (iii) an oscillatory lower-saddle of the (m i )th-order and (iv) an oscillatory upper-saddle of the (m i )th-order. Shaded areas are stable zones. (yk = xk − xk∗ and i ∗ ). ci i T i T yk+1 = xk+1 − xk+1 k+1 = (v ) · yk+1 and ck = (v ) · yk

44

1 Local Stability and Bifurcations

presented in Fig. 1.9iii. The nonlinear curve is tangential to and below the line of i = −cki with the (m i )th order, and the one branch is in the stable zone and ck+1 another branch is in the unstable zone. Finally, an oscillatory upper-saddle of the (m i )th order is presented in Fig. 1.9iv. For clear illustrations, oscillatory stability of fixed-points with higher-order singularity for the two-time iterations are presented in Fig. 1.10.

ckT+1 = −cki

cki +1 = cki

v iT ⋅ y k +1 = − v iT ⋅ y k

( v i )T ⋅ x∗k

cki +1 = cki

( v i )T ⋅ x∗k

( v i )T ⋅ x k +1

( v i )T ⋅ x k +1 ( v i )T ⋅ x k

( v i )T ⋅ x k

(i) cki +1 = −cki

(ii) cki +1 = cki

cki +1 = −cki

( v i )T ⋅ x∗k

cki +1 = cki

( v i )T ⋅ x∗k

( v i )T ⋅ x k +1

( v i )T ⋅ x k +1 ( v i )T ⋅ x k

( v i )T ⋅ x k

(iii)

(iv)

Fig. 1.10 Oscillatory stability of fixed-points with higher-order singularity for the two-time iterations: (i) an oscillatorilly stable node (an oscillatory sink) of the (m i )th-order, (ii) an oscillatorilly unstable node (an oscillatory source) of the (m i )th-order, (iii) an oscillatory lower-saddle of the (m i )th-order and (iv) an oscillatory upper-saddle of the (m i )th-order. Shaded areas are stable zones. i (yk = xk − xk∗ and yk+1 = xk+1 − xk∗ ). ck+1 = (vi )T · yk+1 and cki = (vi )T · yk

1.3 Stability Switching

45

1.3.2 Spiral Stability Definition 1.27 Consider a nonlinear discrete system xk+1 = f(xk , p) ∈ R n in Eq. (1.4) with a fixed-point xk∗ . The corresponding solution is given by xk+ j = f(xk+ j−1 , p) with j ∈ Z. Suppose there is a neighborhood of the fixed-point xk∗ (i.e., Uk (xk∗ ) ⊂ ), and f(xk , p) is C r (r ≥ 1)-continuous in Uk (xk∗ ) with Eq. (1.28). ∗ The linearized system is yk+ j+1 = Df(xk∗ , p)yk+ j (yk+ j = xk+ j − xk∗ ) in √Uk (xk ). Consider a pair of complex eigenvalues αi ± iβi (2i − 1, 2i ∈ N , i = −1 and N = {1, 2, . . . , n}) of matrix Df(xk∗ , p) with a pair of eigenvectors v2i−1 ± iv2i . On the invariant plane of (v2i−1 , v2i ), the contravariants are defined as ck2i−1 = (v2i−1 )T · yk , ck2i = (v2i )T · yk

(1.109)

Dxk f · v2i−1 = αi v2i−1 − βi v2i , Dxk f · v2i = βi v2i−1 + αi v2i .

(1.110)

with

The next step recurrence iterations of the contravariants are 2i−1 ck+1 = G 2i−1 (xk , p) = (v2i−1 )T · f(xk , p)

=

∞  1 (ri ) jr j j G (2i−1)( j1 j2 ··· jr ) (xk∗ , p)(ck1 ck2 · · · ck i ), 2i−1 r! r =1 i i

2i = G 2i (xk , p) = (v2i )T · f(xk , p) ck+1

(1.111)

∞  1 (ri ) jr j j G (2i)( j1 j2 ··· jr ) (xk∗ , p)(ck1 ck2 · · · ck i ), = i r ! r =1 i i

where for jrk ∈ {1, 2, . . . , n}, rk = 1, 2, . . . , (ri ) ∗ 2i−1 T ) · D (r( ij1)) G (2i−1)( j1 j2 ··· jr ) (xk , p) = (v ck

i

(j )

( jri )

ck 2 ··· ck

f(xk∗ , p)

= (v2i−1 )T · Dx(rki ) f(xk∗ , p)v j1 v j2 · · · v jri , (ri ) ∗ 2i T k) G (r (2i)( j1 j2 ··· jr ) (x , p) = (v ) · D ( j1 ) ck

i

(j ) ck 2

( jr ) ··· ck i

f(xk∗ , p)

(1.112)

= (v ) · Dx(rki ) f(xk∗ , p)v j1 v j2 · · · v jri . 2i T

Thus, the next step recurrence iterations of contravariants in Eq. (1.111) become

46

1 Local Stability and Bifurcations

2i−1 ck+1



2i ck+1



αi βi = −βi αi



ck2i−1



ck2i ⎫ ⎧ (ri ) ∗ ∞ ⎬ ⎨ (x , p) G  k j ··· j ) (2i−1)( j 1 jr 1 2 ri j j + (ck1 ck2 · · · ck i ). ∗ ⎭ ri ! ⎩ G (ri ) (x , p) ri =2

(2i)( j1 j2 ··· jri )

(1.113)

k

The radial variables are defined as ρki =



(ck2i−1 )2 + (ck2i )2 , tanθki =

ck2i

(1.114)

ck2i−1

with ck2i−1 = ρki cos θki , ck2i = ρki sin θk(i) .

(1.115)

Thus  2i ck+1 2i−1 2 i 2i (ck+1 ) + (ck+1 )2 , tanθk+1 = 2i−1 ck

i ρk+1 =

(1.116)

with 2i−1 i i 2i i i ck+1 = ρk+1 cos θk+1 , ck+1 = ρk+1 sin θk+1

(1.117)

and i ρk+1 =



" =

 2i−1 2 2i (ck+1 ) + (ck+1 )2 = !

∞ 

m i =2



G (2) ρk(i) !1 + (G (2) )−1 ρi ρi k+1

(i) = arctan θk+1

k+1

(ρki )m i G ρ(mi i ) (θki )

∞  m i =3

k+1

(ρki )m i −2 G ρ(mi i ) (θki )

(1.118)

k+1

2i ck+1 2i−1 ck+1

where G ρ(mi i ) (θki ) = k+1

1 (ρki )m i

mi  mi  ri =1 si =1

j1 j2 (ri ) 1 1 [G (2i−1)( j1 j2 ··· jri ) (ck ck ri ! si !

jr

· · · ck i )

jsi jri j1 j2 j1 j2 (si ) (ri ) (1.119) ×G (2i−1)( j1 j2 ··· js ) (ck ck · · · ck ) + G (2i)( j1 j2 ··· jr ) (ck ck · · · ck ) i

(si ) 1 2 i (ri +si ) ×G (2i)( . j1 j2 ··· js ) (ck ck · · · ck )]δm i j

j

js

i

i

From the foregoing definition, consider the first order terms of G-function

1.3 Stability Switching

47

(2i−1) ∗ 1 ck+1 ≈ G (1) (2i−1)( j1 ) (xk , p)ck j

= (v2i−1 )T · Dx(1) f(xk∗ , p)v j1 ck1 k j

= (v2i−1 )T · Dx(1) f(xk∗ , p)v2i−1 ck2i−1 + (v2i−1 )T · Dx(1) f(xk∗ , p)v2i ck2i k k = (v2i−1 )T · (αi v2i−1 − βi v2i )ck2i−1 + (v2i−1 )T · (βi v2i−1 + αi v2i )ck2i = αi ck2i−1 + βi ck2i ; (2i) ∗ 1 ≈ G (1) ck+1 (2i)( j1 ) (x , p)ck j

= (v2i )T · Dx(rki ) f(xk∗ , p)v j1 ck1 j

= (v2i )T · Dx(1) f(xk∗ , p)v2i−1 ck2i−1 + (v2i )T · Dx(1) f(xk∗ , p)v2i ck2i k k = (v2i )T · (αi v2i−1 − βi v2i )ck2i−1 + (v2i )T · (βi v2i−1 + αi v2i )ck2i = −βi ck2i−1 + αi ck2i . (1.120) Thus 1 j j j1 (1) j1 [G (1) c 1 G (1) c 1 + G (1) (2i)( j1 ) ck G (2i)( j1 ) ck ] (ρki )2 (2i−1)( j1 ) k (2i−1)( j1 ) k 1 = i 2 [(αi ck2i−1 + βi ck2i )2 + (−βi ck2i−1 + αi ck2i )2 ] (ρk )

G (2)(i) (θki ) = ρk+1

(1.121)

= αi2 + βi2 , i tan θk+1 =

2i ck+1 2i ck+1

≈

−βi ck2i−1 + αi ck2i αi ck2i−1 + βi ck2i

=

−βi cos θki + αi sin θki . αi cos θki + βi sin θki

Furthermore, Eq. (1.118) gives i i ρk+1 = ri ρki + o(ρki ) and θk+1 = θki − ϕi + o(ρki )

(1.122)

where ϕi = arctan(βi /αi ) and ri =



αi2 + βi2 .

(1.123)

As ρki 0, the stability of the fixed-point xk∗ on the invariant plane of (v2i−1 , v2i ) can be determined. (i)

xk+ j in a neighborhood of fixed-point xk∗ on the plane of (v2i−1 , v2i ) is spirally stable if i ρk+1 − ρki < 0.

(1.129)

(ii) xk+ j in a neighborhood of fixed-point xk∗ on the plane of (v2i−1 , v2i ) is spirally unstable if i ρk+1 − ρki > 0.

(1.130)

(iii) xk+ j in a neighborhood of fixed-point xk∗ on the plane of (v2i−1 , v2i ) is stable with the (m i )th-order singularity if for θk(i) ∈ [0, 2π ] ri = G



(ski ) i ρk+1

αi2 + βi2 = 1,

(θki ) = 0 for sk(i) = 1, 2, . . . , m i − 1

i ρk+1 − ρki < 0.

(1.131)

1.3 Stability Switching

49

(iv) xk+ j in a neighborhood of fixed-point xk∗ on the plane of (v2i−1 , v2i ) is spirally unstable with the (m i )th-order singularity if for θk(i) ∈ [0, 2π ] ri = G



(ski ) i ρk+1

αi2 + βi2 = 1,

(θki ) = 0 for sk(i) = 1, 2, . . . , m i − 1

(1.132)

i − ρki > 0. ρk+1

(v) xk+ j in a neighborhood of fixed-point xk∗ on the plane of (ui , vi ) is circular if for θk(i) ∈ [0, 2π ] i ρk+1 − ρki = 0.

(1.133)

(vi) xk+ j in a neighborhood of fixed-point xk∗ on the plane of (v2i−1 , v2i ) is degenerate in the direction of v2i−1 if i βi = 0 and θk+1 − θki = 0.

(1.134)

Theorem 1.7 Consider a nonlinear discrete system xk+1 = f(xk , p) ∈ Rn in Eq. (1.4) with a fixed-point xk∗ . The corresponding solution is given by xk+ j = f(xk+ j−1 , p) with j ∈ Z. Suppose there is a neighborhood of the fixed-point xk∗ (i.e., Uk (xk∗ ) ⊂ Ω), and f(xk , p) is C r (r ≥ 1)-continuous in Uk (xk∗ ) with Eq. (1.28). The linearized ∗ system is yk+ j+1 = Df(xk∗ , p)yk+ j (yk+ j = xk+ j − xk∗ ) in Uk (x√ k ). Consider a pair of complex eigenvalue αi ± iβi (i ∈ N = {1, 2, . . . , n 1 }, i = −1 and n 1 ∈ {n/2, . . . , n −1}) of matrix Df(xk∗ , p) with a pair (v2i−1 , v2i ) of eigenvectors v2i−1 ± iv2i . On the invariant plane of (v2i−1 , v2i ), consider ck2i−1 = (v2i−1 )T · yk , and ck2i = (v2i )T · yk with Eq. (1.109). For an arbitrarily small ε > 0, the stability of the fixed-point xk∗ on an invariant plane of (v2i−1 , v2i ) can be determined. (i)

xk+ j in a neighborhood of fixed-point xk∗ on the plane of (v2i−1 , v2i ) is spirally stable if and only if ri =

 αi2 + βi2 < 1.

(1.135)

(ii) xk+ j in a neighborhood of fixed-point xk∗ on the plane of (v2i−1 , v2i ) is spirally unstable if and only if ri =

 αi2 + βi2 > 1.

(1.136)

(iii) xk+ j in a neighborhood of fixed-point xk∗ on the plane of (v2i−1 , v2i ) is stable with the (m i )th-order singularity if and only if for θk(i) ∈ [0, 2π ]

50

1 Local Stability and Bifurcations

ri = G G



(ski ) i ρk+1 (m i ) i ρk+1

αi2 + βi2 = 1,

(θki ) = 0 for ski = 1, 2, . . . , m i − 1 (θki )

(1.137)

< 0.

(iv) xk+ j in a neighborhood of fixed-point xk∗ on the plane of (v2i−1 , v2i ) is spirally unstable with the (m i )th-order singularity if and only if for θk(i) ∈ [0, 2π ] ri =



αi2 + βi2 = 1,

(s i )

G ρ ik (θki ) = 0 for ski = 1, 2, . . . , m i − 1 k+1

G ρ(mi i ) (θki ) k+1

(1.138)

> 0.

(v) xk+ j in a neighborhood of fixed-point xk∗ on the plane of (v2i−1 , v2i ) is circular if and only if for θk(i) ∈ [0, 2π ] ri = G



(ski ) i ρk+1

αi2 + βi2 = 1,

(θki ) = 0 for ski = 1, 2, . . . .

(1.139)

Proof Since

2i−1 ck+1 2i ck+1



 =

αi βi −βi αi



ck2i−1 ck2i



⎧ (r ) ⎫ ∗ i ∞  1 ⎨G (2i−1)( j1 j2 ··· jri ) (xk , p)⎬ j1 j2 jr (ck ck · · · ck i ). + ri ! ⎩G (ri ) (x∗ , p) ⎭ ri =2

(2i)( j1 j2 ··· jri )

k

(2i−1) (2i) For xk+1 = xk = xk∗ , ρki = 0. The first order approximation of ck+1 and ck+1 in the Taylor series expansion gives

2i−1 ck+1 2i ck+1



2i−1  ck αi βi = + o(||¯cki ||) −βi αi ck2i  2i−1  ck cos ϕi sin ϕi = ri + o(||¯cki ||). − sin ϕi cos ϕi dk2i 

 where ri = αi2 + βi2 , ϕi = arctan(βi /αi ) and c¯ ki = (ck2i−1 , ck2i )T . From the foregoing equation, we have i i = ri ρki + o(ρki ) and θk+1 = θki − ϕi + o(ρki ). ρk+1

where

1.3 Stability Switching

51

ρki

 ck2i = (ck2i−1 )2 + (ck2i )2 , tanθki = 2i−1 ck

As ρki 1, then i ρk+1 > ρki

which implies xk+ j in a neighborhood of fixed-point xk∗ on the plane of (v2i−1 , v2i ) is spirally stable, vice versa. (iii) If for θki ∈ [0, 2π ) the following conditions exist " ri =

G (2)(i) = ρk+1

(s i ) G ρ ik (θki ) k+1 G ρ(mi i ) (θki ) k+1

 αi2 + βi2 = 1,

= 0 for ski = 1, 2, . . . , m i − 1 (s i )

= 0, and |G ρ ik (θki )| < ∞ for ski = m i + 1, m + 2 . . . , k+1

then the higher terms can be ignored, i.e., " i ρk+1 = ρki 1 + (ρki )m i −2 G ρ(mi i ) (θki ). k+1

If G ρ(mi i ) (θki ) < 0, then k+1

i ρk+1 < ρki .

In other words, which implies xk+ j in a neighborhood of fixed-point xk∗ on the plane of (v2i−1 , v2i ) is spirally stable with (m i )th order singularity, and vice versa. (iv) If G ρ(mi i ) (θki ) > 0, then k+1

52

1 Local Stability and Bifurcations i ρk+1 > ρki .

That is, xk+ j in a neighborhood of fixed-point xk∗ on the plane of (v2i−1 , v2i ) is spirally unstable with the (m i )th-order singularity, and vice versa. (v) If for θki ∈ [0, 2π ) the following conditions exist (s i )

G ρ ik (θki ) = 0 for ski = 1, 2, . . . , k+1

then i ρk+1 = ρki ,

vice versa. Therefore xk+ j in a neighborhood of fixed-point xk∗ on the plane of  (v2i−1 , v2i ) is circular. This theorem is proved.

1.4 Local Bifurcations In this section, the local bifurcation of fixed-points in nonlinear discrete systems on a specific eigenvector is discussed. Definition 1.29 Consider a nonlinear discrete system xk+1 = f(xk , p) ∈ R n in Eq. (1.4) with a fixed-point xk∗ . The corresponding solution is given by xk+ j = f(xk+ j−1 , p) with j ∈ Z. Suppose there is a neighborhood of the fixed-point xk∗ (i.e., Uk (xk∗ ) ⊂ Ω), and f(xk , p) is C r (r ≥ 1)-continuous in Uk (xk∗ ) with Eq. (1.28). The linearized system is yk+ j+1 = Df(xk∗ , p)yk+ j (yk+ j = xk+ j − xk∗ ) in Uk (xk∗ ) and there are n linearly independent vectors vi (i = 1, 2, . . . , n). For a perturbation of i vi . fixed-point yk = xk − xk∗ , let yk = cki vi and yk+1 = ck+1 cki = (vi )T · yk = (vi )T · (xk − xk∗ ).

(1.140)

i = (vi )T · yk+1 = (vi )T · [f(xk , p) − xk∗ ]. ck+1

(1.141)

∗ , p0 ), (vi )T · f(xk , p) can be expended for (0 < θl < In the vicinity of point (xk0 1, l = 1, 2, . . .) and γ = (γ1 , γ2 , . . . , γn )T with 0 < γ j < 1 ( j = 1, 2, . . . , n) as

1.4 Local Bifurcations

53

∗ ∗ ∗ (vi )T · [f(xk , p) − xk0 ] = (vi )T · [f(xk0 + yk , p0 + p) − xk0 ]

= ai(i) z ki + bTk · (p − p0 ) + +

q mi   1 r (q−r,r ) j j j Cq ai( j1 j2 ··· jq−r ) z k1 z k2 · · · z kq−r (p − p0 )r q! q=2 r =0 m i +1  r =0

1 j j (θ1 z k1 )(θ2 z k2 ) C r a(m i +1−r,r ) (m i + 1)! m i +1 k( j1 j2 ··· jmi +1−r ) jm i +1−r

· · · (θm i +1−r z k

)(γT · (p − p0 ))r , (1.142)

where s (s = 1, 2, . . . , n), z ks = (vs )T · yks = cks − ck0 p= p − p0 ;  biT = (vi )T · ∂p f(xk , p)(x∗ ,p0 ) ,  k0  i T ai( j1 ) = (v ) · ∂c j1 f(xk , p) ∗ k (x ,p0 )  k0 = (vi )T · ∂xk f(xk , p)(x∗ ,p0 ) v j1 k0

(1) ∗ = G i( j1 ) (x0 , p0 ), (r,s) ai( j1 j2 ··· jr )

(1.143)

= (v ) · = (v ) ·

∂x(ri ) ∂p(s) f(xk , p)(x∗ ,p ) v j1 v j2 k0 0



∗ (xk0 ,p0 )

· · · v jr ;   = (vi )T · ∂ (rj1) j2 jr f(xk , p) ck ck ···ck ∗ (xk0 ,p0 )  i T (r )  = (v ) · ∂xk f(xk , p) (x∗ ,p ) v j1 v j2 · · · v jr i T

ai((r,0) j1 j2 ··· jr )

 

∂ (rj1) j2 jr ∂p(s) f(xk , p) ck ck ···ck

i T

k0

=

0

(r ) ∗ G i( j1 j2 ... jr ) (xk0 , p0 ).

1.4.1 Monotonic Bifurcations Definition 1.30 Consider a nonlinear discrete system xk+1 = f(xk , p) ∈ R n in Eq. (1.4) with a fixed-point xk∗ . The corresponding solution is given by xk+ j = f(xk+ j−1 , p) with j ∈ Z. Suppose there is a neighborhood of the fixed-point xk∗ (i.e., Uk (xk∗ ) ⊂ ), and f(xk , p) is C r (r ≥ 1)-continuous in Uk (xk∗ ) with Eq. (1.28). The linearized system is yk+ j+1 = Df(xk∗ , p)yk+ j (yk+ j = xk+ j − xk∗ ) in Uk (xk∗ ) and

54

1 Local Stability and Bifurcations

there are n linearly independent vectors vi (i = 1, 2, . . . , n). For a perturbation of i vi . fixed-point yk = xk − xk∗ , let yk = cki vi and yk+1 = ck+1 For a specific j = i (i, j ∈ {1, 2, . . . , n} ∪ {∅}), the corresponding fixed-point equation in the vicinity of (xk0 , p0 ) is ∗ ∗ (v j )T · [f(xk0 + yk∗ , p0 + p) − xk0 ] = z k+1 = z k j∗ T (or (λ j0 − 1)z k + b j · (p − p0 ) ≈ 0), ( j ∈ {1, 2, . . . , n} ∪ {∅}, but j = i, ), λ j0 = 1. j∗

j∗

(1.144)

For a specific i (i ∈ {1, 2, . . . , n} ∪ {∅}), if ai(i) = λi0 = 1 at (xk0 , p0 ), the corresponding fixed-point equation is i∗ ∗ ∗ + yk∗ , p0 + p) − xk0 ] = z k+1 =z ki∗ with λi0 = 1; (vi )T · [f(xk0 i∗ T (or (λi0 − 1)z k + bi · (p − p0 ) q si   j ∗ j ∗ j ∗ 1 r (q−r,r ) + C a z 1 z k1 · · · z kq−r (p − p0 )r ≈ 0) q! q i( j1 j2 ··· jq−r ) k q=2 r =0

equivalent to Ai0 (z ki )si + Ai1 (z ki )si −1 + · · · + Ais1 = 0 Ail = Ail (z k1∗ , z k2∗ , . . . , z ki−1∗ , z ki+1∗ , . . . , z kn∗ , p), (l = 0, 1, 2, . . . , si ) i ∈ {1, 2, . . . , n} ∪ {∅}.

(1.145)

li αs = Equations (1.144) and (1.145) give li -fixed-points z ki∗ (0 < li ≤ si ) with s=1 m i ≤ si for αs -repeated fixed-point on the eigenvector vi direction.

(i) If (ri ) ∗ λi0 = 1, G i( j1 j2 ... jr ) (xk0 , p0 ) = 0(ri = 2, 3, . . . , m i − 1), k

(m i ) 1 1 ∗ G i( j1 j2 ... jm ) (xk0 , p0 )z k z k · · · z k j

j

jm i

i

 (1.146)

> 0,

∗ , p0 ) is called a monotonic, the bifurcation of fixed-point xk∗ at point (xk0 increasing saddle-node (or a monotonic-upper-saddle-node) bifurcation of the ∗ , p0 ) on the eigen(m i )th order on the eigenvector vi . The bifurcation point (xk0 vector vi is a monotonic-increasing-saddle (or a monotonic-upper-saddle) of the (m i )th order

(ii) If (ri ) ∗ λi0 = 1, G i( j1 j2 ... jr ) (xk0 , p0 ) = 0(ri = 2, 3, . . . , m i − 1), k

(m i ) 1 1 ∗ G i( j1 j2 ... jm ) (xk0 , p0 )z k z k · · · z k j

i

j

jm i

< 0,

 (1.147)

∗ , p0 ) is called a monotonicthe bifurcation of fixed-point xk∗ at point (xk0 decreasing saddle-node (or a monotonic-lower-saddle-node) bifurcation of the

1.4 Local Bifurcations

55

∗ (m i )th order on the eigenvector vi . The bifurcation point (xk0 , p0 ) on the eigenvector vi is a monotonic-decreasing-saddle (or a monotonic-lower-saddle) of the (m i )th order

(iii) If (ri ) ∗ λi0 = 1, G i( j1 j2 ... jr ) (xk0 , p0 ) = 0(ri = 2, 3, . . . , m k − 1), i

(m i ) 1 2 ∗ z ki × G i( j1 j2 ... jm ) (xk0 , p0 )z k z k · · · z k j

j

jm i

i

 (1.148)

< 0,

∗ , p0 ) is called a monotonic-sink the bifurcation of fixed-point xk∗ at point (xk0 bifurcation of the (m i )th order on the eigenvector vi . The bifurcation point ∗ , p0 ) on the eigenvector vi is a monotonic sink of the (m i )th order. (xk0

(iv) If (ri ) ∗ λ0i = 1, G i( j1 j2 ... jr ) (xk0 , p0 ) = 0(ri = 2, 3, . . . , m k − 1), i

(m i ) 1 2 ∗ z ki × G i( j1 j2 ... jm ) (xk0 , p0 )z k z k · · · z k j

i

j

jm i

> 0,

 (1.149)

∗ , p0 ) is called a monotonic-source the bifurcation of fixed-point xk∗ at point (xk0 bifurcation of the (m i )th order on the eigenvector vi . The bifurcation point ∗ , p0 ) on the eigenvector vi is a monotonic source of the (m i )th order. (xk0

The following discussions are given for a few special cases of monotonic bifurcations. Definition 1.31 Consider a nonlinear discrete system xk+1 = f(xk , p) ∈ R n in Eq. (1.4) with a fixed-point xk∗ . The corresponding solution is given by xk+ j = f(xk+ j−1 , p) with j ∈ Z. Suppose there is a neighborhood of the fixed-point xk∗ (i.e., Uk (xk∗ ) ⊂ ), and f(xk , p) is C r (r ≥ 1)-continuous in Uk (xk∗ ) with Eq. (1.28). The linearized system is yk+ j+1 = Df(xk∗ , p)yk+ j (yk+ j = xk+ j − xk∗ ) in Uk (xk∗ ) and there are n linearly independent vectors vi i = 1, 2, . . . , n. For a perturbation of i vi . Three special cases of fixed-point yk = xk − xk∗ , let yk = cki vi and yk+1 = ck+1 bifurcations are defined as follows. (i) Consider λ j0 = 1( j, i = 1, 2, . . . , n but j = i)

(1.150)

with ∗ ∗ (v j )T · [f(xk0 + yk∗ , p0 + p) − xk0 ] = z k+1 = z k j∗ T (or (λ j0 − 1)z k + b j · (p − p0 ) ≈ 0); j∗

j∗

(1.151)

56

1 Local Stability and Bifurcations i∗ ∗ ∗ (vi )T · [f(xk0 + yk∗ , p0 + p) − xk0 ] = z k+1 =z ki∗ with λi0 = 1; j ∗ (1,1) (102) i∗ T 2 (or (λi0 − 1)z k + bi · (p − p0 ) + ai( j1 ) (p − p0 )z k1 + 2!1 ai( j1 ) (p − p0 ) j1 ∗ j2 ∗ + 2!1 ai((2,0) j1 j2 ) z k z k = 0) equivalent to ⎫ i∗ = z ki∗ + Ai0 (z ki∗ )2 + Ai1 (z ki∗ ) + Ai2 ⎪ z ki∗ = z k+1 ⎪ ⎬ ⇒ Ai0 (z ki )2 + Ai1 (z ki ) + Ai2 = 0, Ail = Ail (z k1∗ , z k2∗ , · · · , z ki−1∗ , z ki+1∗ , · · · , z kn∗ , p), ⎪ ⎪ ⎭ (l = 0, 1, 2) 2 i = Bi1 − 4Ci1 ≥ 0 with Bi1 = AAi0i1 , Ci1 = AAi0i2 , (2,0) = 0, Ci1 ≤ 0; Ai1 = 0, Ai0 = ai(ii) (1.152)

where   (2) i T (0) ai((2,0) j1 j2 ) = (v ) · ∂c j1 c j2 ∂p f(xk , p)

∗ (xk0 ,p0 )

  = (vi )T · ∂c(2) j1 c j2 f(xk , p)

∗ (xk0 ,p0 )

 ∗ = (vi )T · ∂x(2) f(xk , p)(v j1 v j1 )(x∗ ,p ) = G (2) k( j1 j2 ) (xk0 , p0 ), k 0 0    (1,1) (1) (1) i T ai( = (vi )T · ∂xk ∂p f(xk , p)v j1 (x∗ ,p0 ) , j1 ) = (v ) · ∂c j1 ∂p f(xk , p) ∗ k0 (xk0 ,p0 )  (2,0) (2) T i T ∗  bi = (v ) · ∂p f(xk , p) (x∗ ,p0 ) , ai(ii) = G i(ii) (xk0 , p0 ) = 0. k0

(1.153) If Eqs. (1.150)–(1.153) have two fixed-point solutions, such a bifurcation at point ∗ (xk0 , p0 ) is called the monotonic-saddle-node bifurcation of the second-order on the eigenvector vi . (i1 ) If (2) 1 2 1 2 ∗ λi0 = 1, ai((2,0) j1 j2 ) z k z k = G i( j1 j2 ) (x0 , p0 )z k z k > 0 for j1 , j2 ∈ {1, 2, . . . , n} j

j

j

j

 (1.154)

∗ such a bifurcation at point (xk0 , p0 ) is called a monotonic, increasing saddle-node (or a monotonic-upper-saddle-node) appearing bifurcation on ∗ , p0 ) is an increasing saddle the eigenvector vi . The bifurcation point at (xk0 (or an upper-saddle) of the second-order on the eigenvector vi . (i2 ) If (2) 1 2 1 2 ∗ λi0 = 1, ai((2,0) j1 j2 ) z k z k = G i( j1 j2 ) (x0 , p0 )z k z k < 0 for j1 , j2 = 1, 2, . . . , n; j

j

j

j

 (1.155)

∗ such a bifurcation at point (xk0 , p0 ) is called a monotonic, decreasing saddle-node (or a monotonic lower-saddle-node) appearing bifurcation of

1.4 Local Bifurcations

57

∗ the second order on the eigenvector vi . The bifurcation point at (xk0 , p0 ) is a decreasing saddle (or a lower-saddle) of the second-order on the eigenvector vi .

(ii) Consider λ j0 = 1 ( j = 1, 2, . . . , n but j = i)

(1.156)

with ∗ ∗ (v j )T · [f(xk0 + yk∗ , p0 + p) − xk0 ] = z k+1 = z k j∗ T (or (λ0 j − 1)z k + b j · (p − p0 ) ≈ 0); j∗

j∗

(1.157)

i∗ ∗ ∗ (vi )T · [f(xk0 + yk∗ , p0 + p) − xk0 ] = z k+1 ≡ z ki∗ with λ0i = 1; j ∗ (1,1) i∗ T (or (λ0i − 1)z k + bi · (p − p0 ) + ai( j1 ) (p − p0 )z k1 + 2!1 ai(0,2) (p − p0 )2 j1 ∗ j2 ∗ 1 (2,0) + 2! ai( j1 j2 ) z k z k = 0) equivalent to ⎫ i∗ = z ki∗ + Ai0 (z ki∗ )2 + Ai1 (z ki∗ ) + Ai2 ⎪ z ki∗ = z k+1 ⎪ ⎬ ⇒ Ai0 (z ki )2 + Ai1 (z ki ) + Ai2 = 0, i−1∗ i+1∗ Ail = Ail (z k1∗ , z k2∗ , · · · , z k , z k , · · · , z kn∗ , p), ⎪ ⎪ ⎭ (l = 0, 1, 2) 2 i = Bi1 − 4Ci1 ≥ 0 with Bi1 = AAi0i1 , Ci1 = AAi2i0 , (2,0) = 0, Ai1 ∈ (−∞, ∞) Ai2 = 0, Ai0 = ai(ii) (1.158)

where   (2) i T (0) = (v ) · ∂ ∂ f(x , p) ai((2,0)  k j j p j1 j2 ) c 1c 2 = (v ) ·

∗ (xk0 ,p0 )



  = (vi )T · ∂c(2) j1 c j2 f(xk , p)

∂x(2) f(xk , p)(v j1 v j1 )(x∗ ,p ) k 0 0

∗ (xk0 ,p0 )

∗ G (2) k( j1 j2 ) (xk0 , p0 ),

=    (1,1) (1) (1) i T ai( = (vi )T · ∂xk ∂p f(xk , p)v j1 (x∗ ,p0 ) , j1 ) = (v ) · ∂c j1 ∂p f(xk , p) ∗ k0 (xk0 ,p0 )  (2,0) (2) T i T ∗ bi = (v ) · ∂p f(xk , p)(x∗ ,p0 ) , ai(ii) = G i(ii) (xk0 , p0 ) = 0. i T

k0

(1.159) If Eqs. (1.156)–(1.159) have two solutions of fixed-points, such a bifurca∗ , p0 ) is called a monotonic saddle transcritical (or switching) tion at point (xk0 bifurcation of the second order on the eigenvector of vi . (ii1 ) If (2) 1 2 1 2 ∗ λi0 = 1, ai((2,0) j1 j2 ) z k z k = G i( j1 j2 ) (x0 , p0 )z k z k > 0 for j1 , j2 = 1, 2, . . . , n, j

j

j

j

 (1.160)

58

1 Local Stability and Bifurcations ∗ such a bifurcation of fixed-point at point (xk0 , p0 ) is called a monotonic, increasing-saddle-node (or a monotonic upper-saddle-node) transcritical switching bifurcation of the second-order on the eigenvector vi . ∗ The bifurcation point at (xk0 , p0 ) is a monotonic, increasing-saddle (or a monotonic-upper-saddle) of the second-order on the eigenvector vi . (ii2 ) If (2) 1 2 1 2 ∗ λi0 = 1, ai((2,0) j1 j2 ) z k z k = G i( j1 j2 ) (x0 , p0 )z k z k < 0 for j1 , j2 = 1, 2, . . . , n; j

j

j

j

 (1.161)

∗ such a bifurcation of fixed-point at (xk0 , p0 ) is called a monotonic, decreasing-saddle (or a lower-saddle-node) transcritical switching bifurcation of the second order on the eigenvector vi . The bifurcation point at ∗ , p0 ) is a monotonic decreasing saddle (or a monotonic-lower-saddle) (xk0 of the second-order on the eigenvector vi .

(iii) Consider λ j0 = 1 ( j = 1, 2, . . . , n but j = i)

(1.162)

with  (λ j0 − 1)z j∗ + bTj · (p − p0 ) = 0, j = 1, 2, . . . , n but j = i; j∗ j∗ ∗ ∗ + y∗ , p0 + p) − xk0 ] = z k+1 ≡ z k with λi0 = 1; (vi )T · [f(xk0 ⎫ q 3   1 r (q−r,r ) ⎪ jq−r ∗ j1 ∗ j2 ∗ T r ⎪ C a z z k · · · z k (p − p0 ) + bk · (p − p0 ) ≈ 0 ⎪ ⎪ q! q i( j1 j2 ··· jq−r ) k ⎪ ⎬ q=2 r =0 equivalent to ⎪ j∗ j∗ j∗ ⎪ ⎪ Ai0 (z k )3 + Ai1 (z k )2 + Ai2 z k + Ai3 = 0, ⎪ ⎪ ⎭ 1∗ 2∗ i−1∗ i+1∗ n∗ ,z , · · · , z , p) (l = 0, 1, 2, 3); Ail = Ail (z , z , · · · , z (3,0) Ai3 = Ai1 = 0, Ai0 = ai(iii) = 0, Ai2 × Ai0 < 0, (1.163) where   (3) i T (0) = (v ) · ∂ ∂ f(x , p) ai((3,0)  k j j j j1 j2 j3 ) c 1c 2c 3 p

∗ (xk0 ,p0 )

(1,1) ai( j1 )

  = (vi )T · ∂c(3) j1 c j2 c j3 f(x,k p)

 (3) ∗ = (vi )T · ∂x(3) f(xk , p)(v j1 v j2 v j3 )(x∗ ,p ) = G i( j1 j2 j3 ) (xk0 , p0 ), k k0 0    (1) = (vi )T · ∂c(1) = (vi )T · ∂c j1 ∂p f(x, p)(x∗ ,p0 ) j1 ∂p f(xk , p) ∗ 0 (x ,p0 )  k0 i T  = (v ) · ∂xk ∂p f(xk , p)v j1 (x∗ ,p0 ) ,

∗ (xk0 ,p0 )

k0

(3,0) ai(iii)

=

(3) ∗ G i(iii) (xk0 , p0 )

= 0. (1.164)

1.4 Local Bifurcations

59

∗ If Eqs. (1.162)–(1.164) have three solutions, such a bifurcation at point (xk0 , p0 ) is called a monotonic pitchfork switching bifurcation on the eigenvector of vi .

(iii1 ) If ⎫ j1 j2 j3 i ⎪ λi0 = 1, z ki × ai((3,0) j1 j2 j3 ) z k z k z k = z k ⎬ j1 j2 j3 (3) ∗ ×G i( j1 j2 j3 ) (xk0 , p0 )z k z k z k < 0 ⎪ ⎭ for j1 , j2 , j3 = 1, 2, . . . , n,

(1.165)

∗ such a bifurcation at point (xk0 , p0 ) is called a monotonically stable pitchfork (or a monotonic sink) switching bifurcation of the third-order ∗ , p0 ) is a monotonic on the eigenvector vi . The bifurcation point at (xk0 sink of the third-order on the eigenvector vi . (iii2 ) If ⎫ j1 j2 j3 i ⎪ λi0 = 1, z ki × ai((3,0) j1 j2 j3 ) z k z k z k = z k ⎬ j1 j2 j3 (3) ∗ (1.166) ×G i( j1 j2 j3 ) (xk0 , p0 )z k z k z k > 0 ⎪ ⎭ for j1 , j2 , j3 = 1, 2, . . . , n, ∗ such a bifurcation at point (xk0 , p0 ) is called the monotonic unstable pitchfork (or a monotonic source) bifurcation of the third-order on the ∗ , p0 ) is a monotonic source eigenvector vi . The bifurcation point at (xk0 of third-order on the eigenvector vi .

1.4.2 One-Step Oscillatory Bifurcations Definition 1.32 Consider a nonlinear discrete system xk+1 = f(xk , p) ∈ R n in Eq. (1.4) with a fixed-point xk∗ . The corresponding solution is given by xk+ j = f(xk+ j−1 , p) with j ∈ Z. Suppose there is a neighborhood of the fixed-point xk∗ (i.e., Uk (xk∗ ) ⊂ ), and f(xk , p) is C r (r ≥ 1)-continuous in Uk (xk∗ ) with Eq. (1.28). The linearized system is yk+ j+1 = Df(xk∗ , p)yk+ j (yk+ j = xk+ j − xk∗ ) in Uk (xk∗ ) and there are n linearly independent vectors vi (i = 1, 2, . . . , n). For a perturbation of i vi . fixed-point yk = xk − xk∗ , let yk = cki vi and yk+1 = ck+1 For a specific j = i (i, j ∈ {1, 2, . . . , n} ∪ {∅}), the fixed-point equation in the vicinity of (xk0 , p0 ) is ∗ ∗ (v j )T · [f(xk0 + yk∗ , p0 + p) − xk0 ] = zk j∗ T (or (λ j0 − 1)z k + b j · (p − p0 ) ≈ 0), ( j = 1, 2, . . . , n but j = i), λ j0 = ±1.

j∗

(1.167)

60

1 Local Stability and Bifurcations

For a specific i, if ai(i) = λ0i = −1 at (xk0 , p0 ), the fixed-point equation is i∗ ∗ ∗ (vi )T · [f(xk0 + yk∗ , p0 + p) − xk0 ] = z k+1 = −z ki∗ with λi0 = −1; i∗ T (or (λi0 + 1)z k + bi · (p − p0 ) q si   j ∗ j ∗ j ∗ 1 r (q−r,r ) + C a z 1 z k1 · · · z kq−r (p − p0 )r ≈ 0) q! q i( j1 j2 ··· jq−r ) k q=2 r =0

equivalent to Ai0 (z ki )si + Ai1 (z ki )si −1 + · · · + Ais1 = 0 Ail = Ail (z k1∗ , z k2∗ , · · · , z ki−1∗ , z ki+1∗ , · · · , z kn∗ , p), (l = 0, 1, 2, . . . , si ); i = 1, 2, . . . , n but i = j.

(1.168)

Equations (1.167) and (1.168) give li -fixed-points z ki∗ (0 < li ≤ si ) with = m i ≤ si for αs -repeated fixed-point on the eigenvector vi direction (i ∈ {1, 2, . . . , n} ∪ {∅}). li s=1 αs

(i) If (ri ) ∗ λi0 = −1, G i( j1 j2 ... jr ) (xk0 , p0 ) = 0 (ri = 2, 3, . . . , m i − 1), k

(m i ) 1 1 ∗ G i( j1 j2 ... jm ) (xk0 , p0 )z k z k · · · z k j

jm i

j

i

 (1.169)

> 0,

∗ , p0 ) is called an oscillatorythe bifurcation of fixed-point xk∗ at point (xk0 increasing-saddle-node (or an oscillatory-upper-saddle-node) bifurcation of the ∗ , p0 ) on the eigen(m i )th order on the eigenvector vi . The bifurcation point (xk0 vector vi is an increasing-saddle (or an oscillatory-upper-saddle) of the (m i )th order

(ii) If (ri ) ∗ λi0 = −1, G i( j1 j2 ... jr ) (xk0 , p0 ) = 0(ri = 2, 3, . . . , m i − 1), k

(m i ) 1 1 ∗ G i( j1 j2 ... jm ) (xk0 , p0 )z k z k · · · z k j

jm i

j

i

 (1.170)

< 0,

∗ , p0 ) is called an oscillatorythe bifurcation of fixed-point xk∗ at point (xk0 decreasing-saddle-node (or an oscillatory-lower-saddle-node) bifurcation of the ∗ , p0 ) on the eigen(m i )th order on the eigenvector vi . The bifurcation point (xk0 vector vi is an oscillatory-decreasing-saddle (or an oscillatory-lower-saddle) of the (m i )th order

(iii) If (ri ) ∗ λi0 = −1, G i( j1 j2 ... jr ) (xk0 , p0 ) = 0(ri = 2, 3, . . . , m k − 1), i

(m i ) 1 2 ∗ z ki × G i( j1 j2 ... jm ) (xk0 , p0 )z k z k · · · z k j

i

j

jm i

> 0,

 (1.171)

1.4 Local Bifurcations

61

∗ the bifurcation of fixed-point xk∗ at point (xk0 , p0 ) is called an oscillatory-sink bifurcation of the (m i )th order on the eigenvector vi . The bifurcation point ∗ , p0 ) on the eigenvector vi is an oscillatory-sink of the (m i )th order. (xk0

(iv) If (ri ) ∗ λi0 = −1, G i( j1 j2 ... jr ) (xk0 , p0 ) = 0(ri = 2, 3, . . . , m k − 1), i

(m i ) 1 2 ∗ z ki × G i( j1 j2 ... jm ) (xk0 , p0 )z k z k · · · z k j

i

j

jm i

< 0,

 (1.172)

∗ , p0 ) is called an oscillatory-source the bifurcation of fixed-point xk∗ at point (xk0 bifurcation of the (m i )th order on the eigenvector vi . The bifurcation point ∗ , p0 ) on the eigenvector vi is an oscillatory-source of the (m i )th order. (xk0

The following discussions are given for a few special cases of oscillatory bifurcations. Definition 1.33 Consider a nonlinear discrete system xk+1 = f(xk , p) ∈ R n in Eq. (1.4) with a fixed-point xk∗ . The corresponding solution is given by xk+ j = f(xk+ j−1 , p) with j ∈ Z. Suppose there is a neighborhood of the fixed-point xk∗ (i.e., Uk (xk∗ ) ⊂ ), and f(xk , p) is C r (r ≥ 1)-continuous in Uk (xk∗ ) with Eq. (1.28). The linearized system is yk+ j+1 = Df(xk∗ , p)yk+ j (yk+ j = xk+ j − xk∗ ) in Uk (xk∗ ) and there are n linearly independent vectors vi (i = 1, 2, . . . , n). For a perturbation of i vi . Three special cases of fixed-point yk = xk − xk∗ , let yk = cki vi and yk+1 = ck+1 bifurcations are defined as follows. (i) Consider λ j0 = ±1( j = 1, 2, . . . , n but j = i)

(1.173)

with ∗ ∗ (v j )T · [f(xk0 + yk∗ , p0 + p) − xk0 ] = z k+1 = z k j∗ T (or (λ j0 − 1)z k + b j · (p − p0 ) ≈ 0), ( j = 1, 2, . . . , n but j = i), λ j0 = ±1; j∗

j∗

(1.174)

i∗ ∗ ∗ (vi )T · [f(xk0 + yk∗ , p0 + p) − xk0 ] = z k+1 ≡ −z ki∗ with λ0i = −1; j1 ∗ (1,1) 1 (0,2) 2 (or (λ0i + 1)z ki∗ + biT · (p − p0 ) + ai( j1 ) (p − p0 )z k + 2! ai( j1 ) (p − p0 ) j1 ∗ j2 ∗ 1 (2,0) + 2! ai( j1 j2 ) z k z k = 0) equivalent to ⎫ i∗ −z ki∗ = z k+1 = −z ki∗ + Ai0 (z ki∗ )2 + Ai1 (z ki∗ ) + Ai2 ⎪ ⎪ ⎬ ⇒ Ai0 (z ki )2 + Ai1 (z ki ) + Ai2 = 0, i−1∗ i+1∗ n∗ Ail = Ail (z k1∗ , z k2∗ , · · · , z k , z k , · · · , z k , p), ⎪ ⎪ ⎭ (l = 0, 1, 2) 2 i = Bi1 − 4Ci1 ≥ 0 with Bi1 = AAi0i1 , Ci1 = AAi2i0 , (2,0) = 0, Ci1 ≤ 0; Ai1 = 0, Ai0 = ai(ii) (1.175)

62

1 Local Stability and Bifurcations

where   (2) i T (0) = (v ) · ∂ ∂ f(x , p) ai((2,0)  k j1 j2 ) c j1 c j2 p

∗ (xk0 ,p0 )

  = (vi )T · ∂c(2) f(x , p)  k j1 c j2

∗ (xk0 ,p0 )

 ∗ = (vi )T · ∂x(2) f(xk , p)(v j1 v j1 )(x∗ ,p ) = G (2) k( j1 j2 ) (xk0 , p0 ), k 0 0    (1,1) (1) (1) i T ai( = (vi )T · ∂xk ∂p f(xk , p)v j1 (x∗ ,p0 ) , j1 ) = (v ) · ∂c j1 ∂p f(xk , p) ∗ k0 (xk0 ,p0 )  (2,0) (2) T i T ∗  bi = (v ) · ∂p f(xk , p) (x∗ ,p0 ) , ai(ii) = G i(ii) (xk0 , p0 ) = 0. k0

(1.176) If Eqs. (1.173)–(1.176) has two fixed-point solutions, such a bifurcation at point ∗ (xk0 , p0 ) is called an oscillatory-saddle-node bifurcation on the eigenvector vi . (i1 ) If (2) 1 2 1 2 ∗ λi0 = −1, ai((2,0) j1 j2 ) z k z k = G i( j1 j2 ) (x0 , p0 )z k z k > 0 for j1 , j2 = 1, 2, . . . , n, j

j

j

j

 (1.177)

∗ such a bifurcation at point (xk0 , p0 ) is called an oscillatory, increasing saddle-node (or an oscillatory-upper-saddle-node) appearing bifurcation of ∗ , p0 ) the second-order on the eigenvector vi . The bifurcation point at (xk0 is an oscillatory increasing saddle (or an oscillatory-upper-saddle) of he second-order on the eigenvector vi . (i2 ) If (2) 1 2 1 2 ∗ λi0 = −1, ai((2,0) j1 j2 ) z k z k = G i( j1 j2 ) (x0 , p0 )z k z k < 0 for j1 , j2 = 1, 2, . . . , n, j

j

j

j

 (1.178)

∗ such a bifurcation at point (xk0 , p0 ) is called an oscillatory, decreasing saddle-node (or an oscillatory lower-saddle-node) appearing bifurcation ∗ , p0 ) of the second order on the eigenvector vi . The bifurcation point at (xk0 is an oscillatory, decreasing saddle (or an oscillatory-lower-saddle) of the second order on the eigenvector vi .

(ii) Consider λ j = ±1( j = 1, 2, . . . , n but j = i)

(1.179)

with ∗ ∗ (v j )T · [f(xk0 + yk∗ , p0 + p) − xk0 ] = z k+1 = z k j∗

j∗

j∗

(or (λ0 j − 1)z k + bTj · (p − p0 ) ≈ 0), ( j = 1, 2, . . . , n but j = k), λ0 j = ±1;

(1.180)

1.4 Local Bifurcations

63

i∗ ∗ ∗ (vi )T · [f(xk0 + yk∗ , p0 + p) − xk0 ] = z k+1 ≡ −z ki∗ with λ0i = −1; j ∗ (1,1) (102) 2 (or (λ0i + 1)z ki∗ + biT · (p − p0 ) + ai( j1 ) (p − p0 )z k1 + 2!1 ai( j1 ) (p − p0 ) j1 ∗ j2 ∗ 1 (2,0) + 2! ai( j1 j2 ) z k z k = 0 equivalent to ⎫ i∗ = −z ki∗ + Ai0 (z ki∗ )2 + Ai1 (z ki∗ ) + Ai2 ⎪ −z ki∗ = z k+1 ⎪ ⎬ ⇒ Ai0 (z ki )2 + Ai1 (z ki ) + Ai2 = 0, i−1∗ i+1∗ n∗ 1∗ 2∗ Ail = Ail (z k , z k , . . . , z k , z k , . . . , z k , p), ⎪ ⎪ ⎭ (l = 0, 1, 2) 2 k1 k = Bk1 − 4Ck1 ≥ 0 with Bk1 = AAk0 , Ck1 = AAk2 , k0 (2,0) Ak2 = 0, Ak0 = ak(kk) = 0, Ak1 ∈ (−∞, ∞), (1.181)

where   (2) i T (0) ai((2,0) j1 j2 ) = (v ) · ∂c j1 c j2 ∂p f(xk , p) = (v ) ·

∗ (xk0 ,p0 )



  = (vi )T · ∂c(2) j1 c j2 f(xk , p)

∂x(2) f(xk , p)(v j1 v j1 )(x∗ ,p ) k 0 0

∗ (xk0 ,p0 )

∗ G (2) k( j1 j2 ) (xk0 , p0 ),

=    (1,1) (1) (1) i T ai( = (vi )T · ∂xk ∂p f(xk , p)v j1 (x∗ ,p0 ) , j1 ) = (v ) · ∂c j1 ∂p f(xk , p) ∗ k0 (xk0 ,p0 )  (2,0) (2) T i T ∗ bi = (v ) · ∂p f(xk , p)(x∗ ,p0 ) , ai(ii) = G i(ii) (xk0 , p0 ) = 0. i T

k0

(1.182) If Eqs. (1.179)–(1.182) have two solutions of fixed-points, such a bifurcation ∗ , p0 ) is called an oscillatory saddle transcritical (or switching) at point (xk0 bifurcation of the second-order on the eigenvector of vi . (ii1 ) If (2) 1 2 1 2 ∗ λi0 = −1, ai((2,0) j1 j2 ) z k z k = G i( j1 j2 ) (x0 , p0 )z k z k > 0 for j1 , j2 = 1, 2, . . . , n, j

j

j

j

 (1.183)

∗ such a bifurcation of fixed-point at point (xk0 , p0 ) is called an oscillatory, increasing-saddle (or an oscillatory-upper-saddle-node) transcritical switching bifurcation of the second order on the eigenvector vi . The ∗ , p0 ) is an oscillatory, increasing saddle (or an bifurcation point at (xk0 oscillatory-upper-saddle) of the second-order on the eigenvector vi . (ii2 ) If (2) 1 2 1 2 ∗ λi0 = −1, ai((2,0) j1 j2 ) z k z k = G i( j1 j2 ) (x0 , p0 )z k z k < 0 for j1 , j2 = 1, 2, . . . , n, j

j

j

j

 (1.184)

64

1 Local Stability and Bifurcations ∗ such a bifurcation of fixed-point at (xk0 , p0 ) is called an oscillatory, decreasing saddle (or an oscillatory-lower-saddle-node) transcritical switching bifurcation of the second-order on the eigenvector vi . The ∗ bifurcation point at (xk0 , p0 ) is an oscillatory decreasing saddle (or an oscillatory-lower-saddle) of the second-order on the eigenvector vi .

(iii) Consider λ j = ±1( j = 1, 2, . . . , n but j = i)

(1.185)

with  (λ j − 1)z j∗ + bTj · (p − p0 ) = 0, j = 1, 2, . . . , n but j = i; j∗ j∗ ∗ ∗ + y∗ , p0 + p) − xk0 ] = z k+1 ≡ −z k with λi0 = −1; ⎫ (vi )T · [f(xk0 q 3   1 r (q−r,r ) ⎪ j ∗ j ∗ j ∗ ⎪ C a z 1 z k2 · · · z kq−r (p − p0 )r + bTk · (p − p0 ) ≈ 0 ⎪ ⎪ q! q i( j1 j2 ··· jq−r ) k ⎪ ⎬ q=2 r =0 equivalent to ⎪ j∗ j∗ j∗ ⎪ ⎪ Ai0 (z k )3 + Ai1 (z k )2 + Ai2 z k + Ai3 = 0, ⎪ ⎪ ⎭ Ail = Ail (z 1∗ , z 2∗ , · · · , z i−1∗ , z i+1∗ , · · · , z n∗ , p) (l = 0, 1, 2, 3); (3,0) Ai3 = Ai1 = 0, Ai0 = ai(iii) = 0, Ai2 × Ai0 < 0, (1.186) where   (3) i T (0) = (v ) · ∂ ∂ f(x , p) ai((3,0)  k j j j j1 j2 j3 ) c 1c 2c 3 p

∗ (xk0 ,p0 )

(1,1) ai( j1 )

  = (vi )T · ∂c(3) j1 c j2 c j3 f(x,k p)

 (3) ∗ = (vi )T · ∂x(3) f(xk , p)(v j1 v j2 v j3 )(x∗ ,p ) = G i( j1 j2 j3 ) (xk0 , p0 ), k k0 0    (1) = (vi )T · ∂c(1) = (vi )T · ∂c j1 ∂p f(x, p)(x∗ ,p0 ) j1 ∂p f(xk , p) ∗ 0 (x ,p0 )  k0 i T  = (v ) · ∂xk ∂p f(xk , p)v j1 (x∗ ,p0 ) ,

∗ (xk0 ,p0 )

k0

(3,0) (3) ∗ = G i(iii) (xk0 , p0 ) = 0. ai(iii)

(1.187) If Eqs. (1.185)–(1.187) have three solutions, such a bifurcation at point ∗ , p0 ) is called an oscillatory pitchfork appearing bifurcation on the (xk0 eigenvector of vi . (iii1 ) If ⎫ j1 j2 j3 i ⎪ λi0 = −1, z ki × ai((3,0) z z z = z k⎬ j1 j2 j3 ) k k k j1 j2 j3 (3) ∗ ×G i( (x , p )z z z > 0 0 k k k j1 j2 j3 ) k0 ⎪ ⎭ for j1 , j2 , j3 = 1, 2, . . . , n,

(1.188)

1.4 Local Bifurcations

65

∗ such a bifurcation at point (xk0 , p0 ) is called an oscillatory stable pitchfork (or an oscillatory sink) bifurcation of the third-order on the eigen∗ , p0 ) is an oscillatory sink of the vector vi . The bifurcation point at (xk0 third-order on the eigenvector vi . (iii2 ) If ⎫ j1 j2 j3 i ⎪ λi0 = −1, z ki × ai((3,0) j1 j2 j3 ) z k z k z k = z k ⎬ j1 j2 j3 (3) ∗ (1.189) ×G i( j1 j2 j3 ) (xk0 , p0 )z k z k z k < 0 ⎪ ⎭ for j1 , j2 , j3 = 1, 2, . . . , n, ∗ such a bifurcation at point (xk0 , p0 ) is called an oscillatory unstable pitchfork (or an oscillatory source) bifurcation of the third-order on the ∗ , p0 ) is an oscillatory source eigenvector vi . The bifurcation point at (xk0 of the third-order on the eigenvector vi .

1.4.3 Two-Step Oscillatory Bifurcations Definition 1.34 Consider a nonlinear discrete system xk+1 = f(xk , p) ∈ R n in Eq. (1.4) with a fixed-point xk∗ . The corresponding solution is given by xk+ j = f(xk+ j−1 , p) with j ∈ Z. Suppose there is a neighborhood of the fixed-point xk∗ (i.e., Uk (xk∗ ) ⊂ ), and f(xk , p) is C r (r ≥ 1)-continuous in Uk (xk∗ ) with Eq. (1.28). The linearized system is yk+ j+1 = Df(xk∗ , p)yk+ j (yk+ j = xk+ j − xk∗ ) in Uk (xk∗ ) and there are n linearly independent vectors vi (i = 1, 2, . . . , n). For a perturbation of i vi . fixed-point yk = xk − xk∗ , let yk = cki vi and yk+1 = ck+1 For a specific j = i (i, j ∈ {1, 2, . . . , n} ∪ {∅}), the fixed-point equation in the vicinity of (xk0 , p0 ) is ∗ ∗ (v j )T · [f(xk0 + yk∗ , p0 + p) − xk0 ] = z k+1 , j∗ j∗ ∗ ∗ ∗ j T (v ) · [f(xk0 + yk+1 , p0 + p) − xk0 ] = z k+2 = z k ; or j∗ j∗ λ j0 z k + bTj · (p − p0 ) ≈ z k+1 , j∗ j∗ j∗ λ j0 z k+1 + bTj · (p − p0 ) ≈ z k+2 = z k ; ( j = 1, 2, . . . , n but j = i), λ j0 = ±1. j∗

(1.190)

For a specific i, if ai(i) = λ0i = −1 at (xk0 , p0 ), the fixed-point equation is

66

1 Local Stability and Bifurcations

 i∗ ∗ ∗ (vi )T · [f(xk0 + yk∗ , p0 + p) − xk0 ] = z k+1 , i∗ ∗ ∗ ∗ + yk+1 , p0 + p) − xk0 ] = z k+2 ≡ z ki∗ (vi )T · [f(xk0 with λi0 = −1; or ⎫ q si   jq−r ∗ j1 ∗ j2 ∗ 1 r (q−r,r ) r ⎪ ⎪ C a z z · · · z (p − p ) ⎪ 0 k k ⎪ q! q i( j1 j2 ··· jq−r ) k ⎪ q=2 r =0 ⎪ ⎪ ⎬ i∗ +λi0 z ki∗ + biT · (p − p0 ) ≈ z k+1 , q si   jq−r ∗ ⎪ j ∗ j ∗ 1 r (q−r,r ) C a z 1 z 2 · · · z k+1 (p − p0 )r ⎪ ⎪ q! q i( j1 j2 ··· jq−r ) k+1 k+1 ⎪ ⎪ q=2 r =0 ⎪ ⎪ ⎭ i∗ i∗ i∗ T +λi0 z k+1 + bi · (p − p0 ) ≈ z k+2 ≡ z k ; equivalent to i∗ λi0 z ki∗ + Ai0 (z ki )si + Ai1 (z ki )si −1 + · · · + Ais1 = z k+1 , i∗ i i i∗ si si −1 + · · · + Ais1 = z k+2 ≡ z ki∗ λi0 z k+1 + Ai0 (z k+1 ) + Ai1 (z k+1 ) i−1∗ i+1∗ n∗ 1∗ 2∗ Ail = Ail (z k , z k , · · · , z k , z k , · · · , z k , p), (l = 0, 1, 2, . . . , si ); i = 1, 2, . . . , n but i = j.

(1.191)

i∗ (0 < Equations (1.190) and (1.191) give li -sets of fixed-points for z ki∗ and z k+1 li li ≤ si ) with s=1 αs = m i ≤ si for αs -repeated fixed-point on the eigenvector vi direction (i ∈ {1, 2, . . . , n} ∪ {∅}).

(i) If (ri ) ∗ λi0 = −1, G i( j1 j2 ... jr ) (xk0 , p0 ) = 0 (ri = 1, 2, 3, . . . , m i − 1), k

(m i ) 1 1 ∗ G i( j1 j2 ... jm ) (xk0 , p0 )z k z k · · · z k j

j

jm i

i

 (1.192)

> 0,

∗ ∗ at point (xk0 , p0 ) is called an the bifurcation of fixed-points xk∗ and xk+1 oscillatory-increasing-saddle-node (or an oscillatory-upper-saddle-node) bifur∗ , p0 ) cation of the (m i )th order on the eigenvector vi . The bifurcation point (xk0 on the eigenvector vi is an increasing saddle (or an oscillatory-upper-saddle) of the (m i )th order

(ii) If (ri ) ∗ λi0 = −1, G i( j1 j2 ... jr ) (xk0 , p0 ) = 0 (ri = 1, 2, 3, . . . , m i − 1), k

(m i ) 1 1 ∗ G i( j1 j2 ... jm ) (xk0 , p0 )z k z k · · · z k j

i

j

jm i

< 0,

 (1.193)

∗ , p0 ) is called an oscillatorythe bifurcation of fixed-point xk∗ at point (xk0 decreasing-saddle-node (or an oscillatory-lower-saddle-node) bifurcation of the ∗ , p0 ) on the eigen(m i )th order on the eigenvector vi . The bifurcation point (xk0 vector vi is an oscillatory-decreasing-saddle (or an oscillatory-lower-saddle) of the (m i )th order

1.4 Local Bifurcations

67

(iii) If (ri ) ∗ λi0 = −1, G i( j1 j2 ... jr ) (xk0 , p0 ) = 0 (ri = 1, 2, 3, . . . , m k − 1), i

(m i ) 1 2 ∗ z ki × G i( j1 j2 ... jm ) (xk0 , p0 )z k z k · · · z k j

j

jm i

i

 (1.194)

> 0,

∗ , p0 ) is called an oscillatorythe bifurcation of fixed-point xk∗ at point (xk0 sink bifurcation the (m i )th order on the eigenvector vi . The bifurcation point ∗ , p0 ) on the eigenvector vi is an oscillatory-sink of the (m i )th order. (xk0

(iv) If (ri ) ∗ λi0 = −1, G i( j1 j2 ... jr ) (xk0 , p0 ) = 0 (ri = 1, 2, 3, . . . , m k − 1), i

(m i ) 1 2 ∗ z ki × G i( j1 j2 ... jm ) (xk0 , p0 )z k z k · · · z k j

i

j

jm i

< 0,

 (1.195)

∗ , p0 ) is called an oscillatory-source the bifurcation of fixed-point xk∗ at point (xk0 bifurcation of the (m i )th order on the eigenvector vi . The bifurcation point ∗ , p0 ) on the eigenvector vi is an oscillatory-source of the (m i )th order. (xk0

The following discussions are given for a few special cases of oscillatory bifurcations. Definition 1.35 Consider a nonlinear discrete system xk+1 = f(xk , p) ∈ R n in Eq. (1.4) with a fixed-point xk∗ . The corresponding solution is given by xk+ j = f(xk+ j−1 , p) with j ∈ Z. Suppose there is a neighborhood of the fixed-point xk∗ (i.e., Uk (xk∗ ) ⊂ ), and f(xk , p) is C r (r ≥ 1)-continuous in Uk (xk∗ ) with Eq. (1.28). The linearized system is yk+ j+1 = Df(xk∗ , p)yk+ j (yk+ j = xk+ j − xk∗ ) in Uk (xk∗ ) and there are n linearly independent vectors vi (i = 1, 2, . . . , n). For a perturbation of i vi . Three special cases of fixed-point yk = xk − xk∗ , let yk = cki vi and yk+1 = ck+1 bifurcations are defined as follows. (i) Consider λ j0 = ±1( j = 1, 2, . . . , n but j = i)

(1.196)

with ∗ ∗ (v j )T · [f(xk0 + yk∗ , p0 + p) − xk0 ] = z k+1 , j∗ j∗ ∗ ∗ ∗ j T (v ) · [f(xk0 + yk+1 , p0 + p) − xk0 ] = z k+2 = z k ; or j∗ j∗ λ j0 z k + bTj · (p − p0 ) ≈ z k+1 , j∗ j∗ λ j0 z k+1 + bTj · (p − p0 ) ≈ z k ; ( j = 1, 2, . . . , n but j = i), λ j0 = 1; j∗

(1.197)

68

1 Local Stability and Bifurcations i∗ ∗ ∗ (vi )T · [f(xk0 + yk∗ , p0 + p) − xk0 ] = z k+1 i∗ ∗ ∗ ∗ i T (v ) · [f(xk0 + yk+1 , p0 + p) − xk0 ] = z k+2 ≡ z ki∗ with λi0 = −1; j1 ∗ (1,1) 1 (0,2) 2 λi0 z ki∗ + biT · (p − p0 ) + ai( j1 ) (p − p0 )z k + 2! ai( j1 ) (p − p0 ) j1 ∗ j2 ∗ 1 (2,0) i∗ + 2! ai( j1 j2 ) z k z k = z k+1 j1 ∗ (1,1) 1 (0,2) i∗ 2 λi0 z k+1 + biT · (p − p0 ) + ai( j1 ) (p − p0 )z k+1 + 2! ai( j1 ) (p − p0 ) j1 ∗ j2 ∗ 1 (2,0) i∗ i∗ + 2! ai( j1 j2 ) z k+1 z k+1 = z k+2 ≡ z k equivalent to ⎫ i∗ λi0 z ki∗ + Ai0 (z ki∗ )2 + Ai1 (z ki∗ ) + Ai2 = z k+1 ⎪ ⎪ ⎪ i∗ i i i∗ λi0 z k+1 + Ai0 (z k+1 )2 + Ai1 (z k+1 ) + Ai2 = z k+2 ≡ z ki∗ , ⎪ ⎪ ⎪ ⎪ i−1∗ i+1∗ ⎪ n∗ 1∗ 2∗ ⎪ Ail = Ail (z k , z k , . . . , z k , z k , . . . , z k , p), ⎬ (l = 0, 1, 2) ⎪ 2 ⎪ − 4Ci1 ≥ 0 with Bi1 = AAi0i1 , Ci1 = AAi2i0 , i = Bi1 ⎪ ⎪ ⎪ ⎪ (2,0) ⎪ Ai1 = 0, Ai0 = ai(ii) = 0, Ci1 ≤ 0; ⎪ ⎪ ⎭

(1.198)

where   (2) i T (0) ai((2,0) = (v ) · ∂ ∂ f(x , p)  k j j j1 j2 ) c 1c 2 p

∗ (xk0 ,p0 )

  = (vi )T · ∂c(2) j1 c j2 f(xk , p)

∗ (xk0 ,p0 )

 (2) ∗ = (vi )T · ∂x(2) f(xk , p)(v j1 v j1 )(x∗ ,p ) = G i( j1 j2 ) (xk0 , p0 ), k 0 0    (1,1) (1) (1) i T ai( = (vi )T · ∂xk ∂p f(xk , p)v j1 (x∗ ,p0 ) , j1 ) = (v ) · ∂c j1 ∂p f(xk , p) ∗ k0 (xk0 ,p0 )  (2,0) (2) ∗ , p0 ) = 0. biT = (vi )T · ∂p f(xk , p)(x∗ ,p0 ) , ai(ii) = G i(ii) (xk0 k0

(1.199) If Eqs. (1.196)–(1.199) have two fixed-point solutions, such a bifurcation at point ∗ (xk0 , p0 ) is called an oscillatory-saddle-node bifurcation on the eigenvector vi . (i1 ) If (2) 1 2 1 2 ∗ λi0 = −1, ai((2,0) j1 j2 ) z k z k = G i( j1 j2 ) (x0 , p0 )z k z k > 0 for j1 , j2 = 1, 2, . . . , n, j

j

j

j

 (1.200)

∗ such a bifurcation at point (xk0 , p0 ) is called an oscillatory, increasing saddle-node (or oscillatory-upper-saddle-node) appearing bifurcation on ∗ , p0 ) is an oscillatory the eigenvector vi . The bifurcation point at (xk0 increasing saddle (or oscillatory-upper-saddle) of the second-order on the eigenvector vi . (i2 ) If (2) 1 2 1 2 ∗ λi0 = −1, ai((2,0) j1 j2 ) z k z k = G i( j1 j2 ) (x0 , p0 )z k z k < 0 for j1 , j2 = 1, 2, . . . , n, j

j

j

j

 (1.201)

1.4 Local Bifurcations

69

∗ such a bifurcation at point (xk0 , p0 ) is called an oscillatory, decreasing saddle-node (or oscillatory lower-saddle-node) appearing bifurcation of ∗ , p0 ) is the second order on the eigenvector vi . The bifurcation point at (xk0 an oscillatory, decreasing saddle (or oscillatory-lower-saddle) of the second order on the eigenvector vi .

(ii) Consider λ j = ±1( j = 1, 2, . . . , n but j = i)

(1.202)

with ∗ ∗ (v j )T · [f(xk0 + yk∗ , p0 + p) − xk0 ] = z k+1 , j∗ j∗ ∗ ∗ ∗ j T (v ) · [f(xk0 + yk+1 , p0 + p) − xk0 ] = z k+2 = z k ; or (1.203) j∗ j∗ λ j0 z k + bTj · (p − p0 ) ≈ z k+1 , j∗ j∗ λ j0 z k+1 + bTj · (p − p0 ) ≈ z k ; ( j = 1, 2, . . . , n but j = i), λ j0 = 1; ⎫ i∗ ∗ ∗ + yk∗ , p0 + p) − xk0 ] = z k+1 (vi )T · [f(xk0 ⎬ i∗ ∗ ∗ ∗ (vi )T · [f(xk0 + yk+1 , p0 + p) − xk0 ] = z k+2 ≡ z ki∗ ⎭ with λ0i = −1; or ⎫ j1 ∗ (1,1) 1 (0,2) 2 λ0i z ki∗ + biT · (p − p0 ) + ai( ⎪ j1 ) (p − p0 )z k + 2! ai( j1 ) (p − p0 ) ⎪ ⎪ j1 ∗ j2 ∗ ⎬ i∗ + 2!1 ai((2,0) z z = z k+1 k j1 j2 ) k j1 ∗ (1,1) 1 (0,2) i∗ 2⎪ λ0i z k+1 + biT · (p − p0 ) + ai( j1 ) (p − p0 )z k+1 + 2! ai( j1 ) (p − p0 ) ⎪ ⎪ ⎭ j1 ∗ j2 ∗ 1 (2,0) i∗ i∗ + 2! ai( j1 j2 ) z k+1 z k+1 = z k+2 ≡ z k equivalent to ⎫ i∗ λ0i z ki∗ + Ai0 (z ki∗ )2 + Ai1 (z ki∗ ) + Ai2 = z k+1 ⎪ ⎪ ⎬ i∗ i i i∗ λ0i z k+1 + Ai0 (z k+1 )2 + Ai1 (z k+1 ) + Ai2 = z k+2 ≡ z ki∗ , i−1∗ i+1∗ 1∗ 2∗ n∗ Ail = Ail (z k , z k , . . . , z k , z k , . . . , z k , p),⎪ ⎪ ⎭ (l = 0, 1, 2) Ai1 Ai2 2 i = Bi1 − 4Ci1 ≥ 0 with Bi1 = Ai0 , Ci1 = Ai0 , (2,0) = 0, Ai1 ∈ (−∞, ∞), Ai2 = 0, Ai0 = ai(ii) (1.204) j∗

where

70

1 Local Stability and Bifurcations

  (2) i T (0) ai((2,0) j1 j2 ) = (v ) · ∂c j1 c j2 ∂p f(xk , p) = (v ) ·

∗ (xk0 ,p0 )



  = (vi )T · ∂c(2) j1 c j2 f(xk , p)

∂x(2) f(xk , p)(v j1 v j1 )(x∗ ,p ) k 0 0

∗ (xk0 ,p0 )

∗ G (2) k( j1 j2 ) (xk0 , p0 ),

=    (1,1) (1) (1) i T ai( = (vi )T · ∂xk ∂p f(xk , p)v j1 (x∗ ,p0 ) , j1 ) = (v ) · ∂c j1 ∂p f(xk , p) ∗ k0 (xk0 ,p0 )  (2,0) (2) T i T ∗ bi = (v ) · ∂p f(xk , p)(x∗ ,p0 ) , ai(ii) = G i(ii) (xk0 , p0 ) = 0. i T

k0

(1.205) If Eqs. (1.202)–(1.205) have two solutions of fixed-points, such a bifurcation at ∗ , p0 ) is called the oscillatory saddle transcritical switching bifurcation point (xk0 of the second-order on the eigenvector of vi . (ii1 ) If (2) 1 2 1 2 ∗ λi0 = −1, ai((2,0) j1 j2 ) z k z k = G i( j1 j2 ) (x0 , p0 )z k z k > 0 for j1 , j2 = 1, 2, . . . , n, j

j

j

j

 (1.206)

∗ such a bifurcation of fixed-point at point (xk0 , p0 ) is called an oscillatory, increasing-saddle (or an oscillatory-upper-saddle-node) transcritical switching bifurcation of the second order on the eigenvector vi . The ∗ , p0 ) is an oscillatory, increasing saddle (or an bifurcation point at (xk0 oscillatory-upper-saddle) of the second-order on the eigenvector vi . (ii2 ) If (2) 1 2 1 2 ∗ λi0 = −1, ai((2,0) j1 j2 ) z k z k = G i( j1 j2 ) (x0 , p0 )z k z k < 0 for j1 , j2 = 1, 2, . . . , n, j

j

j

j

 (1.207)

∗ such a bifurcation of fixed-point at (xk0 , p0 ) is called an oscillatory, decreasing-saddle (or an oscillatory-lower-saddle-node) transcritical switching bifurcation of the second order on the eigenvector vi . ∗ , p0 ) is an oscillatory decreasing saddle (or The bifurcation point at (xk0 oscillatory-lower-saddle) of an second order on the eigenvector vi .

(iii) Consider λ j = ±1( j = 1, 2, . . . , n but j = i)

(1.208)

with ⎫ j∗ j∗ ⎪ λ j0 z k + bTj · (p − p0 ) = z k+1 , ⎬ j∗ j∗ j∗ T λ j0 z k+1 + b j · (p − p0 ) = z k+2 ≡ z k ⎪ ⎭ j = 1, 2, . . . , n but j = i;

(1.209)

1.4 Local Bifurcations

71

⎫ j∗ ∗ ∗ vi )T · [f(xk0 + yk∗ , p0 + p) − xk0 ] = z k+1 ⎬ j∗ j∗ ∗ ∗ ∗ (vi )T · [f(xk0 + yk+1 , p0 + p) − xk0 ] = z k+2 ≡ −z k ⎭ with λi0 = −1; ⎫ q 3   j ∗ ⎪ j ∗ j ∗ 1 r (q−r,r ) C a z 1 z k2 · · · z kq−r (p − p0 )r ⎪ ⎪ q! q i( j1 j2 ··· jq−r ) k ⎪ ⎪ q=2 r =0 ⎪ ⎪ ⎬ j∗ j∗ T +λi0 z k + bk · (p − p0 ) ≈ z k+1 , q 3   ⎪ jq−r ∗ j ∗ j ∗ 1 r (q−r,r ) ⎪ C a z 1 z 2 · · · z k+! (p − p0 )r ⎪ ⎪ q! q i( j1 j2 ··· jq−r ) k+1 k+1 ⎪ ⎪ q=2 r =0 ⎪ ⎭ j∗ j∗ j∗ T +λi0 z k+1 + bk+1 · (p − p0 ) ≈ z k+2 ≡ z k , equivalent to ⎫ j∗ j∗ j∗ j∗ j∗ ⎪ λi0 z k + Ai0 (z k )3 + Ai1 (z k )2 + Ai2 z k + Ai3 = z k+1 , ⎪ ⎬ j∗ j∗ 3 j∗ 2 j∗ j∗ j∗ ⎪ λi0 z k+1 + Ai0 (z k+1 ) + Ai1 (z k+1 ) + Ai2 z k+1 + Ai3 = z k+2 ≡ z k , Ail = Ail (z 1∗ , z 2∗ , . . . , z i−1∗ , z i+1∗ , . . . , z n∗ , p) (l = 0, 1, 2, 3); ⎪ ⎪ ⎪ (3,0) ⎭ = 0, Ai2 × Ai0 < 0, Ai3 = Ai1 = 0, Ai0 = ai(iii) (1.210) where   (3) i T (0) ai((3,0) j1 j2 j3 ) = (v ) · ∂c j1 c j2 c j3 ∂p f(xk , p)

∗ (xk0 ,p0 )

= (v ) ·

∂x(3) f(xk , p)(v j1 v j2 v j3 )(x∗ ,p ) k k0 0

∗ (xk0 ,p0 )

(3) ∗ G i( j1 j2 j3 ) (xk0 , p0 ),

=    (1) i T j1 ∂p f(x, p) ∗ = (vi )T · ∂c(1) = (v ) · ∂ j1 ∂p f(xk , p) c (x0 ,p0 ) (x∗ ,p0 )  k0 i T  = (v ) · ∂xk ∂p f(xk , p)v j1 (x∗ ,p0 ) , i T

(1,1) ai( j1 )



  = (vi )T · ∂c(3) j1 c j2 c j3 f(x,k p)

k0

(3,0) (3) ∗ = G i(iii) (xk0 , p0 ) = 0. ai(iii)

(1.211) ∗ , p0 ) If Eqs. (1.208)–(1.210) have three solutions, such a bifurcation at point (xk0 is called an oscillatory pitchfork appearing bifurcation on the eigenvector of vi .

(iii1 ) If ⎫ j1 j2 j3 i ⎪ λi0 = −1, z ki × ai((3,0) j1 j2 j3 ) z k z k z k = z k ⎬ j1 j2 j3 (3) ∗ ×G i( j1 j2 j3 ) (xk0 , p0 )z k z k z k > 0 ⎪ ⎭ for j1 , j2 , j3 = 1, 2, . . . , n,

(1.212)

∗ , p0 ) is called an oscillatory stable pitchsuch a bifurcation at point (xk0 fork (or an oscillatory sink) bifurcation of the third order on the eigen∗ , p0 ) is an oscillatory sink of the vector vi . The bifurcation point at (xk0 third-order on the eigenvector vi

72

1 Local Stability and Bifurcations

(iii2 ) If ⎫ j1 j2 j3 i ⎪ λi0 = −1, z ki × ai((3,0) j1 j2 j3 ) z k z k z k = z k ⎬ j1 j2 j3 (3) ∗ ×G i( j1 j2 j3 ) (xk0 , p0 )z k z k z k < 0 ⎪ ⎭ for j1 , j2 , j3 = 1, 2, . . . , n,

(1.213)

∗ , p0 ) is called an oscillatory unstable such a bifurcation at point (xk0 pitchfork (or an oscillatory source) bifurcation of the third order on the ∗ , p0 ) is an oscillatory source eigenvector vi . The bifurcation point at (xk0 of the third-order on the eigenvector vi .

1.4.4 Neimark Bifurcations Definition 1.36 Consider a nonlinear discrete system xk+1 = f(xk , p) ∈ R n ∈ R n in Eq. (1.4) with a fixed-point xk∗ . The corresponding solution is given by xk+ j = f(xk+ j−1 , p) with j ∈ Z. Suppose there is a neighborhood of the fixed-point xk∗ (i.e., Uk (xk∗ ) ⊂ ), and f(xk , p) is C r (r ≥ 1)-continuous in Uk (xk∗ ) with Eq. (1.28). ∗ The linearized system is yk+ j+1 = Df(xk∗ , p)yk+ j (yk+ j = xk+ j − xk∗ ) in √Uk (xk ). Consider a pair of complex eigenvalues αi ± iβi (2i − 1, 2i ∈ N , i = −1 and N = {1, 2, . . . , n}) of matrix Df(xk∗ , p) with a pair of eigenvectors v2i−1 ± iv2i . On the invariant plane of (v2i−1 , v2i ), the contravariants are defined as ck2i−1 = (v2i−1 )T · yk , ck2i = (v2i )T · yk

(1.214)

Dxk f · v2i−1 = αi v2i−1 − βi v2i , Dxk f · v2i = βi v2i−1 + αi v2i .

(1.215)

with

Consider a polar coordinate of (rk , θk ) defined by = rki cos θki , and ck2i = rki sin θki ; ck2i−1  ρki = (ck2i−1 )2 + (ck2i )2 , and θki = arctan ck2i /ck2i−1 .

(1.216)

2i−1 ∗ = G 2i−1 (xk , p) = (v2i−1 )T · [f(xk , p) − xk0 ], ck+1 2i ∗ 2i T ck+1 = G 2i (xk , p) = (v ) · [f(xk , p) − xk0 ].

(1.217)

Thus

where

1.4 Local Bifurcations

73

∗ G 2i−1 (xk , p) = (v2i−1 )T · [f(xk , p) − xk0 ] T · (p − p0 ) + αi z k2i−1 + βi z k2i = a2i−1

+

m i >1  ri =0

(m i −ri ,ri ) ∗ ri 1 2 Cmri i G2i−1( j1 j2 ··· jm −r ) (xk0 , p0 )(p − p0 ) (z k z k · · · z k j

i

jm i −ri

j

i

)

+ (v2i−1 )T · [z k ∂xk (·)v j + (p − p0 )∂p (·))]m i +1 ∗ × f(xk0 + θ xk , p0 + θ p), j

(1.218) ∗ ] G 2i (xk , p) = (v2i )T · [f(xk , p) − xk0 T · (p − p0 ) − βi z k2i−1 + αi z k2i = a2i m i >1 jm i −ri j1 j2 (m i −ri ,ri ) ∗ ri + Cmri i G2i( ) j1 j2 ··· jm −r ) (xk0 , p0 )(p − p0 ) (z k z k · · · z k ri =0

+ (v ) · 2i T

i

j [z k ∂xk (·)v j

i

+ (p − p0 )∂p (·)]m i +1

∗ × f(xk0 + θ xk , p0 + θ p),

(1.219) and  (s,r ) ∗ 2i−1 T ) · ∂xsk ∂pr f(xk , p)(x∗ ,p ) (v j1 v j2 · · · v js ), G2i−1( j1 j2 ··· js ) (xk(0) , p0 ) = (v k0 0  G(s,r ) (x∗ , p0 ) = (v2i )T · ∂ s ∂ r f(xk , p) ∗ (v j v j · · · v j ), 2i( j1 j2 ··· js )

xk p

k(0)

(xk0 ,p0 )

1

2

s

T ∗ T ∗ = (v2i−1 )T · ∂p f(xk0 , p), a2i = (v2i )T · ∂p f(xk0 , p); a2i−1  (s) (s,0) 2i−1 T s ) · ∂xk f(xk , p)(x∗ ,p ) (v j1 v j2 · · · v js ), G 2i−1( j1 j2 ··· js ) = G2i−1( j1 j2 ··· js ) = (v k0 0  = G(s,0) = (v2i )T · ∂ s f(xk , p) ∗ (v j v j · · · v j ). G (s) 2i( j1 j2 ··· js )

2i( j1 j2 ··· js )

xk

(xk0 ,p0 )

1

2

s

(1.220) Suppose a2i−1 = 0 and a2i = 0

(1.221)

λ j = 1( j = 1, 2, . . . , n but j = 2i − 1, 2i)

(1.222)

then

with ∗ ∗ (v j )T · [f(xk0 + yk∗ , p0 + p) − xk0 ] = z k+1 = z k j∗ j∗ j∗ T λ j0 z k + b j · (p − p0 ) ≈ z k+1 = z k , ( j = 1, 2, . . . , n but j = 2i − 1, 2i), λ j0 = 1; j∗

and

j∗

(1.223)

74

1 Local Stability and Bifurcations

i ρk+1

 ∞   2i−1 2 2i = (ck+1 ) + (ck+1 )2 = ! (ρki )m G (m) ρi " =

i θk+1

=

k+1

m=2

 G (2,0) ρki !1 + γ2i + ρi k+1

∞ 

(1.224) γmi (ρki )m−2

m=3

2i−1 2i arctan(ck+1 /ck+1 )

where = G (2,0) + G (1,1) and γ2i = G (1,1) /G (2,0) with G (2) ρi ρi ρi ρi ρi k+1

G (2,0) i ρk+1

k+1

=

αi2

+

k+1

k+1

k+1

βi2

(1.225)

and γmi = G (m) /G (2,0) with i i ρk+1 ρk+1 m m   1 = Cmri i Cmsi i G (m) i i m{ ρ (ρ ) k+1

k m i =0 m j =0 jm −r j j (m i −ri ,ri ) [G2i−1( j1 j2 ··· jm −r ) (θki , p0 ) · (p − p0 )ri (z k1 z k2 · · · z k i i ) i i jm i −si (m j −s j ,s j ) j1 j2 i sj ×G2i−1( ) j1 j2 ··· jm−si ) (θk , p0 ) · (p − p0 ) (z k z k · · · z k jm i −ri j1 j2 (m i −ri ,ri ) i ri +[G2i( j1 j2 ··· jm−r ) (θk , p0 ) · (p − p0 ) (z k z k · · · z k ) i jm i −ri (m j −s j ,s j ) j1 j2 i sj ×G2i( j1 j2 ··· jm−r ) (θk , p0 ) · (p − p0 ) (z k z k · · · z k )}δm(2m i −ri +si ) . i

(1.226)

= αi2 + βi2 = 1 and p = p0 , the stability of current fixed-point xk∗ If G (2,0) i ρk+1 on an eigenvector plane of (v2i−1 , v2i ) changes from stable to unstable state (or from unstable to stable state). The bifurcation manifold on the eigenvector plane of (v2i−1 , v2i ) is determined by γ2i (ρki )2 +

M 

γmi (ρki )m = 0.

(1.227)

m=3

There are si -branches of solutions of ρki (si < M − 1). Such a bifurcation at the ∗ , p0 ) is called a generalized Neimark bifurcation on the eigenvector fixed-point (xk0 plane of (ui , vi ). (i) If G (2,0) = αi2 + βi2 = 1, ρi k+1

k) G (r (θki ) = γrik = 0 for rk = 2, 3, . . . , si − 1, ρi k+1

G ρ(sii ) (θki ) k+1

=

γsii

< 0,

(1.228)

1.4 Local Bifurcations

75

∗ the Neimark Bifurcation at (xk0 , p0 ) on the plane of (v2i−1 , v2i ) is called a stable Neimark bifurcation of the (si )th-order for θk(i) ∈ [0, 2π ].

(ii) If = αi2 + βi2 = 1, G (2,0) ρi k+1

k) G (r (θki ) = γrik = 0 for rk = 2, 3, . . . , si − 1, ρi k+1

G ρ(sii ) (θki ) k+1

=

γsii

(1.229)

> 0,

∗ , p0 ) on the plane of (v2i−1 , v2i ) is called an The Neimark Bifurcation at (xk0 unstable Neimark bifurcation of the (si )th-order for θk(i) ∈ [0, 2π ].

For a special case, if (ρki )2 [γ2i + γ4i (ρki )2 ] = 0, for γ2i × γ4i < 0 and γ3i = 0

(1.230)

such a bifurcation at point (x0∗ , p0 ) is called a Neimark bifurcation on the eigenvector plane of (ui , vi ). (i) If G (2,0) = αi2 + βi2 = 1, ρi k+1

G (3) (θki ) = γ3i = 0, ρi k+1

G (4) (θki ) i ρk+1

=

γ4i

(1.231)

< 0,

∗ , p0 ) on the plane of (v2i−1 , v2i ) is called a stable the Neimark Bifurcation at (xk0 Neimark bifurcation of the fourth-order for θk(i) ∈ [0, 2π ].

(ii) If G (2,0) = αi2 + βi2 = 1, ρi k+1

G (3) (θki ) = γ3i = 0, ρi k+1

G (4) (θki ) i ρk+1

=

γ4i

(1.232)

> 0,

∗ , p0 ) on the plane of (v2i−1 , v2i ) is called an The Neimark Bifurcation at (xk0 unstable Neimark bifurcation of the fourth-order for θk(i) ∈ [0, 2π ].

76

1 Local Stability and Bifurcations

References Guckenhiemer J, Holmes P (1990) Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, New-York Luo ACJ (2011) Regularity and complexity in dynamical systems. Springer, New York Nitecki Z (1971) Differentiable dynamics: an introduction to the orbit structures of diffeomorphisms. MIT Press, Cambridge, MA

Chapter 2

Low-Dimensional Discrete Systems

In this Chapter, as in Luo (2019, 2020) one-dimensional and two-dimensional nonlinear discrete systems are discussed from the local theory of stability and bifurcations in Chap. 1. The linear discrete systems are discussed first, and the higher order singularity and stability for one-dimensional and two-dimensional nonlinear discrete systems are presented. A few special cases in 1-dimensional maps are discussed for a better understanding of the local theory.

2.1 One-Dimensional Discrete Systems Definition 2.1 Consider a 1-dimensional, nonlinear discrete system xk+1 = f (xk , p)

(2.1)

with a fixed-point xk∗ . Suppose there is a neighborhood of the fixed-point xk∗ (i.e., Uk (xk∗ ) ⊂ ), and f (xk , p) is C r (r ≥ 1)-continuous in Uk (xk∗ ) = (xk∗ − δ, xk∗ + δ) for an arbitrary δ > 0. The corresponding solution is given by xk+ j = f (xk+ j−1 , p) with j ∈ Z. The linearized system in Uk (xk∗ ) is yk+ j+1 = D f (xk∗ , p)yk+ j

(2.2)

where yk+ j = xk+ j − xk∗ . Define the following functions G(xk , p) = f (xk , p) − xk∗

(2.3)

and G (1) (xk , p) = Dxk f (xk , p) =

d f (xk , p) , d xk

© Higher Education Press 2020 A. C. J. Luo, Bifurcation and Stability in Nonlinear Discrete Systems, Nonlinear Physical Science, https://doi.org/10.1007/978-981-15-5212-0_2

(2.4) 77

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2 Low-Dimensional Discrete Systems

G (m) (x, p) = D (m) f (xk , p), m = 1, 2, . . . .

(2.5)

2.2 Monotonic and Oscillatory Stability Definition 2.2 Consider a 1-dimensional nonlinear discrete system xk+1 = f (xk , p) in Eq. (2.1) with a fixed-point xk∗ . The corresponding solution is given by xk+ j = f (xk+ j−1 , p) with j ∈ Z. Suppose there is a neighborhood of the fixed-point xk∗ (i.e., Uk (xk∗ ) ⊂ ), and f (xk , p) is C r (r ≥ 1)-continuous in Uk (xk∗ ) = (xk∗ − δ, xk∗ + δ) for an arbitrary δ > 0. For a perturbation of fixed-point yk = xk − xk∗ . (i) xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is monotonically stable if 0 < (xk+1 − xk∗ ) < (xk − xk∗ ), for (xk − xk∗ ) > 0, 0 > (xk+1 − xk∗ ) > (xk − xk∗ ), for (xk − xk∗ ) < 0

(2.6)

for xk ∈ U (xk∗ ) ⊂ . The fixed-point xk∗ is called a monotonic sink (or a monotonically stable node). (ii) xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is monotonically unstable if (xk+1 − xk∗ ) > (xk − xk∗ ), for (xk − xk∗ ) > 0, (xk+1 − xk∗ ) < (xk − xk∗ ), for (xk − xk∗ ) < 0

(2.7)

for xk ∈ U (xk∗ ) ⊂ . The fixed-point xk∗ is called a monotonic source (or a monotonic stable node). (iii) xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is monotonically invariant if (xk+1 − xk∗ ) = (xk − xk∗ ), for (xk − xk∗ ) > 0, (xk+1 − xk∗ ) = (xk − xk∗ ), for (xk − xk∗ ) < 0

(2.8)

for xk ∈ U (xk∗ ) ⊂ . The fixed-point xk∗ is called an invariant degenerate case. (iv) xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is zero-stable if (xk+1 − xk∗ ) = 0, for (xk − xk∗ ) > 0, (xk+1 − xk∗ ) = 0, for (xk − xk∗ ) < 0

(2.9)

for xk ∈ U (xk∗ ) ⊂ . The fixed-point xk∗ is called a zero-sink (or zero-stable node).

2.2 Monotonic and Oscillatory Stability

79

(v) xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is oscillatorilly stable if 0 > (xk+1 − xk∗ ) > −(xk − xk∗ ), for (xk − xk∗ ) > 0, 0 < (xk+1 − xk∗ ) < −(xk − xk∗ ), for (xk − xk∗ ) < 0

(2.10)

for xk ∈ U (xk∗ ) ⊂ . The fixed-point xk∗ is called an oscillatory sink (or an oscillatorilly stable node). (vi) xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is oscillatorilly unstable if (xk+1 − xk∗ ) < −(xk − xk∗ ), for (xk − xk∗ ) > 0, (xk+1 − xk∗ ) > −(xk − xk∗ ), for (xk − xk∗ ) < 0

(2.11)

for xk ∈ U (xk∗ ) ⊂ . The fixed-point xk∗ is called an oscillatory source (or an oscillatorilly unstable node). (vii) xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is symmetrically flipped if (xk+1 − xk∗ ) = −(xk − xk∗ ), for (xk − xk∗ ) > 0, (xk+1 − xk∗ ) = −(xk − xk∗ ), for (xk − xk∗ ) < 0

(2.12)

for xk ∈ U (xk∗ ) ⊂ . The fixed-point xk∗ is called a flipped degenerate case. (viii) xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is monotonically unstable of the second-order lower-saddle if 0 < (xk+1 − xk∗ ) < (xk − xk∗ ), for (xk − xk∗ ) > 0, (xk+1 − xk∗ ) < (xk − xk∗ ), for (xk − xk∗ ) < 0

(2.13)

for xk ∈ U (xk∗ ) ⊂ . The fixed-point xk∗ is called a monotonic lower-saddle of the second-order. (ix) xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is monotonically unstable of the second-order upper-saddle if (xk+1 − xk∗ ) > (xk − xk∗ ), for (xk − xk∗ ) > 0, 0 > (xk+1 − xk∗ ) > (xk − xk∗ ), for (xk − xk∗ ) < 0

(2.14)

for xk ∈ U (xk∗ ) ⊂ . The fixed-point xk∗ is called a monotonic upper-saddle of the second-order.

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2 Low-Dimensional Discrete Systems

(x) xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is oscillatorilly unstable of the second-order lower-saddle if (xk+1 − xk∗ ) < −(xk − xk∗ ), for (xk − xk∗ ) > 0, 0 < (xk+1 − xk∗ ) < −(xk − xk∗ ), for (xk − xk∗ ) < 0

(2.15)

for xk ∈ U (xk∗ ) ⊂ . The fixed-point xk∗ is called an oscillatory lower-saddle of the second-order. (xi) xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is oscillatorilly unstable of the second-order upper-saddle if 0 > (xk+1 − xk∗ ) > −(xk − xk∗ ), for (xk − xk∗ ) > 0, (xk+1 − xk∗ ) > −(xk − xk∗ ), for (xk − xk∗ ) < 0

(2.16)

for xk ∈ U (xk∗ ) ⊂ . The fixed-point xk∗ is called an oscillatory upper-saddle of the second-order. Theorem 2.1 Consider a 1-dimensional, nonlinear discrete system xk+1 = f (xk , p) in Eq. (2.1) with a fixed-point xk∗ . The corresponding solution is given by xk+ j = f (xk+ j−1 , p) with j ∈ Z. Suppose there is a neighborhood of the fixed-point xk∗ (i.e., Uk (xk∗ ) ⊂ ), and f (xk , p) is C r (r ≥ 1)-continuous in Uk (xk∗ ) = (xk∗ − δ, xk∗ + δ) for an arbitrary δ > 0. The linearized system in Uk (xk∗ ) is yk+ j+1 = D f (xk∗ , p)yk+ j in Eq. (2.2). For a perturbation of fixed-point yk = xk − xk∗ . (i) xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is monotonically stable if and only if G (1) (xk∗ , p) = D f (xk∗ , p) = λ ∈ (0, 1)

(2.17)

for Uk (xk∗ ) ⊂ . (ii) xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is monotonically unstable if and only if G (1) (xk∗ , p) = D f (xk∗ , p) = λ ∈ (1, ∞)

(2.18)

for Uk (xk∗ ) ⊂ . (iii) xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is invariant if and only if G (1) (xk∗ , p) = D f (xk∗ , p) = λ = 1 and G (xk∗ , p) = D (m) f (xk∗ , p) = 0 m = 2, 3, . . . (m)

for Uk (xk∗ ) ⊂ .

(2.19)

2.2 Monotonic and Oscillatory Stability

81

(iv) xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is zero-stable if and only if G (1) (xk∗ , p) = D f (xk∗ , p) = λ = 0

(2.20)

for Uk (xk∗ ) ⊂ . (v) xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is oscillatorilly stable if and only if G (1) (xk∗ , p) = D f (xk∗ , p) = λ ∈ (−1, 0)

(2.21)

for Uk (xk∗ ) ⊂ . (vi) xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is oscillatorilly unstable if and only if G (1) (xk∗ , p) = D f (xk∗ , p) = λ ∈ (−∞, −1)

(2.22)

for Uk (xk∗ ) ⊂ . (vii) xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is flipped if and only if G (1) (xk∗ , p) = D f (xk∗ , p) = λ = −1 and G (xk∗ , p) = D (m) f (xk∗ , p) = 0 m = 2, 3, . . . (m)

(2.23)

for Uk (xk∗ ) ⊂ . (viii) xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is monotonically unstable of the lower-saddle of the second-order if and only if G (1) (xk∗ , p) = D f (xk∗ , p) = 1, G (2) (xk∗ , p) = D (2) f (xk∗ , p) < 0

(2.24)

for Uk (xk∗ ) ⊂ . (ix) xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is monotonically unstable of the upper-saddle of the second-order if and only if G (1) (xk∗ , p) = D f (xk∗ , p) = 1, G (2) (xk∗ , p) = D (2) f (xk∗ , p) > 0 for Uk (xk∗ ) ⊂ .

(2.25)

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2 Low-Dimensional Discrete Systems

(x) xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is oscillatorilly unstable of the lower-saddle of the second-order if and only if G (1) (xk∗ , p) = D f (xk∗ , p) = −1, G (2) (xk∗ , p) = D (2) f (xk∗ , p) < 0

(2.26)

for Uk (xk∗ ) ⊂ . (xi) xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is oscillatorilly unstable of the upper-saddle of the second-order if and only if G (1) (xk∗ , p) = D f (xk∗ , p) = −1, G (2) (xk∗ , p) = D (2) f (xk∗ , p) > 0

(2.27)

for Uk (xk∗ ) ⊂ . Proof Consider xk+1 − xk∗ = f (xk , p) − xk∗ = f (xk∗ , p) + D f (xk∗ , p)(xk − xk∗ ) − xk∗ + o(xk − xk∗ ), xk+1 − xk∗ = f (xk , p) − xk∗ = f (xk∗ , p) +

m  1 (r ) D f (xk∗ , p)(xk − xk∗ )r − xk∗ + o((xk − xk∗ )m+1 ) r ! r =1

because the fixed-point xk∗ is determined by ∗ = xk∗ . f (xk∗ , p) = xk+1

Thus, we have xk+1 − xk∗ = D f (xk∗ , p)(xk − xk∗ ) = G (1) (xk∗ , p)(xk − xk∗ ) = λ(xk − xk∗ ); xk+1 − xk∗ = =

m  1 (r ) D f (xk∗ , p)(xk − xk∗ )r r ! r =1 m  1 (r ) G f (xk∗ , p)(xk − xk∗ )r . r ! r =1

2.2 Monotonic and Oscillatory Stability

(i1 )

83

For (xk − xk∗ ) > 0, if 0 < (xk+1 − xk∗ ) < (xk − xk∗ ), then, the first-order approximation of f (xk , p) gives 0 < (xk+1 − xk∗ ) = G (1) (xk∗ , p)(xk − xk∗ ) = λ(xk − xk∗ ) < (xk − xk∗ ), so, G (1) (xk∗ , p) = D f (xk∗ , p) = λ ∈ (0, 1).

(i2 )

For (xk − xk∗ ) < 0, if 0 > (xk+1 − xk∗ ) > (xk − xk∗ ), then, the first-order approximation of f (xk , p) 0 > (xk+1 − xk∗ ) = G (1) (xk∗ , p)(xk − xk∗ ) = λ(xk − xk∗ ) > (xk − xk∗ ), so, G (1) (xk∗ , p) = D f (xk∗ , p) = λ ∈ (0, 1). Thus, xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is monotonically stable if and only if G (1) (xk∗ , p) = D f (xk∗ , p) = λ ∈ (0, 1) for Uk (xk∗ ) ⊂ .

(ii1 )

For (xk − xk∗ ) > 0, if (xk+1 − xk∗ ) > (xk − xk∗ ), then, the first-order approximation of f (xk , p) gives (xk+1 − xk∗ ) = G (1) (xk∗ , p)(xk − xk∗ ) = λ(xk − xk∗ ) > (xk − xk∗ ), so, G (1) (xk∗ , p) = D f (xk∗ , p) = λ ∈ (1, ∞).

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2 Low-Dimensional Discrete Systems

(ii2 )

For (xk − xk∗ ) < 0, if (xk+1 − xk∗ ) < (xk − xk∗ ), then, the first-order approximation of f (xk , p) gives (xk+1 − xk∗ ) = G (1) (xk∗ , p)(xk − xk∗ ) = λ(xk − xk∗ ) < (xk − xk∗ ), so, G (1) (xk∗ , p) = D f (xk∗ , p) = λ ∈ (1, ∞). Thus, xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is monotonically unstable if and only if G (1) (xk∗ , p) = D f (xk∗ , p) = λ ∈ (1, ∞) for Uk (xk∗ ) ⊂ .

(iii1 )

For (xk − xk∗ ) < 0, if (xk+1 − xk∗ ) = (xk − xk∗ ), then (xk+1 − xk∗ ) = G (1) (xk∗ , p)(xk − xk∗ ) = λ(xk − xk∗ ) = (xk − xk∗ ), (xk+1 −

xk∗ )

= λ(xk −

xk∗ )

∞  1 (r ) ∗ G (xk , p)(xk − xk∗ )r = (xk − xk∗ ), + r ! r =2

so, G (1) (xk∗ , p) = D f (xk∗ , p) = λ = 1, G (r ) (xk∗ , p) = D (r ) f (xk∗ , p) = 0, r = 2, 3, . . . . (iii2 )

For (xk − xk∗ ) < 0, if (xk+1 − xk∗ ) = (xk − xk∗ ), then, the Taylor series expansion of f (xk , p) gives (xk+1 − xk∗ ) = λ(xk − xk∗ ) +

∞  1 (r ) ∗ G (xk , p)(xk − xk∗ )r = (xk − xk∗ ), r ! r =2

2.2 Monotonic and Oscillatory Stability

85

so, for arbitrary increment of xk − xk∗ , we have G (1) (xk∗ , p) = D f (xk∗ , p) = λ = 1, G (r ) (xk∗ , p) = D (r ) f (xk∗ , p) = 0, r = 2, 3, . . . . Thus, xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is invariant if and only if G (1) (xk∗ , p) = D f (xk∗ , p) = λ = 1, G (r ) (xk∗ , p) = D (r ) f (xk∗ , p) = 0, r = 2, 3, . . . for Uk (xk∗ ) ⊂ . (iv1 )

For (xk − xk∗ ) < 0, if (xk+1 − xk∗ ) = 0, then, the Taylor series expansion of f (xk , p) gives (xk+1 − xk∗ ) = λ(xk − xk∗ ) +

∞  1 (r ) ∗ G (xk , p)(xk − xk∗ )r = 0, r ! r =2

so, for arbitrary increment of xk − xk∗ , we have G (1) (xk∗ , p) = D f (xk∗ , p) = λ = 0, G (r ) (xk∗ , p) = D (r ) f (xk∗ , p) = 0, r = 2, 3, . . . . (iv2 )

For (xk − xk∗ ) < 0, if (xk+1 − xk∗ ) = 0, then the Taylor series expansion of f (xk , p) gives (xk+1 −

xk∗ )

= λ(xk −

xk∗ )

∞  1 (r ) ∗ G (xk , p)(xk − xk∗ )r = 0, + r ! r =2

so, for arbitrary increment of xk − xk∗ , we have G (1) (xk∗ , p) = D f (xk∗ , p) = λ = 0, G (r ) (xk∗ , p) = D (r ) f (xk∗ , p) = 0, r = 2, 3, . . . .

86

2 Low-Dimensional Discrete Systems

Thus, xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is zero-sink if and only if G (1) (xk∗ , p) = D f (xk∗ , p) = λ = 0, G (r ) (xk∗ , p) = D (r ) f (xk∗ , p) = 0, r = 2, 3, . . . . for Uk (xk∗ ) ⊂ . (v1 )

For (xk − xk∗ ) > 0, if 0 > (xk+1 − xk∗ ) > −(xk − xk∗ ), then, the first-order approximation of f (xk , p) gives 0 > (xk+1 − xk∗ ) = G (1) (xk∗ , p)(xk − xk∗ ) = λ(xk − xk∗ ) > −(xk − xk∗ ), so, G (1) (xk∗ , p) = D f (xk∗ , p) = λ ∈ (−1, 0).

(v2 )

For (xk − xk∗ ) < 0, if (xk+1 − xk∗ ) < −(xk − xk∗ ), then the first-order approximation of f (xk , p) gives (xk+1 − xk∗ ) = G (1) (xk∗ , p)(xk − xk∗ ) = λ(xk − xk∗ ) < −(xk − xk∗ ), so, G (1) (xk∗ , p) = D f (xk∗ , p) = λ ∈ (−1, 0). Thus, xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is oscillatorilly stable if and only if G (1) (xk∗ , p) = D f (xk∗ , p) = λ ∈ (−1, 0). for Uk (xk∗ ) ⊂ .

(vi1 )

For (xk − xk∗ ) > 0, if (xk+1 − xk∗ ) > −(xk − xk∗ ),

2.2 Monotonic and Oscillatory Stability

87

then the first-order approximation of f (xk , p) gives (xk+1 − xk∗ ) = G (1) (xk∗ , p)(xk − xk∗ ) = λ(xk − xk∗ ) > −(xk − xk∗ ), so, G (1) (xk∗ , p) = D f (xk∗ , p) = λ ∈ (−∞, −1). (vi2 )

For (xk − xk∗ ) < 0, if (xk+1 − xk∗ ) < (xk − xk∗ ), then the first-order approximation of f (xk , p) gives (xk+1 − xk∗ ) = G (1) (xk∗ , p)(xk − xk∗ ) = λ(xk − xk∗ ) < (xk − xk∗ ), so, G (1) (xk∗ , p) = D f (xk∗ , p) = λ ∈ (−∞, −1). Thus, xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is oscillatorilly unstable if and only if G (1) (xk∗ , p) = D f (xk∗ , p) = λ ∈ (−∞, −1) for Uk (xk∗ ) ⊂ .

(vii1 )

For (xk − xk∗ ) < 0, if (xk+1 − xk∗ ) = −(xk − xk∗ ), then the Taylor series expansion of f (xk , p) gives (xk+1 − xk∗ ) = λ(xk − xk∗ ) +

∞  1 (r ) ∗ G (xk , p)(xk − xk∗ )r = −(xk − xk∗ ), r ! r =2

so, for an arbitrary increment of xk − xk∗ , we have G (1) (xk∗ , p) = D f (xk∗ , p) = λ = −1, G (r ) (xk∗ , p) = D (r ) f (xk∗ , p) = 0, r = 2, 3, . . . .

88

2 Low-Dimensional Discrete Systems

(vii2 )

For (xk − xk∗ ) < 0, if (xk+1 − xk∗ ) = −(xk − xk∗ ), then the Taylor series expansion of f (xk , p) gives (xk+1 − xk∗ ) = λ(xk − xk∗ ) +

∞  1 (r ) ∗ G (xk , p)(xk − xk∗ )r = −(xk − xk∗ ), r ! r =2

so, for an arbitrary increment of xk − xk∗ , we have G (1) (xk∗ , p) = D f (xk∗ , p) = λ = −1, G (r ) (xk∗ , p) = D (r ) f (xk∗ , p) = 0, r = 2, 3, . . . . Thus, xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is oscillatorilly flipped if and only if G (1) (xk∗ , p) = D f (xk∗ , p) = λ = −1, G (r ) (xk∗ , p) = D (r ) f (xk∗ , p) = 0, r = 2, 3, . . . for Uk (xk∗ ) ⊂ . (viii1 ) For (xk − xk∗ ) > 0, if 0 < (xk+1 − xk∗ ) < (xk − xk∗ ), then 0 < (xk+1 − xk∗ ) = G(xk∗ , p)(xk − xk∗ ) = λ(xk − xk∗ ) < (xk − xk∗ ), so, G (1) (xk∗ , p) = D f (xk∗ , p) = λ ∈ (0, 1). (viii2 ) For (xk − xk∗ ) < 0, if (xk+1 − xk∗ ) < (xk − xk∗ ), then (xk+1 − xk∗ ) = G (1) (xk∗ , p)(xk − xk∗ ) = λ(xk − xk∗ ) < (xk − xk∗ ),

2.2 Monotonic and Oscillatory Stability

89

so, G (1) (xk∗ , p) = D f (xk∗ , p) = λ ∈ (1, ∞). Thus, because of the C r -continuity (r ≥ 1) of f (xk , p), we have G (1) (xk∗ , p) = D f (xk∗ , p) = λ = 1. Through the second-order approximations of f (xk , p), consider (xk+1 − xk∗ ) = λ(xk − xk∗ ) +

1 (2) ∗ G (xk , p)(xk − xk∗ )2 < (xk − xk∗ ), 2!

and we have 1 (2) ∗ G (xk , p)(xk − xk∗ )2 < 0. 2! Thus, G (2) (xk∗ , p) = D (2) f (xk∗ , p) < 0. Therefore, xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is monotonically unstable of the lower-saddle of the second-order if and only if G (1) (xk∗ , p) = D f (xk∗ , p) = 1, G (2) (xk∗ , p) = D (2) f (xk∗ , p) < 0 for Uk (xk∗ ) ⊂ . (ix1 )

For (xk − xk∗ ) > 0, if (xk+1 − xk∗ ) > (xk − xk∗ ), then (xk+1 − xk∗ ) = G (1) (xk∗ , p)(xk − xk∗ ) = λ(xk − xk∗ ) > (xk − xk∗ ), so, G (1) (xk∗ , p) = D f (xk∗ , p) = λ ∈ (1, ∞).

90

2 Low-Dimensional Discrete Systems

(ix2 )

For (xk − xk∗ ) < 0, if 0 > (xk+1 − xk∗ ) > (xk − xk∗ ), then 0 > (xk+1 − xk∗ ) = G (1) (xk∗ , p)(xk − xk∗ ) = λ(xk − xk∗ ) > (xk − xk∗ ), so, G (1) (xk∗ , p) = D f (xk∗ , p) = λ ∈ (0, 1). Thus, because of the C r -continuity (r ≥ 1) of f (xk , p), we have G (xk∗ , p) = D f (xk∗ , p) = λ = 1. Through the second-order approximations of f (xk , p), consider (xk+1 − xk∗ ) = λ(xk − xk∗ ) +

1 (2) ∗ G (xk , p)(xk − xk∗ )2 > (xk − xk∗ ), 2!

and we have 1 (2) ∗ G (xk , p)(xk − xk∗ )2 > 0. 2! Thus, G (2) (xk∗ , p) = D (2) f (xk∗ , p) > 0. Therefore, xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is monotonically unstable of the upper-saddle of the second-order if and only if G (1) (xk∗ , p) = D f (xk∗ , p) = 1, G (2) (xk∗ , p) = D (2) f (xk∗ , p) > 0 for Uk (xk∗ ) ⊂ . (x1 )

For (xk − xk∗ ) > 0, if

2.2 Monotonic and Oscillatory Stability

91

(xk+1 − xk∗ ) < −(xk − xk∗ ), then (xk+1 − xk∗ ) = G (1) (xk∗ , p)(xk − xk∗ ) = λ(xk − xk∗ ) < −(xk − xk∗ ), so, G (1) (xk∗ , p) = D f (xk∗ , p) = λ ∈ (−∞, −1). (x2 )

For (xk − xk∗ ) < 0, if 0 < (xk+1 − xk∗ ) < −(xk − xk∗ ), then 0 < (xk+1 − xk∗ ) = G (1) (xk∗ , p)(xk − xk∗ ) = λ(xk − xk∗ ) < −(xk − xk∗ ), so, G (1) (xk∗ , p) = D f (xk∗ , p) = λ ∈ (−1, 0). Thus, because of the C r -continuity (r ≥ 1) of f (xk , p), we have G (1) (xk∗ , p) = D f (xk∗ , p) = λ = −1. Through the second-order approximations of f (xk , p), consider xk+1 − xk∗ = λ(xk − xk∗ ) +

1 (2) ∗ G (xk , p)(xk − xk∗ )2 < −(xk − xk∗ ), 2!

and we have 1 (2) ∗ G (xk , p)(xk − xk∗ )2 < 0. 2! Thus, G (2) (xk∗ , p) = D (2) f (xk∗ , p) < 0.

92

2 Low-Dimensional Discrete Systems

Therefore, xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is oscillatorilly unstable of the lower-saddle of the second-order if and only if G (1) (xk∗ , p) = D f (xk∗ , p) = −1, G (2) (xk∗ , p) = D (2) f (xk∗ , p) < 0 (xi1 )

for Uk (xk∗ ) ⊂ . For (xk − xk∗ ) > 0, if 0 > (xk+1 − xk∗ ) > −(xk − xk∗ ), then 0 > (xk+1 − xk∗ ) = G (1) (xk∗ , p)(xk − xk∗ ) = λ(xk − xk∗ ) > −(xk − xk∗ ), so, G (1) (xk∗ , p) = D f (xk∗ , p) = λ ∈ (−1, 0).

(xi2 )

For (xk − xk∗ ) < 0, if (xk+1 − xk∗ ) > −(xk − xk∗ ), then (xk+1 − xk∗ ) = G (1) (xk∗ , p)(xk − xk∗ ) = λ(xk − xk∗ ) < −(xk − xk∗ ), so, G (1) (xk∗ , p) = D f (xk∗ , p) = λ ∈ (−∞, −1). Thus, because of the C r -continuity (r ≥ 1) of f (xk , p), we have G (1) (xk∗ , p) = D f (xk∗ , p) = λ = −1. Through the second-order approximations of f (xk , p), consider (xk+1 − xk∗ ) = λ(xk − xk∗ ) +

1 (2) ∗ G (xk , p)(xk − xk∗ )2 > −(xk − xk∗ ), 2!

2.2 Monotonic and Oscillatory Stability

93

and we have 1 (2) ∗ G (xk , p)(xk − xk∗ )2 > 0. 2! Thus, G (2) (xk∗ , p) > 0. Therefore, xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is oscillatory unstable of the upper-saddle of the second-order if and only if G (1) (xk∗ , p) = D f (xk∗ , p) = −1, G (2) (xk∗ , p) = D (2) f (xk∗ , p) > 0 for Uk (xk∗ ) ⊂ . The theorem is proved.



From the definition, in Fig. 2.1i–xi, shown are a monotonic sink, a monotonic source, a monotonic invariant node, an invariant sink, an oscillatory sink, an oscillatory source, an oscillatory node (a flipped node), a monotonic lower-saddle of the second-order, a monotonic upper-saddle, an oscillatory lower-saddle, an oscillatory upper-saddle of the second-order, respectively. From a selected initial point, the iteration of discrete systems in the vicinity of the fixed-points are sketched for all eleven cases. For a monotonic sink fixed-point, any initial points on both sides of the vicinity of the fixed-point will approach the corresponding fixed-point, as shown in Fig. 2.1i. For a monotonic source fixed-point, any initial points on both sides of the vicinity of the fixed-point will move away from the corresponding fixed-point, as shown in Fig. 2.1ii. For an invariant case of fixed-point, any initial points on both sides of the vicinity of the fixed-point will be invariant, as shown in Fig. 2.1iii. For a zero-sink fixed-point, any initial points on both sides of the vicinity of the fixed-point will return to the corresponding fixed-point, as shown in Fig. 2.1iv. For an oscillatory sink fixed-point, any initial points will oscillatorilly jump on both sides of the vicinity of the fixed-point and then to the corresponding fixed-point, as shown in Fig. 2.1v. For an oscillatory source fixed-point, any initial points will oscillatorilly jump on both sides of the vicinity of the fixed-point and then will move away from the corresponding fixed-point, as shown in Fig. 2.1vi. For an oscillatorilly flipped case of fixed-point, any initial points will jump back-and-forth on both sides of the vicinity of the fixed-point, as shown in Fig. 2.1vii. For a monotonic lower-saddle fixed-point, any initial points on the positive side of the vicinity of the fixed-point will monotonically approach the corresponding fixed-point, but any initial points on the negative side of the vicinity of the fixed-point will monotonically move away the corresponding fixed-point, as shown in Fig. 2.1viii. For a monotonic upper-saddle

94

(i)

(iii)

2 Low-Dimensional Discrete Systems

xk∗

(ii)

xk∗ (iv)

xk∗

xk∗

xk∗

(v)

xk∗

(vi)

xk∗ (vii)

(viii)

xk∗

(ix)

xk∗

xk∗

(x)

xk∗

(xi)

Fig. 2.1 1-dimensional discrete system: i a monotonic sink, ii a monotonic source, iii a monotonic invariant node, iv an invariant sink, v an oscillatory sink, vi an oscillatory source, vii an oscillatory node (a flipped node), viii a monotonic lower-saddle of the second-order, ix a monotonic uppersaddle of the second-order, x an oscillatory lower-saddle of the second-order, xi an oscillatory upper-saddle of the second-order

2.2 Monotonic and Oscillatory Stability

95

fixed-point, any initial points on the positive side of the vicinity of the fixed-point will monotonically move away the corresponding fixed-point, but any initial points on the negative side of the vicinity of the fixed-point will monotonically approach the corresponding fixed-point, as shown in Fig. 2.1ix. For a oscillatory lower-saddle fixed-point, any initial points on the positive side of the vicinity of the fixed-point will oscillatorilly move away the corresponding fixed-point, but any initial points on the negative side of the vicinity of the fixed-point will oscillatorilly approach the corresponding fixed-point, as shown in Fig. 2.1x. For an oscillatory upper-saddle fixed-point, any initial points on the positive side of the vicinity of the fixed-point will oscillatorilly approach the corresponding fixed-point, but any initial points on the negative side of the vicinity of the fixed-point will oscillatorilly move away from the corresponding fixed-point, as shown in Fig. 2.1xi. The stability of fixed-points in 1-dimensional discrete systems is presented in Fig. 2.2. The solid curve is xk+1 = f (xk , p). The circular symbol is fixed-point. The shaded regions are stable. The horizontal solid line is for a degenerate case. The vertical solid line is for a line with infinite slope. A monotonically stable node (a monotonic sink) is presented in Fig. 2.2i. The dashed and dotted lines are for yk+1 = yk and yk+1 = −yk where yk = xk − xk∗ and yk+1 = xk+1 − xk∗ for a fixedpoint xk∗ . The iterative responses approach the fixed-point. However, a monotonically unstable node (a monotonic source) is presented in Fig. 2.2ii. The iterative responses go away from the fixed-point. Similarly, an oscillatorilly stable node (an oscillatory sink) after iteration with a flip at the line of yk+1 = −yk is presented in Fig. 2.2iii. The dashed and dotted lines are for yk+1 = −yk and yk+1 = yk , respectively. In a similar fashion, an oscillatory unstable node (an oscillatory source) is presented in Fig. 2.2iv. This illustration can be easily observed the stability of fixed-points. In Fig. 2.2v, vi, oscillatory stable and unstable nodes are presented as usual through the two iterations. The saddle fixed-points in 1-dimensional discrete systems are presented in Fig. 2.3. A monotonic lower-saddle of the second-order is presented in Fig. 2.3i. The dashed and dotted lines are for yk+1 = yk and yk+1 = −yk , respectively. The increments of yk = xk − xk∗ and yk+1 = xk+1 − xk∗ are for a fixed-point xk∗ . The upper part is monotonically stable, and the lower part is monotonically unstable. A monotonic upper-saddle of the second-order is presented in Fig. 2.3ii. The iterative responses go away from the fixed-point. The upper part is monotonically unstable, and the lower part is monotonically stable. Similarly, an oscillatory lower-saddle of the second-order based on the flipped line of yk+1 = −yk is presented in Fig. 2.3iii. The dashed and dotted lines are for yk+1 = −yk and yk+1 = yk , respectively. Based on the flipped line of yk+1 = −yk , it is clearly observed that the upper part is oscillatorilly stable, and the lower part is oscillatorilly unstable. In a similar fashion, an oscillatory upper-saddle of the second-order is presented in Fig. 2.3iv. Based on the line of yk+1 = −yk , the upper part is oscillatorilly unstable, and the lower part is oscillatorilly stable. This illustration can be easily observed the stability of fixedpoints. In Fig. 2.3v, vi, the oscillatory lower- and upper-saddle are presented through the two iterations.

96

2 Low-Dimensional Discrete Systems

xk +1

xk +1

yk +1 = yk

xk

xk

xk∗

∗ k

x

yk +1 = yk yk +1 = − yk

(i)

(ii) yk +1 = yk

yk +1 = − yk

yk +1 = yk

xk∗

v iT ⋅ x∗k

xk +1

xk +1 T i

T i

v ⋅ y k +1 = − v ⋅ y k

xk

xk

(iii)

(iv) yk +1 = − yk

yk = yk +1

yk = yk +1

xk∗ v iT ⋅ x∗k

xk +1

yk +1 = − yk

xk +1 xk

xk

(v)

(vi)

Fig. 2.2 Stability of fixed-points: i monotonically stable node (sink), ii monotonically unstable node (source); iii oscillatory stable node (sink) and iv oscillatory unstable node (sink); v oscillatory stable node (sink) and vi oscillatory unstable node (sink). Shaded areas are stable zones. (yk = xk − xk∗ and yk+1 = xk+1 − xk∗ )

2.2 Monotonic and Oscillatory Stability xk +1

97 xk +1

yk +1 = yk

xk

xk

xk∗

xk∗

yk +1 = yk

yk +1 = − yk

yk +1 = yk

(i) yk +1 = − yk

(ii) yk +1 = yk

yk +1 = − yk

yk +1 = yk

xk∗

xk∗

xk +1

xk +1 xk

xk

(iii) yk +1 = − yk

(iv) yk +1 = yk

xk∗

xk +1

yk +1 = yk

yk +1 = − yk

xk∗

xk +1 xk

xk

(v)

(vi)

Fig. 2.3 Saddle fixed-points: i monotonically lower-saddle of the second-order and ii monotonically upper-saddle of the second-order; iii oscillatory lower-saddle of the second-order and iv oscillatory upper-saddle of the second-order. v oscillatory lower-saddle of the second-order and vi oscillatory upper-saddle of the second-order from two iterations. Shaded areas are stable zones. (yk = xk − xk∗ ∗ and yk+1 = xk+1 − xk+1 )

98

2 Low-Dimensional Discrete Systems

Definition 2.3 Consider a 1-dimensional, nonlinear discrete system xk+1 = f (xk , p) in Eq. (2.1) with a fixed-point xk∗ . The corresponding solution is given by xk+ j = f (xk+ j−1 , p) with j ∈ Z. Suppose there is a neighborhood of the fixed-point xk∗ (i.e., Uk (xk∗ ) = (xk∗ − δ, xk∗ + δ) for an arbitrary δ > 0), and f (xk , p) is C r -continuous (r ≥ 1) in Uk (xk∗ ). For a perturbation of fixed-point yk = xk − xk∗ . (i) xk+ j ( j ∈ Z) at the fixed-point xk∗ is monotonically stable of the (2m +1)th-order sink if G (1) (xk∗ , p) = D f (xk∗ , p) = λ = 1, G (r ) (xk∗ , p) = 0 for r = 2, 3, . . . , 2m, G (2m+1) (xk∗ , p) = 0,  0 < (xk+1 − xk∗ ) < (xk − xk∗ ) for (xk − xk∗ ) > 0, 0 > (xk+1 − xk∗ ) > (xk − xk∗ ) for (xk − xk∗ ) < 0

(2.28)

for xk ∈ U (xk∗ ) ⊂ . The fixed-point xk∗ is called a monotonic sink (or a monotonically stable node) of the (2m + 1)th-order. (ii) xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is monotonically unstable of the (2m + 1)th-order source if G (1) (xk∗ , p) = D f (xk∗ , p) = λ = 1, G (r ) (xk∗ , p) = 0 for r = 2, 3, . . . , 2m, G (2m+1) (xk∗ , p) = 0,  (xk+1 − xk∗ ) > (xk − xk∗ ) > 0 for (xk − xk∗ ) > 0, (xk+1 − xk∗ ) < (xk − xk∗ ) < 0 for (xk − xk∗ ) < 0

(2.29)

for xk ∈ U (xk∗ ) ⊂ . The fixed-point xk∗ is called a monotonic source (or a monotonically unstable node) of the (2m + 1)th-order. (iii) xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is monotonically unstable of the (2m)th-order, lower-saddle if G (1) (xk∗ , p) = D f (xk∗ , p) = λ = 1, G (r ) (xk∗ , p) = 0 for r = 2, 3, . . . , 2m − 1, G (2m) (xk∗ , p) = 0,  0 < (xk+1 − xk∗ ) < (xk − xk∗ ) for (xk − xk∗ ) > 0, (xk+1 − xk∗ ) < (xk − xk∗ ) < 0 for (xk − xk∗ ) < 0

(2.30)

for xk ∈ U (xk∗ ) ⊂ . The fixed-point xk∗ is called a monotonic, lower-saddle of the (2m)th-order.

2.2 Monotonic and Oscillatory Stability

99

(iv) xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is monotonically unstable of the (2m)th-order, upper-saddle if G (1) (xk∗ , p) = D f (xk∗ , p) = λ = 1, G (r ) (xk∗ , p) = 0 for r = 2, 3, . . . , 2m − 1, G (2m) (xk∗ , p) = 0,  (xk+1 − xk∗ ) > (xk − xk∗ ) > 0 for (xk − xk∗ ) > 0, 0 > (xk+1 − xk∗ ) > (xk − xk∗ ) for (xk+1 − xk∗ ) < 0

(2.31)

for xk ∈ U (xk∗ ) ⊂ . The fixed-point xk∗ is called a monotonic, upper-saddle of the (2m i )th-order. (v) xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is oscillatorilly of the (2m + 1)th-order sink if G (1) (xk∗ , p) = D f (xk∗ , p) = λ = −1, G (r ) (xk∗ , p) = 0 for r = 2, 3, . . . , 2m, G (2m+1) (xk∗ , p) = 0,  0 > (xk+1 − xk∗ ) > −(xk − xk∗ ) for (xk − xk∗ ) > 0, 0 < (xk+1 − xk∗ ) < −(xk − xk∗ ) for (xk − xk∗ ) < 0

(2.32)

for xk ∈ U (xk∗ ) ⊂ . The fixed-point xk∗ is called an oscillatory sink (or an oscillatorilly stable node) of the (2m i + 1)th-order. (vi) xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is oscillatorilly unstable of the (2m + 1)th-order source if G (1) (xk∗ , p) = D f (xk∗ , p) = λ = −1, G (r ) (xk∗ , p) = 0 for r = 2, 3, . . . , 2m, G (2m+1) (xk∗ , p) = 0,  (xk+1 − xk∗ ) < −(xk − xk∗ ) < 0 for (xk − xk∗ ) > 0, (xk+1 − xk∗ ) > −(xk − xk∗ ) > 0 for (xk − xk∗ ) < 0

(2.33)

for xk ∈ U (xk∗ ) ⊂ . The fixed-point xk∗ is called an oscillatory source (or an oscillatorilly unstable node) of the (2m + 1)th-order. (vii) xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is oscillatorilly unstable of the (2m)th-order, lower-saddle if G (1) (xk∗ , p) = D f (xk∗ , p) = λ = −1, G (r ) (xk∗ , p) = 0 for r = 2, 3, . . . , 2m − 1, G (2m+1) (xk∗ , p) = 0,  (xk+1 − xk∗ ) < −(xk − xk∗ ) < 0 for (xk − xk∗ ) > 0, 0 < (xk+1 − xk∗ ) < −(xk − xk∗ ) for (xk − xk∗ ) < 0

(2.34)

100

2 Low-Dimensional Discrete Systems

for xk ∈ U (xk∗ ) ⊂ . The fixed-point xk∗ is called an oscillatory lower-saddle of the (2m)th-order. (viii) xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is oscillatorilly unstable of the (2m)th-order, upper-saddle if G (1) (xk∗ , p) = D f (xk∗ , p) = λ = −1, G (r ) (xk∗ , p) = 0 for r = 2, 3, . . . , 2m − 1, G (2m) (xk∗ , p) = 0,  0 > (xk+1 − xk∗ ) > −(xk − xk∗ ) for (xk − xk∗ ) > 0, (xk+1 − xk∗ ) > −(xk − xk∗ ) > 0 for (xk − xk∗ ) < 0

(2.35)

for xk ∈ U (xk∗ ) ⊂ . The fixed-point xk∗ is called an oscillatory, upper saddle of the (2m)th-order. The monotonic stability of fixed-points with higher-order singularity for a 1dimensional discrete system is presented in Fig. 2.4. The solid curve is xk+1 = f (xk , p). The circular symbol is for a fixed-point xk∗ . In the vicinity of the fixed∗ . The dashed and dotted lines are point xk∗ , yk = xk − xk∗ and yk+1 = xk+1 − xk+1 for yk+1 = yk and yk+1 = −yk , respectively. The shaded regions are stable. The horizontal solid line is also for the degenerate case. The vertical solid line is for a line with infinite slope. A monotonically stable node (a monotonic sink) of the (2m + 1)th order is sketched in Fig. 2.4a. The nonlinear curve lies in the upper and lower stable zones, and the iterative responses approach the fixed-point. However, a monotonically unstable node (source) of the (2m+1)th order is presented in Fig. 2.4b. The nonlinear curve lies in both upper and lower unstable zones, and the iterative responses go away from the fixed-point. A monotonically lower saddle of the (2m)th order is presented in Fig. 2.4c. The nonlinear curve is tangential to and below the line of yk+1 = yk with the (2m)th order, and the one branch is in the upper stable zone and another branch is in the lower unstable zone. Similarly, a monotonically upper-saddle of the (2m)th order is presented in Fig. 2.4d. The nonlinear curve is also tangential to and above the line of yk+1 = yk with the (2m)th order, and the one branch is in the lower stable zone and another branch is in the upper unstable zone. In a similar fashion, the oscillatory stability of fixed-points with higher-order singularity for a 1-dimensional discrete systems is presented in Fig. 2.5. The circular symbol is for a fixed-point xk∗ . In the vicinity of the fixed-point xk∗ , yk = xk − ∗ . An oscillatorilly stable node (an oscillatory sink) of xk∗ and yk+1 = xk+1 − xk+1 the (2m + 1)th order is sketched in Fig. 2.5a. The dashed and dotted lines are for yk+1 = −yk and yk+1 = yk , respectively. The nonlinear curve lies in both stable zones, and the iterative responses oscillatorilly approach the fixed-point. However, an oscillatorilly unstable node (an oscillatory source) of the (2m + 1)th order is presented in Fig. 2.5b. The nonlinear curve lies in both unstable zones, and the iterative responses go away oscillatorilly from the fixed-point. An oscillatory lowersaddle of the (2m i )th order is presented in Fig. 2.5c. The nonlinear curve is tangential to and below the line of yk+1 = −yk with the (2m i )th order, and the one branch is in

2.2 Monotonic and Oscillatory Stability xk +1

yk +1 = yk

101 xk +1

xk

xk

xk∗

xk∗

yk +1 = − yk

yk +1 = yk

(a)

yk +1 = − yk

(b)

xk +1

yk +1 = yk xk

xk +1

xk

xk∗

xk∗

yk +1 = − yk

yk +1 = − yk

yk +1 = yk

(c)

(d)

Fig. 2.4 Monotonic stability of fixed-points with higher-order singularity: a monotonically stable node (sink) of (2m + 1)th-order, b monotonically unstable node (source) of (2m + 1)th-order, c monotonically lower saddle of (2m)th-order and d monotonically upper saddle of (2m)th-order. ∗ Shaded areas are stable zones. (yk = xk − xk∗ and yk+1 = xk+1 − xk+1 )

the upper stable zone and another branch is in the lower unstable zone. Finally, an oscillatory upper-saddle of the (2m i )th order is presented in Fig. 2.5d. The nonlinear curve is also tangential to and above the line of yk+1 = −yk with the (2m i )th order, and the one branch is in the lower stable zone and another branch is in the upper unstable zone. For clear illustrations, oscillatory stability of fixed-points with higherorder singularity for the two iterations are presented in Fig. 2.6. Oscillatory sink and source of the (2m i + 1)th order for the fixed-point are presented in Fig. 2.6a, b, respectively. Oscillatory lower-saddle and upper-saddle of the (2m i )th order is presented in Fig. 2.6c, d, respectively. Theorem 2.2 Consider a 1-dimensional, nonlinear discrete system xk+1 = f (xk , p) in Eq. (2.1) with a fixed-point xk∗ . The corresponding solution is given by xk+ j =

102

2 Low-Dimensional Discrete Systems yk +1 = − yk

yk +1 = yk

yk +1 = yk

xk∗

xk∗

xk +1 xk +1

yk +1 = − yk

xk

xk

(a)

(b)

yk +1 = − yk

yk +1 = − yk

yk +1 = yk

yk +1 = yk

xk∗

xk∗

xk +1

xk +1 xk

xk

(c)

(d)

Fig. 2.5 The oscillatory stability of fixed-points with higher-order singularity: a an oscillatory stable node (an oscillatory sink) of (2m + 1)th-order, b an oscillatory unstable node (an oscillatory source) of (2m + 1)th-order, c an oscillatory lower-saddle of (2m)th-order and d an oscillatory ∗ ) upper-saddle of (2m)th-order. Shaded areas are stable zones. (yk = xk −xk∗ and yk+1 = xk+1 −xk+1

f (xk+ j−1 , p) with j ∈ Z. Suppose there is a neighborhood of the fixed-point xk∗ (i.e., Uk (xk∗ ) = (xk∗ − δ, xk∗ + δ) for an arbitrary δ > 0), and f (xk , p) is C r -continuous (r ≥ 1) in Uk (xk∗ ). For a perturbation of fixed-point yk = xk − xk∗ . (i) xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is monotonically stable of the (2m + 1)th-order sink if and only if G (1) (xk∗ , p) = D f (xk∗ , p) = λ = 1, G (r ) (xk∗ , p) = D (r ) f (xk∗ , p) = 0 for r = 2, 3, . . . , 2m, G

(2m+1)

(xk∗ , p)

for xk ∈ U (xk∗ ) ⊂ .

=D

(2m+1)

f (xk∗ , p)

0

(2.37)

104

2 Low-Dimensional Discrete Systems

(iii) xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is monotonically unstable of the (2m)th-order, lower-saddle if and only if G (1) (xk∗ , p) = D f (xk∗ , p) = λ = 1, G (r ) (xk∗ , p) = D (r ) f (xk∗ , p) = 0 for r = 2, 3, . . . , 2m − 1, G (2m) (xk∗ , p) = D (2m) f (xk∗ , p) < 0 monotonically stable for yk > 0, G (2m) (xk∗ , p) = D (2m) f (xk∗ , p) < 0 monotonically unstable for yk < 0 (2.38) for xk ∈ U (xk∗ ) ⊂ . (iv) xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is monotonically unstable of the (2m)th-order upper-saddle if and only if G (1) (xk∗ , p) = D f (xk∗ , p) = λ = 1, G (r ) (xk∗ , p) = D (r ) f (xk∗ , p) = 0 for r = 2, 3, . . . , 2m − 1, G (2m) (xk∗ , p) = D (2m) f (xk∗ , p) > 0 monotonically unstable for yk > 0, G (2m) (xk∗ , p) = D (2m) f (xk∗ , p) > 0 monotoically stable for yk < 0 (2.39) for xk ∈ U (xk∗ ) ⊂ . (v) xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is oscillatorilly stable of the (2m + 1)th-order sink if and only if G (1) (xk∗ , p) = D f (xk∗ , p) = λ = −1, G (r ) (xk∗ , p) = D (r ) f (xk∗ , p) = 0 for r = 2, 3, . . . , 2m, G

(2m+1)

(xk∗ , p)

=D

(2m+1)

f (xk∗ , p)

(2.40)

>0

for xk ∈ U (xk∗ ) ⊂ . (vi) xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is oscillatorilly unstable of the (2m + 1)th-order source if and only if G (1) (xk∗ , p) = D f (xk∗ , p) = λ = −1, G (r ) (xk∗ , p) = D (r ) f (xk∗ , p) = 0 for r = 2, 3, . . . , 2m, G

(2m+1)

(xk∗ , p)

for xk ∈ U (xk∗ ) ⊂ .

=D

(2m+1)

f (xk∗ , p)

0, G (2m) (xk∗ , p) = D (2m) f (xk∗ , p) < 0 oscillatorilly stable for yk < 0 (2.42) for xk ∈ U (xk∗ ) ⊂ . (viii) xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is oscillatorilly unstable of the (2m)th-order upper-saddle if and only if G (1) (xk∗ , p) = D f (xk∗ , p) = λ = −1, G (r ) (xk∗ , p) = D (r ) f (xk∗ , p) = 0 for r = 2, 3, . . . , 2m − 1, G (2m) (xk∗ , p) = D (2m) f (xk∗ , p) > 0 oscillatorilly stable for yk > 0, G (2m) (xk∗ , p) = D (2m) f (xk∗ , p) > 0 oscillatorilly unstable for yk < 0 (2.43) for xk ∈ U (xk∗ ) ⊂ . Proof Consider xk+1 − xk∗ = f (xk , p) − xk∗ = f (xk∗ , p) − xk∗ + +

2m  1 (2m) D f (xk∗ , p)(xk − xk∗ )r r ! r =1

1 D (2m+1) f (xk∗ , p)(xk − xk∗ )2m+1 + o((xk − xk∗ )2m+1 ) (2m + 1)!

xk+1 − xk∗ = f (xk , p) − xk∗ = f (xk∗ , p) − xk∗ +

2m−1  r =1

1 (2m−1) D f (xk∗ , p)(xk − xk∗ )r r!

1 D (2m) f (xk∗ , p)(xk − xk∗ )2m + o((xk − xk∗ )2m ) + (2m)! because the fixed-point xk∗ is determined by ∗ = xk∗ f (xk∗ , p) = xk+1

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2 Low-Dimensional Discrete Systems

thus, we have (xk+1 − xk∗ ) =

2m  1 (2m) ∗ (xk , p)(xk − xk∗ )r G r ! r =1

1 G (2m+1) (xk∗ , p)(xk − xk∗ )2m+1 , (2m + 1)! 2m−1  1 (xk+1 − xk∗ ) = G (2m−1) (xk∗ , p)(xk − xk∗ )r r ! r =1 +

+

1 G (2m) (xk∗ , p)(xk − xk∗ )2m . (2m)!

(a) Consider the following conditions G (1) (xk∗ , p) = D f (xk∗ , p) = λ = 1, G (r ) (xk∗ , p) = 0 for r = 2, 3, . . . , 2m, G (2m+1) (xk∗ , p) = 0. Thus, (xk+1 − xk∗ ) = (xk − xk∗ ) +

1 G (2m+1) (xk∗ , p)(xk − xk∗ )2m+1 . (2m + 1)!

(b) Consider G (1) (xk∗ , p) = D f (xk∗ , p) = λ = 1, G (r ) (xk∗ , p) = 0 for r = 2, 3, . . . , 2m − 1, G (2m) (xk∗ , p) = 0. Thus, (xk+1 − xk∗ ) = (xk − xk∗ ) +

1 G (2m) (xk∗ , p)(xk − xk∗ )2m . (2m + 1)!

(c) Consider the following conditions G (1) (xk∗ , p) = D f (xk∗ , p) = λ = −1, G (r ) (xk∗ , p) = 0 for r = 2, 3, . . . , 2m, G (2m+1) (xk∗ , p) = 0.

2.2 Monotonic and Oscillatory Stability

107

Thus, (xk+1 − xk∗ ) = −(xk − xk∗ ) +

1 G (2m+1) (xk∗ , p)(xk − xk∗ )2m+1 . (2m + 1)!

(d) Consider G (1) (xk∗ , p) = D f (xk∗ , p) = λ = −1, G (r ) (xk∗ , p) = 0 for r = 2, 3, . . . , 2m − 1, G (2m) (xk∗ , p) = 0. Thus, (xk+1 − xk∗ ) = −(xk − xk∗ ) +

1 G (2m) (xk∗ , p)(xk − xk∗ )2m . (2m + 1)!

(i) From the definition, a monotonically stable fixed-point of the (2m + 1)th-order sink is based on the following two cases. (i1 ) For (xk − xk∗ ) > 0, if G (1) (xk∗ , p) = D f (xk∗ , p) = λ = 1, G (r ) (xk∗ , p) = 0 for r = 2, 3, . . . , 2m, G (2m+1) (xk∗ , p) = 0; 0 < (xk+1 − xk∗ ) < (xk − xk∗ ), then 0 < (xk+1 − xk∗ ) = (xk − xk∗ ) + < (xk − xk∗ ),

1 G (2m+1) (xk∗ , p)(xk − xk∗ )2m+1 (2m + 1)!

so, 1 G (2m+1) (xk∗ , p)(xk − xk∗ )2m+1 < 0. (2m + 1)! Owing to (xk − xk∗ ) > 0, we have G (2m+1) (xk∗ , p) = D (2m+1) f (xk∗ , p) < 0.

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2 Low-Dimensional Discrete Systems

(i2 ) For (xk − xk∗ ) < 0, if G (1) (xk∗ , p) = D f (xk∗ , p) = λ = 1, G (r ) (xk∗ , p) = 0 for r = 2, 3, . . . , 2m, G (2m+1) (xk∗ , p) = 0; 0 > (xk+1 − xk∗ ) > (xk − xk∗ ), then 0 > (xk+1 − xk∗ ) = (xk − xk∗ ) + > (xk − xk∗ ),

1 G (2m+1) (xk∗ , p)(xk − xk∗ )2m+1 (2m + 1)!

so, 1 G (2m+1) (xk∗ , p)(xk − xk∗ )2m+1 > 0. (2m + 1)! Owing to (xk − xk∗ ) < 0, we have G (2m+1) (xk∗ , p) = D (2m+1) f (xk∗ , p) < 0. In summary, xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is monotonically stable of the (2m + 1)th-order sink if G (1) (xk∗ , p) = D f (xk∗ , p) = λ = 1, G (r ) (xk∗ , p) = 0 for r = 2, 3, . . . , 2m, G (2m+1) (xk∗ , p) = D (2m+1) f (xk∗ , p) < 0 for Uk (xk∗ ) ⊂ , vice versa. (ii) From the definition, a monotonically unstable fixed-point of the (2m+1)th-order source is based on the following two cases. (ii1 ) For (xk − xk∗ ) > 0, if G (1) (xk∗ , p) = D f (xk∗ , p) = λ = 1, G (r ) (xk∗ , p) = 0 for r = 2, 3, . . . , 2m, G (2m+1) (xk∗ , p) = 0; (xk+1 − xk∗ ) > (xk − xk∗ ) > 0, then

2.2 Monotonic and Oscillatory Stability

xk+1 − xk∗ = (xk − xk∗ ) + > (xk − xk∗ ),

109

1 G (2m+1) (xk∗ , p)(xk − xk∗ )2m+1 (2m + 1)!

so, 1 G (2m+1) (xk∗ , p)(xk − xk∗ )2m+1 > 0. (2m + 1)! Owing to (xk − xk∗ ) > 0, we have G (2m+1) (xk∗ , p) = D (2m+1) f (xk∗ , p) > 0. (ii2 ) For (xk − xk∗ ) < 0, if G (1) (xk∗ , p) = D f (xk∗ , p) = λ = 1, G (r ) (xk∗ , p) = 0 for r = 2, 3, . . . , 2m, G (2m+1) (xk∗ , p) = 0; (xk+1 − xk∗ ) < (xk − xk∗ ) < 0, then 1 G (2m+1) (xk∗ , p)(xk − xk∗ )2m+1 (2m + 1)! < (xk − xk∗ ) < 0,

(xk+1 − xk∗ ) = (xk − xk∗ ) +

so, 1 G (2m+1) (xk∗ , p)(xk − xk∗ )2m+1 < 0. (2m + 1)! Owing to (xk − xk∗ ) < 0, we have G (2m+1) (xk∗ , p) = D (2m+1) f (xk∗ , p) > 0. In summary, xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is monotonically unstable of the (2m + 1)th-order source if G (1) (xk∗ , p) = D f (xk∗ , p) = λ = 1, G (r ) (xk∗ , p) = 0 for r = 2, 3, . . . , 2m, G (2m+1) (xk∗ , p) = D (2m+1) f (xk∗ , p) > 0 for Uk (xk∗ ) ⊂ , vice versa.

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2 Low-Dimensional Discrete Systems

(iii) From the definition, a monotonically unstable fixed-point of the (2m)th-order lower-saddle is based on the following two cases. (iii1 ) For (xk − xk∗ ) > 0, if G (1) (xk∗ , p) = D f (xk∗ , p) = λ = 1, G (r ) (xk∗ , p) = 0 for r = 2, 3, . . . , 2m − 1, G (2m) (xk∗ , p) = 0; 0 < xk+1 − xk∗ < (xk − xk∗ ), then (xk+1 − xk∗ ) = (xk − xk∗ ) + < (xk − xk∗ ),

1 G (2m) (xk∗ , p)(xk − xk∗ )2m (2m)!

so, 1 G (2m) (xk∗ , p)(xk − xk∗ )2m < 0. (2m)! Thus G (2m) (xk∗ , p) = D (2m) f (xk∗ , p) < 0. (iii2 ) For (xk − xk∗ ) < 0, if G (1) (xk∗ , p) = D f (xk∗ , p) = λ = 1, G (r ) (xk∗ , p) = 0 for r = 2, 3, . . . , 2m − 1, G (2m) (xk∗ , p) = 0; (xk+1 − xk∗ ) < (xk − xk∗ ) < 0, then (xk+1 − xk∗ ) = (xk − xk∗ ) + < (xk − xk∗ ),

1 G (2m) (xk∗ , p)(xk − xk∗ )2m (2m + 1)!

so, 1 G (2m) (xk∗ , p)(xk − xk∗ )2m < 0. (2m + 1)!

2.2 Monotonic and Oscillatory Stability

111

Thus, G (2m) (xk∗ , p) = D (2m) f (xk∗ , p) < 0. In summary, xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is monotonically unstable of the (2m)th-order, lower-saddle if G (1) (xk∗ , p) = D f (xk∗ , p) = λ = 1, G (r ) (xk∗ , p) = 0 for r = 2, 3, . . . , 2m − 1, G (2m) (xk∗ , p) = D (2m) f (xk∗ , p) < 0 for Uk (xk∗ ) ⊂ , vice versa. (iv) From the definition, a monotonically unstable fixed-point of the (2m)th-order upper-saddle is based on the following two cases. (iv1 ) For (xk − xk∗ ) > 0, if G (1) (xk∗ , p) = D f (xk∗ , p) = λ = 1, G (r ) (xk∗ , p) = 0 for r = 2, 3, . . . , 2m − 1, G (2m) (xk∗ , p) = 0; (xk+1 − xk∗ ) > (xk − xk∗ ) > 0, then (xk+1 − xk∗ ) = (xk − xk∗ ) + > (xk − xk∗ ),

1 G (2m) (xk∗ , p)(xk − xk∗ )2m (2m)!

so, 1 G (2m) (xk∗ , p)(xk − xk∗ )2m > 0. (2m)! Thus G (2m) (xk∗ , p) = D (2m) f (xk∗ , p) > 0. (iv2 ) For (xk − xk∗ ) < 0, if G (1) (xk∗ , p) = D f (xk∗ , p) = λ = 1, G (r ) (xk∗ , p) = 0 for r = 2, 3, . . . , 2m − 1, G (2m) (xk∗ , p) = 0;

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2 Low-Dimensional Discrete Systems

0 > (xk+1 − xk∗ ) > (xk − xk∗ ), then (xk+1 − xk∗ ) = (xk − xk∗ ) + > (xk − xk∗ ),

1 G (2m) (xk∗ , p)(xk − xk∗ )2m (2m + 1)!

so, 1 G (2m) (xk∗ , p)(xk − xk∗ )2m > 0. (2m + 1)! Thus, G (2m) (xk∗ , p) = D (2m) f (xk∗ , p) > 0. In summary, xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is monotonically unstable of the (2m)th-order, upper-saddle if G (1) (xk∗ , p) = D f (xk∗ , p) = λ = 1, G (r ) (xk∗ , p) = 0 for r = 2, 3, . . . , 2m − 1, G (2m) (xk∗ , p) = D (2m) f (xk∗ , p) > 0 for Uk (xk∗ ) ⊂ , vice versa. (v) From the definition, an oscillatorilly stable fixed-point of the (2m + 1)th-order sink is based on the following two cases. (v1 ) For (xk − xk∗ ) > 0, if G (1) (xk∗ , p) = D f (xk∗ , p) = λ = −1, G (r ) (xk∗ , p) = 0 for r = 2, 3, . . . , 2m, G (2m+1) (xk∗ , p) = 0; 0 > (xk+1 − xk∗ ) > −(xk − xk∗ ), then (xk+1 − xk∗ ) = −(xk − xk∗ ) + > −(xk − xk∗ ),

1 G (2m+1) (xk∗ , p)(xk − xk∗ )2m+1 (2m + 1)!

so, 1 G (2m+1) (xk∗ , p)(xk − xk∗ )2m+1 > 0. (2m + 1)!

2.2 Monotonic and Oscillatory Stability

113

Owing to (xk − xk∗ ) > 0, we have G (2m+1) (xk∗ , p) = D (2m+1) f (xk∗ , p) > 0. (v2 ) For (xk − xk∗ ) < 0, if G (1) (xk∗ , p) = D f (xk∗ , p) = λ = −1, G (r ) (xk∗ , p) = 0 for r = 2, 3, . . . , 2m, G (2m+1) (xk∗ , p) = 0; 0 < (xk+1 − xk∗ ) < −(xk − xk∗ ), then (xk+1 − xk∗ ) = −(xk − xk∗ ) + < −(xk − xk∗ ),

1 G (2m+1) (xk∗ , p)(xk − xk∗ )2m+1 (2m + 1)!

so, 1 G (2m+1) (xk∗ , p)(xk − xk∗ )2m+1 < 0. (2m + 1)! Owing to (xk − xk∗ ) < 0, we have G (2m+1) (xk∗ , p) = D (2m+1) f (xk∗ , p) > 0. In summary, xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is oscillatorilly stable of the (2m + 1)th order sink if G (1) (xk∗ , p) = D f (xk∗ , p) = λ = −1, G (r ) (xk∗ , p) = 0 for r = 2, 3, . . . , 2m, G (2m+1) (xk∗ , p) = D (2m+1) f (xk∗ , p) > 0 for Uk (xk∗ ) ⊂ , vice versa. (vi) From the definition, an oscillatorilly unstable fixed-point of the (2m+1)th-order source is based on the following two cases. (vi1 ) For (xk − xk∗ ) > 0, if G (1) (xk∗ , p) = D f (xk∗ , p) = λ = −1, G (r ) (xk∗ , p) = 0 for r = 2, 3, . . . , 2m, G (2m+1) (xk∗ , p) = 0; (xk+1 − xk∗ ) < −(xk − xk∗ ) < 0,

114

2 Low-Dimensional Discrete Systems

then (xk+1 − xk∗ ) = −(xk − xk∗ ) + < −(xk − xk∗ ),

1 G (2m+1) (xk∗ , p)(xk − xk∗ )2m+1 (2m + 1)!

so, 1 G (2m+1) (xk∗ , p)(xk − xk∗ )2m+1 < 0. (2m + 1)! Owing to (xk − xk∗ ) > 0, we have G (2m+1) (xk∗ , p) = D (2m+1) f (xk∗ , p) < 0. (vi2 ) For (xk − xk∗ ) < 0, if G (1) (xk∗ , p) = D f (xk∗ , p) = λ = −1, G (r ) (xk∗ , p) = 0 for r = 2, 3, . . . , 2m, G (2m+1) (xk∗ , p) = 0; (xk+1 − xk∗ ) > −(xk − xk∗ ) > 0, then 1 G (2m+1) (xk∗ , p)(xk − xk∗ )2m+1 (2m + 1)! > −(xk − xk∗ ) > 0,

(xk+1 − xk∗ ) = −(xk − xk∗ ) +

so, 1 G (2m+1) (xk∗ , p)(xk − xk∗ )2m+1 > 0. (2m + 1)! Owing to (xk − xk∗ ) < 0, we have G (2m+1) (xk∗ , p) = D (2m+1) f (xk∗ , p) < 0. In summary, xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is oscillatorilly unstable of the (2m + 1)th-order source if G (1) (xk∗ , p) = D f (xk∗ , p) = λ = −1, G (r ) (xk∗ , p) = 0 for r = 2, 3, . . . , 2m, G (2m+1) (xk∗ , p) = D (2m+1) f (xk∗ , p) < 0 for Uk (xk∗ ) ⊂ , vice versa.

2.2 Monotonic and Oscillatory Stability

115

(vii) From the definition, an oscillatorilly unstable fixed-point of the (2m)th-order lower-saddle is based on the following two cases. (vii1 ) For (xk − xk∗ ) > 0, if G (1) (xk∗ , p) = D f (xk∗ , p) = λ = −1, G (r ) (xk∗ , p) = 0 for r = 2, 3, . . . , 2m − 1, G (2m) (xk∗ , p) = 0; (xk+1 − xk∗ ) < −(xk − xk∗ ) < 0, then (xk+1 − xk∗ ) = −(xk − xk∗ ) + < −(xk − xk∗ ),

1 G (2m) (xk∗ , p)(xk − xk∗ )2m (2m)!

so, 1 G (2m) (xk∗ , p)(xk − xk∗ )2m < 0. (2m)! Thus G (2m) (xk∗ , p) = D (2m) f (xk∗ , p) < 0. (vii2 ) For (xk − xk∗ ) < 0, if G (1) (xk∗ , p) = D f (xk∗ , p) = λ = −1, G (r ) (xk∗ , p) = 0 for r = 2, 3, . . . , 2m − 1, G (2m) (xk∗ , p) = 0; 0 < (xk+1 − xk∗ ) < −(xk − xk∗ ), then (xk+1 − xk∗ ) = −(xk − xk∗ ) + < −(xk − xk∗ ),

1 G (2m) (xk∗ , p)(xk − xk∗ )2m (2m + 1)!

so, 1 G (2m) (xk∗ , p)(xk − xk∗ )2m < 0. (2m + 1)! Thus, G (2m) (xk∗ , p) = D (2m) f (xk∗ , p) < 0.

116

2 Low-Dimensional Discrete Systems

In summary, xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is oscillatorilly unstable of the (2m)th-order, lower-saddle if G (1) (xk∗ , p) = D f (xk∗ , p) = λ = −1, G (r ) (xk∗ , p) = 0 for r = 2, 3, . . . , 2m − 1, G (2m) (xk∗ , p) = D (2m) f (xk∗ , p) < 0 for Uk (xk∗ ) ⊂ Ω , vice versa. (viii) From the definition, an oscillatorilly unstable fixed-point of the (2m)th-order upper-saddle is based on the following two cases. (viii1 ) For (xk − xk∗ ) > 0, if G (1) (xk∗ , p) = D f (xk∗ , p) = λ = −1, G (r ) (xk∗ , p) = 0 for r = 2, 3, . . . , 2m − 1, G (2m) (xk∗ , p) = 0; 0 > (xk+1 − xk∗ ) > −(xk − xk∗ ), then (xk+1 − xk∗ ) = −(xk − xk∗ ) + > −(xk − xk∗ ),

1 G (2m) (xk∗ , p)(xk − xk∗ )2m (2m)!

so, 1 G (2m) (xk∗ , p)(xk − xk∗ )2m > 0. (2m)! Thus G (2m) (xk∗ , p) = D (2m) f (xk∗ , p) > 0. (viii2 ) For (xk − xk∗ ) < 0, if G (1) (xk∗ , p) = D f (xk∗ , p) = λ = −1, G (r ) (xk∗ , p) = 0 for r = 2, 3, . . . , 2m − 1, G (2m) (xk∗ , p) = 0; (xk+1 − xk∗ ) > −(xk − xk∗ ) > 0, then (xk+1 − xk∗ ) = −(xk − xk∗ ) + > −(xk − xk∗ ),

1 G (2m) (xk∗ , p)(xk − xk∗ )2m (2m + 1)!

2.2 Monotonic and Oscillatory Stability

117

so, 1 G (2m) (xk∗ , p)(xk − xk∗ )2m > 0. (2m + 1)! Thus, G (2m) (xk∗ , p) = D (2m) f (xk∗ , p) > 0. In summary, xk+ j ( j ∈ Z) in a neighborhood of the fixed-point xk∗ is oscillatorilly unstable of the (2m)th-order, upper-saddle if G (1) (xk∗ , p) = D f (xk∗ , p) = λ = −1, G (r ) (xk∗ , p) = 0 for r = 2, 3, . . . , 2m − 1, G (2m) (xk∗ , p) = D (2m) f (xk∗ , p) > 0 for Uk (xk∗ ) ⊂ , vice versa. 

The theorem is proved.

2.3 Bifurcations In this section, monotonic and oscillatory bifurcations for 1-dimensional nonlinear discrete systems are discussed.

2.3.1 Monotonic Bifurcations Definition 2.4 Consider a 1-dimensional, nonlinear discrete system xk+1 = f (xk , p) in Eq. (2.1) with a fixed-point xk∗ . The corresponding solution is given by xk+ j = f (xk+ j−1 , p) with j ∈ Z. Suppose there is a neighborhood of the fixed-point xk∗ (i.e., Uk (xk∗ ) = (xk∗ − δ, xk∗ + δ) for an arbitrary δ > 0), and f (xk , p) is C r -continuous (r ≥ 1) in Uk (xk∗ ). For a perturbation of fixed-point yk = xk − xk∗ . In the vicinity of ∗ , p0 ), f (xk , p) can be expended for (0 < θ < 1) as point (xk0 ∗ ∗ f (xk , p) − xk(0) = λ0 (xk − xk(0) ) + bT · (p − p0 )

+

q m>1  q=2 r =0

1 r (q−r,r ) ∗ C a (xk − xk(0) )q−r (p − p0 )r q! q i

1 ∗ [(xk − xk(0) + )∂s (i) + (p − p0 )∂p ]m+1 k (m + 1)! ∗ × f (xk(0) + θ xk , p0 + θ p))

(2.44)

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2 Low-Dimensional Discrete Systems

where  λ0 = ∂xk f (xk , p)(x∗

 , bT = ∂p f (xk , p)(x ∗ ,p0 ) , k(0)  (r ) (s)  = ∂xk ∂p f (xk , p) (x ∗ ,p ) . k(0) ,p0 )

a(r,s)

k(0)

(2.45)

0

If λ0 = 1 and p = p0 with yk+1 = +yk , the stability of fixed-point xk∗ changes from stable to unstable state (or from unstable to stable state). The bifurcation manifold of fixed-points is determined by bT · (p − p0 ) +

m>1 

∗ Cmr a(m−r,r ) (xk∗ − xk(0) )m−r (p − p0 )r = 0,

r =0

(2.46)

equivalent to A0 (yk∗ )m

+

A1 (yk∗ )m−1 + (m,0) m

with A0 = a

··· +

Am−1 yk∗

+ Am = 0

= D f = 0.

∗ In the neighborhood of (xk0 , p0 ) Eq. (2.46) possesses s-branch solutions of equi∗ librium yk (0 < s ≤ m). Such s-branch solutions are called the bifurcation solutions ∗ ∗ , p0 ). Such a bifurcation at point (xk0 , p0 ) of fixed-point xk∗ in the neighborhood of (xk0 is called a bifurcation of mth-order.

(i) For m = 2l, there are (2l) fixed-points in Eq. (2.46). If ⎫ λ0 = 1, ⎪ ⎪ ⎬ ∗ ∗ , p0 ) = D (r ) f (xk0 , p0 ) = 0 G (r ) (xk0 ⎪ (r = 2, 3, . . . , 2l − 1), ⎪ ⎭ ∗ , p0 ) > 0, G (2l) = D (2l) f (xk0

(2.47)

∗ the bifurcation of fixed-point xk∗ at point (xk0 , p0 ) is called a monotonic-upper∗ , p0 ) is a saddle-node) bifurcation of the (2l)th order. The bifurcation point (xk0 monotonic-upper-saddle of the (2l)th order

(ii) For m = 2l, there are (2l) fixed-points in Eq. (2.46). If ⎫ λ0 = 1, ⎪ ⎪ ⎬ ∗ ∗ , p0 ) = D (r ) f (xk0 , p0 ) = 0 G (r ) (xk0 ⎪ (r = 2, 3, . . . , 2l − 1), ⎪ ⎭ ∗ (2l) (2l) G = D f (xk0 , p0 ) < 0,

(2.48)

∗ the bifurcation of fixed-point xk∗ at point (xk0 , p0 ) is called a monotonic-lower∗ , p0 ) is saddle-node) bifurcation of the (2l)th order. The bifurcation point (xk0 a monotonic-lower-saddle of the (2l)th order

2.3 Bifurcations

119

(iii) For m = 2l + 1, there are (2l + 1) fixed-points in Eq. (2.46). If ⎫ λ0 = 1, ⎪ ⎪ ⎬ ∗ ∗ , p0 ) = D (r ) f (xk0 , p0 ) = 0 G (r ) (xk0 ⎪ (r = 2, 3, . . . , 2l), ⎪ ⎭ ∗ , p0 ) < 0, G (2l+1) = D (2l+1) f (xk0

(2.49)

∗ the bifurcation of fixed-point xk∗ at point (xk0 , p0 ) is called a monotonic∗ , p0 ) is a sink bifurcation of the (2l + 1)th order. The bifurcation point (xk0 monotonic-sink of the (2l + 1)th order.

(iv) For m = 2l + 1, there are (2l + 1) fixed-points in Eq. (2.46). If ⎫ λ0 = 1, ⎪ ⎪ ⎬ ∗ ∗ , p0 ) = D (r ) f (xk0 , p0 ) = 0 G (r ) (xk0 ⎪ (r = 2, 3, . . . , 2l), ⎪ ⎭ ∗ , p0 ) > 0, G (2l+1) = D (2l+1) f (xk0

(2.50)

∗ the bifurcation of fixed-point xk∗ at point (xk0 , p0 ) is called a monotonic∗ , p0 ) is a source bifurcation of the (2l + 1)th order. The bifurcation point (xk0 monotonic-source of (2l + 1)th order.

Consider a quadratic, monotonic bifurcation manifold with λ0 = 1, yk+1 = yk as A2 + A1 yk∗ + A0 (yk∗ )2 = 0

(2.51)

where A0 = 21 a (2,0) , A1 = a(1,1) (p − p0 ), A2 = bT · (p − p0 ) + 21 a(0,2) (p − p0 )2 ;   = ∂ (2) f (xk , p) a (2,0) = ∂ (2) ∂ (0) f (xk , p) ∗ xk

= a(1,1) =

p

(xk0 ,p0 )

∗ (xk0 ,p0 )

xk

(2)

∗ ∗ D f (xk0 , p0 ) = G (2) (xk(0) , p0 )  (1) (1) ∂xk ∂p f (xk , p)(x ∗ ,p ) ,

= 0,

(2.52)

k0   a(0,2) = ∂p(2) f (xk , p)(x ∗ ,p ) , b = ∂p f (xk , p)(x∗ ,p0 ) . 0

k0

0

k0

(a) If the bifurcation manifold is A1 = 0 and A2 + A0 (yk∗ )2 = 0

(2.53)

λ0 = 1, A0 × A2 < 0,

(2.54)

for

120

2 Low-Dimensional Discrete Systems ∗ such a bifurcation at point (xk0 , p0 ) is called a monotonic-saddle-node appearing bifurcation. ∗ ∗ ∗ , p0 ) = D (2) f (xk0 , p0 ) < 0, the bifurcation at point (xk0 , p0 ) (a1 ) If G (2) (xk0 is called a monotonic-lower-saddle-node appearing bifurcation of the second-order. ∗ ∗ ∗ , p0 ) = D (2) f (xk0 , p0 ) > 0, The bifurcation at point (xk0 , p0 ) (a2 ) If G (2) (xk0 is called a monotonic-upper-saddle-node appearing bifurcation of the second-order.

(b) If the bifurcation manifold is λ0 = 1, A2 = 0 and A1 yk∗ + A0 (yk∗ )2 = 0,

(2.55)

∗ , p0 ) is called a monotonic-saddle-node switching such a bifurcation at point (xk0 (transcritical) bifurcation of the second-order. ∗ ∗ ∗ , p0 ) = D (2) f (xk0 , p0 ) < 0, the bifurcation at point (xk0 , p0 ) (b1 ) If G (2) (xk0 is called a monotonic lower-saddle-node switching bifurcation of the second-order. ∗ ∗ ∗ , p0 ) = D (2) f (xk0 , p0 ) > 0, The bifurcation at point (xk0 , p0 ) (b2 ) If G (2) (xk0 is called a monotonic upper-saddle-node switching bifurcation.

Consider a cubic polynomial monotonic bifurcation manifold as A3 + A2 yk∗ + A1 (yk∗ )2 + A0 (yk∗ )3 = 0

(2.56)

where   ∂p(0) f (xk , p)(x ∗ ,p ) = 16 ∂x(3) f (xk , p)(x ∗ ,p A0 = 16 a (3,0) = 16 ∂x(3) k k k0

= A1 = A3 = A2 =

0

∗ ∗ 1 (3) D f (xk0 , p0 ) = 16 G (3) (xk0 , p0 ) = 0, 6 2 (1,1) 1 (1,2) a (p − p0 ) + a (p − p0 ), 2 1 (0,2) b · (p − p0 ) + 2 a (p − p0 )2 + 16 a(0,3) (p 1 (2,0) a + 21 a(2,1) (p − p0 ). 2

k0

0)

(2.57) − p0 ) , 3

(a) If the monotonic bifurcation manifold is A2 yk∗ + A0 (yk∗ )3 = 0 with A3 = 0 and A1 = 0

(2.58)

λ0 = 1 and A0 × A2 < 0,

(2.59)

for

2.3 Bifurcations

121

∗ such a bifurcation at point (xk0 , p0 ) is called a monotonic-node switching bifurcation of the third-order. ∗ ∗ ∗ , p0 ) = D (3) f (xk0 , p0 ) < 0, the bifurcation at point (xk0 , p0 ) (a1 ) If G (3) (xk0 is called a monotonic-sink switching bifurcation of the third-order. ∗ ∗ ∗ , p0 ) = D (3) f (xk0 , p0 ) > 0, the bifurcation at point (xk0 , p0 ) (a2 ) If G (3) (xk0 is called a monotonic-source switching bifurcation of the third-order.

The more detailed discussion on the bifurcation can be found later.

2.3.2 One-Step Oscillatory Bifurcations Definition 2.5 Consider a discrete, nonlinear system xk+1 = f (xk , p) in Eq. (2.1) with a fixed-point xk∗ . The corresponding solution is given by xk+ j = f (xk+ j−1 , p) with j ∈ Z. Suppose there is a neighborhood of the fixed-point xk∗ (i.e., Uk (xk∗ ) = (xk∗ − δ, xk∗ + δ) for an arbitrary δ > 0), and f (xk , p) is C r -continuous (r ≥ 1) in Uk (xk∗ ). For a perturbation of fixed-point yk = xk − xk∗ . In the vicinity of point ∗ , p0 ), f (xk , p) can be expended for (0 < θ < 1) as (xk0 ∗ ∗ xk+1 − xk0 = f (xk , p) − xk0 ∗ = λ0 (xk − xk0 ) + bT · (p − p0 )

+

q m>1  q=2 r =0

1 r (q−r,r ) ∗ q−r C a (xk − xk0 ) (p − p0 )r q! q i

(2.60)

1 ∗ [(xk − xk0 )∂xk + (p − p0 )∂p ]m+1 (m + 1)! ∗ × f (xk0 + θ xk , p0 + θ p)) +

where   λ0 = ∂xk f (xk , p)(x∗ ,p0 ) , bT = ∂p f (xk , p)(x ∗ ,p0 ) , k0 k0  a(r,s) = ∂x(rk ) ∂p(s) f (xk , p)(x ∗ ,p ) . k0

(2.61)

0

If λ0 = −1 and p = p0 with yk+1 = −yk , the stability of fixed-point xk∗ changes from stable to unstable state (or from unstable to stable state). The oscillatory bifurcation manifold of fixed-points is determined by bT · (p − p0 ) +

m>1 

∗ Cmr a(m−r,r ) (xk∗ − xk(0) )m−r (p − p0 )r = 0,

r =0

(2.62)

equivalent to A0 (yk∗ )m + A1 (yk∗ )m−1 + · · · + Am−1 yk∗ + ∗ , A0 = a (m,0) = D m f with yk∗ = xk∗ − xk(0)

Am = 0 = 0.

122

2 Low-Dimensional Discrete Systems

∗ In the neighborhood of (xk0 , p0 ) Eq. (2.62) possesses s-branch solutions of equi∗ librium yk (0 < s ≤ m). Such s-branch solutions are called the bifurcation solutions ∗ , p0 ). of fixed-point xk∗ in the neighborhood of (xk0

(i) For m = 2l, there are (2l) fixed-points in Eq. (2.62). If ⎫ λ0 = −1, ⎪ ⎪ ⎬ ∗ ∗ , p0 ) = D (r ) f (xk0 , p0 ) = 0 G (r ) (xk0 ⎪ (r = 2, 3, . . . , 2l − 1), ⎪ ⎭ ∗ , p0 ) > 0, G (2l) = D (2l) f (xk0

(2.63)

∗ the bifurcation of fixed-point xk∗ at point (xk0 , p0 ) is called an oscillatory-upper∗ , p0 ) is saddle-node) bifurcation of the (2l)th-order. The bifurcation point (xk0 an oscillatory-upper-saddle of the (2l)th-order.

(ii) For m = 2l, there are 2l fixed-points in Eq. (2.62). If ⎫ λ0 = −1, ⎪ ⎪ ⎬ ∗ ∗ , p0 ) = D (r ) f (xk0 , p0 ) = 0 G (r ) (xk0 ⎪ (r = 2, 3, . . . , 2l − 1), ⎪ ⎭ ∗ (2l) (2l) G = D f (xk0 , p0 ) < 0,

(2.64)

∗ the bifurcation of fixed-point xk∗ at point (xk0 , p0 ) is called an oscillatory-lower∗ , p0 ) is saddle-node) bifurcation of the (2l)th-order. The bifurcation point (xk0 an oscillatory-lower-saddle of the (2l)th-order.

(iii) For m = 2l + 1, there are (2l + 1) fixed-points in Eq. (2.62). If ⎫ λ0 = −1, ⎪ ⎪ ⎬ ∗ ∗ , p0 ) = D (r ) f (xk0 , p0 ) = 0 G (r ) (xk0 ⎪ (r = 2, 3, . . . , 2l), ⎪ ⎭ ∗ (2l+1) (2l+1) =D f (xk0 , p0 ) > 0, G

(2.65)

∗ the bifurcation of fixed-point xk∗ at point (xk0 , p0 ) is called an oscillatory∗ , p0 ) is an sink bifurcation of the (2l + 1)th-order. The bifurcation point (xk0 oscillatory-sink of the (2l + 1)th-order.

(iv) For m = 2l + 1, there are (2l + 1) fixed-points in Eq. (2.62). If ⎫ λ0 = −1, ⎪ ⎪ ⎬ ∗ ∗ , p0 ) = D (r ) f (xk0 , p0 ) = 0 G (r ) (xk0 ⎪ (r = 2, 3, . . . , 2l), ⎪ ⎭ ∗ , p0 ) < 0, G (2l+1) = D (2l+1) f (xk0

(2.66)

2.3 Bifurcations

123

∗ the bifurcation of fixed-point xk∗ at point (xk0 , p0 ) is called an oscillatory∗ , p0 ) is an source bifurcation of the (2l + 1)th order. The bifurcation point (xk0 oscillatory-source of the (2l + 1)th order.

Consider a quadratic polynomial oscillatory bifurcation manifold with λ0 = −1 and yk+1 = −yk as A2 + A1 (yk∗ )2 + A0 (yk∗ )2 = 0

(2.67)

where A0 = 21 a (2,0) , A1 = a(1,1) (p − p0 ), A2 = bT · (p − p0 ) + 21 a(0,2) (p − p0 )2   a (2,0) = ∂ (2) ∂ (0) f (xk , p) ∗ = ∂ (2) f (xk , p) xk

a(1,1) a(0,2)

p

(xk0 ,p0 )

xk

∗ (xk0 ,p0 )

∗ ∗ = D (2) f (xk0 , p0 ) = G (2) (xk0 , p0 ) = 0,  (1) (1)  = ∂xk ∂p f (xk , p) (x ∗ ,p ) , k0 0   (2)  = ∂p f (xk , p) (x ∗ ,p ) , b = ∂p f (xk , p)(x∗ ,p0 ) . k0

0

(2.68)

k0

(a) If the oscillatory bifurcation manifold is A1 = 0 and A2 + A0 (yk∗ )2 = 0

(2.69)

λ0 = −1, A0 × A2 < 0,

(2.70)

with

∗ such a bifurcation at point (xk(0) , p0 ) is called an oscillatory-saddle-node bifurcation of the second-order. ∗ ∗ ∗ , p0 ) = D (2) f (xk0 , p0 ) < 0, the bifurcation at point (xk0 , p0 ) (a1 ) If G (2) (xk0 is called an oscillatory lower-saddle-node bifurcation of the second-order. ∗ ∗ ∗ , p0 ) = D (2) f (xk0 , p0 ) > 0, The bifurcation at point (xk0 , p0 ) (a2 ) If G (2) (xk0 is called an oscillatory-upper-saddle-node bifurcation of the second-order.

(b) If the oscillatory bifurcation manifold is λ0 = −1, A2 = 0 and A1 yk∗ + A0 (yk∗ )2 = 0

(2.71)

∗ , p0 ) is called an oscillatory-saddle-node such a bifurcation at point (xk0 switching (transcritical) bifurcation of the second-order. ∗ ∗ ∗ (b1 ) If G (2) (xk0 , p0 ) = D (2) f (xk0 , p0 ) < 0, the bifurcation at point (xk0 , p0 ) is called an oscillatory-lower-saddle-node switching bifurcation of the second-order.

124

2 Low-Dimensional Discrete Systems ∗ ∗ ∗ (b2 ) If G (2) (xk0 , p0 ) = D (2) f (xk0 , p0 ) > 0, The bifurcation at point (xk0 , p0 ) is called an oscillatory-upper-saddle-node switching bifurcation of the second-order.

Consider a cubic polynomial oscillatory bifurcation manifold as A3 + A2 yk∗ + A1 (yk∗ )2 + A0 (yk∗ )3 = 0

(2.72)

where   ∂p(0) f (xk , p)(x ∗ ,p ) = 16 ∂x(3) f (xk , p)(x ∗ ,p A0 = 16 a (3,0) = 16 ∂x(3) k k k0

0

k0

0)

∗ ∗ = 16 D (3) f (xk0 , p0 ) = 16 G (3) (xk0 , p0 ) = 0,

A1 = 21 a(1,2) (p − p0 )2 + a(1,1) (p − p0 ),

(2.73)

A3 = b + 21 a(0,2) (p − p0 )2 + 16 a(0,3) (p − p0 )3 , A2 = 21 a (2,0) + 21 a(2,1) (p − p0 ). (a) If the oscillatory manifold is A0 (yk∗ )3 + A2 yk∗ = 0 wtih A3 = 0 and A1 = 0

(2.74)

λ0 = −1, A0 × A2 < 0,

(2.75)

with

∗ such a bifurcation at point (xk0 , p0 ) is called an oscillatory-node bifurcation of the third-order. ∗ ∗ ∗ (a1 ) If G (3) (xk0 , p0 ) = D (3) f (xk0 , p0 ) > 0, the bifurcation at point (xk0 , p0 ) is called an oscillatory-sink bifurcation of the third-order. ∗ ∗ ∗ , p0 ) = D (3) f (xk0 , p0 ) < 0, the bifurcation at point (xk0 , p0 ) (a2 ) If G (3) (xk0 is called an oscillatory-source bifurcation of the third-order.

The detailed discussion on the oscillatory bifurcation manifold can be found later.

2.3.3 Two-Step Oscillatory Bifurcations Definition 2.6 Consider a 1-dimensional, nonlinear system xk+1 = f (xk , p) in Eq. (2.1) with a fixed-point xk∗ . The corresponding solution is given by xk+ j = f (xk+ j−1 , p) with j ∈ Z. Suppose there is a neighborhood of the fixed-point xk∗ (i.e., Uk (xk∗ ) = (xk∗ − δ, xk∗ + δ) for an arbitrary δ > 0), and f (xk , p) is C r -continuous (r ≥ 1) in Uk (xk∗ ). For a perturbation of fixed-point yk = xk − xk∗ . In the vicinity of ∗ , p0 ), f (xk , p) can be expended for (0 < θ < 1) as point (xk0

2.3 Bifurcations

125

∗ ∗ xk+1 − xk0 = f (xk , p) − xk0 ∗ = λ0 (xk − xk0 ) + bT · (p − p0 )

+

q m>1  q=2 r =0

1 r (q−r,r ) ∗ q−r (xk − xk0 ) (p − p0 )r C a q! q i

(2.76)

1 ∗ )∂xk + (p − p0 )∂p ]m+1 [(xk − xk0 (m + 1)! ∗ × f (xk0 + θ xk , p0 + θ p)) +

and ∗ ∗ = f (xk+1 , p) − xk0 xk+2 − xk0 ∗ = λ0 (xk+1 − xk0 ) + bT · (p − p0 )

+

q m>1  q=2 r =0

1 r (q−r,r ) ∗ q−r C a (xk+1 − xk0 ) (p − p0 )r q! q i

(2.77)

1 ∗ )∂xk + (p − p0 )∂p ]m+1 [(xk+1 − xk0 (m + 1)! ∗ × f (xk0 + θ xk , p0 + θ p)) +

If λ0 = −1 and p = p0 , the stability of fixed point xk∗ from stable to unstable state (or from unstable to stable state). The bifurcation manifold is determined by − (xk −

∗ xk0 )

+ b · (p − p0 ) + T

q m>1  q=2 r =0

1 r (q−r,r ) ∗ q−r C a (xk − xk0 ) (p − p0 )r q! q i

∗ = xk+1 − xk0 , ∗ − (xk+1 − xk0 ) + bT · (p − p0 ) +

q m>1  q=2 r =0

= xk −

1 r (q−r,r ) ∗ q−r C a (xk+1 − xk0 ) (p − p0 )r q! q i

∗ xk0 ;

equivalen to ∗ − yk∗ + A0 (yk∗ )m + A1 (yk∗ )m−1 + · · · + Am−1 yk∗ + Am = yk+1 , ∗ ∗ ∗ ∗ ∗ − yk+1 + A0 (yk+1 )m + A1 (yk+1 )m−1 + · · · + Am−1 yk+1 + Am = yk+2 = yk∗ , ∗ , A0 = a (m,0) = D m f = 0 with yk∗ = xk∗ − xk(0)

(2.78) ∗ , p0 ) Eq. (2.78) possesses s-branch solutions of equiIn the neighborhood of (xk0 ∗ ∗ librium yk and yk+1 (0 < s ≤ m). Such s-branch solutions are called the bifurcation ∗ , p0 ). solutions of fixed-point xk∗ in the neighborhood of (xk0

126

2 Low-Dimensional Discrete Systems

(i) For m = 2l, there are (2l) fixed-points in Eq. (2.78). If ⎫ ∗ ∗ λ0 = −1, G (r ) (xk0 , p0 ) = D (r ) f (xk0 , p0 ) = 0 ⎬ (r = 2, 3, . . . , 2l − 1), ⎭ ∗ , p0 ) > 0, G (2l) = D (2l) f (xk0

(2.79)

∗ , p0 ) is called an oscilthe bifurcation of period-1 fixed-point xk∗ at point (xk0 latory-upper-saddle-node period-doubling bifurcation of the (2l)th order. The ∗ , p0 ) is an oscillatory-upper-saddle of the (2l)th order. bifurcation point (xk0

(ii) For m = 2l, there are 2l fixed-points in Eq. (2.78). If ⎫ λ0 = −1, ⎪ ⎪ ⎬ ∗ ∗ , p0 ) = D (r ) f (xk0 , p0 ) = 0 G (r ) (xk0 ⎪ (r = 2, 3, . . . , 2l − 1), ⎪ ⎭ ∗ , p0 ) < 0, G (2l) = D (2l) f (xk0

(2.80)

∗ the bifurcation of period-1 fixed-point xk∗ at point (xk0 , p0 ) is called an oscillatory-lower-saddle-node period-doubling bifurcation of the (2l)th order. The ∗ , p0 ) is an oscillatory-lower-saddle of the (2l)th order bifurcation point (xk0

(iii) For m = 2l + 1, there are (2l + 1) fixed-points in Eq. (2.78). If ⎫ λ0 = −1, ⎪ ⎪ ⎬ ∗ ∗ , p0 ) = D (r ) f (xk0 , p0 ) = 0 G (r ) (xk0 ⎪ (r = 2, 3, . . . , 2l), ⎪ ⎭ ∗ (2l+1) (2l+1) =D f (xk0 , p0 ) > 0, G

(2.81)

∗ the bifurcation of period-1 fixed-point xk∗ at point (xk0 , p0 ) is called an oscillatory-sink period-doubling bifurcation of the (2l + 1)th order. The bifurcation ∗ , p0 ) is an oscillatory-sink of the (2l + 1)th order. point (xk0

(iv) For m = 2l + 1, there are (2l + 1) fixed-points in Eq. (2.78). If ⎫ λ0 = −1, ⎪ ⎪ ⎬ ∗ ∗ , p0 ) = D (r ) f (xk0 , p0 ) = 0 G (r ) (xk0 ⎪ (r = 2, 3, . . . , 2l), ⎪ ⎭ ∗ , p0 ) < 0, G (2l+1) = D (2l+1) f (xk0

(2.82)

∗ the bifurcation of period-1 fixed-point xk∗ at point (xk0 , p0 ) is called an oscillatory-source period-doubling bifurcation of the (2l +1)th order. The bifurcation ∗ , p0 ) is an oscillatory-source of the (2l + 1)th order. point (xk0

2.3 Bifurcations

127

At the bifurcation point, period-1 and period-2 fixed-points are the same. However, the bifurcation conditions are different, which are determined by d r xk+1 /d xkr |(xk0∗ ,p0 ) and d r xk+2 /d xkr |(xk0∗ ,p0 ) (r = 1, 2, . . .). Consider a special case as ∗ −yk∗ + A3 + A2 yk∗ + A1 (yk∗ )2 + A0 (yk∗ )3 = yk+1 , ∗ ∗ ∗ ∗ ∗ −yk+1 + A3 + A2 yk+1 + A1 (yk+1 )2 + A0 (yk+1 )3 = yk+2 = yk∗

(2.83)

where   ∂p(0) f (xk , p)(x ∗ ,p ) = 16 ∂x(3) f (xk , p)(x ∗ ,p A0 = 16 a (3,0) = 16 ∂x(3) k k k0

= A1 = A3 = A2 =

0

k0

0)

∗ ∗ 1 (3) D f (xk0 , p0 ) = 16 G (3) (xk0 , p0 ) = 0, 6 2 (1,1) 1 (1,2) a (p − p0 ) + a (p − p0 ), 2 2 1 (0,2) 1 (0,3) b + 2a (p − p0 ) + 6 a (p − p0 )3 , 1 (2,0) a + 21 a(2,1) (p − p0 ). 2

(2.84)

If the oscillatory manifold is ∗ , −yk∗ + A0 (yk∗ )3 + A2 yk∗ = yk+1 ∗ ∗ ∗ ∗ 3 ≡ yk∗ −yk+1 + A0 (yk+1 ) + A2 yk+1 = yk+2 wtih A3 = 0 and A1 = 0

(2.85)

λ0 = −1, A0 × A2 < 0,

(2.86)

for

∗ such a bifurcation at point (xk0 , p0 ) is called a period-doubling bifurcation. ∗ ∗ ∗ , p0 ) = D (3) f (xk0 , p0 ) > 0, the bifurcation at point (xk0 , p0 ) is (a) If G (3) (xk0 called an oscillatory-sink period-doubling bifurcation of the third-order for the period-1 fixed point. ∗ ∗ ∗ , p0 ) = D (3) f (xk0 , p0 ) < 0, the bifurcation at point (xk0 , p0 ) is (b) If G (3) (xk0 called an oscillatory-source period-doubling bifurcation of the third-order for the period-1 fixed point.

2.4 Sample Bifurcation Analysis Four types of special cases can be discussed through 1-dimensional discrete systems and intuitive illustrations are presented for a better understanding of bifurcation for nonlinear discrete maps.

128

2 Low-Dimensional Discrete Systems

2.4.1 Saddle-Node Appearing Bifurcation Consider a 1-dimensional discrete system as xk+1 = f (xk , p) ≡ xk + p − xk2 .

(2.87)

√ For xk+1 = xk , the fixed-points of the foregoing equation are xk∗ = ± p( p ≥ 0) and no any fixed-points exist for p < 0. In the vicinity of the fixed-point xk∗ with yk = xk − xk∗ , the linearized equation of Eq. (2.87) is yk+1 = D f (xk∗ , p)yk = (1 − 2xk∗ )yk .

(2.88)

√ √ (i) For the branch of xk∗ = + p( p > 0),the fixed-point xk∗ = + p is • monotonically stable for p ∈ (0, 41 ) due to 0 < yk+1 < yk or 0 > yk+1 > yk , • zero stable for p = 14 due to yk+1 = 0 for all yk , • oscillatorilly stable for p ∈ ( 41 , 1) due to 0 > yk+1 > −yk or 0 < yk+1 < −yk , • flipped for p = 1 due to yk+1 = −yk , • oscillatorilly unstable for p ∈ (1, ∞) due to yk+1 < −yk < 0 or yk+1 > −yk > 0. (i1 )

For p = 1, the higher-order Taylor series of Eq. (2.87) is yk+1 = D f (xk∗ , p)yk +

(i2 )

1 D2 2!

f (xk∗ , p)yk2 = −yk − yk2 .

(2.89)

√ Thus, the fixed-point xk∗ = + p at p = 1 with yk+1 = −yk is an oscillatory lower-saddle of the second-order. √ The period-doubling of xk∗ = + p for p ≥ 1 is determined by √ √ yk+1 = (1 − 2 p)yk − yk2 = yk − yk (yk + 2 p), (2.90) √ √ 2 yk+2 = (1 − 2 p)yk+1 − yk+1 = yk+1 − yk+1 (yk+1 + 2 p), or √ √ √ yk+2 = yk − yk (yk + 2 p)[yk2 − 2(1 − p)yk + 2(1 − p)]. (2.91) For yk+2 = yk , we have ∗ yk∗ = 0, yk+1 = 0, √ √ ∗ ∗ = −2 p yk = −2 p, yk+1



⇒

√ √ ∗ = xk∗ = p, xk∗ = p, xk+1 √ √ ∗ ∗ ∗ xk = − p, xk+1 = xk = − p; (2.92)

Such period-2 fixed-points is trivial, which is equivalent to the period-1 fixed-points.

2.4 Sample Bifurcation Analysis

129

∗ ∗ (i2a ) The trivial period-2 fixed-point of yk∗ = yk+1 = yk+2 = 0 with p > 1 and

dyk+1 √ √ | y ∗ = 1 − 2 p − 2yk∗ = 1 − 2 p < −1, dyk k dyk+2 √ √ ∗ | y ∗ = 1 − 2 p − 2yk+1 = 1 − 2 p < −1, dyk+1 k+1

(2.93)

is oscillatorilly unstable. √ ∗ ∗ = yk+2 = −2 p with (i2b ) The trivial period-2 fixed-point of yk∗ = yk+1 p > 1 and dyk+1 √ √ | y ∗ = 1 − 2 p − 2yk∗ = 1 + 2 p > 0, dyk k dyk+2 √ √ ∗ | y ∗ = 1 − 2 p − 2yk+1 = 1 + 2 p > 0, dyk+1 k+1

(2.94)

is monotonically unstable. (i2c ) The non-trivial period-2 fixed-points have two fixed-points as

(2.95)

dyk+1 dyk+2 dyk+2 ∗ | yk∗ = | yk+1 |y∗ dyk dyk+1 dyk k √ √ ∗ = (1 − 2 p − 2yk+1 )(1 − 2 p − 2yk∗ )

√ √ = (−1 ± 2 −(1 − p)(1 + p)(−1

√ √ ∓ 2 −(1 − p)(1 + p) √ √ = 1 − 4( p − 1)(1 + p) = 5 − 4 p.

(2.96)

∗ yk+1

√



√ √ −(1 − p)(1 + p),

√ √ √ = (1 − p) ∓ −(1 − p)(1 + p);

yk∗ = (1 −

p) ±

and

∗ ( p > 1) are The two non-trivial fixed-points of yk∗ = yk+2

• • • • •

monotonically stable for p ∈ (1, 45 ), zero stable for p = 54 , oscillatorilly stable for p ∈ ( 45 , 23 ), flipped for p = 23 due to yk+1 = −yk , oscillatorilly unstable for p ∈ ( 23 , ∞).

130

2 Low-Dimensional Discrete Systems ∗ For yk∗ = yk+2 = 0 ( p = 1), we have

dyk+2  √ ∗ p)[yk∗2 (yk∗ , p)=(0,1) = 1 − 2(yk + dyk √ √ − 2(1 − p)yk∗ + 2(1 − p)] √ √ − 2yk∗ (yk∗ + 2 p)[yk∗ − (1 − p)]

(2.97)

= 1, and d 2 yk+2  √ √ ∗2 p)yk∗ + 2(1 − p)] (yk∗ , p)=(0,1) = −2[yk − 2(1 − dyk2 √ √ − 8(yk∗ + p)[yk∗ − (1 − p)] √ − 2yk∗ (yk∗ + 2 p) = 0,

(2.98)

and d 3 yk+2  √ √ ∗ p)] − 12(yk∗ + p) (yk∗ , p)=(0,1) = −12[yk − (1 − dyk3

(2.99)

= −12 < 0. Thus, the period-2 fixed-point of (yk∗ , p) = (0, 1) is a monotonic sink of the third-order. The corresponding bifurcation is the monotonic sink bifurcation of the third-order for the period-2 fixed-point. ∗ = 0 is For p = 23 , the non-trivial period-2 fixed-point of yk∗ = yk+2 yk∗ = (1 −

√

p) ±

√ √ −(1 − p)(1 + p)

(2.100)

and dyk+2  √ √ ∗ p)[yk∗2 − 2(1 − p)yk∗ (yk∗ , p)=(yk∗ ,3/2) = 1 − 2(yk + dyk √ + 2(1 − p)] (2.101) √ √ − 2yk∗ (yk∗ + 2 p)[yk∗ − (1 − p)] = 5 − 4 p = −1, and

2.4 Sample Bifurcation Analysis

131

d 2 yk+2  √ √ ∗2 p)yk∗ + 2(1 − p)] (yk∗ , p)=(yk∗ ,3/2) = −2[yk − 2(1 − dyk2 √ √ − 8(yk∗ + p)[yk∗ − (1 − p)] √ − 2yk∗ (yk∗ + 2 p) √ = −4( 2 + 1) < 0, (2.102) Thus, such a non-trivial period-2 fixed-point is an oscillatory lower-saddle of the second-order. Similarly, the corresponding period-doubling of the period-2 fixed-points can be developed as presented before. √ √ (ii) For the branch of xk∗ = − p ( p > 0), the fixed-point xk∗ = − p is monotonically unstable because √ yk+1 = D f (xk∗ , p)yk = (1 + 2 p)yk , √ D f (xk∗ , p) = (1 + 2 p) > 1.

(2.103)

∗ ∗ (iii) For p = p0 = 0, at xk∗ = xk0 = 0 and D f (xk0 , p0 ) = 1. ∗ yk+1 = D f (xk0 , p0 )yk = yk .

(2.104)

∗ Since D 2 f (xk0 , p0 ) = −2 < 0, we have

yk+1 = yk +

1 D2 2!

∗ f (xk(0) , p0 )yk2 = (1 − yk )yk .

(2.105)

∗ At (xk(0) , p0 ) = (0, 0), 0 < yk+1 < yk for yk > 0 and yk+1 < yk < 0 for ∗ , p0 ) = (0, 0) is bifurcation point, which is a yk < 0. The fixed-point (xk(0) monotonic lower-saddle of the second order.

(iv) For p < 0, from Eq. (2.87), p − (xk∗ )2 < 0. Thus, no any fixed-point exists. The fixed-point xk∗ varying with parameter p is sketched in Fig. 2.7a. On the left side of xk -axes, no fixed-point exists. So only the vector field of the map in Eq. (2.87) is presented. The afore-discussed saddle-node bifurcation is a monotonic lower-saddlenode bifurcation on the left side. The other three saddle-node bifurcations are the upper-saddle-node bifurcation on the left side, the lower-saddle-node bifurcation on the right side and the upper-saddle-node bifurcation on the right side, which can be similarly discussed. Readers can complete such analysis for the other three cases as excises. All four cases of the saddle-node bifurcations are sketched in Fig. 2.7a– d. The corresponding discrete systems for saddle-node appearing bifurcations are tabulated in Table 2.1.

132

2 Low-Dimensional Discrete Systems xk

xk

3rd mSI

P-1

P-2 oLS

mUS

mLS

p

p

oLS P-2 P-1

3rd mSI

(a) 3rd mSI

(b)

xk

xk

P-1

P-2 oLS

mLS

mLS

p

p

oUS P-2 3rd mSI

P-1

(c)

(d)

Fig. 2.7 Bifurcation diagrams for saddle-node appearing bifurcations: a left monotonic lowersaddle-node bifurcation, b left monotonic upper-saddle-node bifurcation, c right monotonic lowersaddle-node bifurcation, d right monotonic upper-saddle-node bifurcation. Solid and dashed curves are for stable and unstable fixed-points. Black and red colors are for period-1 and period-2 fixedpoints. The vector fields are represented by lines with arrows. If no fixed-points exist, such a region is shaded. P-1: period-1 fixed-points. P-2: period-2 fixed points. mSI: monotonic sink, oLS: oscillatory lower-saddle, oUS: oscillatory upper-saddle, mLS: monotonic lower-saddle, mUS: monotonic upper-saddle

2.4.2 Saddle-Node Switching Bifurcations Consider a saddle-node switching (or transcritical) bifurcation via a 1-dimensional nonlinear discrete system as xk+1 = f (xk , p) ≡ xk + pxk − xk2

(2.106)

The fixed-points of the map in the foregoing equation are xk∗ = 0, p. From Eq. (2.106), the linearized equation in the vicinity of the fixed-points with yk = xk − xk∗ is

2.4 Sample Bifurcation Analysis

133

Table 2.1 Saddle-node appearing bifurcations and period-2 fixed-points Figure 2.7

1–D discrete systems

(a)

xk+1 = xk + p −

(b)

xk+1 = xk − p +

(c)

xk+1 = xk − p −

(d)

xk+1 = xk + p +

Bifurcations xk2 xk2 xk2 xk2

p = 0 (P-1)

p = 1(P-1)

p = 1(P-2)

(left) mLS

oLS

3rd mSI

(left) mUS

oUS

3rd mSI

(right) mLS

oLS

3rd mSI

(right) mUS

oUS

3rd mSI

Notice P-1: period-1 fixed-points. P-2: period-2 fixed points. mSI: monotonic sink, oLS: oscillatory lower-saddle, oUS: oscillatory upper-saddle, mLS: monotonic lower-saddle, mUS: monotonic upper-saddle

yk+1 = D f (xk∗ , p)yk = (1 + p − 2xk∗ )yk .

(2.107)

(i) For the branch of xk∗ = 0( p > 0), the fixed-point is monotonically unstable with yk+1 > yk > 0 and yk+1 < yk < 0 because yk+1 = D f (xk∗ , p)yk = (1 + p − 2xk∗ )yk ,  with D f (x ∗ , p)(x ∗ , p)=(0, p) = 1 + p > 1. k

(2.108)

k

(ii) For the branch of xk∗ = p( p > 0), we have yk+1 = D f (xk∗ , p)yk = (1 + p − 2xk∗ )yk ,  with D f (x ∗ , p)(x ∗ , p)=( p, p) = 1 − p < 1. k

(2.109)

k

Such a fixed-point is • monotonically stable for p ∈ (0, 1) due to 0 < yk+1 < yk or 0 > yk+1 > yk , • zero stable for p = 1 due to yk+1 = 0 for all yk , • oscillatorilly stable for p ∈ (1, 2) due to 0 > yk+1 > −yk or 0 < yk+1 < −yk , • flipped for p = 2 due to yk+1 = −yk , • oscillatorilly stable for p ∈ (2, ∞) due to yk+1 < −yk < 0 or yk+1 > −yk > 0. (ii1 )

For p = 2, the higher-order Taylor series of Eq. (2.106) is yk+1 = D f (xk∗ , p)yk + D 2 f (xk∗ , p) = −2 < 0.

1 D2 2!

f (xk∗ , p)yk2 = −yk − yk2 ,

(2.110)

Thus, the period-1 fixed-point of xk∗ = p at p = 2 with yk+1 = −yk is an oscillatory lower-saddle. The corresponding bifurcation is the oscillatory lower-saddle bifurcation of the second-order for the period-1 fixed-point.

134

2 Low-Dimensional Discrete Systems

(ii2 )

The period-doubling of xk∗ = p for p ≥ 2 is determined by yk+1 = (1 − p)yk − yk2 = yk − yk (yk + p), 2 yk+2 = (1 − p)yk+1 − yk+1 = yk+1 − yk+1 (yk+1 + p),

(2.111)

or yk+2 = yk − yk (yk + p)[yk2 − (2 − p)yk + (2 − p)].

(2.112)

For yk+2 = yk , we have ∗ yk∗ = 0, yk+1 = 0, ∗ ∗ = −p yk = − p, yk+1



⇒

∗ = xk∗ = p, xk∗ = p, xk+1 ∗ ∗ xk = 0, xk+1 = xk∗ = 0;

(2.113)

Such period-2 fixed-points is trivial, which is equivalent to the period-1 fixed-points. ∗ ∗ = yk+2 = 0 with p > 2 (ii2a ) The trivial period-2 fixed-point of yk∗ = yk+1 and dyk+1   = 1 − p − 2yk∗ = 1 − p < −1, dyk yk∗ dyk+2  ∗ = 1 − p < −1,  ∗ = 1 − p − 2yk+1 dyk+1 yk+1

(2.114)

is oscillatorilly unstable. ∗ ∗ (ii2b ) The trivial period-2 fixed-point of yk∗ = yk+1 = yk+2 = − p with p > 1 and dyk+1   = 1 − p − 2yk∗ = 1 + p > 0, dyk yk∗ dyk+2  ∗ = 1 + p > 0,  ∗ = 1 − p − 2yk+1 dyk+1 yk+1

(2.115)

is monotonically unstable. (ii2c ) The non-trivial period-2 fixed-points have two fixed-points as yk∗ = 21 [(1 − p) ± ∗ yk+1

and

=

1 [(1 2



−(2 − p)(2 + p)],  − p) ∓ −(2 − p)(2 + p)];

(2.116)

2.4 Sample Bifurcation Analysis

135

   dyk+2  dyk+1  dyk+2  = dyk  yk∗ dyk+1  yk+1 dyk  yk∗ ∗ ∗ = (1 − p − 2yk+1 )(1 − p − 2yk∗ )   = (−1 ± −(2 − p)(2 + p)(−1 ∓ −(2 − p)(2 + p)

= 1 − ( p − 2)(2 + p) = 5 − p 2 . (2.117) ∗ ( p > 2) are The two non-trivial fixed-points of yk∗ = yk+2

• • • • •

√ monotonically stable√for p ∈ (2, 5), zero stable for p = 5, √ √ oscillatorilly stable √ for p ∈ ( 5, 6), flipped for p = 6 due to yk+1√ = −yk , oscillatorilly unstable for p ∈ ( 6, ∞).

∗ For yk∗ = yk+2 = 0 ( p = 2), we have

dyk+2  ∗ ∗2 ∗ (yk∗ , p)=(0,2) = 1 − 2(yk + p)[yk − (2 − p)yk + (2 − p)] dyk − yk∗ (yk∗ + p)[2yk∗ − (2 − p)] = 1, (2.118) and d 2 yk+2  ∗2 ∗ (yk∗ , p)=(0,2) = −2[yk − (2 − p)yk + (2 − p)] dyk2 − 2(2yk∗ + p)[2yk∗ − (2 − p)] − 2yk∗ (yk∗ + p)

(2.119)

= 0, and d 3 yk+2  ∗ ∗ (yk∗ , p)=(0,2) = −6[2yk − (2 − p)] − 6(2yk + p) dyk3

(2.120)

= −12 < 0. Thus, the period-2 fixed-point of (yk∗ , p) = (0, 2) is a monotonic sink of the third-order. The corresponding bifurcation is the monotonic sink bifurcation √ of the third-order. ∗ = 0 is For p = 6 , the non-trivial period-2 fixed-point of yk∗ = yk+2 yk∗ = 21 [(2 − p) ±



−(2 − p)(2 + p)]

(2.121)

136

2 Low-Dimensional Discrete Systems

and dyk+2  √ = 1 − (2yk∗ + p)[yk∗2 − (2 − p)yk∗ + (2 − p)] ∗  ∗ dyk (yk , p)=(yk , 6) − yk∗ (yk∗ + p)[2yk∗ − (2 − p)] = 5 − p 2 = −1, (2.122) and d 2 yk+2  √ = −2[yk∗2 − (2 − p)yk∗ + (2 − p)] ∗  ∗ dyk2 (yk , p)=(yk , 6) − 2(2yk∗ + p)[2yk∗ − (2 − p)] −

(2.123)

2yk∗ (yk∗

+ p) √ = −2(3 2 + 2) < 0. Thus, such a non-trivial period-2 fixed-point is an oscillatory lowersaddle of the second-order. (iii) For the branch of xk∗ = p ( p < 0), such a fixed-point is monotonically unstable with yk+1 > yk > 0 and yk+1 < yk < 0 because of yk+1 = D f (xk∗ , p)yk = (1 + p − 2xk∗ )yk ,  with D f (x ∗ , p)(x ∗ , p)=(0, p) = 1 − p > 1. k

(2.124)

k

(iv) For the branch of xk∗ = 0 ( p < 0), we have yk+1 = D f (xk∗ , p)yk = (1 + p − 2xk∗ )yk ,  with D f (x ∗ , p)(x ∗ , p)=(0, p) = 1 + p < 1. k

(2.125)

k

Such a fixed-point is • monotonically stable for p ∈ (−1, 0) due to 0 < yk+1 < yk or 0 > yk+1 > yk , • zero stable for p = −1 due to yk+1 = 0 for all yk , • oscillatorilly stable for p ∈ (−2, −1) due to 0 > yk+1 > −yk or 0 < yk+1 < −yk , • flipped for p = −2 due to yk+1 = −yk , • oscillatorilly unstable for p ∈ (−∞, −2) due to yk+1 < −yk < 0 or yk+1 > −yk > 0. (iv1 )

For p = −2, the higher-order Taylor series of Eq. (2.106) is yk+1 = D f (xk∗ , p)yk + D 2 f (xk∗ , p) = −2 < 0.

1 D2 2!

f (xk∗ , p)yk2 = −yk − yk2 ,

(2.126)

2.4 Sample Bifurcation Analysis

(iv2 )

137

Thus, the period-1 fixed-point of xk∗ = 0 at p = −2 with yk+1 = −yk is an oscillatory lower-saddle. The corresponding bifurcation is the oscillatory lower-saddle bifurcation of the second-order for the period-1 fixed-point. The period-doubling of xk∗ = 0 for p ≤ −2 is determined by yk+1 = (1 + p)yk − yk2 = yk − yk (yk − p), 2 yk+2 = (1 + p)yk+1 − yk+1 = yk+1 − yk+1 (yk+1 − p),

(2.127)

or yk+2 = yk − yk (yk − p)[yk2 − (2 + p)yk + (2 + p)].

(2.128)

For yk+2 = yk , we have ∗ yk∗ = 0, yk+1 = 0, ∗ ∗ =p yk = p, yk+1



⇒

∗ = xk∗ = 0, xk∗ = 0, xk+1 ∗ ∗ xk = p, xk+1 = xk∗ = p;

(2.129)

Such period-2 fixed-points is trivial, which is equivalent to the period-1 fixed-points. ∗ ∗ ∗ = yk+2 = 0 (i.e., xk+1 = (iv2a ) The trivial period-2 fixed-point of yk∗ = yk+1 ∗ xk = 0) with p < −2 and dyk+1 | y ∗ = 1 + p − 2yk∗ = 1 + p < −1, dyk k dyk+2 ∗ | y ∗ = 1 + p − 2yk+1 = 1 + p < −1, dyk+1 k+1

(2.130)

is oscillatorilly unstable. ∗ ∗ ∗ = yk+2 = p (i.e., xk+1 = (iv2b ) The trivial period-2 fixed-point of yk∗ = yk+1 ∗ xk = p) with p < −2 and dyk+1 | y ∗ = 1 + p − 2yk∗ = 1 − p > 1, dyk k dyk+2 ∗ | y ∗ = 1 + p − 2yk+1 = 1 − p > 1, dyk+1 k+1

(2.131)

is monotonically unstable. (iv2c ) The non-trivial period-2 fixed-points have two fixed-points as  −(2 + p)(2 − p)],  = 21 [(1 + p) ∓ −(2 + p)(2 − p)];

yk∗ = 21 [(1 + p) ± ∗ yk+1

(2.132)

138

2 Low-Dimensional Discrete Systems

and    dyk+1  dyk+2  dyk+2  = dyk  yk∗ dyk+1  yk+1 dyk  yk∗ ∗ ∗ = (1 + p − 2yk+1 )(1 + p − 2yk∗ )   = (−1 ± −(2 + p)(2 − p)(−1 ∓ −(2 + p)(2 − p)

= 1 − ( p − 2)(2 + p) = 5 − p 2 . (2.133) ∗ The two non-trivial fixed-points of yk∗ = yk+2 ( p < −2) are

• • • • •

√ monotonically stable for √ p ∈ (− 5, −2), zero stable for p = − 5, √ √ oscillatorilly stable√for p ∈ (− 6, − 5), flipped for p = − 6 due to yk+1 = −y√k , oscillatorilly unstable for p ∈ (−∞, − 6).

∗ For yk∗ = yk+2 = 0( p = −2), we have

dyk+2  ∗ ∗2 ∗ (yk∗ , p)=(0,−2) = 1 − 2(yk + p)[yk − (2 + p)yk + (2 + p)] dyk − yk∗ (yk∗ + p)[2yk∗ − (2 + p)] = 1, (2.134) and d 2 yk+2  ∗2 ∗ (yk∗ , p)=(0,−2) = −2[yk − (2 + p)yk + (2 + p)] dyk2 − 2(2yk∗ − p)[2yk∗ − (2 + p)] − 2yk∗ (yk∗ − p)

(2.135)

= 0, and d 3 yk+2  ∗ ∗ (yk∗ , p)=(0,−2) = −6[2yk − (2 + p)] − 6(2yk − p) dyk3

(2.136)

= −12 < 0. Thus, the period-2 fixed-point of (yk∗ , p) = (0, −2) is a monotonic sink of the third-order. The corresponding bifurcation is the monotonic sink bifurcation of the third-order.

2.4 Sample Bifurcation Analysis

139

√ ∗ For p = − 6 , the non-trivial period-2 fixed-point of yk∗ = yk+2 = 0 is yk∗ = 21 [(2 − p) ±

 −(2 − p)(2 + p)]

(2.137)

and dyk+2  √ = 1 − (2yk∗ − p)[yk∗2 − (2 + p)yk∗ + (2 + p)] ∗  ∗ dyk (yk , p)=(yk ,− 6) − yk∗ (yk∗ − p)[2yk∗ − (2 + p)] = 5 − p 2 = −1, (2.138) and d 2 yk+2  √ = −2[yk∗2 − (2 + p)yk∗ + (2 + p)] ∗  ∗ dyk2 (yk , p)=(yk ,− 6) − 2(2yk∗ − p)[2yk∗ − (2 + p)] − 2yk∗ (yk∗ − p) √ = −2(3 2 + 2) < 0.

(2.139)

Thus, such a non-trivial period-2 fixed-point is an oscillatory lowersaddle of the second-order. Similarly, the corresponding perioddoubling of the period-2 fixed- points can be developed as presented before. ∗ ∗ (v) For p = p0 = 0, xk∗ = xk(0) = 0 and D f (xk(0) , p0 ) = 1 are obtained. ∗ , p0 )yk = yk . yk+1 = D f (xk0

(2.140)

∗ Since D 2 f (xk0 , p0 ) = −2 < 0, we have

yk+1 = yk +

1 D2 2!

∗ f (xk(0) , p0 )yk2 = (1 − yk )yk .

(2.141)

∗ At (xk(0) , p0 ) = (0, 0), |yk+1 | < |yk | for yk > 0 and |yk+1 | > |yk | for yk < 0. ∗ , p0 ) = (0, 0) is bifurcation point, which is a monotonic The fixed-point (xk(0) lower-saddle of the second order.

The fixed-point varying with parameter p is sketched in Fig. 2.8a. The aforediscussed saddle-node bifurcation is a monotonic lower-saddle-node bifurcation with a positive slope line. The other three saddle-node bifurcations are the monotonic upper-saddle-node bifurcation with a positive slope line, the monotonic lower-saddlenode bifurcation with a negative slope line and the monotonic upper-saddle-node bifurcation with a negative slope line, which can be similarly discussed. Readers

140

2 Low-Dimensional Discrete Systems xk

xk

3rd mSI

P-1

P-2 P-2

3rd mSI

oLS P-1

oLS

mUS

P-1

p

mLS

P-2

oUS

p rd

3 mSI

oUS P-2

P-1

3rd mSI

(a) 3rd mSI

(b) xk

xk

P-1 P-2 3rd mSI P-2

oLS P-1 mLS

P-2

oLS

P-1

mUS

p

oLS

p

3rd mSI

oUS P-2

P-1 3rd mSI

(c)

(d)

Fig. 2.8 saddle-node switching bifurcation diagrams: a monotonic lower-saddle-node switching bifurcation with a positive slope diagonal line, b monotonic upper-saddle-node switching bifurcation with a positive slope diagonal line, c monotonic lower-saddle-node switching bifurcation with a negative slope diagonal line, d monotonic upper-saddle-node switching bifurcation with a negative slope diagonal line. Solid and dashed curves are for stable and unstable fixed-points. Black and red colors are for period-1 and period-2 fixed-points. P-1: period-1 fixed-points. P-2: period-2 fixed points. mSI: monotonic sink, oLS: oscillatory lower-saddle, oUS: oscillatory upper-saddle, mLS: monotonic lower-saddle, mUS: monotonic upper-saddle

can complete the other three cases as excises. All four cases of the saddle-node bifurcations are sketched in Fig. 2.8a–d. The corresponding discrete systems for saddle-node switching bifurcations are tabulated in Table 2.2.

2.4.3 Third-Order Monotonic Sink Pitchfork Bifurcations Consider a 1-dimensional, cubic nonlinear discrete system for the third-order monotonic sink pitchfork bifurcation for period-1 fixed-points as xk+1 = xk + pxk − xk3 .

(2.142)

2.4 Sample Bifurcation Analysis

141

Table 2.2 Saddle-node switching bifurcations and period-2 fixed-points Figure 2.8

1-dimensional systems

Bifurcations

(a)

xk+1 = xk + pxk −

(b)

xk+1 = xk − pxk +

(c)

xk+1 = xk − pxk −

(d)

xk+1 = xk + pxk +

xk2 xk2 xk2 xk2

p = 0 (P-1)

p = ±2(P-1)

p = ±2(P-2)

(+) mLS

oLS

3rd mSI

(–) mUS

oUS

3rd mSI

(+) mLS

oLS

3rd mSI

(–) mUS

oUS

3rd mSI

Notice P-1: period-1 fixed-points. P-2: period-2 fixed points. mSI: monotonic sink, oLS: oscillatory lower-saddle, oUS: oscillatory upper-saddle, mLS: monotonic lower-saddle, mUS: monotonic upper-saddle

√ For xk+1 = xk = xk∗ , the corresponding fixed-point are xk∗ = 0, ± p ( p > 0) and xk∗ = 0 ( p ≤ 0). From Eq. (2.142), the linearized equation in vicinity of the fixed-point with yk = xk − xk∗ is yk+1 = D f (xk∗ , p)yk = [1 + p − 3(xk∗ )2 ]yk .

(2.143)

(i) For the branch of xk∗ = 0( p > 0), the fixed-point is monotonically unstable and the variational equation is yk+1 = D f (xk∗ , p)yk = (1 + p)yk with D f (xk∗ , p) = 1 + p > 1.

(2.144)

√ (ii) For the branches of xk∗ = ± p( p > 0), the variational equation is yk+1 = D f (xk∗ , p)yk = (1 − 2 p)yk with D f (xk∗ , p) = 1 − 2 p < 1.

(2.145)

Such two fixed-points are • monotonically stable for p ∈ (0, 21 ) due to 0 < yk+1 < yk or 0 > yk+1 > yk , • zero stable for p = 21 due to yk+1 = 0 for all yk , • oscillatorilly stable for p ∈ ( 21 , 1) due to 0 > yk+1 > −yk or 0 < yk+1 < −yk , • flipped for p = 1 due to yk+1 = −yk , • oscillatorilly unstable for p ∈ (1, ∞) due to yk+1 < −yk < 0 or yk+1 > −yk > 0. √ (ii1 ) For p = 1 and xk∗ = ± p , the higher-order Taylor series of Eq. (2.142) is yk+1 = D f (xk∗ , p)yk + = −yk ∓ 3yk2 , D 2 f (xk∗ , p) = −6xk∗ = ∓6.

1 D2 2!

f (xk∗ , p)yk2 (2.146)

142

2 Low-Dimensional Discrete Systems

√ Thus, the period-1 fixed-point of xk∗ = p at p = 1 with yk+1 = −yk is an oscillatory lower-saddle. The corresponding bifurcation is the oscillatory lowersaddle bifurcation of the second-order for the period-1 fixed-point. the period-1 √ fixed-point of xk∗ = − p at p = 1 with yk+1 = −yk is an oscillatory uppersaddle. The corresponding bifurcation is the oscillatory upper-saddle bifurcation of the second-order for the period-1 fixed-point. A. Local analysis of period-doubling fixed-points √ (ii2 ) The local period-doubling of xk∗ = ± p for p ≥ 1 is determined by √ yk+1 ≈ (1 − 2 p)yk ∓ 3 pyk2 √ √ = yk ∓ 3 pyk (yk − 23 p), √ 2 yk+2 ≈ (1 − 2 p)yk+1 ∓ 3 pyk+1 √ √ = yk+1 ∓ 3 pyk+1 (yk+1 − 23 p),

(2.147)

√ √ √ yk+2 ≈ yk ∓ 3 pyk (yk ± 23 p)[(3 pyk )2 √ ∓ 2(1 − p)(3 pyk ) + 2(1 − p)].

(2.148)

or

For yk+2 = yk , we have two trivial fixed-points  ∗ = 0, yk∗ = 0, yk+1 √ √ ∗ = ∓ 23 p yk∗ = ∓ 23 p, yk+1 ∗ √ √ ∗ xk = ± p, xk+1 = xk∗ = ± p, ⇒ √ √ ∗ = xk∗ = ± 13 p (not fixed - point); xk∗ = ± 13 p, xk+1

(2.149)

∗ Such two period-2 fixed-points are trivial, but only yk∗ = yk+1 = 0 (xk∗ = √ ± p) is equivalent to the period-1 fixed-points. This is because the local period-doubling analysis. ∗ ∗ = yk+2 = 0 with p > 1 and (ii2a ) The trivial period-2 fixed-point of yk∗ = yk+1

dyk+1 √ | yk∗ (xk∗ =±√ p) = 1 − 2 p ∓ 6 pyk∗ = 1 − 2 p < −1, dyk dyk+2 √ ∗ | y ∗ (x ∗ =±√ p) = 1 − 2 p ∓ 6 pyk+1 = 1 − 2 p < −1, dyk+1 k+1 k

(2.150)

is oscillatorilly unstable. √ (ii2b ) The non-trivial period-2 fixed-points have two fixed-points for xk∗ = ± p as  yk∗ = ± 3√1 p [(1 − p) ± −(1 − p)(1 + p)], (2.151)  ∗ yk+1 = ± 3√1 p [(1 − p) ∓ −(1 − p)(1 + p)];

2.4 Sample Bifurcation Analysis

143

and dyk+1 dyk+2 dyk+2  √ ∗ | yk+1 |y∗ yk∗ (xk∗ =± p) = dyk dyk+1 dyk k √ ∗ √ = (1 − 2 p ∓ 6 pyk+1 )(1 − 2 p ∓ 6 pyk∗ )  (2.152) = (−1 ± 2 −(1 − p)(1 + p)(−1  ∓ 2 −(1 − p)(1 + p) = 1 − 4( p − 1)(1 + p) = 5 − 4 p 2 . ∗ The two non-trivial fixed-points of yk∗ = yk+2 = 0( p > 1) related to √ ∗ xk = ± p are

• monotonically stable for p ∈ (1, √ • zero stable for p = 25 , √

√

5 ), 2

√

• oscillatorilly stable for p ∈ ( 25 , 26 ), √ • flipped for p = 26 due to yk+1 = −yk , • oscillatorilly unstable for p ∈ (

√ 6 , ∞). 2

√ ∗ For yk∗ = yk+2 = 0( p = 1) with xk∗ = ± p , we have dyk+2  √ ∗ 2√ p)[9 pyk∗2 (yk∗ , p)=(0,1) = 1 ∓ 3 p(2yk ± 3 dyk √ ∓ 6(1 − p) pyk∗ + 2(1 − p)] √ √ √ ∓ 3 pyk∗ (yk∗ ± 23 p)[18 pyk∗ ∓ 6(1 − p) p] = 1, (2.153) and d 2 yk+2  √ √ ∗ ∗2 (yk∗ , p)=(0,1) = ∓6 p[9 pyk ∓ 6(1 − p) pyk + 2(1 − p)] dyk2 √ √ √ ∓ 6 p(2yk∗ ± 23 p)[18 pyk∗ ∓ 6(1 − p) p] √ √ ∓ 54 p pyk∗ (yk∗ ± 23 p) = 0, (2.154) and d 3 yk+2  √ ∗ 2√ p] (yk∗ , p)=(0,1) = ∓3 × 54 p p[2yk ± 3 3 dyk √ √ ∓ 3 × 6 p(18 pyk∗ ∓ 6 p(1 − p)) = −108 < 0.

(2.155)

144

2 Low-Dimensional Discrete Systems

√ Thus, the period-2 fixed-point of (yk∗ , p) = (0, 1) for xk∗ = ± p is a monotonic sink of the third-order. The corresponding bifurcation is the monotonic√sink bifurcation of the third-order. For p = 26 , the approximate non-trivial period-2 fixed-point of yk∗ = √ ∗ = 0 relative to xk∗ = ± p is yk+2 yk∗ = ± 3√1 p [(1 − p) ±



−(1 − p)(1 + p)]

(2.156)

and  dyk+2  √ √ √ = 1 ∓ 3 p(2yk∗ ± 23 p)[9 pyk2 dyk (yk∗ , p)=(yk∗ , 26 ) √ ∓ 6(1 − p) pyk + 2(1 − p)] √ √ √ ∓ 3 pyk (yk ± 23 p)[18 pyk∗ ∓ 6(1 − p) p] = 5 − 4 p 2 = −1, (2.157) and  d 2 yk+2  √ √ √ = ∓6 p[9 pyk∗2 ∓ 6(1 − p) pyk∗ + 2(1 − p)] dyk2 (yk∗ , p)=(yk∗ , 26 ) √ √ √ ∓ 6 p(2yk∗ ± 23 p)[18 pyk∗ ∓ 6(1 − p) p] √ √ ∓ 18 p pyk∗ (yk∗ ± 23 p) √ = ∓2(3 2 + 2). (2.158) √ Thus, such a non-trivial period-2 fixed-point for xk∗ = p is an oscillatory lower√ saddle of the second-order, and such a non-trivial period-2 fixed-point for xk∗ = − p is an oscillatory upper-saddle of the second-order. The above local analysis of the period-doubling on the two branches of xk∗ = √ ± p is presented, which gives an approximate analysis. Such local analysis is only valid for the small perturbation of yk = xk − xk∗ (yk 1 and

d xk+1 |x ∗ = 1 + p − 3xk∗2 = 1 + p > 1, d xk k d xk+2 ∗2 |x ∗ = 1 + p − 3xk+1 = 1 + p > 1, d xk+1 k+1

(2.162)

is monotonically unstable. √ ∗ ∗ (ii3b ) The trivial period-2 fixed-point of xk∗ = xk+1 = xk+2 = ± p with p > 1 and d xk+1 |x ∗ = 1 + p − 3xk∗2 = 1 − 2 p < −1, d xk k d xk+2 ∗2 |x ∗ = 1 + p − 3xk+1 = 1 − 2 p < −1, d xk+1 k+1

(2.163)

is oscillatorilly unstable. (ii3c ) The non-trivial period-2 fixed-points have maximum six fixed-points given by (2 + p) − ( p 2 + 3 p + 3)xk2 + (3 + 2 p)xk4 − xk6 = 0, or (xk2

(2.164) −

1 (3 3

+ 2 p)) + 3

P(xk2

−

1 (3 3

+ 2 p)) + Q = 0

146

2 Low-Dimensional Discrete Systems

with P = − 13 p( p + 3), Q =

1 (2 p 3 27

+ 9 p 2 − 27).

(2.165)

The corresponding non-trivial period-2 fixed-points for p ≥ 1 are    xk∗ ∈ {± b1 , ± b2 , ± b3 } if bi > 0 for i = 1, 2, 3; √ √ b1 = (− 21 Q + )1/3 + (− 21 Q − )1/3 + 13 (3 + 2 p) = p + 2,

√ 1/3 √ ) + ω2 (− 21 Q − )1/3 + 13 (3 + 2 p) (2.166)  = 21 ( p + 1 + ( p + 3)( p − 1)), √ √ b3 = ω2 (− 21 Q + )1/3 + ω(− 21 Q − )1/3 + 13 (3 + 2 p)  = 21 ( p + 1 − ( p + 3)( p − 1));

b2 = ω(− 21 Q +

where  = 41 Q 2 + ω=

√

1 P3 27

−1+i 3 , ω2 2

=

=

1 (− p 2 ( p + 4)2 27×4 √ √ −1−i 3 , i = −1. 2

− 2 p 2 + 27),

(2.167)

For p = 1, we have xk∗2 = b2 = b3 = 1, ⇒ xk∗ = ±1.

(2.168)

√ xk∗2 = b1 = 3, ⇒ xk∗ = ± 3.

(2.169)

and

Thus, the non-trivial period-doubled fixed-points are   xk∗ = ± b2 and xk∗ = ± b3

(2.170)

which are related to the period-1 fixed-points. However, there are two period-2 fixed-points independent of the period-1 fixed-points as  xk∗ = ± b1 .

(2.171)

Thus the period-2 discrete system related to Eq. (2.160) is xk+2 = xk +

9 i=1

(xk − ai ).

   √ {a1 , a2 . . . , a9 } = sort{0, ± p, ± b1 , ± b2 , ± b3 } with ai < ai+1 .

(2.172)

2.4 Sample Bifurcation Analysis

147

The derivative of xk+2 with respect to xk is 9 d xk+2  ∗ (a j − ai ). x =a = 1 + d xk k j i=1,i= j

(2.173)

with 9

(a j − ai ) = 0 for p = 1,

i=1,i= j 9

(a j − ai ) = −1 for p = β1 ,

(2.174)

i=1,i= j 9

(a j − ai ) = −2 for p = β2 .

i=1,i= j

The non-trivial period-2 fixed-points of xk∗ = a j are • • • • •

monotonically stable for p ∈ (1, β1 ) zero stable for p = β1 , oscillatorilly stable for p ∈ (β1 , β2 ) flipped for p = β2 due to xk+1 = −xk , oscillatorilly unstable for p ∈ (β2 , ∞). √ ∗ (ii4a ) For xk∗ = xk+2 = ± p ( p = 1), we have xk+2 = xk − xk (xk + 1)(xk − 1)[3 − 7xk2 + 5xk4 − xk6 ] √ √ = xk + xk (xk + 1)3 (xk − 1)3 (xk − 3)(xk + 3)

(2.175)

and d xk+2  3 (xk∗ , p)=(1,1) = 1 + 3x k (x k + 1) (x k d xk  √ √  − 3)(xk + 3)(xk − 1)2  d 2 xk+2  3 (xk∗ , p)=(1,1) = 6x k (x k + 1) (x k d xk2  √ √  − 3)(xk + 3)(xk − 1)

xk∗ =1

= 1,

(2.176) xk∗ =1

= 0,

√ √  d 3 xk+2  3 ∗ 3)(x = 3!x (x + 1) (x − + 3) ∗ (x , p)=(1,1) k k k k k xk =1 d xk3 = 3! × 1 × 23 × (−2) = −96 < 0.

148

2 Low-Dimensional Discrete Systems

Thus, the period-2 fixed-point of (xk∗ , p) = (1, 1) is a monotonic sink of the third-order. The corresponding bifurcation is the monotonic sink bifurcation of the third-order. (ii4b ) Similarly, for (xk∗ , p) = (−1, 1), we have d xk+2  3 (xk∗ , p)=(−1,1) = 1 + 3x k (x k − 1) (x k d xk  √ √  − 3)(xk + 3)(xk + 1)2  d 2 xk+2  3 (xk∗ , p)=(1,1) = 6x k (x k − 1) (x k d xk2  √ √  − 3)(xk + 3)(xk + 1)

xk∗ =−1

xk∗ =−1

= 1,

= 0,

√ √  d 3 xk+2  3 ∗ = 3!x (x − 1) (x − 3)(x + 3) ∗ (x , p)=(1,1) k k k k k xk =−1 d xk3 = 3! × (−1) × (−2)3 × (−2) = −96 < 0. (2.177)

(ii5 )

Thus, the period-2 fixed-point of (xk∗ , p) = (−1, 1) is a monotonic sink of the third-order. The corresponding bifurcation is the monotonic sink bifurcation of the third-order. ∗ For xk∗ = xk+2 = 0 ( p = −2), we have xk+2 = xk + xk3 (xk2 + 2)(1 + xk2 + xk4 ).

(2.178)

with  d xk+2  2 2 2 4  (xk∗ , p)=(0,−2) = 1 + 3x k (x k + 2)(1 + x k + x k ) x ∗ =0 = 1, k d xk 2  d xk+2  2 2 4  (xk∗ , p)=(0,−2) = 3 × 2x k (x k + 2)(1 + x k + x k ) x ∗ =0 = 0, 2 k d xk  d 3 xk+2  2 2 4  (xk∗ , p)=(0,−2) = 3!(x k + 2)(1 + x k + x k ) x ∗ =0 = 3! 3 k d xk × 2 × 1 = 12 > 0. (2.179) Thus, the period-2 fixed-point of (xk∗ , p) = (0, −2) is a monotonic source of the third-order. The corresponding bifurcation is the monotonic source bifurcation of the third-order.

2.4 Sample Bifurcation Analysis

149

(iii) For the branch of xk∗ = 0 ( p = 0), the fixed-point is monotonically unstable and the variational equation is yk+1 = D f (xk∗ , p)yk = (1 + p)yk = yk , yk+1 = D f (xk∗ , p)yk + = (1 + p)yk +

1 D 2 f (xk∗ , p)yk2 2! 1 (3xk∗2 )yk2 = yk , 2!

(2.180)

with D f (xk∗ , p) = 1 + p − 3xk∗2 = 1, D 2 f (xk∗ , p) = −6xk∗ = 0.

(2.181)

Thus yk+1 = D f (xk∗ , p)yk +

1 D2 2!

f (xk∗ , p)yk2 +

1 D3 3!

f (xk∗ , p)yk3

= (1 + p)yk − yk3

(2.182)

with D 3 f (xk∗ , p) = −6 < 0.

(2.183)

Thus, the period-1 fixed-point of (xk∗ , p) = (0, 0) is a monotonic sink of the third-order. The corresponding bifurcation is the monotonic sink bifurcation of the third-order. (iv) For the branch of xk∗ = 0 ( p < 0), the variational equation is yk+1 = D f (xk∗ , p)yk = (1 + p)yk with D f (xk∗ , p) = 1 + p < 1.

(2.184)

Such a fixed-point is • monotonically stable for p ∈ (−1, 0) due to 0 < yk+1 < yk or 0 > yk+1 > yk , • zero stable for p = −1 due to yk+1 = 0 for all yk , • oscillatorilly stable for p ∈ (−2, −1) due to 0 > yk+1 > −yk or 0 < yk+1 < −yk , • flipped for p = −2 due to yk+1 = −yk , • oscillatorilly unstable for p ∈ (−∞, −2) due to yk+1 < −yk < 0 or yk+1 > −yk > 0.

150

2 Low-Dimensional Discrete Systems

(iv1 ) For p = −2, the higher-order Taylor series of Eq. (2.145) is yk+1 = D f (xk∗ , p)yk +

1 D2 2!

f (xk∗ , p)yk2 +

1 D3 3!

f (xk∗ , p)yk3

= (1 + p)yk − yk3

(2.185)

with D f (xk∗ , p) = 1 + p − 3xk∗2 = −1, D 2 f (xk∗ , p) = −6xk∗ = 0,

(2.186)

D 3 f (xk∗ , p) = −6 < 0. Thus, the period-1 fixed-point of xk∗ = 0 at p = −2 with yk+1 = −yk is an oscillatory source of the third order. The corresponding bifurcation is the oscillatory source bifurcation of the third-order for the period-1 fixed-point. (iv2 ) The period-doubling of xk∗ = 0 for p ≤ −2 is determined by yk+1 = (1 + p)yk − yk3 = yk − yk (yk2 − p), 3 2 yk+2 = (1 + p)yk+1 − yk+1 = yk+1 − yk+1 (yk+1 − p),

(2.187)

or yk+2 = yk − yk (yk2 − p)[(2 + p) − ( p 2 + 3 p + 3)yk2 + (3 + 2 p)yk4 − yk6 ].

(2.188)

For yk+2 = yk and p < 0 ∗ ∗ yk∗ = 0, yk+1 = 0, ⇒ xk∗ = 0, xk+1 = xk∗ = 0

(2.189)

Such period-2 fixed-points is trivial, which is equivalent to the period-1 fixed-points. For non-trivial period-2 fixed-points, we have (2 + p) − ( p 2 + 3 p + 3)yk∗2 + (3 + 2 p)yk∗4 − yk∗6 = 0.

(2.190)

At p = −2, −yk∗2 − yk∗4 − yk∗6 = 0 ⇒ yk∗2 = 0 and 1 + yk∗2 + yk∗4 = 0

(2.191)

From Eq. (2.190), the derivative with respect to p gives  dyk2  1 ∗2 = . dp (yk , p)=(0,−2) 6

(2.192)

2.4 Sample Bifurcation Analysis

151

Thus, yk∗2 < 0 for p < −2, yk∗2 = 0 for p = −2, yk∗2

(2.193)

> 0 for p > −2.

Therefore, no more non-trivial period-2 fixed-points exist for p < −2. Two non-trivial period-2 fixed-points exist for p > −2. With a trivial fixed-point, yk∗ = 0 for p = −2 is the triple-repeated fixedpoint, as discussed in (ii5 ) and (iv1 ). From the period-1 fixed-point, the corresponding bifurcation is an oscillatory source bifurcation of the third-order. However, based on the non-trivial period-2 fixed-point, the corresponding bifurcation is a monotonic source bifurcation of the third-order. From the above analysis, the bifurcation diagram for the pitchfork bifurcation with stable-symmetry of the period-1 fixed-points and the corresponding period-2 fixed-points are presented through the fixed-point varying with parameter p, as shown in Fig. 2.9. The third-order monotonic sink of the period-1 fixed point on the right side for a 1-dimensional cubic nonlinear discrete system is presented in Fig. 2.9a. However, The third-order monotonic sink of the period-1 fixed point on the left side for a 1-dimensional cubic nonlinear discrete system is presented in Fig. 2.9b, and the corresponding analysis can be completed similarly. Readers can complete such analysis as an excise. The corresponding discrete systems for 3rd order sink switching bifurcations are tabulated in Table 2.3.

xk

oLS

3rd oSO

P-2

P-2

P-1

P-1

P-2

P-2

3rd mSI

3rd mSI P-1

P-1

3rd mSI

xk

oLS

3rd oSO P-1

P-1

p

3 mSI

3 mSO

oUS

P-1 P-2

(a)

3rd mSO

3rd mSI P-2

p

3rd mSI

rd

rd

P-2 P-1

oUS

P-2

(b)

Fig. 2.9 Pitchfork bifurcation diagrams: a the 3rd order monotonic sink bifurcation for stablesymmetry on the left side, b the 3rd order monotonic sink bifurcation for stable-symmetry on the right side. Solid and dashed curves are for stable and unstable fixed-points. Black and red colors are for period-1 and period-2 fixed-points. P-1: period-1 fixed-points. P-2: period-2 fixed points. mSI: monotonic sink, oSO: oscillatory source, oLS: oscillatory lower-saddle, oUS: oscillatory upper-saddle

152

2 Low-Dimensional Discrete Systems

Table 2.3 3rd order sink pitchfork switching bifurcations and period-2 fixed-points Figure 2.9

1- dimensional systems

(a)

xk+1 = xk + pxk − xk3

(b)

Bifurcations

xk+1 = xk + pxk + xk3

p = 0 (P-1)

p = −1(P-1)

p = −1(P-2)

3rd mSI

3rd oSO

3rd mSO

p = 1(P-1)

p = 1(P-1)

p = 1(P-2)

oLS (upper)

oUS (lower)

3rd mSI

p = 0 (P-1)

p = 1(P-1)

p = 1(P-2)

3rd mSI

3rd oSO

3rd mSO

p = −1(P-1)

p = −1(P-1)

p = −1 (P-2)

oLS (upper)

oUS (lower)

3rd mSI

Notice P-1: period-1 fixed-points. P-2: period-2 fixed points. mSI: monotonic sink, oSO: oscillatory source, oLS: oscillatory lower-saddle, oUS: oscillatory upper-saddle

2.4.4 Third-Order Monotonic Source Pitchfork Bifurcations Consider a 1-dimensional, cubic nonlinear discrete system for the third-order monotonic source pitchfork bifurcation for period-1 fixed-points as xk+1 = xk − pxk + xk3 .

(2.194)

√ For xk+1 = xk = xk∗ , the corresponding fixed-point are xk∗ = 0, ± p ( p > 0) ∗ and xk = 0 ( p ≤ 0). From Eq. (2.194), the linearized equation in vicinity of the fixed-point with yk = xk − xk∗ is yk+1 = D f (xk∗ , p)yk = [1 − p + 3(xk∗ )2 ]yk .

(2.195)

√ (i) For the branches of xk∗ = ± p ( p > 0), the two fixed-points are monotonically unstable and the variational equation is yk+1 = D f (xk∗ , p)yk = (1 + 2 p)yk with D f (xk∗ , p) = 1 + 2 p > 1.

(2.196)

(ii) For the branch of xk∗ = 0 ( p > 0), the fixed-point is monotonically stable and the variational equation is yk+1 = D f (xk∗ , p)yk = (1 − p)yk with D f (xk∗ , p) = 1 − p < 1. Such a fixed-point is

(2.197)

2.4 Sample Bifurcation Analysis

153

• monotonically stable for p ∈ (0, 1) due to 0 < yk+1 < yk or 0 > yk+1 > yk , • zero stable for p = 1 due to yk+1 = 0 for all yk , • oscillatorilly stable for p ∈ (1, 2) due to 0 > yk+1 > −yk or 0 < yk+1 < −yk , • flipped for p = 2 due to yk+1 = −yk , • oscillatorilly unstable for p ∈ (2, ∞) due to yk+1 < −yk < 0 or yk+1 > −yk > 0. (ii1 )

For p = 2 and xk∗ = 0, the higher-order Taylor series of Eq. (2.194) is yk+1 = D f (xk∗ , p)yk +

1 D2 2!

f (xk∗ , p)yk2 +

1 D3 3!

f (xk∗ , p)yk3

= −yk + yk3 ,

(2.198)

with D f (xk∗ , p) = 1 − p D 2 f (xk∗ , p) = −6xk∗ = 0,

(2.199)

D 3 f (xk∗ , p) = +6 > 0,

(ii2 )

Thus, the period-1 fixed-point of xk∗ = 0 at p = 2 with yk+1 = −yk is an oscillatory sink of the third order. The corresponding bifurcation is the oscillatory sink bifurcation of the third-order for the period-1 fixed-point. The period-doubling of xk∗ = 0 for p ≥ 2 is determined by yk+1 = (1 − p)yk + yk3 = yk + yk (yk2 − p), 3 2 yk+2 = (1 − p)yk+1 + yk+1 = yk+1 + yk+1 (yk+1 − p),

(2.200)

or yk+2 = yk + yk (yk2 − p)[(2 − p) + ( p 2 − 3 p + 3)yk2 + (3 − 2 p)yk4 + yk6 ].

(2.201)

For xk+2 = xk , we have ⎫ ⎫ ∗ ∗ ∗ ∗ yk∗ = yk+1 = yk+2 = 0, = xk+2 = 0, xk∗ = xk+1 ⎬ ⎬ √ √ ∗ ∗ ∗ ∗ ⇒ xk∗ = xk+1 = yk+2 = p, = xk+2 = p, yk∗ = yk+1 √ ⎭ √ ⎭ ∗ ∗ ∗ ∗ = yk+2 =− p = xk+2 = − p. yk∗ = yk+1 xk∗ = xk+1 (2.202) Such period-2 fixed-points are trivial, which are the period-1 fixed-points.

154

2 Low-Dimensional Discrete Systems

√ ∗ ∗ (ii2a ) The trivial period-2 fixed-point of yk∗ = yk+1 = yk+2 = ± p with p > 2 and dyk+1 | y ∗ = 1 − p + 3yk∗2 = 1 + p > 1, dyk k dyk+2 ∗2 |x ∗ = 1 − p + 2yk+1 = 1 + p > 1, dyk+1 k+1

(2.203)

is monotonically unstable. ∗ ∗ = yk+2 = 0 with p > 2 (ii3b ) The trivial period-2 fixed-point of yk∗ = yk+1 and dyk+1 |x ∗ = 1 − p + 3yk∗2 = 1 − p < −1, dyk k dyk+2 ∗2 |x ∗ = 1 − p + 3yk+1 = 1 − p < −1, dyk+1 k+1

(2.204)

is oscillatory unstable. (ii3c ) The non-trivial period-2 fixed-points have maximum six fixed-points given by (2 − p) + ( p 2 − 3 p + 3)yk2 + (3 − 2 p)yk4 + yk6 = 0, or

(2.205)

(yk2 + 13 (3 − 2 p))3 + P(yk2 + 13 (3 − 2 p)) + Q = 0 with 1 1 P = − p( p − 3), Q = (2 p 3 − 9 p 2 + 27). 3 27

(2.206)

The corresponding non-trivial period-2 fixed-points for p ≥ 2 are    yk∗ ∈ {± b1 , ± b2 , ± b3 } if bi ≥ 0 for i = 1, 2, 3; √ √ 1 1 b1 = (− 21 Q + )1/3 + (− Q − )1/3 − (3 − 2 p) 2 3 = p − 2, √ √ 1 1 1 b2 = ω(− Q + )1/3 + ω2 (− Q − )1/3 − (3 − 2 p) 2 2 3  1 = ( p − 1 + ( p − 3)( p + 1)), 2 √ √ 1 1 1 b3 = ω2 (− Q + )1/3 + ω(− Q − )1/3 − (3 − 2 p) 2 2 3  = 21 ( p − 1 + ( p − 3)( p + 1)); (2.207)

2.4 Sample Bifurcation Analysis

155

where  = 41 Q 2 + ω=

√

=

1 P3 27

−1+i 3 , ω2 2

=

1 (− p 2 ( p − 4)2 27×4 √ √ −1−i 3 , i = −1. 2

− 2 p 2 + 27),

(2.208)

Consider =

1 (− p 2 ( p 27×4

− 4)2 − 2 p 2 + 27) = 0

(2.209)

p = −1.0, 3.0

(2.210)

we have

For p ∈ (−1.0, 3.0),  > 0 which implies there are only one solution of yk∗2 = bi (i ∈ {1, 2, 3}) in Eq. (2.205). For p ∈ (−∞, −1.0) ∪ (3, ∞),  < 0 which implies three of yk∗2 = bi (i = 1, √ 2, 3) in Eq. (2.205). If bi ≥ 0 (i ∈ {1, 2, 3}), then we have yk∗ = ± bi for the non-trivial period-2 fixed-points. (ii3d ) For p = 2, Eq. (2.205) give yk∗ = 0 for the double-repeated period-2 fixed-points, which is also equal to the period-1 fixed-point of yk∗ = 0. Thus, the point of yk∗ = 0 at p = 2 is the triple-repeated points. Therefore, for p = 2, Eq. (2.201) become yk+2 = yk + yk3 (yk2 − 2)[(yk2 − 21 )2 + 43 ].

(2.211)

Further, we have  dyk+2  2 2 2 1 2 3  (yk∗ , p)=(0,2) = 1 + 3yk (yk − 2)[(yk − 2 ) + 4 ] (yk∗ , p)=(0,2) = 1, dyk  d 2 yk+2  2 2 1 2 3  (yk∗ , p)=(0,2) = 3 × 2yk (yk − 2)[(yk − 2 ) + 4 ] (yk∗ , p)=(0,2) = 0, 2 dyk 3  d yk+2  2 2 1 2 3  (yk∗ , p)=(0,2) = 3!(yk − 2)[(yk − 2 ) + 4 ] (yk∗ , p)=(0,2) = −6 < 0 3 dyk (2.212) Thus, point (yk∗ , p) = (0, 2) is a monotonic sink of the third-order for period-2 fixed-points, and the corresponding bifurcation is a monotonic sink bifurcation of the third -order for the period-2 fixed-point. From Eq. (2.205), we have dyk2  (y ∗ , p)=(0,2) = 1 > 0. dp k

(2.213)

Thus, for p < 2, in a neighborhood of (yk∗ , p) = (0, 2), there is a relation of yk∗2 < 0 for the period-2 fixed-point, which implies no any

156

2 Low-Dimensional Discrete Systems

period-2 fixed-point exists for such a discrete system. For p > 2, in a neighborhood of (yk∗ , p) = (0, 2), there is a relation of yk∗2 > 0 for the period-2 fixed-point, which implies√two period-2 fixed-points exist for such a discrete system, i.e., yk∗ = ± b1 . (ii3e ) For p = 3,  = 0. Equation (2.201) becomes yk+2 = yk + yk (yk2 − 3)[−1 + 3yk2 − 3yk4 + yk6 ] = yk + yk (yk2 − 3)(yk − 1)3 (yk + 1)3 .

(2.214)

Further, we have dyk+2  2 2 (yk∗ , p)=(1,3) = 1 + 3yk (yk − 3)(yk − 1) dyk  (yk + 1)3 (yk∗ , p)=(1,3) = 1, d 2 yk+2  2 (yk∗ , p)=(1,3) = 3 × 2yk (yk − 3)(yk − 1) dyk2  (yk + 1)3 (yk∗ , p)=(1,3) = 0,  d 3 yk+2  2 3 ∗ ∗ = 3!y (y − 3)(y + 1) (y , p)=(1,3) k k (yk , p)=(1,3) k k dyk3

(2.215)

= −96 < 0 Thus, point (yk∗ , p) = (1, 3) is a monotonic sink of the third-order for period-2 fixed-points, and the corresponding bifurcation is the monotonic sink bifurcation of the third-order for the period-2 fixed-point. ∗2 (ii3f ) For p > 3, there are three roots √ of yk = bi > 0 (i = 1, 2, 3). ∗ Therefore, we have yk = ± bi (i = 1, 2, 3) for perod-2 fixedpoints. Thus the period-2 fixed-point of discrete system related to Eq. (2.194) is yk+2 = yk + yk (yk2 − p)[(2 − p) + ( p 2 − 3 p + 3)yk2 + (3 − 2 p)yk4 + yk6 ] = yk +

9

(yk − ai ).

i=1

   √ {a1 , a2 . . . , a9 } = sort{0, ± p, ± b1 , ± b2 , ± b3 } with ai < ai+1 . (2.216) The derivative of yk+2 with respect to yk is 9 dyk+2  ∗ = 1 + (a j − ai ). y =a dyk k j i=1,i= j

with

(2.217)

2.4 Sample Bifurcation Analysis

157 9

(a j − ai ) = 0 for p = 3,

i=1,i= j 9

(a j − ai ) = −1 for p = β1

(2.218)

i=1,i= j 9

(a j − ai ) = −2 for p = β2

i=1,i= j

The nontrivial period-2 fixed-points of yk∗ = a j are • monotonically stable for p ∈ (1, β1 ) with zero stable for p = β1 , • oscillatorilly stable for p ∈ (β1 , β2 ) with • flipped for p = β2 due to yk+1 = −yk , • oscillatorilly unstable for p ∈ (β2 , ∞). (ii3g ) For p = −1,  = 0. Equation (2.201) becomes yk+2 = yk + yk (yk2 + 1)3 (yk2 + 3).

(2.219)

From the foregoing equation, for the period-2 fixed-point, only yk∗ = 0 is real, and other roots are complex. In addition, yk∗ = 0 is for xk∗ = 0. From Eq. (2.219), there are two sets for triple-repeated complex, non-trivial period-2 fixed-points plus a pair of simple conjugate complex, non-trivial period-2 fixed-points. In real space, such complex, non-trivial period-2 fixed-points do not exist. In conclusion, for p ∈ (−1.0, 2.0),  > 0 which implies one of yk∗2 = b1 < 0. Thus, except for the trivial period-2 fixed-points, no non-trivial period-2 fixedpoints exist. For p ∈ (−∞, −1.0),  < 0 which implies three of yk∗2 = bi < 0 (i = 1, 2, 3) in Eq. (2.205). Therefore, for p < 2, non-trivial period-2 fixed-points do not exist, and only trivial period-2 fixed-points exist. (iii) For the branch of xk∗ = 0 ( p < 0), the fixed-point is monotonically unstable and the variational equation is yk+1 = D f (xk∗ , p)yk = (1 − p)yk with D f (xk∗ , p) = 1 − p > 1.

(2.220)

(iv) For the branch of xk∗ = 0 ( p = 0), the fixed-point is monotonically unstable and the variational equation is yk+1 = D f (xk∗ , p)yk + = yk + yk3

1 D2 2!

f (xk∗ , p)yk2 +

1 D3 3!

f (xk∗ , p)yk3

(2.221)

158

2 Low-Dimensional Discrete Systems

and D f (xk∗ , p) = 1, D 2 f (xk∗ , p) = −6xk∗ = 0, D

3

f (xk∗ ,

(2.222)

p) = +6 > 0.

The point of (xk∗ , p) = (0, 0) is a monotonic source of the third-order, and the corresponding bifurcation is a monotonic source bifurcation of the third-order. From the above analysis, the bifurcation diagram for the pitchfork bifurcation with unstable-symmetry of the period-1 fixed-points and the corresponding period-2 fixed-points are presented through the fixed-point varying with parameter p, as shown in Fig. 2.10. A monotonic source bifurcation of the third-order (3rd mSO) for period1 fixed-point is on the left side, as shown in Fig. 2.10a. At p = 0, the bifurcation of period-1 fixed point is a monotonic source bifurcation of the third-order. At p = 2, the bifurcation of period-1 fixed-point is an oscillatory sink bifurcation of the third-order (3rd oSI), and the bifurcation of period-2 fixed-point is a monotonic sink bifurcation of the third-order (3rd mSI). At p = 3, the bifurcation of period-2 fixed-point is a monotonic sink bifurcation of the third-order (3rd mSI) for three period-2 fixed point. A monotonic source bifurcation of the third-order (3rd mSO) for period-1 fixed-point is on the right side, as shown in Fig. 2.10b. At p = 0, the bifurcation of period-1 fixed point is a monotonic source bifurcation of the third-order. At p = −2, the bifurcation of period-1 fixed-point is an oscillatory sink bifurcation of the third-order P-1

P-1 xk

xk

3rd oSI

3rd mSI

P-2

P-2

P-2

P-2

rd 3rd mSI 3 oSI

P-1

P-1 p

rd

3 mSO 3rd mSI

3rd mSI P-2 P-2

P-1

(a)

3rd mSI P-2

3rd mSO

p

rd

3 mSI

P-2

P-1

(b)

Fig. 2.10 3rd order source pitchfork switching bifurcation diagrams: a the 3rd order monotonic source bifurcation for unstable-symmetry on the left side, b the 3rd order monotonic source bifurcation for unstable-symmetry on the right side. Solid and dashed curves are for monotonic stable and unstable fixed-points. Black and red colors are for period-1 and period-2 fixed-points.: P-1: period-1 fixed-points. P-2: period-2 fixed points. mSI: monotonic sink, mSO: monotonic source, oSI: oscillatory sink

2.4 Sample Bifurcation Analysis

159

Table 2.4 3rd order source pitchfork switching bifurcations and period-2 fixed-points Figure 2.10

1-dimensional Systems

(a)

xk+1 = xk − pxk + xk3

(b)

Bifurcations

xk+1 = xk − pxk − xk3

p = 0(P-1)

p = 2(P-1)

p = 2, 3(P-2)

3rd mSO

3rd oSI

3rd mSI

p = 0(P-1)

p = −2(P-1)

p = −2, −3 (P-2)

3rd mSO

3rd oSI

3rd mSI

Notice P-1: period-1 fixed-points. P-2: period-2 fixed points. mSI: monotonic sink, mSO: monotonic source, oSI: oscillatory sink

(3rd oSI), and, the bifurcation of period-2 fixed-point is a monotonic sink bifurcation of the third-order (3rd mSI). At p = −3, the bifurcation of period-2 fixed-point is a monotonic sink bifurcation of the third-order (3rd mSI) for three period-2 fixed point. Readers can complete the similar stability and bifurcation analysis as an excise. The corresponding discrete systems for 3rd order source pitchfork switching bifurcations are tabulated in Table 2.4.

2.5 Two-Dimensional Discrete Systems In this section, the stability and bifurcation for two-dimensional discrete systems are discussed.

2.5.1 Stability and Singularity Consider a two-dimensional discrete system as xk+1 = f(xk , p) ∈ R 2

(2.223)

where f = ( f 1 , f 2 )T , xk = (xk1 , xk2 )T , p = ( p1 , p2 . . . , pm )T .

(2.224)

The equilibrium xk∗ is determined by f(xk∗ , p) − xk∗ = 0 ⇒



∗ ∗ ∗ , xk2 , p) − xk1 = 0, f 1 (xk1 ∗ ∗ ∗ = 0. f 2 (xk1 , xk2 , p) − xk2

In vicinity of xk∗ , the linearized equation is

(2.225)

160

2 Low-Dimensional Discrete Systems

yk+1 = Df(xk∗ , p)yk ,

(2.226)

where yk = xk − xk∗ with ||yk || < δ for a small δ > 0

Df(xk∗ , p)

=

∂ f1 ∂ x1 ∂ f2 ∂ x1

∂ f1 ∂ x2 ∂ f2 ∂ x2

 xk∗

 a11 a12 . = a21 a22 

(2.227)

The corresponding eigenvalues are determined by    a11 − λ a12   = 0,  − λI2×2 | = 0 ⇒  a21 a22 − λ 

(2.228)

λ2 − Tr(Df)λ + Det(Df) = 0,

(2.229)

  a a  Tr(Df) = a11 + a12 , Det(Df) =  11 12 . a21 a22

(2.230)

|Df(xk∗ , p) and

where

So the eigenvalues are λ1,2

Tr(Df) ± = 2

√ 

,

(2.231)

Where  = (Tr(Df))2 − 4Det(Df).

(2.232)

The stability and bifurcations of the fixed-point xk∗ in the two-dimensional nonlinear discrete are given as follows: (i) (ii) (iii) (iv) (v) (vi)

∗ The   fixed-point xk is a saddle for real eigenvalues of |λi | < 1 (i ∈ {1, 2}) and λ j  > 1 ( j ∈ {1, 2} and j = i); The fixed-point xk∗ is a stable node for real eigenvalues of |λi | < 1 (i = 1, 2); The fixed-point xk∗ is an unstable node for real eigenvalues of |λi | > 1 (i = 1, 2); The fixed-point xk∗ is a stable focus for complex eigenvalues of |λi | < 1 (i = 1, 2); The fixed-point xk∗ is an unstable focus for complex eigenvalues of |λi | > 1 (i = 1, 2); The fixed-point xk∗ has a Neimark bifurcation for complex eigenvalues of |λi | = 1 (i = 1, 2), i.e.,

2.5 Two-Dimensional Discrete Systems

161

det(Df) = 1.

(2.233)

∗ (vii) The fixed-point   xk has a saddle-stable node bifurcation for real eigenvalues of λi = 1 and λ j  < 1 (i, j ∈ {1, 2} and j = i), i.e.,

tr(Df) = 1 + det(Df) for j = {1, 2} |λ j | < 1 for j ∈ {1, 2} and j = i;

(2.234)

(viii) The fixed-point xk∗ has a saddle-unstable node bifurcation for real eigenvalues of λi = 1 and λ j  > 1 (i, j ∈ {1, 2} and j = i), i.e., tr(Df) = 1 + det(Df) for j = {1, 2} |λ j | > 1 for j ∈ {1, 2} and j = i;

(2.235)

(ix) The fixed-point xk∗ has a saddle-stable   node bifurcation of the second kind for real eigenvalues of λi = −1 and λ j  < 1 (i, j ∈ {1, 2} and j = i), i.e., tr(Df) + det(Df) + 1 = 0 for j = {1, 2} |λ j | < 1 for j ∈ {1, 2} and j = i;

(2.236)

(x) The fixed-point xk∗ has a saddle-unstable   node bifurcation of the second kind for real eigenvalues of λi = −1 and λ j  < 1 ( j ∈ {1, 2} and j = i), i.e., tr(Df) + det(Df) + 1 = 0 for j = {1, 2} |λ j | > 1 for j ∈ {1, 2} and j = i;

(2.237)

(xi) The fixed-point xk∗ has saddle-node boundary of the third kind for real eigenvalues of λi = −1 and λ j = 1 (i, j ∈ {1, 2} and j = i), i.e., tr(Df) = 0 and det(Df) = −1

(2.238)

from which there are eight possibility; (xii) The fixed-point xk∗ has a degenerate fixed-point for det(Df) = 0. The summarization of stability and bifurcation of the fixed-point xk∗ of the 2dimensional nonlinear discrete system in Eq. (2.223) is illustrated in Fig. 2.11 through the complex plane of eigenvalue through the linearized discrete system in Eq. (2.226). The shaded area is for stable nodes and stable focus. The area above the shaded area is for unstable node, and the area below the shade area is for stable node. The left area of the axis of tr(Df) outside of the shaded area is for saddle. The vertical line is for center with det(Df) = 1 and |tr(Df)| < 2, which is also called the Neimark bifurcation. For det(Df) > 1, the area between the dashed curves are for unstable focus. The dashed parabolic curve is a boundary of complex and real eigenvalues. The

162

2 Low-Dimensional Discrete Systems Im

Im Re

tr( Df )

Re Im

Im

Im Re

Re

Im Im

Re Saddle-node bifurcation

Re

Im

Re

Im

Im Re

Re

Re

Im

Re Re

Im

Im Re

Im

Re

Im

Im Re

Im

Im

Re

Im

Neimark Bifurcation

Im

Re

Im

Re Re

Period-doubling bifurcation

Re

Re

Im Im

det( Df )

Im

Im Re

Re

Re

Re

Re

Re

Re

Im

Im Im

Re Re

Im Im

Im

Im

Im Im

Im

Re Re

Re

Re

Im Re

Im

Repeated eigenvalues Im Im Re Re Im

Re

Re

Fig. 2.11 Stability and bifurcations of the fixed-point xk∗ through trace tr(Df) and determinant det(Df) of the linearized discrete systems

2.5 Two-Dimensional Discrete Systems

163

upper line is the monotonic saddle-node bifurcation. The lower line is the oscillatory saddle-node bifurcation (or period-doubling bifurcation). The left point of the shaded triangle is the saddle-node bifurcation of the third kind. From the eigenvalues of the linearized discrete system at the fixed-point xk∗ , the eigenvectors v j (j = 1, 2) are determined by (Df(xk∗ , p) − λ j I2×2 )v j = 0 ⇒ Df(xk∗ , p)v j = λ j v j ( j = 1, 2).

(2.239)

or 

a12 a11 − λ j a21 a22 − λ j



v j1 v j2

 =

 0 ( j = 1, 2). 0

(2.240)

Thus a new transformation based on the covariant eigenvectors v j ( j = 1, 2) is defined as j

xk = ck v j = ck1 v1 + ck2 v2 ,

(2.241)

and the matrix of the covariant eigenvectors is P = (v1 , v2 ),

(2.242)

and the contravariant eigenvectors v j ( j = 1, 2) are defined as v1 = (b11 , b12 )T , v2 = (b21 , b22 )T and Q = (v1 , v2 ),

(2.243)

Q T = P −1 = (bi j )2×2 .

(2.244)

ck = (v j )T · xk , ck = (v j )T · xk∗ ( j = 1, 2),

(2.245)

where

Thus, j

j∗

and j

ck+1 = (v j )T · xk+1 = (v j )T · f(x, p).

(2.246)

Let ∗j

z k = ck − ck = (v j )T · (xk − xk∗ ), j

j

∗j

z k+1 = ck+1 − ck = (v j )T · (xk+1 − xk∗ ) = (v j )T · [f(xk∗ + xk , p) − xk∗ ] ( j = 1, 2). (2.247) j

j

164

2 Low-Dimensional Discrete Systems

In the neighborhood of equilibrium xk∗ , the linearized equation on the covariant eigenvector v j ( j = 1, 2) is 1 ∗ j1 j z k+1 = G (1) j ( j1 ) (xk , p)z k = λ j1 δ j1 z = λ j z (summation on j1 but not j),

j

j

j

(2.248)

where G j ( j1 ) (xk∗ , p) = (v j )T · Df(xk∗ , p)v j1 = (v j )T · λ j1 v j1 = λ j1 δ j1 . j

(2.249)

Thus j

j

z k+l = (λ j )l z 0 for l = 1, 2, . . . . (i) (ii)

(2.250)

monotonically

If λ j ∈ (1, ∞), z k+l −−−−−−−→ ±∞ as l → ∞. Thus, the fixed-point xk∗ on the covariant eigenvector v j ( j = 1, 2) is a monotonically unstable node. j j If λ j = 1, z k+l = z 0 as l → ∞. Thus, the fixed-point xk∗ on the covariant eigenvector v j ( j = 1, 2) is a monotonical critical case. For linear discrete systems, this is forever invariant. For a nonlinear discrete system, in vicinity of fixedpoint xk∗ , the higher-order variational equations on the covariant eigenvector v j ( j ∈ {1, 2}) should be considered. j

monotonically

(iii) If λ j ∈ (0, 1), z k+l −−−−−−−→ 0 as l → ∞. Thus, the fixed-point xk∗ on the covariant eigenvector v j ( j ∈ {1, 2}) is a monotonically stable node. j (iv) If λ j = 0, z k+l = 0 as l = 1, 2, . . .. Thus, the fixed-point xk∗ on the covariant eigenvector v j ( j ∈ {1, 2}) is a zero-stable node. j

oscillatorilly

If λ j ∈ (−1, 0), z k+l −−−−−−−→ 0 as l → ∞. Thus, the fixed-point xk∗ on the covariant eigenvector vk (k ∈ {1, 2}) is a monotonically stable node. j j (vi) If λ j = −1, z k+l = (−1)l z 0 for l = 1, 2, . . . . as l = 1, 2, . . .. Thus, the fixed∗ point xk on the covariant eigenvector v j ( j ∈ {1, 2}) is oscillatorilly flipped. For linear discrete systems, this is forever flipped. For a nonlinear discrete system, in vicinity of fixed-point xk∗ , the higher-order variational equations on the covariant eigenvector v j ( j ∈ {1, 2}) should be considered. j

(v)

oscillatorilly

(vii) If λ j ∈ (−∞, −1), z k+l −−−−−−−→ ± ∞ as l → ∞. Thus, the fixed-point xk∗ on the covariant eigenvector v j ( j ∈ {1, 2}) is an oscillatorilly unstable node. j

In the neighborhood of equilibrium xk∗ , the higher-order variational equation on the covariant eigenvector v j ( j = 1, 2) is (r )

G j (jj1 j2 ... jr ) = 0, r j = 1, 2, . . . , s j − 1 j

j z k+1

(s )

j

= G j (jj1 ) z k1 +

js 1 (s j ) j j (x∗ , p)z k1 z k2 · · · z k j G m j ! j ( j1 j2 ··· js j ) k

(summation on j1 , j2 , · · · , js j = 1, 2),

(2.251)

2.5 Two-Dimensional Discrete Systems

165

where (s )

G j (jj1 j2 ··· js ) (xk∗ , p) = (v j )T · D (s j ) f(xk∗ , p)v j1 v j2 · · · v js j . k

(2.252)

For s j = 2, we have G (1) j ( j1 ) = λ j1 δ j1 = λ j = 1, 1 j j j j z k+1 = z k + G (2) (x∗ , p)z k1 z k2 2! j ( j1 j2 ) k 1 j ∗ 1 2 = z k + [G (2) (x∗ , p)z k1 z k1 + G (2) j (12) (xk , p)z k z k 2! j (11) k (2) ∗ 2 1 ∗ 2 2 + G (2) j (21) (xk , p)z k z k + G j (22) (xk , p)z k z k ], j

(2.253)

( j = 1, 2) where ∗ j T 2 ∗ G (2) j ( j1 j2 ) (xk , p) = (v ) · D f(xk , p)v j1 v j2

( j1 , j2 = 1, 2).

(2.254)

(i) If ∗ 1 2 G (2) j ( j1 j2 ) (xk , p)z k z k > 0, j

j

(2.255)

then the fixed-point xk∗ on the covariant eigenvector v j ( j ∈ {1, 2}) is a monotonic increasing-saddle (or a monotonic upper-saddle) of the second order. (ii) If ∗ 1 2 G (2) j ( j1 j2 ) (xk , p)z k z k < 0, j

j

(2.256)

then the fixed-point xk∗ on the covariant eigenvector v j ( j ∈ {1, 2}) is a monotonic deceasing-saddle (or a monotonic lower-saddle) of the second order (iii) If ∗ 1 2 z k × G (2) j ( j1 j2 ) (xk , p)z k z k > 0, j

j

j

(2.257)

then the fixed-point xk∗ on the covariant eigenvector v j ( j ∈ {1, 2}) is a monotonic of the second order. (iv) If ∗ 1 2 z k × G (2) j ( j1 j2 ) (xk , p)z k z k < 0, j

j

j

(2.258)

166

2 Low-Dimensional Discrete Systems

then the fixed-point xk∗ on the covariant eigenvector v j ( j ∈ {1, 2}) is a monotonic of the second order. (v) If ∗ G (2) j ( j1 j2 ) (xk , p) = 0,

(2.259)

then the fixed-point xk∗ on the covariant eigenvector v j ( j ∈ {1, 2}) is a critical case of the second order. For this case, the higher order singularity should be considered for s j = 3. For s j = 3, we have (2) ∗ G (1) j ( j1 ) = λ j1 δ j1 = λ j = 1, G j ( j1 j2 ) (xk , p) = 0, 1 j j j j j z k+1 = z k + G (3) (x∗ , p)z k1 z k2 z k3 3! j ( j1 j2 j3 ) k 1 j ∗ 1 1 2 = z k + [G (3) (x∗ , p)z k1 z k1 z k1 + G (3) j (112) (xk , p)z k z k z k 3! j (111) k (3) ∗ 1 2 1 ∗ 1 2 2 + G (3) j (121) (xk , p)z k z k z k + G j (122) (xk , p)z k z k z k j

(2.260)

(3) ∗ 2 1 1 ∗ 2 1 2 + G (3) j (211) (xk , p)z k z k z k + G j (212) (xk , p)z k z k z k (3) ∗ 2 2 1 ∗ 2 2 2 + G (3) j (221) (xk , p)z k z k z k + G j (222) (xk , p)z k z k z k ],

where ∗ j T 3 ∗ G (3) j ( j1 j2 j3 ) (xk , p) = (v ) · D f(xk , p)v j1 v j2 v j3

( j1 , j2 , j3 = 1, 2).

(2.261)

(i) If ∗ 1 2 3 G (3) j ( j1 j2 j3 ) (xk , p)z k z k z k > 0, j

j

j

(2.262)

then the fixed-point xk∗ on the covariant eigenvector v j ( j ∈ {1, 2}) is a monotonic increasing-saddle (or a monotonic upper-saddle) of the third order. (ii) If ∗ 1 2 3 G (3) j ( j1 j2 j3 ) (xk , p)z k z k z k < 0, j

j

j

(2.263)

then the fixed-point xk∗ on the covariant eigenvector v j ( j ∈ {1, 2}) is a monotonic deceasing-saddle (or a monotonic lower-saddle) of the third order. (iii) If ∗ 1 2 3 z k × G (3) j ( j1 j2 j3 ) (xk , p)z k z k z k > 0, j

j

j

j

(2.264)

2.5 Two-Dimensional Discrete Systems

167

then the fixed-point xk∗ on the covariant eigenvector v j ( j ∈ {1, 2}) is a monotonic source of the third order (iv) If ∗ 1 2 3 z k × G (3) j ( j1 j2 j3 ) (xk , p)z k z k z k < 0, j

j

j

j

(2.265)

then the fixed-point xk∗ on the covariant eigenvector v j ( j ∈ {1, 2}) is a monotonic sink of the third order Similarly, for s j = 2, we have G (1) j ( j1 ) = λ j1 δ j1 = λ j = −1 ( j = 1, 2) 1 j j j j z k+1 = −z k + G (2) (x∗ , p)z k1 z k2 2! j ( j1 j2 ) k 1 j ∗ 1 2 = −z k + [G (2) (x∗ , p)z k1 z k1 + G (2) j (12) (xk , p)z k z k 2! j (11) k (2) ∗ 2 1 ∗ 2 2 + G (2) j (21) (xk , p)z k z k + G j (22) (xk , p)z k z k ]. j

(2.266)

(i) If ∗ 1 2 G (2) j ( j1 j2 ) (xk , p)z k z k > 0, j

j

(2.267)

then the fixed-point xk∗ on the covariant eigenvector v j ( j ∈ {1, 2}) is an oscillatory increasing-saddle (or an oscillatory upper-saddle of the second order. (ii) If ∗ 1 2 G (2) j ( j1 j2 ) (xk , p)z k z k < 0, j

j

(2.268)

then the fixed-point xk∗ on the covariant eigenvector v j ( j ∈ {1, 2}) is an oscillatory deceasing-saddle (or an oscillatory lower-saddle) of the second order. (iii) If ∗ 1 2 z k × G (2) j ( j1 j2 ) (xk , p)z k z k < 0, j

j

j

(2.269)

then the fixed-point xk∗ on the covariant eigenvector v j ( j ∈ {1, 2}) is an oscillatory source of the second order (iv) If ∗ 1 2 z k × G (2) j ( j1 j2 ) (xk , p)z k z k > 0, j

j

j

(2.270)

then the fixed-point xk∗ on the covariant eigenvector v j ( j ∈ {1, 2}) is an oscillatory sink of the second order.

168

2 Low-Dimensional Discrete Systems

(v) If ∗ G (2) j ( j1 j2 ) (xk , p) = 0,

(2.271)

then the fixed-point xk∗ on the covariant eigenvector v j ( j ∈ {1, 2}) is an oscillatory critical case of the second order. For this case, the higher order singularity should be considered for s j = 3. For s j = 3, we have (2) ∗ G (1) j ( j1 ) = λ j1 δ j1 = λ j = −1, G j ( j1 j2 ) (xk , p) = 0, j

j

1 (3) j j j G (x∗ , p)z k1 z k2 z k3 3! j ( j1 j2 j3 ) k 1 j ∗ 1 1 2 = −z k + [G (3) (x∗ , p)z k1 z k1 z k1 + G (3) j (112) (xk , p)z k z k z k 3! j (111) k (3) ∗ 1 2 1 ∗ 1 2 2 + G (3) j (121) (xk , p)z k z k z k + G j (122) (xk , p)z k z k z k j

z k+1 = −z k +

(2.272)

(3) ∗ 2 1 1 ∗ 2 1 2 + G (3) j (211) (xk , p)z k z k z k + G j (212) (xk , p)z k z k z k (3) ∗ 2 2 1 ∗ 2 2 2 + G (3) j (221) (xk , p)z k z k z k + G j (222) (xk , p)z k z k z k ].

(i) If ∗ 1 2 3 G (3) j ( j1 j2 j3 ) (xk , p)z k z k z k > 0, j

j

j

(2.273)

then the fixed-point xk∗ on the covariant eigenvector v j ( j ∈ {1, 2}) is an oscillatory increasing-saddle (or an oscillatory upper-saddle) of the third order. (ii) If ∗ 1 2 3 G (3) j ( j1 j2 j3 ) (xk , p)z k z k z k < 0, j

j

j

(2.274)

then the fixed-point xk∗ on the covariant eigenvector v j ( j ∈ {1, 2}) is an oscillatory deceasing-saddle (or an oscillatory lower-saddle) of the third order. (iii) If ∗ 1 2 3 z k × G (3) j ( j1 j2 j3 ) (xk , p)z k z k z k < 0, j

j

j

j

(2.275)

then the fixed-point xk∗ on the covariant eigenvector v j ( j ∈ {1, 2}) is an oscillatory source of the third order. (iv) If ∗ 1 2 3 z k × G (3) j ( j1 j2 j3 ) (xk , p)z k z k z k > 0, j

j

j

j

(2.276)

2.5 Two-Dimensional Discrete Systems

169

then the fixed-point xk∗ on the covariant eigenvector v j ( j ∈ {1, 2}) is an oscillatory sink of the third order.

2.5.2 An Example Consider a discrete system as xk+1 = α[1 + λ + a(xk2 + yk2 )]xk + β[1 + λ + a(xk2 + yk2 )]yk , yk+1 = −β[1 + λ + a(xk2 + yk2 )]xk + α[1 + λ + a(xk2 + yk2 )]yk .

(2.277)

Setting rk2 = xk2 + yk2 with xk = rk cos θk and yk = rk sin θk ,

(2.278)

we have rk+1 =



2 2 xk+1 + yk+1 = ρrk (1 + λ + ark2 ),

−β cos θk + α sin θk = θk − ϑ β sin θk + α cos θk  β ρ = α 2 + β 2 and ϑ = arctan α

θk+1 = arctan

(2.279)

If ρ = 1, the fixed-point is ∗ rk(1) = 0 for λ ∈ (−∞, +∞), ∗ rk(2) = (−λ/a)1/2 for λ × a < 0.

(2.280)

If λ = 0, we have D fr (rk∗ , λ) = 1 + λ + 3a(rk∗ )2 , the variational equation is sk+1 = D fr (rk∗ , λ)sk = [1 + λ + 3a(rk∗ )2 ]sk with sk = rk − rk∗ .

(2.281)

∗ = 0, D fr = 1 + λ. This fixed-point is stable for αλ < 0 due to sk+1 < sk For rk(1) or unstable for λ > 0 owing to sk+1 > sk . The fixed-point is a critical point for ∗ = (−λ/a)1/2 requires aλ < 0. For a > 0, such a λ = 0. The fixed-point of rk(2) fixed-point exists for λ < 0. For a < 0, the fixed-point existence condition is λ > 0. From D fr = 1 − 2λ, the fixed-point is stable for λ > 0 owing to sk+1 < sk and ∗ = 0 and unstable for λ < 0 owing to sk+1 > sk . For λ = 0, we have rk(0)

D frk (rk∗ , λ) = 1 and Dλ D frk (rk∗ , α) = 1 = 0.

(2.282)

170

2 Low-Dimensional Discrete Systems yk

yk xk

xk

λ

λ

(a)

(b)

Fig. 2.12 Neimark bifurcations: a supercritical (a < 0) and b subcritical (a > 0) ∗ For rk(0) = 0 and λ = 0, D fr (rk∗ , λ) = 1 and D 2 fr (rk∗ , λ) = 6ark∗ = 0 exists. So we have D 3 fr (rk∗ , λ) = 6a. The variational equation is given by sk+1 = (1 + ask2 )sk . ∗ , λ) = (0, 0) is spirally stable of the third For a < 0, sk+1 < sk , the fixed-point (rk(0) ∗ , λ) = (0, 0) is the Neimark bifurcation. order. The bifurcation of the fixed-point (rk(0) The Neimark bifurcation with stable focus (a < 0) is called a supercritical case. For ∗ , λ) = (0, 0) is spirally unstable of the third a > 0, sk+1 > sk , the fixed-point (rk(0) ∗ , λ) = (0, 0) is the Neimark bifurcation. order. The bifurcation of the fixed-point (rk(0) The Neimark bifurcation with unstable focus (a > 0) is called a subcritical case. The supercritical and subcritical Neimark bifurcation is shown in Fig. 2.12a, b. The solid lines and curves represent stable fixed-point. The dashed lines and curves represent ∗ = 0, unstable fixed-point. The phase shift is determined by θk+1 = θk − ϑ and rk(2) one get a stable or unstable fixed-point on the circle. From the foregoing analysis of the Neimark bifurcation, the Neimark bifurcation points possess the higher-order singularity of the fixed-point in discrete system in the radial direction. For the stable Neimark bifurcation, the mth order singularity of the fixed-point at the bifurcation point exists as a sink of the mth-order in the radial direction. For the unstable Neimark bifurcation, the mth-order singularity of the fixed-point at the bifurcation point exists as a source of the mth-order in the radial direction.

References Luo ACJ (2019) On stability and bifurcation of equilibriums in nonlinear dynamical systems. Journal of Vibration Testing and System Dynamics 3(2):147–312 Luo ACJ (2020) Bifurcation and stability in nonlinear dynamical systems. Springer, New York

Chapter 3

Global Stability of 1-D Discrete Systems

In this Chapter, as in Luo (2019, 2020), a global analysis of fixed-point stability in 1-dimensional nonlinear discrete systems is presented. The classification of discrete systems is discussed first, and infinite-fixed-point discrete systems are presented. The 1-dimensional discrete systems with a single fixed-point are discussed first. The 1-dimensional discrete systems with two and three fixed-points are discussed. Simple fixed-points and higher-order fixed-points in 1-dimensional discrete systems are analyzed, and herein a higher-order fixed-point is a fixed-point with higher-order singularity. The discrete flow of fixed-points in 1-dimensional systems in alike phase space are illustrated for a better understanding of the global stability of fixed-points in 1-dimensional discrete systems.

3.1 Discrete System Classifications Definition 3.1 Consider a discrete system as xk+1 = xk + f(xk , p) for xk ∈  and p ∈ 

(3.1)

f = ( f 1 , f 2 , . . . , f n )T

(3.2)

where

(i) The discrete system is invariant if f(xk , p) ≡ 0 for xk ∈ . (ii) The discrete system is with constant adding if f(xk , p) ≡ constant for xk ∈ . (iii) The discrete system is independent of discrete states if f(xk , p) ≡ g(p) for k ∈ Z and xk ∈ . (iv) The discrete system is with fixed-points if f(xk , p) ≡ g(xk , p) and © Higher Education Press 2020 A. C. J. Luo, Bifurcation and Stability in Nonlinear Discrete Systems, Nonlinear Physical Science, https://doi.org/10.1007/978-981-15-5212-0_3

171

172

3 Global Stability of 1-D Discrete Systems

g(xk , p)|xk =xk∗ = 0

(3.3)

for xk , xk∗ ∈ . (v) The discrete system is without fixed-point if f(xk , p) ≡ g(xk , p) and g(xk , p)|xk =xk∗ = 0

(3.4)

for xk , xk∗ ∈ . Definition 3.2 Consider a discrete system as xk+1 = xk + g(xk , p) for xk ∈  and p ∈ 

(3.5)

g = (g1 , g2 , . . . , gn )T .

(3.6)

where

(i)

Such a discrete system is called a discrete system of the rth order without fixed-points on domain  if for all xk∗ ∈ ,  g j (xk , p)xk =x∗ = 0 for j ∈ {l1 , l2 , . . . , lr } ⊆ {1, 2, . . . , n}. k

(3.7)

(ii) Such a discrete system is called a discrete system fully without fixed-points if for all xk∗ ∈ ,  g j (xk , p)xk =x∗ = 0 for j = 1, 2, . . . , n. k

(3.8)

(iii) Such a discrete system is called a discrete system of the rth order partially with fixed-points if for all xk∗ ∈ ,  g j (xk , p)xk =x∗ = 0 for j ∈ {l1 , l2 , . . . , lr } ⊆ {1, 2, . . . , n}. k

(3.9)

(iv) Such a discrete system is called a discrete system with fixed-points if for xk∗ ∈ ,  g j (xk , p)xk =x∗ = 0 for j = 1, 2, . . . , n. (3.10) k Definition 3.3 Consider a discrete system as xk+1 = xk + g(xk , p) and g(xk , p)|xk =xk∗ = 0 for xk ∈  and p ∈ 

(3.11)

3.1 Discrete System Classifications

173

where  ∂g  . g = (g1 , g2 , . . . , gn ) , A = ∂x x=x∗ T

(3.12)

(i) Such a discrete system is called a discrete system with finite-fixed-points if

g(xk∗ , p) = 0, and det A = 0.

(3.13)

(ii) Such a discrete system is called a discrete system with infinite-fixed-points if

g(xk∗ , p) = 0, and det A = 0.

(3.14)

(iii) Such a discrete system is called a discrete system with at least r-sets of infinitefixed-points if g(xk∗ , p) = 0 and det A = 0 with rank(A) = n − r.

(3.15)

Definition 3.4 Consider a discrete system as xk+1(i) = xk(i) + gi1 (xk , pi1 )gi2 (xk , pi2 ) . . . giri (xk , piri ) (i = 1, 2, . . . , n), ri ≥ 1

(3.16)

gi (xk , p) ≡ gi1 (xk , pi1 )gi2 (xk , pi2 ) . . . giri (xk , piri ).

(3.17)

where

(i)

Such a discrete system is called a discrete system without fixed-points in the direction of the ith component if gi (xk , p) = 0 or gi j (xk , pi j ) = 0 for i ∈ {1, 2, . . . , n}, j = 1, 2, . . . , ri .

(3.18)

(ii) Such a discrete system is called a discrete system with fixed-points in the direction of the ith component if gi (xk , p) = 0 or gi j (xk , pi j ) = 0 for i ∈ {1, 2, . . . , n}, j ∈ {1, 2, . . . , ri }.

(3.19)

174

3 Global Stability of 1-D Discrete Systems

(iii) There is an infinite-fixed-point surface of the zeroth order in such a discrete system if gi (xk , p) = 0 for i = 1, 2, . . . , n; gi1 j1 (x, pi1 j1 ) ≡ gi2 j2 (x, pi2 j2 ) = ϕ(xk , q) = 0 for i 1 , i 2 ∈ {1, 2, . . . , n}, i 1 = i 2 ;     j1 ∈ 1, 2, . . . , ri1 , j2 ∈ 1, 2, . . . , ri2 .

(3.20)

(iv) There is an infinite-fixed-point surface of the first order in such a discrete system as ϕ(xk , q) = 0 with gi (xk , p) = 0 for i = 1, 2, . . . , n

(3.21)

if lim

ϕ(xk ,q)→0

gi1 j1 (xk , pi1 j1 ) = 0,

lim

ϕ(xk ,q)→0

gi2 j2 (xk , pi2 j2 ) = 0,

d gi1 j1 (xk , pi1 j1 )  = 0, ϕ(xk ,q)→0 ϕ(xk ,q)→0 dϕ d lim Dϕ gi2 j2 (xk , pi2 j2 ) = lim gi2 j2 (xk , pi2 j2 )  = 0 ϕ(xk ,q)→0 ϕ(xk ,q)→0 dϕ gi1 j1 (xk , pi1 j1 ) Dϕ gi1 j1 (xk , pi1 j1 ) with lim = lim = , ϕ(xk ,q)→0 gi 2 j2 (xk , pi 2 j2 ) ϕ(xk ,q)→0 Dϕ gi 2 j2 (xk , pi 2 j2 ) lim

Dϕ gi1 j1 (xk , pi1 j1 ) =

lim

(3.22)

0  = || < ∞ for i 1 , i 2 ∈ {1, 2, . . . , n}, i 1  = i 2 ; j1 , j2 ∈ {1, 2, . . . , ri }.

(v) There is an infinite-fixed-point surface of the rth order in such a discrete system as ϕ(xk , q) = 0 with gi (xk , p) = 0 for i = 1, 2, . . . , n if

(3.23)

3.1 Discrete System Classifications

lim

gi1 j1 (xk , pi1 j1 ) = 0,

lim

Dϕs gi1 j1 (xk , pi1 j1 ) =

ϕ(xk ,q)→0

ϕ(xk ,q)→0

175

lim

ϕ(xk ,q)→0

gi2 j2 (xk , pi2 j2 ) = 0

ds gi1 j1 (xk , pi1 j1 ) = 0, ϕ(xk ,q)→0 dϕ s lim

( s = 1, 2, . . . , r − 1) lim

ϕ(xk ,q)→0

Dϕs gi2 j2 (xk , pi2 j2 ) =

ds gi2 j2 (xk , pi2 j2 ) = 0, ϕ(xk ,q)→0 dϕ s lim

(s = 1, 2, . . . , r − 1) dr gi1 j1 (xk , pi1 j1 ) = 0, ϕ(xk ,q)→0 ϕ(xk ,q)→0 dϕ r dr gi2 j2 (xk , pi2 j2 ) = 0 lim Dϕr gi2 j2 (xk , pi2 j2 ) = lim ϕ(xk ,q)→0 ϕ(xk ,q)→0 dϕ r Dϕr gi1 j1 (xk , pi1 j1 ) gi1 j1 (xk , pi1 j1 ) = lim = , with lim rg ϕ(xk ,q)→0 gi 2 j2 (xk , pi 2 j2 ) ϕ(xk ,q)→0 Dϕ i 2 j2 (xk , pi 2 j2 ) lim

Dϕr gi1 j1 (xk , pi1 j1 ) =

lim

0 = || < ∞, for i 1 , i 2 ∈ {1, 2, . . . , n}, i 1 = i 2 ; j1 , j2 ∈ {1, 2, . . . , ri }. (3.24)

3.2 Fixed-Point Global Stability Consider a discrete system as xk+1 = σ xk + g(xk , p) = σ xk + g0 (xk )g1 (xk − a1 )g2 (xk − a2 ) . . . gm−1 (xk − am−1 )gm (xk − am ) (3.25) where σ = 1 for monotonic case, σ = −1 for oscillatory case; g0 (xk ) = 0 for xk ∈ R,

(3.26)

∗ g j (xk − a j ) = 0 with xk( j) = a j ,

( j = 1, 2, . . . , m). Thus, such a discrete system has fixed-points as ∗ xk( j) = a j for j = 1, 2, . . . , m.

(3.27)

Note that g j (x −a j ) can be any type of functions but not necessary to be a polynomial function. Consider a polynomial discrete system as

176

3 Global Stability of 1-D Discrete Systems

xk+1 = σ xk + g(xk , p) = σ xk + g0 (xk )(xk − a1 )r1 (xk − a2 )r2 . . . (xk − am−1 )rm−1 (xk − am )rm (3.28) where g0 (xk ) = 0 for xk ∈ R = (−∞, ∞), ak < ak+1 with − ∞ < a1 and am < ∞, ai = a j and ri ∈ {1, 2, . . .} for i, j ∈ {1, 2, . . . , m}.

(3.29)

Thus, such a discrete system has fixed-points as ∗ th xk( j) = a j with the r j order roots for j = 1, 2, . . . , m.

(3.30)

For the case of g0 (xk ) > 0 and σ = 1, there are four types of monotonic fixedpoints. (i)

∗ For a fixed-point xk( j) ( j = 1, 2, . . . , m), if m m α= j+1 r α = 2l j1 and α= j r α = 2l j2 for l j1 , l j2 ∈ {1, 2, 3, . . .},

(3.31)

then such a fixed-point is a monotonic-upper-saddle of the r th j order because ∗ ∗ ∗ ∗ , x ) and (x , x xk+1 − xk > 0 always for xk ∈ (xk( j−1) k( j) k( j) k( j+1) ). ∗ ( j = 1, 2, . . . , m), if (ii) For a fixed-point xk( j) m m α= j+1 r α = 2l j1 − 1 and α= j r α = 2l j2 − 1 for l j1 , l j2 ∈ {1, 2, 3, . . .}, (3.32)

then such a fixed-point is a monotonic-lower saddle of the r th j order because ∗ ∗ ∗ ∗ , x ) and (x , x xk+1 − xk < 0 always for xk ∈ (xk( j−1) k( j) k( j) k( j+1) ). ∗ (iii) For a fixed-point xk( j) ( j = 1, 2, . . . , m), if m m α= j+1 r α = 2l j1 and α= j r α = 2l j2 − 1 for l j1 , l j2 ∈ {1, 2, 3, . . .}, (3.33)

then such a fixed-point is a monotonic source of the r th j order because x k+1 − ∗ ∗ ∗ ∗ xk > 0 for xk ∈ (xk( j−1) , x k( j) ) and x k+1 − x k < 0 for x k ∈ (x k( j) , x k( j+1) ). ∗ ( j = 1, 2, . . . , m), if (iv) For a fixed-point xk( j) m m α= j+1 r α = 2l j1 − 1 and α= j r α = 2l j2 for l j1 , l j2 ∈ {1, 2, 3, . . .}, (3.34)

3.2 Fixed-Point Global Stability

177

then such a fixed-point is a monotonic sink of the r th j order because x k+1 −x k < 0 ∗ ∗ ∗ ∗ , x ) and x − x > 0 for x for xk ∈ (xk( k+1 k k ∈ (x k( j) , x k( j+1) ). If r j = 1, j−1) k( j) such a fixed-point can be one of the following states from a monotonic sink to oscillatory source. For the case of g0 (xk ) < 0 and σ = 1, similarly, there are four types of monotonic fixed-points. (i)

∗ For a fixed-point xk( j) ( j = 1, 2, . . . , m), if m m α= j+1 r α = 2l j1 and α= j r α = 2l j2 for l j1 , l j2 ∈ {1, 2, 3, . . .},

(3.35)

then such a fixed-point is a monotonic-lower-saddle of the r th j order because ∗ ∗ ∗ ∗ , x ) and (x , x xk+1 − xk < 0 always for xk ∈ (xk( j−1) k( j) k( j) k( j+1) ). ∗ ( j = 1, 2, . . . , m), if (ii) For a fixed-point xk( j) m m α= j+1 r α = 2l j1 − 1 and α= j r α = 2l j2 − 1 for l j1 , l j2 ∈ {1, 2, 3, . . .}, (3.36)

then such a fixed-point is a monotonic-upper-saddle of the r th j order because ∗ ∗ ∗ ∗ , x ) and (x , x xk+1 − xk > 0 always for xk ∈ (xk( j−1) k( j) k( j) k( j+1) ). ∗ ( j = 1, 2, . . . , m), if (iii) For a fixed-point xk( j) m m α= j+1 r α = 2l j1 and α= j r α = 2l j2 − 1 for l j1 , l j2 ∈ {1, 2, 3, . . .}, (3.37)

then such a fixed-point is a monotonic sink of the r th j order because x k+1 −x k < 0 ∗ ∗ ∗ ∗ , x ) and x − x > 0 for x ∈ (xk( for xk ∈ (xk( k+1 k k j−1) k( j) j) , x k( j+1) ). For r j = 1, such a fixed-point can be one of the following states from a monotonic sink to oscillatory source. ∗ (iv) For a fixed-point xk( j) ( j = 1, 2, . . . , m), if m m α= j+1 r α = 2l j1 − 1 and α= j r α = 2l j2 for l j1 , l j2 ∈ {1, 2, 3, . . .}, (3.38)

then such a fixed-point is a monotonic source of the r th j order because x k+1 − ∗ ∗ ∗ ∗ xk > 0 for xk ∈ (xk( j−1) , x k( j) ) and x k+1 − x k < 0 for x k ∈ (x k( j) , x k( j+1) ). For σ = 1, from local analysis, fixed-points of Eq. (3.28) are presented in Fig. 3.1. In Fig. 3.1a, b, g0 (xk ) > 0 and g0 (xk ) < 0. The acronyms mLS, mUS, mSI and mSO represent monotonic lower-saddle, monotonic upper-saddle, monotonic sink (monotonic stable node), and monotonic source (monotonic unstable node), respectively. The circular symbols are fixed-points. For the case of g0 (xk ) > 0 and σ = −1, there are four types of oscillatory fixed-points.

178

3 Global Stability of 1-D Discrete Systems

Fig. 3.1 Possible distributions of monotonic stability of fixed-points of the 1-dimensional discrete system (σ = 1): a g0 > 0 and b g0 < 0. mLS: monotonic-lower-saddle, mUS: monotonicupper-saddle, mSI: monotonic-sink (monotonic-stable node), mSO: monotonic-source (monotonicunstable node). mSI represents mSI-oSO for r j = 1 for j = 1, 2, . . . , m

(i)

∗ For a fixed-point xk( j) ( j = 1, 2, . . . , m), if m m α= j+1 r α = 2l j1 and α= j r α = 2l j2 for l j1 , l j2 ∈ {1, 2, 3, . . .},

(3.39)

then such a fixed-point is an oscillatory upper-saddle of the r th j order because ∗ ∗ ∗ ∗ , x ) and x ∈ (x , x xk+1 + xk > 0 for xk ∈ (xk( k j) k( j+1) k( j−1) k( j) ). ∗ ( j = 1, 2, . . . , m), if (ii) For a fixed-point xk( j) m m α= j+1 r α = 2l j1 − 1 and α= j r α = 2l j2 − 1 for l j1 , l j2 ∈ {1, 2, 3, . . .}, (3.40)

then such a fixed-point is an oscillatory lower-saddle of the r th j order because ∗ ∗ , x ) and x + x xk+1 + xk < 0 for xk ∈ (xk( k+1 k < 0 for x k ∈ j) k( j+1) ∗ ∗ , x ). (xk( j−1) k( j) ∗ (iii) For a fixed-point xk( j) ( j = 1, 2, . . . , m), if

3.2 Fixed-Point Global Stability

179

m m α= j+1 r α = 2l j1 − 1 and α= j r α = 2l j2 for l j1 , l j2 ∈ {1, 2, 3, . . .}, (3.41)

then such a fixed-point is an oscillatory source of the rkth order because xk+1 + ∗ ∗ ∗ ∗ xk < 0 for xk ∈ (xk( j) , x k( j+1) ) and x k+1 + x k > 0 for x k ∈ (x k( j−1) , x k( j) ). ∗ ( j = 1, 2, . . . , m), if (iv) For a fixed-point xk( j) m m α= j+1 r α = 2l j1 and α= j r α = 2l j2 − 1 for l j1 , l j2 ∈ {1, 2, 3, . . .}, (3.42)

then such a fixed-point is an oscillatory sink of the r th j order because x k+1 +x k > ∗ ∗ ∗ ∗ , x ) and x + x < 0 for xk ∈ (xk( 0 for xk ∈ (xk( k+1 k j) k( j+1) j−1) , x k( j) ). For r j = 1, such a fixed-point can be one of the following states from an oscillatory sink to monotonic source. For the case of g0 (xk ) < 0 and σ = −1, similarly, there are four types of oscillatory fixed-points (i)

∗ For a fixed-point xk( j) ( j = 1, 2, . . . , m), if m m α= j+1 r α = 2l j1 and α= j r α = 2l j2 for l j1 , l j2 ∈ {1, 2, 3, . . .},

(3.43)

then such a fixed-point is an oscillatory-lower-saddle of the r th j order because ∗ ∗ ∗ ∗ , x ) and (x , x xk+1 + xk < 0 always for xk ∈ (xk( j−1) k( j) k( j) k( j+1) ). ∗ ( j = 1, 2, . . . , m), if (ii) For a fixed-point xk( j) m m α= j+1 r α = 2l j1 − 1 and α= j r α = 2l j2 − 1 for l j1 , l j2 ∈ {1, 2, 3, . . .}, (3.44)

then such a fixed-point is an oscillatory-upper-saddle of the r th j order because ∗ ∗ ∗ ∗ , x ) and (x , x xk+1 + xk > 0 always for xk ∈ (xk( j−1) k( j) k( j) k( j+1) ). ∗ ( j = 1, 2, . . . , m), if (iii) For a fixed-point xk( j) m m α= j+1 r α = 2l j1 − 1 and α= j r α = 2l j2 for l j1 , l j2 ∈ {1, 2, 3, . . .}, (3.45)

then such a fixed-point is an oscillatory sink of the r th j order because x k+1 +x k > ∗ ∗ ∗ ∗ , x ) and x + x < 0 for xk ∈ (xk( 0 for xk ∈ (xk( k+1 k j−1) k( j) j) , x k( j+1) ). such a fixed-point can be one of the following states from an oscillatory sink to monotonic source. ∗ (iv) For a fixed-point xk( j) ( j = 1, 2, . . . , m), if m m α= j+1 r α = 2l j1 and α= j r α = 2l j2 − 1 for l j1 , l j2 ∈ {1, 2, 3, . . .}, (3.46)

180

3 Global Stability of 1-D Discrete Systems

then such a fixed-point is an oscillatory source of the r th j order because x k+1 + ∗ ∗ ∗ ∗ , x ) and x + x > 0 for x xk < 0 for xk ∈ (xk( k+1 k k ∈ (x k( j) , x k( j+1) ). j−1) k( j) For σ = −1, from local analysis, oscillatory fixed-points of Eq. (3.28) are presented in Fig. 3.2. In Fig. 3.2a, b, g0 (xk ) > 0 and g0 (xk ) < 0. The acronyms oLS, oUS, oSI and oSO represent oscillatory lower-saddle, oscillatory-upper-saddle, oscillatory-sink (oscillatory stable node) and oscillatory source (oscillatory unstable node), respectively. The circular symbols are fixed-points. From the above discussions, the properties of distributed fixed-points can be determined. Thus, a few examples are presented as follows.

Fig. 3.2 Possible distributions of oscillatory stability of fixed-points of the 1-dimensional discrete system (σ = −1): a g0 > 0 and b g0 < 0. oLS: oscillatory-lower-saddle, oUS: oscillatory-uppersaddle, oSI: oscillatory-sink (oscillatory-stable node), oSO: oscillatory-source oscillatory-unstable node). oSI represents oSI-mSO for r j = 1 for j = 1, 2, . . . , m

3.3 One Fixed-Point Systems

181

3.3 One Fixed-Point Systems In this section, a discrete system of one fixed-point will be discussed.

3.3.1 Monotonic Stability Consider a 1-dimensional discrete system with one fixed-point as xk+1 = xk + g(xk , p) = xk + a0 (xk − a1 )r with r = 2m − 1, 2m for m = 1, 2, . . .

(3.47)

where a0 = 0. The rth-order fixed-point of xk∗ = a1 is obtained easily and the corresponding stability is discussed. Thus, we have d xk+1 = 1 + Dg(xk , p) = 1 + ra0 (xk − a1 )r −1 d xk for xk = xk∗ = a1

(3.48)

and g(xk∗ , p) = a0 (xk∗ − a1 )r = 0, D ( j) g(xk∗ , p) = a0 r (r − 1) . . . (r − j + 1)(xk∗ − a1 )r − j = 0, j = 1, 2, . . . , r, ˙ 2, . . . . D (r ) g(xk∗ , p) = a0 r ! with r = 2m − 1, 2m for m = 1, (3.49) From Eqs. (3.48) and (3.49), the stability of (xk+1 − xk ) = g(xk , p) is tabulated in Table 3.1. The flow direction in a phase space of (xk , xk+1 − xk ) is presented through the following discussion. (i) For a0 > 0, we have • if xk > a1 , (xk+1 − xk ) = a0 (xk − a1 )r > 0 for r = 2m, 2m − 1; • if xk < a1 , there are two cases – (xk+1 − xk ) = a0 (xk − a1 )r > 0 for r = 2m but Table 3.1 The monotonic stability of a one-fixed-point discrete system xk+1 − xk = a0 (xk − a1 )r xk∗ = a1 a0 > 0 − 1)th

a0 < 0

r = 2m − 1

(2m

r = 2m

(2m)th mUS

mSO

(2m − 1)th mSI (2m)th mLS

mLS: monotonic-lower-saddle, mUS: monotonic-upper-saddle, mSI: monotonic-sink, mSO: monotonic-source

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3 Global Stability of 1-D Discrete Systems

– (xk+1 − xk ) = a0 (xk − a1 )r < 0 for r = 2m − 1. (ii) For a0 < 0, we have • if xk > a1 , (xk+1 − xk ) = a0 (xk − a1 )r < 0 for r = 2m, 2m − 1; • if xk < a1 , there are two cases: – (xk+1 − xk ) = a0 (xk − a1 )r < 0 for r = 2m but – (xk+1 − xk ) = a0 (xk − a1 )r > 0 for r = 2m − 1. Because Dr g = a0 r ! with r = 2m − 1, 2m (m = 1, 2, . . .), Dr g = a0 r ! > 0 if a0 > 0 and Dr g = a0 r ! < 0 if a0 < 0. Thus, there are four monotonic fixed-points: a monotonic-source of the (2m −1)th order, a monotonic-sink of the (2m −1)th order, a monotonic-upper-saddle of the (2m)th order, and a monotonic-lower saddle of the (2m − 1)th order. In a plane of (xk , xk+1 − xk ), the fixed-point and phase trajectory are presented in Fig. 3.3.

a0>0

(2m-1)th order

mSO

Function, xk+1xk= g(xk,p)

Function, xk+1xk= g(xk,p)

a0 0, b a monotonic sink of the (2m − 1)th order (mSI) for a0 < 0, c a monotonic upper saddle of the (2m)th order (mUS) for a0 > 0, d a monotonic lower saddle of the (2m)th order (mLS) for a0 < 0. (m = 1, 2, . . .). mLS: monotonic-lower-saddle, mUS: monotonic-upper-saddle, mSI: monotonic-sink (monotonic stable node), mSO-monotonic-source (monotonic unstable node)

3.3 One Fixed-Point Systems

183

3.3.2 Oscillatory Stability Consider a 1-dimensional discrete system with one oscillatory fixed-point as xk+1 = −xk + g(xk , p) = −xk + a0 (xk − a1 )r with r = 2m − 1, 2m for m = 1, 2, . . .

(3.50)

where a0 = 0. The rth-order oscillatory fixed-point of xk∗ = a1 is obtained easily and the corresponding stability is discussed. Thus, we have d xk+1 = −1 + Dg(xk , p) = −1 + ra0 (xk − a1 )r −1 d xk for xk = xk∗ = a1 .

(3.51)

From Eq. (3.51), the stability of (xk+1 + xk ) = g(xk , p) is tabulated in Table 3.2 and the flow direction in a phase space of (xk , xk+1 + xk ) is presented through the following discussion. (i) For a0 > 0, we have • if xk > a1 , (xk+1 + xk ) = a0 (xk − a1 )r > 0 for r = 2m, 2m − 1; • if xk < a1 , there are two cases – (xk+1 + xk ) = a0 (xk − a1 )r > 0 for r = 2m but – (xk+1 + xk ) = a0 (xk − a1 )r < 0 for r = 2m − 1. (ii) For a0 < 0, we have • if xk > a1 , (xk+1 + xk ) = a0 (xk − a1 )r < 0 for r = 2m, 2m − 1; • if xk < a1 , there are two cases: – (xk+1 + xk ) = a0 (xk − a1 )r < 0 for r = 2m but – (xk+1 + xk ) = a0 (xk − a1 )r > 0 for r = 2m − 1. Because Dr g = a0 r ! with r = 2m − 1, 2m (m = 1, 2, . . .), Dr g = a0 r ! > 0 if a0 > 0 and Dr g = a0 r ! < 0 if a0 < 0. Thus, there are four oscillatory fixedpoints: an oscillatory-source (oscillatory-unstable-node) of the (2m − 1)th order, an oscillatory-sink (oscillatory-stable-node) of the (2m − 1)th order, an oscillatoryupper-saddle of the (2m)th order, and a oscillatory-lower saddle of the (2m − 1)th order. In a plane of (xk , xk+1 +xk ), the fixed-points and phase trajectory are presented in Fig. 3.4.

184

3 Global Stability of 1-D Discrete Systems a00

(2m-1)thorder

oSI

(2m-1)thorder

oSO

a1

a1

Variable, xk

Variable, xk

(a)

(b) a00

(2m)thorder

oUS

oLS

a1

a1

Variable, xk

Variable, xk

(c)

(d)

Fig. 3.4 The oscillatory stability of a single fixed-point in a one-fixed-point discrete system: a an oscillatory-sink of the (2m −1)th order (oSI) for a0 > 0, b an oscillatory-sink of the (2m −1)th order (oSO) for a0 < 0, c an oscillatory-sink upper-saddle of the (2m)th order (oUS) for a0 > 0, d an oscillatory-lower saddle of the (2m)th order (oLS) for a0 < 0. (m = 1, 2, . . .). oLS: oscillatorylower-saddle, oUS: oscillatory-upper-saddle, oSI: oscillatory-sink (oscillatory-stable node), oSO: oscillatory-source (monotonic unstable node) Table 3.2 The oscillatory stability of a one-fixed-point discrete system xk+1 + xk = a0 (xk − a1 )r xk∗ = a1 a0 > 0

a0 < 0

r = 2m − 1

(2m − 1)th oSI

(2m − 1)th oSO

r = 2m

(2m)th

(2m)th oLS

oUS

oLS: oscillatory lower-saddle, oUS: oscillatory-upper-saddle, oSI: oscillatory-sink, oSO: oscillatory source

3.4 Two-Fixed-Point Systems

185

3.4 Two-Fixed-Point Systems In this section, consider a discrete system with two-fixed-points. Such two fixedpoints can be simple or repeated.

3.4.1 Monotonic Two-Fixed-Point Systems Consider a 1-dimensional discrete system with two fixed-points as xk+1 = xk + g(xk , p) = xk + a0 (xk − a1 )r1 (xk − a2 )r2 , a0 = 0, ai ∈ R with a1 < a2 , for ri = 2m i − 1, 2m i , i ∈ {1, 2} and m i = 1, 2, . . . .

(3.52)

Thus, the monotonic stability can be determined by d xk+1 = 1 + a0 r1 (xk − a1 )r1 −1 (xk − a2 )r2 + a0 r2 (xk − a1 )r1 (xk − a2 )r2 −1 d xk (3.53) = 1 + a0 (r1 + r2 )(xk − a1 )r1 −1 (xk − a12 )(xk − a2 )r2 −1 where a12 =

r 1 a2 + r 2 a1 with a1 < a12 < a2 . r1 + r2

(3.54)

Using xk+1 − xk = g(xk , p), trajectories and fixed-points are presented. If a1 = a2 = a12 , the monotonic bifurcation point exists. The two-fixed-point system becomes one fixed-point system with the (r1 +r2 )th order singularity. This monotonic bifurcation generates two fixed-points with the r1th and r2th order singularity. For a0 > 0, the (2m 1 −1 : 2m 2 −1)-fixed-points (i.e., r1 = 2m 1 −1, r2 = 2m 2 −1) possess the (2m 1 − 1)th order and (2m 2 − 1)th order singularity for xk∗ = a1 and xk∗ = a2 , respectively. From xk+1 − xk = a0 (xk − a1 )2m 1 −1 (xk − a2 )2m 2 −1 , • (xk+1 − xk ) > 0 for xk > a2 , • (xk+1 − xk ) < 0 for a1 < xk < a2 , • (xk+1 − xk ) > 0 for xk < a1 . Thus, the fixed-point of xk∗ = a2 is a monotonic source (monotonically unstable node) of the (2m 2 − 1)th order (mSO) and the fixed-point of xk∗ = a1 is a monotonic sink (monotonically stable node) of the (2m 1 − 1)th order (mSI), as shown in Fig. 3.5a. For a0 > 0, the (2m 1 − 1 : 2m 2 )-fixed-points (i.e., r1 = 2m 1 − 1, r2 = 2m 2 ) possess the (2m 1 − 1)th order and (2m 2 )th order singularity for xk∗ = a1 and xk∗ = a2 , respectively. From xk+1 − xk = a0 (xk − a1 )2m 1 −1 (xk − a2 )2m 2 ,

186

3 Global Stability of 1-D Discrete Systems

(2m1-1)th order

(2m2-1)th order

mSI

mSO

a1

(2m1-1:2m2) Function, xk+1xk= g(xk,p)

Function, xk+1xk= g(xk,p)

(2m1-1:2m2-1)

a2

(2m1-1)th order

mUS

a1

a2

Variable, xk

Variable, xk

(a)

(b) (2m1:2m2)

(2m2-1)th order

(2m1)th order

mSO

mLS

a1

a2

Variable, xk

(c)

Function, xk+1xk= g(xk,p)

(2m1:2m2-1) Function, xk+1xk= g(xk,p)

(2m2)th order

mSO

(2m2)th order

(2m1)th order

mUS

mUS a1

a2

Variable, xk

(d)

Fig. 3.5 The Monotonic stability of two fixed-points (a0 > 0) in a two-fixed-point discrete systems: a ((2m 1 −1)th mSI: (2m 2 −1)th mSO), b ((2m 1 −1)th mSO: (2m 2 )th mUS), c ((2m 1 )th mLS:(2m 2 − 1)th mSO), d ((2m 1 )th mUS: (2m 2 )th mUS). mLS: monotonic lower-saddle, mUS: monotonicupper-saddle, mSI: monotonic sink (stable node), mSO: monotonic source (unstable node)

• (xk+1 − xk ) > 0 for xk > a2 , • (xk+1 − xk ) > 0 for a1 < xk < a2 , • (xk+1 − xk ) < 0 for xk < a1 . Thus, the fixed-point of xk∗ = a2 is a monotonic upper-saddle of the (2m 2 )th order (mUS) and the fixed-point of xk∗ = a1 is a monotonic source (monotonically unstable node) of the (2m 1 − 1)th order (mSO), as shown in Fig. 3.5b. For a0 > 0, the (2m 1 : 2m 2 − 1)-fixed-points (i.e., r1 = 2m 1 , r2 = 2m 2 − 1) possess the (2m 1 )th order and (2m 2 − 1)th order singularity for xk∗ = a1 and xk∗ = a2 , respectively. From xk+1 − xk = a0 (xk − a1 )2m 1 (xk − a2 )2m 2 −1 , • (xk+1 − xk ) > 0 for xk > a2 , • (xk+1 − xk ) < 0 for a1 < xk < a2 , • (xk+1 − xk ) < 0 for xk < a1 . Thus, the fixed-point of xk∗ = a2 is a monotonic source (monotonically unstable node) of the (2m 2 − 1)th order (mSO) and the fixed-point of xk∗ = a1 is a monotonic lower-saddle of the (2m 1 )th order (mLS), as shown in Fig. 3.5c.

3.4 Two-Fixed-Point Systems

187

Table 3.3 The monotonic stability of two fixed-points for a two-fixed-point system (xk+1 = xk + a0 (xk − a1 )r1 (xk − a2 )r2 , a0 > 0) (r1 : r2 )

xk∗ = a1

xk∗ = a2

(2m 1 − 1 : 2m 2 − 1)

(2m 1 − 1)th mSI

(2m 2 − 1)th mSO

(2m 1

− 1)th

(2m 1 − 1 : 2m 2 ) (2m 1 : 2m 2 − 1)

(2m 1

)th

(2m 1 : 2m 2 )

(2m 1 )th mUS

mSO

mLS

(2m 2 )th mUS (2m 2 − 1)th -mSO (2m 2 )th mUS

mLS: monotonic-lower-saddle, mUS: monotonic-upper-saddle, mSI: monotonic-sink, mSO: monotonic source

For a0 > 0, the (2m 1 : 2m 2 )-fixed-points (i.e., r1 = 2m 1 , r2 = 2m 2 ) possess the (2m 1 )th order and (2m 2 )th order singularity for xk∗ = a1 and xk∗ = a2 , respectively. From xk+1 − xk = a0 (xk − a1 )2m 1 (xk − a2 )2m 2 , • (xk+1 − xk ) > 0 for xk > a2 , • (xk+1 − xk ) > 0 for a1 < x < a2 , • (xk+1 − xk ) > 0 for x < a1 . Thus, as shown in Fig. 3.5d, the fixed-points of xk∗ = a1 and xk∗ = a2 are two monotonic upper-saddles of the (2m 1 )th order and (2m 2 )th order (mUS), respectively. The stability and higher-order singularity of the two-fixed-point system for a0 > 0 are summarized in Table 3.3. If the (2m i − 1)th order monotonic sink (i = 1, 2) is a monotonic sink of the first order (2m i − 1 = 1), the monotonic sink represents for monotonic-sink to oscillatory-source. Similarly, for a0 < 0, the stability and higherorder singularity of the two-fixed-point discrete system are discussed as follows. For a0 < 0, the (2m 1 −1 : 2m 2 −1)-fixed-points (i.e., r1 = 2m 1 −1, r2 = 2m 2 −1) possess the (2m 1 − 1)th order and (2m 2 − 1)th order singularity for xk∗ = a1 and xk∗ = a2 , respectively. From xk+1 − xk = a0 (xk − a1 )2m 1 −1 (xk − a2 )2m 2 −1 , • (xk+1 − xk ) < 0 for xk > a2 , • (xk+1 − xk ) > 0 for a1 < xk < a2 , • (xk+1 − xk ) < 0 for xk < a1 . Thus, the fixed-point of xk∗ = a2 is a monotonic sink (monotonically stable node) of the (2m 2 − 1)th order (mSI) and the fixed-point of xk∗ = a1 is a monotonic source (monotonically unstable node) of the (2m 1 −1)th order (mSO), as shown in Fig. 3.6a. For a0 < 0, the (2m 1 − 1 : 2m 2 )-fixed-points (i.e., r1 = 2m 1 − 1, r2 = 2m 2 ) possess the (2m 1 − 1)th order and (2m 2 )th order singularity for xk∗ = a1 and xk∗ = a2 , respectively. From xk+1 − xk = a0 (xk − a1 )2m 1 −1 (xk − a2 )2m 2 , • (xk+1 − xk ) < 0 for xk > a2 , • (xk+1 − xk ) < 0 for a1 < xk < a2 , • (xk+1 − xk ) > 0 for xk < a1 .

188

3 Global Stability of 1-D Discrete Systems (2m1-1:2m2)

(2m2-1)th order

(2m1-1)th order

mSO

mSI

Function, xk+1xk= g(xk,p)

Function, xk+1xk= g(xk,p)

(2m1-1:2m2-1)

(2m2)th order

mSI

mLS

a1

a2

a1

(2m1-1)th order

a2

Variable, xk

Variable, xk

(a)

(b) (2m1:2m2)

(2m1)th order

(2m2-1)th order

mSI

mUS

a2

a1

Function, xk+1xk= g(xk,p)

Function, xk+1xk= g(xk,p)

(2m1:2m2-1)

(2m2)th order

(2m1)th order

mLS

mLS

a2

a1

Variable, xk

Variable, xk

(c)

(d)

Fig. 3.6 The monotonic stability of two fixed-points (a0 < 0) in a two-fixed-point discrete systems. a ((2m 1 − 1)th mSO: (2m 2 − 1)th mSI), b ((2m 1 − 1)th mSI: (2m 2 )th mLS), c ((2m 1 )th mUS: (2m 2 −1)th mSI), d ((2m 1 )th mLS: (2m 2 )th mLS). mLS: monotonic-lower-saddle, mUS: monotonicupper-saddle, mSI: monotonic-sink, mSO: monotonic-source

Thus, the fixed-point of xk∗ = a2 is a monotonic lower-saddle of the (2m 2 )th order (mLS) and the fixed-point of xk∗ = a1 is a monotonic sink (monotonically stable node) of the (2m 1 − 1)th order (mSI), as shown in Fig. 3.6b. For a0 < 0, the (2m 1 : 2m 2 − 1)-fixed-points (i.e., r1 = 2m 1 , r2 = 2m 2 − 1) possess the (2m 1 )th order and (2m 2 − 1)th order singularity for xk∗ = a1 and xk∗ = a2 , respectively. From xk+1 − xk = a0 (xk − a1 )2m 1 (xk − a2 )2m 2 −1 , • (xk+1 − xk ) < 0 for xk > a2 , • (xk+1 − xk ) > 0 for a1 < xk < a2 , • (xk+1 − xk ) > 0 for xk < a1 . Thus, the fixed-point of xk∗ = a2 is a monotonic sink (monotonically stable node) of the (2m 2 − 1)th order (mSI) and the fixed-point of xk∗ = a1 is a monotonic upper-saddle of the (2m 1 )th order (mUS), as shown in Fig. 3.6c. For a0 < 0, the (2m 1 : 2m 2 )-fixed-points (i.e., r1 = 2m 1 , r2 = 2m 2 ) possess the (2m 1 )th order and (2m 2 )th order singularity for xk∗ = a1 and xk∗ = a2 , respectively. From xk+1 − xk = a0 (xk − a1 )2m 1 (xk − a2 )2m 2 ,

3.4 Two-Fixed-Point Systems

189

Table 3.4 The monotonic stability of two fixed-points for a two-fixed-point system (xk+1 = xk + a0 (xk − a1 )r1 (xk − a2 )r2 , a0 < 0) (r1 : r2 )

xk∗ = a1

xk∗ = a2

(2m 1 − 1 : 2m 2 − 1)

(2m 1 − 1)th mSO

(2m 2 − 1)th mSI

(2m 1

− 1)th

(2m 1 − 1 : 2m 2 ) (2m 1 : 2m 2 − 1)

(2m 1

)th

(2m 1 : 2m 2 )

(2m 1 )th mLS

mSI

mUS

(2m 2 )th mLS (2m 2 − 1)th mSI (2m 2 )th mLS

mLS: monotonic-lower-saddle, mUS: monotonic-upper-saddle, mSI: monotonic-sink, mSO: monotonic-source

• (xk+1 − xk ) < 0 for xk > a2 , • (xk+1 − xk ) < 0 for a1 < x < a2 , • (xk+1 − xk ) < 0 for x < a1 . Thus, as shown in Fig. 3.6d, the fixed-points of xk∗ = a1 and xk∗ = a2 are two monotonic lower-saddles of the (2m 1 )th order and (2m 2 )th order (mLS), respectively. For a0 < 0, the stability and higher-order singularity of two fixed-points of xk+1 = xk + a0 (xk − a1 )r1 (xk − a2 )r2 are summarized in Table 3.4. If (2m i − 1)th order monotonic sink (i = 1, 2) is a monotonic sink of the first order (2m i − 1 = 1), the monotonic sink represents for monotonic-sink to oscillatory-source. The odd order singularity is relative to the monotonically stable and unstable nodes (i.e., monotonic sink and monotonic source). The even order singularity is relative to the monotonic upper- and lower-saddles.

3.4.2 Oscillatory Two-Fixed-Point Systems Consider a 1-dimensional system with two oscillatory fixed-points as xk+1 = −xk + g(xk , p) = −xk + a0 (xk − a1 )r1 (xk − a2 )r2 , a0 = 0, ai ∈ R with a1 < a2 , for ri = 2m i − 1, 2m i , i ∈ {1, 2} and m i = 1, 2, . . . .

(3.55)

Thus the stability can be determined by d xk+1 = −1 + a0 r1 (xk − a1 )r1 −1 (xk − a2 )r2 + a0 r2 (xk − a1 )r1 (xk − a2 )r2 −1 d xk = −1 + a0 (r1 + r2 )(xk − a1 )r1 −1 (xk − a12 )(xk − a2 )r2 −1 . (3.56)

190

3 Global Stability of 1-D Discrete Systems

Using xk+1 + xk = g(xk , p), trajectories and fixed-points are presented. If a1 = a2 = a12 , the oscillatory bifurcation point exists. The two-fixed-point system becomes one fixed-point system with the (r1 +r2 )th order singularity. This oscillatory bifurcation generates two fixed-points with the r1th and r2th order singularity. For a0 > 0, the (2m 1 −1 : 2m 2 −1)-fixed-points (i.e., r1 = 2m 1 −1, r2 = 2m 2 −1) possess the (2m 1 − 1)th order and (2m 2 − 1)th order singularity for xk∗ = a1 and xk∗ = a2 , respectively. From xk+1 + xk = a0 (xk − a1 )2m 1 −1 (xk − a2 )2m 2 −1 , • (xk+1 + xk ) > 0 for xk > a2 , • (xk+1 + xk ) < 0 for a1 < xk < a2 , • (xk+1 + xk ) > 0 for xk < a1 . Thus, the fixed-point of xk∗ = a2 is an oscillatory sink (oscillatorilly stable node) of the (2m 2 − 1)th order (oSI) and the fixed-point of xk∗ = a1 is an oscillatory source (oscillatory unstable node) of (2m 1 − 1)th order (oSO), as shown in Fig. 3.7a. For a0 > 0, the (2m 1 − 1 : 2m 2 )-fixed-points (i.e., r1 = 2m 1 − 1, r2 = 2m 2 ) possess the (2m 1 − 1)th order and (2m 2 )th order singularity for xk∗ = a1 and xk∗ = a2 , respectively. From xk+1 + xk = a0 (xk − a1 )2m 1 −1 (xk − a2 )2m 2 ,

(2m1-1)th order

(2m2-1)th order

oSO

oSI

a1

(2m1-1:2m2) Function, xk+1xk= g(xk,p)

Function, xk+1xk= g(xk,p)

(2m1-1:2m2-1)

a2

(2m1-1)th order

oUS

a1

a2

Variable, xk

Variable, xk

(a)

(b) (2m1:2m2)

(2m2-1)th order

(2m1)th order

oSI

oLS

a1

a2

Variable, xk

(c)

Function, xk+1xk= g(xk,p)

(2m1:2m2-1) Function, xk+1+xk= g(xk,p)

(2m2)th order

oSI

(2m2)th order

(2m1)th order

oUS

oUS a1

a2

Variable, xk

(d)

Fig. 3.7 The oscillatory stability of two fixed-points (a0 > 0) in a two-fixed-point discrete systems. a ((2m 1 −1)th oSO: (2m 2 −1)th oSI), b ((2m 1 −1)th oSI: (2m 2 )th oUS), c ((2m 1 )th oLS: (2m 2 −1)th oSI), d ((2m 1 )th oUS: (2m 2 )th oUS). oLS: oscillatory-lower-saddle, oUS: oscillatory-upper-saddle, oSI: oscillatory sink (stable node), oSO: oscillatory source (unstable node)

3.4 Two-Fixed-Point Systems

191

• (xk+1 + xk ) > 0 for xk > a2 , • (xk+1 + xk ) > 0 for a1 < xk < a2 , • (xk+1 + xk ) < 0 for xk < a1 . Thus, the fixed-point of xk∗ = a2 is an oscillatory upper-saddle of the (2m 2 )th order (oUS) and the fixed-point of xk∗ = a1 is an oscillatory sink (oscillatorilly stable node) of the (2m 1 − 1)th order (oSI), as shown in Fig. 3.7b. For a0 > 0, the (2m 1 : 2m 2 − 1)-fixed-points (i.e., r1 = 2m 1 , r2 = 2m 2 − 1) possess the (2m 1 )th order and (2m 2 − 1)th order singularity for xk∗ = a1 and xk∗ = a2 , respectively. From xk+1 + xk = a0 (xk − a1 )2m 1 (xk − a2 )2m 2 −1 , • (xk+1 + xk ) > 0 for xk > a2 , • (xk+1 + xk ) < 0 for a1 < xk < a2 , • (xk+1 + xk ) < 0 for xk < a1 . Thus, the fixed-point of xk∗ = a2 is an oscillatory sink (oscillatorilly stable node) of the (2m 2 − 1)th order (oSI) and the fixed-point of xk∗ = a1 is an oscillatory lower-saddle of the (2m 1 )th order (oLS), as shown in Fig. 3.7c. For a0 > 0, the (2m 1 : 2m 2 )-fixed-points (i.e., r1 = 2m 1 , r2 = 2m 2 ) possess the (2m 1 )th order and (2m 2 )th order singularity for xk∗ = a1 and xk∗ = a2 , respectively. From xk+1 + xk = a0 (xk − a1 )2m 1 (xk − a2 )2m 2 , • (xk+1 + xk ) > 0 for xk > a2 , • (xk+1 + xk ) > 0 for a1 < x < a2 , • (xk+1 + xk ) > 0 for x < a1 . Thus, as shown in Fig. 3.7d, the fixed-points of xk∗ = a1 and xk∗ = a2 are two oscillatory upper-saddles of the (2m 1 )th order and (2m 2 )th order (oUS), respectively. The stability and higher-order singularity of the two-fixed-point system for a0 > 0 are summarized in Table 3.5. If (2m i − 1)th order oscillatory sink (i = 1, 2) is an oscillatory sink of the first order (2m i − 1 = 1), the oscillatory sink represents for an oscillatory-sink to monotonic-source. Similarly, for a0 < 0, the stability and higher-order singularity of the two-fixed-point discrete system are discussed as follows. Table 3.5 The oscillatory stability of two fixed-points for a two-fixed-point system (xk+1 = −xk + a0 (xk − a1 )r1 (xk − a2 )r2 , a0 > 0) (r1 : r2 )

xk∗ = a1

xk∗ = a2

(2m 1 − 1 : 2m 2 − 1)

(2m 1

(2m 1 − 1 : 2m 2 )

(2m 1 − 1)th oSI

(2m 1 : 2m 2 − 1) (2m 1 : 2m 2 )

− 1)th

oSO

(2m 2 − 1)th oSI (2m 2 )th oUS

(2m 1

)th

oLS

(2m 2 − 1)th -oSI

(2m 1

)th

oUS

(2m 2 )th oUS

oLS: oscillatory-lower-saddle, oUS: oscillatory-upper-saddle, oSI: oscillatory-sink, oSO: oscillatory-source

192

3 Global Stability of 1-D Discrete Systems

For a0 < 0, the (2m 1 −1 : 2m 2 −1)-fixed-points (i.e., r1 = 2m 1 −1, r2 = 2m 2 −1) possess the (2m 1 − 1)th order and (2m 2 − 1)th order singularity for xk∗ = a1 and xk∗ = a2 , respectively. From xk+1 + xk = a0 (xk − a1 )2m 1 −1 (xk − a2 )2m 2 −1 , • (xk+1 + xk ) < 0 for xk > a2 , • (xk+1 + xk ) > 0 for a1 < xk < a2 , • (xk+1 + xk ) < 0 for xk < a1 . Thus, the fixed-point of xk∗ = a2 is an oscillatory source (oscillatory unstable node) of the (2m 2 − 1)th order (oSO) and the fixed-point of xk∗ = a1 is an oscillatory sink (oscillatory stable node) of (2m 1 − 1)th order (oSI), as shown in Fig. 3.8a. For a0 < 0, the (2m 1 − 1 : 2m 2 )-fixed-points (i.e., r1 = 2m 1 − 1, r2 = 2m 2 ) possess the (2m 1 − 1)th order and (2m 2 )th order singularity for xk∗ = a1 and xk∗ = a2 , respectively. From xk+1 + xk = a0 (xk − a1 )2m 1 −1 (xk − a2 )2m 2 , • (xk+1 + xk ) < 0 for xk > a2 , • (xk+1 + xk ) < 0 for a1 < xk < a2 , • (xk+1 + xk ) > 0 for xk < a1 .

(2m2-1)th order

(2m1-1)th order

oSO

oSI

(2m1-1:2m2) Function, xk+1xk= g(xk,p)

Function, xk+1xk= g(xk,p)

(2m1-1:2m2-1)

(2m2)th order

oSO

oLS

a1

a2

a1

(2m1-1)th order

a2

Variable, xk

Variable, xk

(a)

(b) (2m1:2m2)

(2m1)th order

(2m2-1)th order

oUS

oSO

a2

a1

Variable, xk

(c)

Function, xk+1xk= g(xk,p)

Function, xk+1xk= g(xk,p)

(2m1:2m2-1)

(2m2)th order

(2m1)th order

oLS

oLS

a2

a1

Variable, xk

(d)

Fig. 3.8 The oscillatory stability of two fixed-points (a0 < 0) in a two-fixed-point discrete systems: a ((2m 1 −1)th oSI: (2m 2 −1)th oSO), b ((2m 1 −1)th oSO:(2m 2 )th oLS), c ((2m 1 )th oUS: (2m 2 −1)th oSO), d ((2m 1 )th oLS: (2m 2 )th oLS). oLS: oscillatory-lower-saddle, oUS: oscillatory-upper-saddle, oSI: oscillatory-sink, oSO: oscillatory-source

3.4 Two-Fixed-Point Systems

193

Thus, the fixed-point of xk∗ = a2 is an oscillatory lower-saddle of the (2m 2 )th order (oLS) and the fixed-point of xk∗ = a1 is an oscillatory source (oscillatorilly unstable node) of the (2m 1 − 1)th order (oSO), as shown in Fig. 3.8b. For a0 < 0, the (2m 1 : 2m 2 − 1)-fixed-points (i.e., r1 = 2m 1 , r2 = 2m 2 − 1) possess the (2m 1 )th order and (2m 2 − 1)th order singularity for xk∗ = a1 and xk∗ = a2 , respectively. From xk+1 + xk = a0 (xk − a1 )2m 1 (xk − a2 )2m 2 −1 , • (xk+1 + xk ) < 0 for xk > a2 , • (xk+1 + xk ) > 0 for a1 < xk < a2 , • (xk+1 + xk ) > 0 for xk < a1 . Thus, the fixed-point of xk∗ = a2 is an oscillatory source (oscillatorilly unstable node) of the (2m 2 − 1)th order (oSO) and the fixed-point of xk∗ = a1 is a monotonic upper-saddle of the (2m 1 )th order (oUS), as shown in Fig. 3.8c. For a0 < 0, the (2m 1 : 2m 2 )-fixed-points (i.e., r1 = 2m 1 , r2 = 2m 2 ) possess the (2m 1 )th order and (2m 2 )th order singularity for xk∗ = a1 and xk∗ = a2 , respectively. From xk+1 + xk = a0 (xk − a1 )2m 1 (xk − a2 )2m 2 , • (xk+1 + xk ) < 0 for xk > a2 , • (xk+1 + xk ) < 0 for a1 < x < a2 , • (xk+1 + xk ) < 0 for x < a1 . Thus, as shown in Fig. 3.6d, the fixed-points of xk∗ = a1 and xk∗ = a2 are two oscillatory lower-saddles of the (2m 1 )th order and (2m 2 )th order (oLS), respectively. For a0 < 0, the stability and higher-order singularity of two fixed-points of xk+1 = −xk + a0 (xk − a1 )r1 (xk − a2 )r2 are summarized in Table 3.6. If (2m i − 1)th order oscillatory sink (i = 1, 2) is an oscillatory sink of the first order (2m i −1 = 1), the oscillatory sink represents for oscillatory-sink to monotonic-source. The odd order singularity is relative to the oscillatorilly stable and unstable nodes (i.e., sink and source). The even order singularity is relative to the oscillatorilly upper- and lower-saddles. Table 3.6 The oscillatory stability of two fixed-points for a two-fixed-point system (xk+1 = −xk + a0 (xk − a1 )r1 (xk − a2 )r2 , a0 < 0) (r1 : r2 )

xk∗ = a1

xk∗ = a2

(2m 1 − 1 : 2m 2 − 1)

(2m 1 − 1)th oSI

(2m 2 − 1)th oSO

(2m 1

− 1)th

(2m 1 − 1 : 2m 2 ) (2m 1 : 2m 2 − 1)

(2m 1

)th

(2m 1 : 2m 2 )

(2m 1 )th oLS

oSO

oUS

(2m 2 )th oLS (2m 2 − 1)th oSO (2m 2 )th oLS

oLS: oscillatory-lower-saddle, oUS: oscillatory-upper-saddle, oSI: oscillatory-sink, oSO: oscillatory-source

194

3 Global Stability of 1-D Discrete Systems

3.5 Three-Fixed-Point Systems In this section, consider a discrete system with three-fixed-points. Such three fixedpoints can be simple or repeated.

3.5.1 Monotonic Three-Fixed-Point Systems Consider a 1-dimensional discrete system with three fixed-points as xk+1 = xk + g(xk , p) = xk + a0 (xk − a1 )r1 (xk − a2 )r2 (xk − a3 )r2 , a0 = 0, ai ∈ R with a1 < a2 < a3 , for ri = 2m i − 1, 2m i , i ∈ {1, 2, 3} and m i = 1, 2, . . . .

(3.57)

Thus d xk+1 = 1 + a0 (r1 + r2 + r3 )(xk2 + a12 xk + a123 ) d xk × (xk − a1 )r1 −1 (xk − a2 )r2 −1 (x − a3 )r3 −1

(3.58)

r1 (a2 + a3 ) + r2 (a1 + a3 ) + r3 (a1 + a2 ) , r1 + r2 + r3 r 1 a2 a3 + r 2 a1 a3 + r 3 a1 a2 = . r1 + r2 + r3

(3.59)

where a12 = − a123

Except for fixed-points, the existence of the three-fixed-point discrete system in Eq. (3.57) requires two points to make d xk+1 /d xk = 1, i.e., xk2 + a12 xk + a123 = 0.

(3.60)

Thus, the existence condition for such three-fixed-point discrete systems is 2 − 4a123 > 0.  = a12

(3.61)

with b1,2 =

a12 √ ∓ , a1 < b1 < a2 < b2 < a3 . 2

(3.62)

3.5 Three-Fixed-Point Systems

195

Table 3.7 The monotonic stability of three fixed-points for a three-fixed-point system (xk+1 = xk + a0 (xk − a1 )r1 (xk − a2 )r2 (xk − a3 )r3 , a0 > 0) (r1 : r2 : r3 )

x ∗ = a1

x ∗ = a2

− 1)th

(2m 1 − 1 : 2m 2 − 1 : 2m 3 − 1)

(2m 1

(2m 1 : 2m 2 − 1 : 2m 3 − 1)

(2m 1 )th mUS

(2m 1 − 1 : 2m 2 : 2m 3 − 1)

(2m 1

(2m 1 − 1 : 2m 2 − 1 : 2m 3 )

(2m 1 − 1)th mSI

(2m 1 : 2m 2 : 2m 3 − 1)

− 1)th

mSO mSI

(2m 1

)th

(2m 1 : 2m 2 − 1 : 2m 3 )

(2m 1

)th

(2m 1 − 1 : 2m 2 : 2m 3 )

(2m 1 − 1)th mSO

(2m 1 : 2m 2 : 2m 3 )

(2m 1

)th

mLS mLS mUS

(2m 2

x ∗ = a3 mSI

(2m 3 − 1)th mSO

(2m 2 − 1)th mSI

(2m 3 − 1)th mSO

(2m 2

− 1)th )th

mLS

(2m 2 − 1)th mSO (2m 2

)th

(2m 2

− 1)th

mLS mSO

(2m 3 − 1)th mSO (2m 3 )th mUS (2m 3 − 1)th mSO (2m 3 )th mUS

(2m 2 )th mUS

(2m 3 )th mUS

)th

(2m 3 )th mUS

(2m 2

mUS

mLS: monotonic-lower-saddle, US: monotonic upper-saddle, mSI: monotonic-sink, mSO: monotonic-source

If b1 = a1 = a2 , the fixed-point is a monotonic bifurcation point with the (r1 +r2 )th order singularity. If b2 = a2 = a3 , the fixed-point is a monotonic bifurcation point with the (r2 + r3 )th order singularity. If b1 = b2 = a2 , the fixed-point of xk∗ = a2 is a bifurcation point with the r2th order singularity from the three-fixed-point systems become a two-fixed-point system. If b1 = b2 = a1 = a2 = a3 , the fixed-point is a monotonic bifurcation point with the (r1 + r2 + r3 )th order singularity from the three-fixed-point systems become a one-fixed-point system. With singularity, the fixed-point stability and singularity for the general cases of the three-fixed-point systems are tabulated in Table 3.7 for a0 > 0. The odd number order singularity is for monotonic sink (monotonically stable node) and monotonic source (monotonically unstable node). The even number order singularity is for lower-saddles and upper-saddles. The sampled monotonic stability and singularity of three fixed-points in the three-fixed-point system are presented in Fig. 3.9 for a0 > 0, which are summarized in Table 3.8. The fixed-point stability and singularity for the general cases of the three-fixedpoint systems are tabulated in Table 3.9 for a0 < 0. The odd number order singularity is for monotonic sink (monotonically stable node) and monotonic source (monotonically unstable node). The even number order singularity is for monotonic lowersaddles and monotonic upper-saddles. The sampled monotonic stability and singularity of three-fixed-points in the three-fixed-point discrete system are presented for a0 < 0 in Fig. 3.10, which are summarized in Table 3.10.

196

3 Global Stability of 1-D Discrete Systems (2:1:1)

1st order

1st order

mSO

a1

1st order

mSO

mSI-oSO

a2

Function, xk+1xk= g(xk,p)

Function, xk+1xk= g(xk,p)

(1:1:1)

a3

mUS

mSI-oSO

2nd order

1st order

a1

a2

1st order

a3

Variable, xk

Variable, xk

(i)

(ii) (1:1:2)

1st order

2nd order

mLS

a2

a1

1st order

mSO

Function, xk+1xk= g(xk,p)

Function, xk+1xk= g(xk,p)

(1:2:1)

mSI-oSO

mSI-oSO

1st order

a1

a3

a2

a3

(2:1:2)

2nd order

mLS

a2

Variable, xk

(v)

1st order

mSO

a3

Function, xk+1xk= g(xk,p)

Function, xk+1xk= g(xk,p)

2nd order

(iv)

(2:2:1)

a1

mUS

Variable, xk

(iii)

mLS

mSO

1st order

Variable, xk

2nd order

mSO

2nd order

1st order

2st order

a2

a3

mLS

a1

mUS

mSO

Variable, xk

(vi)

Fig. 3.9 The sampled monotonic stability of three-fixed-points in a three-fixed-point discrete systems (a0 > 0): i (1:1:1), ii (2:1:1), iii (1:2:1), iv (1:1:2), v (2:2:1), vi (2:1:2), vii (1:2:2), viii (3:1:1), ix (1:3:1), x (1:1:3). mLS: monotonic-lower-saddle, mUS: monotonic-upper-saddle, mSI: monotonic-sink, mSO: monotonic-source

3.5 Three-Fixed-Point Systems

197 (3:1:1)

mUS

mUS

mSO 1st order

2nd order

2nd order

a1

a2

a3

Function, xk+1xk= g(xk,p)

Function, xk+1xk= g(xk,p)

(1:2:2)

3rd order

mSO

a1

1st order

mSI-oSO

a2

Variable, xk

Variable, xk

(vii)

(viii)

a1

mSO

mSI

a2

Function, xk+1xk= g(xk,p)

Function, xk+1xk= g(xk,p)

1st order

3rd order

mSO

a3

(1:1:3)

(1:3:1)

1st order

1st order

mSO

1st order

1st order

mSO

a3

a1

mSI

a2

Variable, xk

Variable, xk

(ix)

(x)

3rd order

mSO

a3

Fig. 3.9 (continued) Table 3.8 The sampled monotonic stability of three fixed-points for a three-fixed-point system (xk+1 = xk + a0 (xk − a1 )r1 (xk − a2 )r2 (xk − a3 )r3 , a0 > 0) (r1 : r2 : r3 ).

x ∗ = a1

x ∗ = a2

x ∗ = a3

(1:1:1)

mSO

mSI-oSO

mSO

(2:1:1)

2nd mUS

mSI-oSO

mSO

(1:2:1)

mSI-oSO

2nd mLS

mSO

(1:1:2)

mSI-oSO

mSO

2nd mUS

(2:2:1)

2nd

2nd

(2:1:2)

2nd mLS

mSO

2nd mUS

(1:2:2)

mSO

2nd mUS

2nd mUS

(3:1:1)

3rd

mSI-oSO

mSO

mLS

mSO

mLS

(1:3:1)

mSO

3rd

(1:1:3)

mSO

mSI-oSO

mSI

mSO

mSO 3rd mSO

mLS: monotonic-lower-saddle, mUS: monotonic-upper-saddle, mSI: monotonic-sink, mSO: monotonic source, mSI-oSO: monotonic sink to oscillatory source

198

3 Global Stability of 1-D Discrete Systems

Table 3.9 The monotonic stability of three fixed-points for a three-fixed-point system (xk+1 = xk + a0 (xk − a1 )r1 (xk − a2 )r2 (xk − a3 )r3 , a0 < 0) x ∗ = a1

(r1 : r2 : r3 ) (2m 1 − 1 : 2m 2 − 1 : 2m 3 − 1) (2m 1 : 2m 2 − 1 : 2m 3 − 1) (2m 1 − 1 : 2m 2 : 2m 3 − 1) (2m 1 − 1 : 2m 2 − 1 : 2m 3 ) (2m 1 : 2m 2 : 2m 3 − 1) (2m 1 : 2m 2 − 1 : 2m 3 ) (2m 1 − 1 : 2m 2 : 2m 3 ) (2m 1 : 2m 2 : 2m 3 )

x ∗ = a2

(2m 1

− 1)th

(2m 1

)th

(2m 1

− 1)th

(2m 1

− 1)th

(2m 1

)th

(2m 1

)th

(2m 1

− 1)th

(2m 1

)th

mSI

mLS mSO mSO

mUS mUS mSI

mLS

x ∗ = a3

(2m 2

− 1)th

mSO

(2m 3 − 1)th mSI

(2m 2

− 1)th

mSO

(2m 3 − 1)th mSI

(2m 2

)th

(2m 2

− 1)th

(2m 2

)th

(2m 2

− 1)th

(2m 2

)th

mLS

(2m 3 )th mLS

(2m 2

)th

mLS

(2m 3 )th mLS

mUS mSI

mUS mSI

(2m 3 − 1)th mSI (2m 3 )th mLS (2m 3 − 1)th mSI (2m 3 )th mLS

mLS: monotonic-lower-saddle, US: monotonic upper-saddle, mSI: monotonic-sink, mSO: monotonic-source

3.5.2 Oscillatory Three-Fixed-Point Systems Consider a 1-dimensional discrete system with three oscillatory fixed-points as xk+1 = −xk + g(xk , p) = −xk + a0 (xk − a1 )r1 (xk − a2 )r2 (xk − a3 )r2 , a0 = 0, ai ∈ R with a1 < a2 < a3 , for ri = 2m i − 1, 2m i , i ∈ {1, 2, 3} and m i = 1, 2, . . . .

(3.63)

Thus d xk+1 = −1 + a0 (r1 + r2 + r3 )(xk2 + a12 xk + a123 ) d xk × (xk − a1 )r1 −1 (xk − a2 )r2 −1 (x − a3 )r3 −1 .

(3.64)

The fixed-point stability and singularity for the general cases of the three-fixedpoint discrete systems are tabulated in Table 3.11 for a0 > 0. The odd number order singularity is for oscillatory sink (oscillatorilly stable node) and oscillatory source (oscillatorilly unstable node). The even number order singularity is for oscillatory lower-saddles and oscillatory upper-saddles. The sampled oscillatory stability and singularity of fixed-points for the three-fixed-point systems are presented in Fig. 3.11 for a0 > 0, which are summarized in Table 3.12. The oscillatory stability and singularity of the fixed point for the general cases of the three-fixed-point systems are tabulated in Table 3.13 for a0 < 0. The odd number order singularity is for oscillatory sinks (oscillatorilly stable nodes) and oscillatory sources (oscillatorilly unstable nodes). The even number order singularity is for oscillatory lower-saddles and oscillatory upper-saddles. The sampled oscillatory stability and singularity of three fixed-points for a0 < 0 are presented in Fig. 3.12, which are summarized in Table 3.14.

3.5 Three-Fixed-Point Systems

199

1st order

1st order

mSO

mSI-oSO

a1

a2

1st order

mSI-oSO

(2:1:1) Function, xk+1xk= g(xk,p)

Function, xk+1xk= g(xk,p)

(1:1:1)

2nd order

a3

1st order

mLS

mSO

a1

a2

Variable, xk

Variable, xk

(i)

(ii)

mUS

mSO 1st order

2nd order

a1

a2

mSI-oSO 1st order

1st order

1st order

mSO

mSI-oSO

a2

a1

a3

Variable, xk

Variable, xk

(iii)

(iv)

a1

mUS

mSI-oSO 1st order

2nd order

a2

a3

a3

(2:1:2) Function, xk+1xk= g(xk,p)

Function, xk+1xk= g(xk,p)

mUS

2nd order

mLS

(2:2:1)

2nd order

a3

(1:1:2) Function, xk+1xk= g(xk,p)

Function, xk+1xk= g(xk,p)

(1:2:1)

1st order

mSI-oSO

mUS

1st order

2nd order

mSI-oSO

2nd order

a1

a2

Variable, xk

Variable, xk

(v)

(vi)

mLS

a3

Fig. 3.10 The sampled monotonic stability of three-fixed-points in a three-fixed-point discrete systems (a0 < 0): i (1:1:1), ii (2:1:1), iii (1:2:1), iv (1:1:2), v (2:2:1), vi (2:1:2), vii (1:2:2), viii (3:1:1), ix (1:3:1), x (1:1:3). mLS: monotonic-lower-saddle, mUS: monotonic-upper-saddle, mSI: monotonic-sink, mSO: monotonic-source

200

3 Global Stability of 1-D Discrete Systems

1st order

2nd order

2nd order

mSI-oSO

mLS

mLS

a1

(3:1:1)

Function, xk+1xk= g(xk,p)

Function, xk+1xk= g(xk,p)

(1:2:2)

3rd order

mSI-oSO

a2

a1

a3

a2

1st order

1st order

mSO

mSI-oSO

Variable, xk

Variable, xk

(vii)

(viii)

a3

1st order

mSI-oSO

1st order

3rd order

mSO

a1

mSI-oSO

a2

(1:1:3) Function, xk+1xk= g(xk,p)

Function, xk+1xk= g(xk,p)

(1:3:1) mSI-oSO 1st order

a3

a1

mSO 1st order

mSI 3rd order

a2

Variable, xk

Variable, xk

(ix)

(x)

a3

Fig. 3.10 (continued) Table 3.10 The sampled monotonic stability of three fixed-points for a three-fixed-point discrete system (xk+1 = xk + a0 (xk − a1 )r1 (xk − a2 )r2 (xk − a3 )r3 , a0 < 0) (r1 : r2 : r3 )

x ∗ = a1

x ∗ = a2

x ∗ = a3

(1:1:1)

mSI-oSO

mSO

mSI-oSO

(2:1:1)

2nd mLS

mSO

mSI-oSO

(1:2:1)

mSO

2nd

mUS

mSI-oSO

(1:1:2)

mSO

mSI-oSO

2nd mLS

(2:2:1)

2nd mUS

2nd mUS

mSI-oSO

(2:1:2)

2nd

mSI-oSO

2nd mLS 2nd mLS

mUS

(1:2:2)

mSI-oSO

2nd

(3:1:1)

3rd mSI

mSO

(1:3:1)

mSI-oSO

3rd

(1:1:3)

mSI-oSO

mSO

mLS mSO

mSI-oSO mSI-oSO 3rd mSI

mLS: monotonic-lower-saddle, mUS: monotonic-upper-saddle, mSI: monotonic-sink, mSO: monotonic source, mSI-oSO: monotonic sink to oscillatory source

3.5 Three-Fixed-Point Systems

201

Table 3.11 The oscillatory stability of three fixed-points for a three-fixed-point system (xk+1 = xk + a0 (xk − a1 )r1 (xk − a2 )r2 (xk − a3 )r3 , a0 > 0) x ∗ = a1

(r1 : r2 : r3 )

x ∗ = a2

− 1)th

(2m 1 − 1 : 2m 2 − 1 : 2m 3 − 1)

(2m 1

(2m 1 : 2m 2 − 1 : 2m 3 − 1)

(2m 1 )th oUS

(2m 1 − 1 : 2m 2 : 2m 3 − 1)

(2m 1

(2m 1 − 1 : 2m 2 − 1 : 2m 3 )

(2m 1 − 1)th oSO

(2m 1 : 2m 2 : 2m 3 − 1)

− 1)th

oSI oSO

(2m 1

)th

(2m 1 : 2m 2 − 1 : 2m 3 )

(2m 1

)th

(2m 1 − 1 : 2m 2 : 2m 3 )

(2m 1 − 1)th oSI

(2m 1 : 2m 2 : 2m 3 )

(2m 1

)th

oLS oLS

(2m 2

oSO

(2m 3 − 1)th oSI

(2m 2 − 1)th oSO

(2m 3 − 1)th oSI

(2m 2

)th

(2m 3 − 1)th oSI

oLS

(2m 2 − 1)th oSI (2m 2

)th

(2m 2

− 1)th

(2m 3 )th oLS (2m 3 − 1)th oSI

oLS

(2m 3 )th oUS

oSI

(2m 2 )th oUS

(2m 3 )th oUS

)th

(2m 3 )th oUS

(2m 2

oUS

x ∗ = a3

− 1)th

oUS

oLS: oscillatory-lower-saddle, oUS: oscillatory-upper-saddle, mSI: oscillatory-sink, oSO: oscillatory-source (2:1:1)

1st order

1st order

oSI-mSO

1st order

oSI-mSO

oSO

a2

a1

Function, xk+1xk= g(xk,p)

Function, xk+1xk= g(xk,p)

(1:1:1)

a3

oUS

oSO

2nd order

1st order

a1

a2

1st order

a3

Variable, xk

Variable, xk

(i)

(ii) (1:1:2)

oSO

a1

2nd order

oLS

1st order

oSI-mSO

a2

Variable, xk

(iii)

a3

Function, xk+1xk= g(xk,p)

Function, xk+1xk= g(xk,p)

(1:2:1) 1st order

oSI-mSO

oSO

oSI-mSO 1st order

1st order

a1

a2

oUS 2nd order

a3

Variable, xk

(iv)

Fig. 3.11 The sampled oscillatory stability of three fixed-points for a three-fixed-point discrete system (a0 > 0): i (1:1:1), ii (2:1:1), iii (1:2:1), iv (1:1:2), v (2:2:1), vi (2:1:2), vii (1:2:2), viii (3:1:1), ix (1:3:1), x (1:1:3). oLS: oscillatory-lower-saddle, oUS: oscillatory-upper-saddle, oSI: oscillatorysink, oSO: oscillatory-source

202

3 Global Stability of 1-D Discrete Systems (2:1:2)

2nd order

2nd order

oLS

oLS

a1

1st order

oSI-mSO

Function, xk+1xk= g(xk,p)

Function, xk+1xk= g(xk,p)

(2:2:1)

2nd order

a3

a2

a1

Variable, xk

1st order

2st order

a2

a3

Variable, xk

(v)

(vi) (3:1:1)

oSI-mSO

oUS

oUS

1st order

2nd order

2nd order

a1

a2

a3

Function, xk+1xk= g(xk,p)

(1:2:2) Function, xk+1xk= g(xk,p)

oUS

oSI-mSO

oLS

3rd order

1st order

oSO

oSI

a1

Variable, xk

a2

oSI-mSO

a3

Variable, xk

(vii)

(viii) (1:1:3)

1st order

oSI-mSO

3rd order

1st order

oSO

oSI-mSO

a2

a1

Variable, xk

(ix) Fig. 3.11 (continued)

a3

Function, xk+1xk= g(xk,p)

(1:3:1) Function, xk+1xk= g(xk,p)

1st order

1st order

1st order

oSI-mSO

a1

oSO

a2

Variable, xk

(x)

3rd order

oSI

a3

3.5 Three-Fixed-Point Systems

203

Table 3.12 The sampled oscillatory stability of three fixed-points for a three-fixed-point discrete system (xk+1 = xk + a0 (xk − a1 )r1 (xk − a2 )r2 (xk − a3 )r3 , a0 > 0) (r1 : r2 : r3 )

x ∗ = a1

x ∗ = a2

x ∗ = a3

(1:1:1)

oSI-mSO

oSO

oSI-mSO

(2:1:1)

2nd oUS

oSO

oSI-mSO

(1:2:1)

oSO

2nd

(1:1:2)

oSO

oSI-mSO

2nd oUS

(2:2:1)

2nd oLS

2nd oLS

oSI-mSO

(2:1:2)

2nd

oSI-mSO

2nd oUS 2nd oUS

oLS

oLS

(1:2:2)

oSI-mSO

2nd

(3:1:1)

3rd oSI

oSO

(1:3:1)

oSI-mSO

3rd

(1:1:3)

oSO

oSI-oSO

oSI-mSO

oUS

oSO-mSO

oSO

oSI-mSO 3rd oSO

oLS: oscillatory-lower-saddle, oUS: oscillatory-upper-saddle, mSI: oscillatory-sink, oSO: oscillatory-source, oSI-mSO: oscillatory sink to monotonic-source Table 3.13 The oscillatory stability of three fixed-points for a three-fixed-point system (xk+1 = xk + a0 (xk − a1 )r1 (xk − a2 )r2 (xk − a3 )r3 , a0 < 0) (r1 : r2 : r3 )

x ∗ = a1

x ∗ = a2

x ∗ = a3

(2m 1 − 1 : 2m 2 − 1 : 2m 3 − 1)

(2m 1 − 1)th oSO

(2m 2 − 1)th oSI

(2m 3 − 1)th oSO

(2m 1

)th

(2m 1 : 2m 2 − 1 : 2m 3 − 1) (2m 1 − 1 : 2m 2 : 2m 3 − 1)

(2m 1

− 1)th

(2m 1 − 1 : 2m 2 − 1 : 2m 3 )

(2m 1 − 1)th oSI

(2m 1 : 2m 2 : 2m 3 − 1)

oLS oSI

(2m 1

)th

(2m 1 : 2m 2 − 1 : 2m 3 )

(2m 1

)th

(2m 1 − 1 : 2m 2 : 2m 3 )

(2m 1 − 1)th oSO

(2m 1 : 2m 2 : 2m 3 )

(2m 1

)th

oUS oUS oLS

(2m 2

− 1)th

(2m 2

)th

oSI

oUS

(2m 2 − 1)th oSO (2m 2

)th

(2m 2

− 1)th

oUS oSO

(2m 3 − 1)th oSO (2m 3 − 1)th oSO (2m 3 )th oUS (2m 3 − 1)th oSO (2m 3 )th oLS

(2m 2 )th oLS

(2m 3 )th oLS

)th

(2m 3 )th oLS

(2m 2

oLS

oLS: oscillatory-lower-saddle, oUS: oscillatory-upper-saddle, mSI: oscillatory-sink, oSO: oscillatory-source

204

3 Global Stability of 1-D Discrete Systems

1st order

1st order

oSI-mSO

oSO

a1

a2

1st order

oSO

(2:1:1) Function, xk+1xk= g(xk,p)

Function, xk+1xk= g(xk,p)

(1:1:1)

2nd order

a3

1st order

oLS

oSI-mSO

a1

a2

(i)

(ii)

oUS

1st order

2nd order

a1

a2

oSO 1st order

(1:1:2) Function, xk+1xk= g(xk,p)

Function, xk+1xk= g(xk,p)

(1:2:1)

1st order

1st order

oSI-mSO

a3

oSO

(iv)

oUS

a1

a2

Variable, xk

(v)

oSO 1st order

a3

(2:1:2) Function, xk+1xk= g(xk,p)

Function, xk+1xk= g(xk,p)

(2:2:1)

2nd order

a3

Variable, xk

(iii)

oUS

2nd order

oLS

a2

a1

Variable, xk

2nd order

a3

Variable, xk

Variable, xk

oSI-mSO

1st order

oSO

oUS

1st order

oSO

2nd order

a1

a2

2nd order

oLS

a3

Variable, xk

(vi)

Fig. 3.12 The sampled oscillatory stability for a three-fixed-point discrete systems (a0 < 0): i (1:1:1), ii (2:1:1), iii (1:2:1), iv (1:1:2), v (2:2:1), vi (2:1:2), vii (1:2:2), viii (3:1:1), ix (1:3:1), x (1:1:3). oLS: oscillatory-lower-saddle, oUS: oscillatory-upper-saddle, oSI: oscillatory-sink, oSO: oscillatory-source

3.5 Three-Fixed-Point Systems

205 (3:1:1)

1st order

2nd order

2nd order

oSO

oLS

oLS

a1

Function, xk+1xk= g(xk,p)

Function, xk+1xk= g(xk,p)

(1:2:2)

3rd order

1st order

oSO

a2

a1

a3

a2

oSI-mSO

1st order

oSO

a3

Variable, xk

Variable, xk

(vii)

(viii)

1st order

1st order

3rd order

oSO

oSO

oSI

a1

a2

(1:1:3) Function, xk+1xk= g(xk,p)

Function, xk+1xk= g(xk,p)

(1:3:1)

oSO

oSI-mSO

1st order

a3

1st order

a1

Variable, xk

a2

oSO 3rd order

a3

Variable, xk

(ix)

(x)

Fig. 3.12 (continued) Table 3.14 The sampled oscillatory stability of three fixed-points for a three-fixed-point discrete system (xk+1 = xk + a0 (xk − a1 )r1 (xk − a2 )r2 (xk − a3 )r3 , a0 < 0) (r1 : r2 : r3 )

x ∗ = a1

x ∗ = a2

x ∗ = a3

(1:1:1)

oSO

oSI-mSO

oSO

(2:1:1)

2nd oLS

oSI-mSO

oSO

(1:2:1)

oSI-mSO

2nd oUS

oSO

(1:1:2)

oSI-mSO

oSO

2nd oLS

(2:2:1)

2nd

2nd

(2:1:2)

2nd oUS

oSO

2nd oLS

(1:2:2)

oSO

2nd oLS

2nd oLS

(3:1:1)

3rd

oSI-mSO

oSO

(1:3:1)

oSO

3rd oSI

oSO

(1:1:3)

oSO

oSI-mSO

3rd oSI

oUS

oSO

oUS

oSO

oLS: oscillatory-lower-saddle, oUS: oscillatory-upper-saddle, mSI: oscillatory-sink, oSO: oscillatory-source, oSI-mSO: oscillatory sink to monotonic source

206

3 Global Stability of 1-D Discrete Systems

For an n-fixed-point discrete system with a (r1 : r2 : . . . : rn )-type, there are 2n -potential singularities of 2m i − 1 and 2m j number combinations for given m i and m j (i, j = 1, 2, . . . , n), and m i , m j ∈ Z+ .

References Luo ACJ (2019) The Global Analysis of Equilibrium Stability in 1-dimensional Systems. J Vib Testing Syst Dyn 3(3):347–367 Luo ACJ (2020) Bifurcation and stability in nonlinear dynamical systems. Springer, New York

Chapter 4

Forward and Backward Discrete Systems

This chapter presents a Yin-Yang theory for nonlinear discrete systems with positive and negative iterations of discrete iterative maps. Based on the Yin-Yang theory, the complete dynamics of discrete dynamical systems are discussed. The critical dynamics of forward and backward discrete systems are discussed from the explicit expressions of discrete systems.

4.1 Yin-Yang Theory of Discrete Systems Definition 4.1 Consider an implicit vector function f:D → D on an open set D ⊂ Rn in an n-dimensional discrete system. For xk , xk+1 ∈ D, there is a discrete relation as f(xk , xk+1 , p) = 0

(4.1)

where the discrete vector function is f = ( f 1 , f 2 , . . . , f n )T ∈ Rn and the discrete variable vector is xk = (xk,1 , xk,2 , . . . , xk,n )T ∈ D with a parameter vector p = ( p1 , p2 , . . . , pm )T ∈ Rm . As in Luo (2010), to symbolically describe the discrete dynamical systems, introduce two discrete sets. Definition 4.2 For a discrete dynamical system in Eq. (4.1), the positive and negative discrete sets are defined by + = {xk+i |xk+i ∈ R n , i ∈ Z+ } ⊂ D, − = {xk−i |xk−i ∈ R n , i ∈ Z+ } ⊂ D

 (4.2)

respectively. The discrete set is © Higher Education Press 2020 A. C. J. Luo, Bifurcation and Stability in Nonlinear Discrete Systems, Nonlinear Physical Science, https://doi.org/10.1007/978-981-15-5212-0_4

207

208

4 Forward and Backward Discrete Systems

 = + ∪ − .

(4.3)

A positive (or forward) mapping is defined as P+ : → + ⇒ P+ :xk → xk+1

(4.4)

and a negative (or backward) mapping is defined by P− : → − ⇒ P− :xk → xk−1 .

(4.5)

Definition 4.3 For a discrete system in Eq. (4.1), consider two points xk ∈ D and xk+1 ∈ D, and there is a specific, differentiable, vector function g ∈ Rn to make g(xk , xk+1 , λ) = 0. The stable solution based on xk+1 = P+ xk for the positive mapping P+ is called the “Yang” (or forward solution) of the discrete dynamical system in Eq. (4.1) ∗ ) of f(xk , xk+1 , p) = 0 and in sense of g(xk , xk+1 , λ) = 0 if solutions (xk∗ , xk+1 g(xk , xk+1 , λ) = 0 exist. (ii) The stable solution based on xk = P− xk+1 for the negative mapping P− is called the “Yin” (or backward solution) of the discrete dynamical system in Eq. (4.1) ∗ ) of f(xk , xk+1 , p) = 0 and in sense of g(xk , xk+1 , λ) = 0 if solutions (xk∗ , xk+1 g(xk , xk+1 , λ) = 0 exist. (iii) The solution based on xk+1 = P+ xk is called “Yin-Yang” for the positive mapping P+ of the discrete dynamical system in Eq. (4.1) in sense ∗ ) of f(xk , xk+1 , p) = 0 and of g(xk , xk+1 , λ) = 0 if solutions (xk∗ , xk+1 g(xk , xk+1 , λ) = 0 exist and the eigenvalues of D P+ (xk∗ ) are distributed inside and outside the unit circle. (iv) The solution based on xk = P− xk+1 is called the “Yin-Yang” for the negative mapping P− of the discrete dynamical system in Eq. (4.1) in sense ∗ ) of f(xk , xk+1 , p) = 0 and of g(xk , xk+1 , λ) = 0 if solutions (xk∗ , xk+1 ∗ ) are distributed g(xk , xk+1 , λ) = 0 exist and the eigenvalues of D P− (xk+1 inside and outside unit circle. (i)

Consider the positive and negative mappings are xk+1 = P+ xk and xk = P− xk+1 .

(4.6)

For the simplest case, consider the constraint condition of g(xk , xk+1 , λ) = xk+1 − xk = 0 . Thus, the positive and negative mappings have, respectively, the constraints of xk+1 = xk and xk = xk+1 .

(4.7)

Both positive and negative mappings are governed by the following discrete relation in Eq. (4.1). In other words, Eq. (4.6) gives

4.1 Yin-Yang Theory of Discrete Systems

209

f(xk , xk+1 , p) = 0 and f(xk , xk+1 , p) = 0

(4.8)

Setting the period-1 solution xk∗ and substitution of Eq. (4.7) into Eq. (4.8) gives f(xk∗ , xk∗ , p) = 0 and f(xk∗ , xk∗ , p) = 0.

(4.9)

From Eq. (4.9), the period-1 solutions for the positive and negative mappings are identical. The two relations for positive and negative mappings are illustrated in Fig. 4.1a, b respectively. To determine the period-1 solution, the fixed points of Eq. (4.8) exist under constraints in Eq. (4.7), which are also shown in Fig. 4.1. The two thick lines on the axis are two sets for the mappings from the starting to final states. The relation in Eq. (4.7) is presented by a solid curve. The intersection points of the curves and straight lines for relations in Eqs. (4.7) and (4.8) give the fixed points of Eq. (4.9), which are period-1 solutions, labeled by the circular symbols. However, the stability and bifurcation of the period-1 solutions for such positive and Fig. 4.1 Period-1 solution for a positive mapping and b negative mapping. The two thick lines on the axis are two sets for the mappings from the starting to final states. The mapping relation is presented by a solid curve. The circular symbols give period-1 solutions for the positive and negative mappings

x k +1 f (x k , x k +1 , p) = 0

Σ+

x k +1 = x k

Σ

xk

(a) xk

f (x k , x k +1 , p) = 0

Σ−

x k = x k +1

Σ

(b)

x k +1

210

4 Forward and Backward Discrete Systems

negative mappings are different. To determine the stability and bifurcation of the positive and negative mappings, consider a small perturbation of fixed-points as Theorem 4.1 For a discrete dynamical system in Eq. (4.1), there are two points xk ∈ D and xk+1 ∈ D, and two positive and negative mappings are xk+1 = P+ xk and xk = P− xk+1

(4.10)

f(xk , xk+1 , p) = 0 and f(xk , xk+1 , p) = 0

(4.11)

with

Suppose a specific, differentiable, vector function g ∈ Rn makes g(xk , xk+1 , λ) = ∗ ) of both f(xk , xk+1 , p) = 0 and g(xk , xk+1 , λ) = 0 0 hold. If the solutions (xk∗ , xk+1 exist, then the following conclusions in the sense of g(xk , xk+1 , λ) = 0 hold. The stable P+ -1 solutions are the unstable P− -1 solutions with all eigenvalues of D P− (xk∗ ) outside the unit circle, vice versa. (ii) The unstable P+ -1 solutions with all eigenvalues of D P+ (xk∗ ) outside the unit circle are the stable P− -1 solutions, vice versa. (iii) For the unstable P+ -1 solutions with eigenvalue distribution of D P+ (xk∗ ) inside and outside the unit circle, the corresponding of P− -1 solution is also unstable with switching the eigenvalue distribution of D P− (xk∗ ) inside and outside the unit circle, vice versa. (iv) All the bifurcations of the stable and unstable P+ -1 solutions are all the bifurcations of the unstable and stable P− -1 solutions, respectively. (i)

Proof Consider the positive and negative mappings with relations in Eq. (4.9). The period-1 solution in sense of g(xk , xk+1 , λ) = 0 is given by f(xk , xk+1 , p) = 0 and g(xk , xk+1 , λ) = 0 ∗ from which the fixed points (xk∗ , xk+1 ) can be determined. Consider a small perturbation

The linearization of mappings in Eq. (4.9) gives ∗ δxk+1 = D P+ (xk∗ )δxk and δxk = D P− (xk+1 )δxk+1

where D P+ (xk∗ ) =



∂xk+1 ∂xk

 xk∗

∗ and D P− (xk+1 )=



∂xk ∂xk+1

 ∗ xk+1

.

4.1 Yin-Yang Theory of Discrete Systems

211

From Eq. (4.10), one obtains 

    ∂f ∂f ∂xk+1 + = 0, ∂xk ∂xk+1 ∂xk ∗ (xk∗ ,xk+1 )

∗ g(xk∗ , xk+1 , λ) = g(xk , xk+1 , λ) = 0;



    ∂f ∂xk ∂f + = 0, ∂xk+1 ∂xk ∂xk+1 (xk∗ ,xk+1 ∗ )

∗ g(xk∗ , xk+1 , λ) = g(xk , xk+1 , λ) = 0.

That is,      ∂f −1 ∂f ∂xk+1  = , xk∗ = − ∂xk ∂xk+1 ∂xk ∗ (xk∗ ,xk+1 )   −1     ∗  ∂f ∂f ∂x k x ∗ = − D P− (xk+1 ) = ∂xk+1 k+1 ∂xk ∂xk+1 ∗

D P+ (xk∗ )



.

∗ (xk ,xk+1 )

Taking the inverse of the second equation in the foregoing equation gives ∗ ) D P−−1 (xk+1



∂xk = ∂xk+1

−1

 x ∗

k+1

    ∂f −1 ∂f =− ∂xk+1 ∂xk (xk+1 ∗ ,xk∗ )

which is identical to D P+ (xk∗ ). Therefore, one obtains ∗ ) = D P+ (xk∗ ). D P−−1 (xk+1

In other words, D P+ (xk∗ ) is the inverse of D P− (xk∗ ). ∗ ) and D P+ (xk∗ ), accordingly. Consider the eigenvalues λ− and λ+ of D P− (xk+1 The following relations hold ∗ ) − λ− I)δxk+1 = 0, (D P− (xk+1

(D P+ (xk∗ ) − λ+ I)δxk = 0. Left multiplication of D P+ (xk∗ ) in the first equation of the foregoing equation, and ∗ division of λ− on both sides and application of D P−−1 (xk+1 ) = D P+ (xk∗ ) gives [D P+ (xk∗ ) − λ−1 − I]δxk+1 = 0. Thus, one can obtain λ+ = λ−1 −

212

4 Forward and Backward Discrete Systems

From the stability and bifurcation theory for P+ -1 and P− -1 solutions for discrete dynamical system in Eq. (4.1), the following conclusions can be given as follows: (i)

The stable P+ -1 solutions are the unstable P− -1 solutions with all eigenvalues of D P− (xk∗ ) outside the unit circle, vice versa. (ii) The unstable P+ -1 solutions with all eigenvalues of D P+ (xk∗ ) outside the unit circle are the stable P− -1 solutions, vice versa. (iii) For the unstable P+ -1 solutions with eigenvalue distribution of D P+ (xk∗ ) inside and outside the unit circle, the corresponding of P− -1 solution is also unstable with switching the eigenvalue distribution of D P− (xk∗ ) inside and outside the unit circle, vice versa. (iv) All the bifurcations of the stable and unstable P+ -1 solutions are all the bifurcations of the unstable and stable P− -1 solutions, respectively. 

This theorem is proved.

From the foregoing theorem, the Yin, Yang and Yin-Yang states in discrete systems exist. To generate the above ideas to P+(N ) -1 and P−(N ) -1 solutions in discrete dynamical systems in sense of g(xk , xk+N , λ) = 0, the mapping structure consisting of Npositive or negative mappings is considered. Definition 4.4 For a discrete dynamical system in Eq. (4.1), the mapping structures of N-mappings for the positive and negative mappings are defined as xk+N = P+ · P+ · · · P+ xk = P+(N ) xk ,



(4.12)

N

xk = P− · P− · · · P− xk+N = P−(N ) xk+N .



(4.13)

N

with f(xk+i−1 , xk+i , p) = 0 for i = 1, 2, . . . , N

(4.14)

where P+(0) = 1 and P−(0) = 1 for N = 0. Definition 4.5 For a discrete dynamical system in Eq. (4.1), consider two points xk+i−1 ∈ D (i = 1, 2, . . . , N ) and xk+N ∈ D, and there is a specific, differentiable, vector function g ∈ Rn to make g(xk , xk+N , λ) = 0. (i)

The stable solution based on xk+N = P+(N ) xk for the positive mapping P+ is called the “Yang” of the discrete dynamical system in Eq. (4.1) in sense of ∗ ∗ , . . . , xk+N ) of Eq. (4.14) with g(xk , xk+N , λ) = 0 if the solutions (xk∗ , xk+1 g(xk , xk+N , λ) = 0 exist.

4.1 Yin-Yang Theory of Discrete Systems

213

(ii) The stable solution based on xk = P− xk+1 for the negative mapping P− is called “Yin” of the discrete dynamical system in Eq. (4.1) in sense of g(xk , xk+N , λ) = ∗ ∗ , . . . , xk+N ) of Eq. (4.14) with g(xk , xk+N , λ) = 0 0 if the solutions (xk∗ , xk+1 exist. (iii) The solution based on xk+N = P+(N ) xk is called “Yin-Yang” for the positive mapping P+ of the discrete dynamical system in Eq. (4.1) in sense of ∗ ∗ ∗ , . . . , xk+N ) of Eq. (4.14) with xk+1 g(xk , xk+N , λ) = 0 if the solutions (xk∗ , xk+1 (N ) ∗ exist and the eigenvalues of D P+ (xk+N ) are distributed inside and outside the unit circle. (iv) The solution based on xk = P−(N ) xk+N is called “Yin-Yang” for the negative mapping P− of the discrete dynamical system in Eq. (4.1) in sense of ∗ ∗ , . . . , xk+N ) of Eq. (4.14) with g(xk , xk+N , λ) = 0 if the solutions (xk∗ , xk+1 ∗ ) are distributed g(xk , xk+N , λ) = 0 exist and the eigenvalues of D P− (xk+1 inside and outside unit circle. To determine the Yin-Yang properties of P+(N ) -1 and P−(N ) -1 in the discrete mapping system in Eq. (4.1), the corresponding theorem is presented as follows. Theorem 4.2 For a discrete dynamical system in Eq. (4.1), there are two points xk ∈ D and xk+N ∈ D, and two positive and negative mappings are xk+N = P+(N ) xk and xk = P−(N ) xk+N .

(4.15)

and xk+i = P+ xk+i−1 and xk+i−1 = P− xk+i can be governed by f(xk+i−1 , xk+i , p) = 0 for i = 1, 2, . . . , N

(4.16)

Suppose a specific, differentiable, vector function of g ∈ Rn makes g(xk , xk+N , λ) = ∗ ∗ , . . . , xk+i ) of Eq. (4.16) with g(xk , xk+N , λ) ≡ 0 0 hold. If the solutions (xk∗ , xk+1 exist, then the following conclusions in the sense of g(xk , xk+N , λ) = 0 hold. (i) The stable P+(N ) -1 solution is the unstable P−(N ) -1 solutions, vice versa. (ii) The unstable P+(N ) -1 solutions with all eigenvalues of D P+(N ) (xk∗ ) outside the unit circle are the stable P−(N ) -1 solutions, vice versa. (iii) For the unstable P+(N ) -1 solution with eigenvalue distribution of D P+(N ) (xk∗ ) inside and outside the unit circle, the corresponding P−(N ) -1 solution is also ∗ ) inside and unstable with switching eigenvalue distribution of D P−(N ) (xk+N outside the unit circle, vice versa. (iv) All the bifurcations of the stable and unstable P+(N ) -1 solution are all the bifurcations of the unstable and stable P−(N ) -1 solution, respectively. Proof Consider positive and negative mappings with relations in Eq. (4.15), i.e., f(xk+i−1 , xk+i , p) = 0 for i = 1, 2, . . . , N

214

4 Forward and Backward Discrete Systems

from which xk+i is a function of xk+i−1 in the positive mapping iteration and xk+i−1 is a function of xk+i in the negative mapping iteration. The periodic solution in sense of g(xk , xk+N , λ) = 0 is given by f(xk+i−1 , xk+i , p) = 0 for i = 1, 2, . . . , N g(xk , xk+N , λ) = 0. ∗ ∗ or xk+i (i = 1, 2, . . . , N ) and the foregoing Setting the period-1 solution be xk+i−1 equation gives ∗ ∗ , xk+i , p) = 0 for i = 0, 1, . . . , N f(xk+i−1 ∗ ∗ g(xk , xk+N , λ) = 0.

for both the positive and negative mapping iterations. The existence condition of the foregoing equation requires det[(Di j ) N ×N ] = 0 where  DN 1 = − 

∂f(xk+N −1 , xk+N , p) ∂xk+N





n×n



∂xk ∂xk+N

−1 n×n



∗ ∗ (xk+N −1 ,xk )

∂f(xk+N −1 , xk+N , p) ∂g(xk+N , xk , p) ∂xk+N ∂xk+N n×n    ∂g(xk+N , xk , p)  × (x∗ ,x∗ ) , k+N −1 k ∂xk n×n    ∂f(xk+N −1 , xk , p) (x∗ ,x∗ ) , = k+N −1 k ∂xk+N −1 n×n =−

DN N

D N j = [0]n×n for j = 2, 3, . . . , N − 1;   ∂f(xk+i−1 , xk+i , p) (x∗ ,x∗ ) , k+i−1 k+i ∂xk+i−1 n×n    ∂f(xk+i−1 , xk+i , p) (x∗ ,x∗ ) = k+i−1 k+i ∂xk+i n×n

 Dii = Di(i+1)

Di j = [0]n×n for i = 1, 2, · · · , N − 1; j = 1, 2, . . . , i − 1; i + 2, i + 3, . . . , N ;

−1

n×n

4.1 Yin-Yang Theory of Discrete Systems

215

∗ ∗ Once xk+i−1 or xk+i (r = 1, 2, · · · , l) is obtained in sense of g(xk , xk+N , λ) ≡ 0, the corresponding stability and bifurcation of the periodic solutions can be determined. However, the stability and bifurcation of the P+(N ) -1 and P−(N ) -1 solutions will be different. Herein, consider a small perturbation from the periodic solution ∗ + δxk+i xk+i = xk+i ∗ xk+i+1 = xk+i+1 + δxk+i+1

 for i = 0, 1, . . . , N

With the foregoing equation, linearization of Eq. (4.15) gives  δxk+N = D P+ · D P+ · · · D P+ δxk xk∗



=

N (N ) ∗ D P+ (xk )δxk

∗ δxk = D P− · D P− · · · D P− (xk+N )δxk+N



=

N (N ) ∗ D P− (xk+N )δxk+N .

On the other hand, for each single positive and negative mappings gives ∗ )δxk+i−1 for i = 1, 2, . . . , N δxk+i = D P+ (xk+i−1 ∗ δxk+i−1 = D P− (xk+i )δxk+i for i = 1, 2, . . . , N

where ∗ )= D P+ (xk+i−1

∗ D P− (xk+i )

 

∂xk+i ∂xk+i−1

∂xk+i−1 = ∂xk+i

 

∗ xk+i−1

∗ xk+i

for i = 1, 2, . . . N

for i = 1, 2, . . . N

and for i = 1, 2, . . . N , linearization of Eq. (4.15) gives ∂xk+i ∂f −1 ∂f ∗ ]xk+i−1 = −([ ] [ ])(x∗ ,x∗ ) , ∂xk+i−1 ∂xk+i ∂xk+i−1 k+i k+i−1 ∂xk+i−1 ∂f ∂f ∗ ∗ ∗ ∗ )=[ ]xk+i = −([ ]−1 [ ])(xk+i D P− (xk+i ,xk+i−1 ). ∂xk+i ∂xk+i−1 ∂xk+i

∗ D P+ (xk+i−1 )=[

Therefore, the resultant Jacobian matrices for P+(N ) -1 and P−(N ) -1 are

216

4 Forward and Backward Discrete Systems

∗ ∗ ∗ ∗ D P+(N ) (xk∗ ) = D P+ (xk+N −1 ) · D P+ (xk+N −2 ) · · · D P+ (xk+1 ) · D P+ (xk )         ∂xk+N ∂xk+2 ∂xk+1 ∂xk+N · ··· · = ∂xk+N −1 xk+N ∂xk+N −1 xk+N ∂xk+1 xk+1 ∂xk xk∗ ∗ ∗ ∗ −1 −2  −1   ∂f ∂f = (−1) N ··· ∂xk+N ∂xk+N −1 ∗ ∗ (xk+N ,xk+N −1 )  −1   ∂f ∂f , ∂xk+1 ∂xk ∗ ∗ (xk+1 ,xk )

∗ ∗ ∗ ∗ ∗ D P−(N ) (xk+N ) = D P− (xk+1 ) · D P− (xk+2 ) · · · D P− (xk+N −1 ) · D P− (xk+N )         ∂xk+1 ∂xk+N −2 ∂xk+N −1 ∂xk · ··· · = ∂xk+1 xk+1 ∂xk+2 xk+2 ∂xk+N −1 xk+N ∂xk+N xk+N ∗ ∗ ∗ ∗ −1     ∂f −1 ∂f = (−1) N ··· ∂xk ∂xk+1 ∗ (xk+1 ,xk∗ )  −1   ∂f ∂f . ∂xk+N −1 ∂xk+N ∗ ∗ (xk+N ,xk+N −1 )

From the two equations, it is very easily proved that the two resultant Jacobian matrices are inverse each other, i.e., ∗ ) = In×n . D P+(N ) (xk∗ ) · D P−(N ) (xk+N ∗ Similarly, consider eigenvalues λ− and λ+ of D P−(N ) (xk+N ) and D P+(N ) (xk∗ ), accordingly. The following relations hold ∗ ) − λ− I)δxk+N = 0, (D P−(N ) (xk+N

(D P+(N ) (xk∗ ) − λ+ I)δxk = 0. Left multiplication of D P+(N ) (xk∗ ) in the first equation of the foregoing equation, dividing λ− on both sides and application of Eq. (4.17) gives [D P+(N ) (xk∗ ) − λ−1 − I]δxk+N = 0 Since δxk+N is arbitrarily selected, compared to (D P+(N ) (xk∗ ) − λ+ I)δxk = 0, one obtains λ+ = λ−1 −

4.1 Yin-Yang Theory of Discrete Systems

217

in the sense of g(xk , xk+N , λ) = 0 hold. From the stability and bifurcation theory for discrete dynamical systems, the following conclusions can be summarized as The stable P+(N ) -1 solution is the unstable P−(N ) -1 solutions with all eigenvalues ∗ ) outside the unit circle, vice versa. of D P−(N ) (xk+N (N ) (ii) The unstable P+ -1 solutions with all eigenvalues of D P+(N ) (xk∗ ) outside the unit circle are the stable P−(N ) -1 solutions, vice versa. (iii) For the unstable P+(N ) -1 solution with eigenvalue distribution of D P+(N ) (xk∗ ) inside and outside the unit circle, the corresponding P−(N ) -1 solution is also ∗ ) inside and unstable with switching eigenvalue distribution of D P−(N ) (xk+N outside the unit circle, vice versa. (iv) All the bifurcations of the stable and unstable P+(N ) -1 solution are all the bifurcations of the unstable and stable P−(N ) -1 solution, respectively. (i)



This theorem is proved. P+(N ) -1

P−(N ) -1

Notice that the number N for the and solutions in the discrete dynamical system can be any integer if such a solution exists in sense of g(xk , xk+N , λ) = 0. Theorem 4.3 For a discrete dynamical system in Eq. (4.1), there are two points xk ∈ D and xk+N ∈ D. If the period-doubling cascade of the P+(N ) -1 and P−(N ) -1 solution occurs, the corresponding mapping structures are given by xk+2N = P+(N ) · P+(N ) xk = P+(2N ) xk and g(xk , xk+2N , λ) = 0; xk+22 N = P+(2N ) · P+(2N ) xk = P+(2 .. . xk+2l N = P+(2

l−1

N)

· P+(2

l−1

N)

2

N)

xk and g(xk , xk+22 N , λ) = 0; (4.17)

xk = P+(2 N ) xk and g(xk , xk+2l N , λ) = 0; l

for positive mappings and xk = P−(N ) · P−(N ) xk+2N = P−(2N ) xk+2N and g(xk , xk+2N , λ) = 0; xk = P−(2N ) · P−(2N ) xk+22 N = P−(2 .. . xk = P−(2

l−1

N)

· P−(2

l−1

N)

2

N)

xk+22 N and g(xk , xk+22 N , λ) = 0; (4.18)

xk+2l N = P−(2 N ) xk+2l N and g(xk , xk+2l N , λ) = 0 l

for negative mapping, then the following statements hold, i.e.,

218

4 Forward and Backward Discrete Systems

The stable chaos generated by the limit state of the stable P+(2 N ) -1 solutions (l → ∞) in sense of g(xk , xk+2l N , λ) = 0 is the unstable chaos generated l by the limit state of the unstable stable P−(2 N ) -1 solution (l → ∞) in sense l of g(xk , xk+2l N , λ) = 0 with all eigenvalue distribution of D P−(2 N ) outside unit circle, vice versa. Such a chaos is the “Yang” chaos in nonlinear discrete dynamical systems. l (ii) The unstable chaos generated by the limit state of the unstable P+(2 N ) -1 solutions (l → ∞) in sense of g(xk , xk+2l N , λ) = 0 with all eigenvalue distribution l of D P+(2 N ) outside the unit circle is the stable chaos generated by the limit l state of the stable P−(2 N ) -1 solution (l → ∞) in sense of g(xk , xk+2l N , λ) = 0, vice versa. Such a chaos is the “Yin” chaos in nonlinear discrete dynamical systems. l (iii) The unstable chaos generated by the limit state of the unstable P+(2 N ) -1 solutions (l → ∞) in sense of g(xk , xk+2l N , λ) = 0 with all eigenvalue distril bution of D P+(2 N ) inside and outside the unit circle is the unstable chaos l generated by the limit state of the unstable P−(2 N ) -1 solution (l → ∞) in sense l of g(xk , xk+2l N , λ) = 0 with switching all eigenvalue distribution of D P+(2 N ) inside and outside the unit circle, vice versa. Such a chaos is the “Yin-Yang” chaos in nonlinear discrete dynamical systems. l

(i)

Proof The proof is similar to the proof of Theorem 4.2, and the chaos is obtained by l → ∞. This theorem is proved. 

4.2 Forward Discrete Systems In this section, the explicit mapping in discrete dynamical system is used to discuss dynamical behaviors under a specific constraint condition. Without losing generality, the critical dynamics of forward discrete systems based on the explicit maps are discussed as follows. Definition 4.6 Consider a discrete system with an explicit mapping P:xk → xk+1 as xk+1 = g(xk , λ) + f(xk , p)

(4.19)

under a constrained condition of xk+1 = g(xk , λ) where

(4.20)

4.2 Forward Discrete Systems

219

f = ( f 1 , f 2 , . . . , f n )T ∈ R n , g = (g1 , g2 , . . . , gn )T ∈ Rn , xk = (x1,k , x2,k , . . . , xn,k )T ∈ Rn ,

(4.21)

p = ( p1 , p2 , . . . , pm ) ∈ R , T

m

λ = (λ1 , λ2 , . . . , λl )T ∈ Rl . Such a discrete system in Eq. (4.19) is called a forward discrete system in sense of xk+1 = g(xk , λ). (i) The forward discrete system in Eq. (4.19) does not have any fixed points in the sense of xk+1 = g(xk , λ) if f(xk∗ , p) = 0.

(4.22)

(ii) The fixed points xk∗ of the forward discrete system in Eq. (4.19) in the sense of xk+1 = g(xk , λ) exist if f(xk∗ , p) = 0.

(4.23)

The corresponding stability of the fixed point xk∗ in the sense of xk+1 = g(xk , λ) is determined through |D P − λI| = 0

(4.24)

where DP =

∂xk+1  x∗ = ∂xk k



 ∂g ∂f  + xk∗ . ∂xk ∂xk

(4.25)

Such stability of the fixed point xk∗ in the sense of xk+1 = g(xk , λ) is classified by o m o ([n m 1 , n 1 ] : [n 2 , n 2 ] : [n 3 : κ3 ] : [n 4 : κ4 ]|n 5 : n 5 : [n 7 , l; κ7 ]).

(4.26)

Definition 4.7 Consider a forward discrete system with an explicit mapping P:xk → xk+1 as xk+1 = Axk + f(xk , p)

(4.27)

under a constrained condition of xk+1 = Axk where

(4.28)

220

4 Forward and Backward Discrete Systems

f = ( f 1 , f 2 , . . . , f n )T ∈ R n , xk = (x1,k , x2,k , . . . , xn,k )T ∈ Rn ,

(4.29)

p = ( p1 , p2 , . . . , pm ) ∈ R . T

m

Suppose   A − λI = 0

(4.30)

with the eigenvalues of o m o ([n m 1 , n 1 ] : [n 2 , n 2 ] : [n 3 : κ3 ] : [n 4 : κ4 ]|n 5 : n 5 : [n 7 , l; κ7 ]).

(4.31)

(i) The forward discrete system in Eq. (4.27) does not have any fixed points in the sense of xk+1 = Axk if f(xk∗ , p) = 0.

(4.32)

(ii) The fixed points xk∗ of the forward discrete system in Eq. (4.27) in the sense of xk+1 = Axk exist if f(xk∗ , p) = 0.

(4.33)

The corresponding stability of the fixed point xk∗ in the sense of xk+1 = Axk is determined through |D P − λI| = 0

(4.34)

where DP =

 ∂xk+1  ∂f  ) xk∗ = (A + Df(xk ))xk∗ . xk∗ = (A + ∂xk ∂xk

(4.35)

(ii1 ) The stability of the fixed point xk∗ in the sense of xk+1 = Axk is invariant if the classification of eigenvalues in Eq. (4.31) is invariant. (ii2 ) The stability of the fixed point xk∗ in the sense of xk+1 = Axk is switched if the classification of eigenvalues in Eq. (4.31) is changed. Thus, the fixed point xk∗ possesses at least a bifurcation or stability switching. Theorem 4.4 Consider a forward discrete system with an explicit mapping P:xk → xk+1 as xk+1 = g(xk , λ) + f(xk , p)

(4.36)

4.2 Forward Discrete Systems

221

under a constraint condition of xk+1 = g(xk , λ).

(4.37)

(i) For the ith component of g(xk , λ) given by xi,k+1 = gi (xk , λ) = xi,k ,

(4.38)

there is a matrix for a constraint of xk+1 = Axk as

(4.39)

(i1 )

The forward discrete system in Eq. (4.36) does not have any fixed in the sense of xk+1 = In×n xk = xk if f(xk∗ , p) = 0.

(i2 )

(4.40)

The fixed points xk∗ of the forward discrete system in Eq. (4.36) in the sense of xk+1 = In×n xk = xk exist if f(xk∗ , p) = 0.

(4.41)

The stability of the fixed point xk∗ in the sense of xk+1 = In×n xk = xk is determined through |D P − λIn×n | = 0

(4.42)

where DP =

 ∂xk+1  ∂f  ) xk∗ = (In×n + Df(xk ))xk∗ . xk∗ = (In×n + ∂xk ∂xk

(4.43)

(i2a ) Such stability of the fixed point xk∗ in the sense of xk+1 = xk is classified by o m o ([n m 1 , n 1 ] : [n 2 , n 2 ] : [n 3 : κ3 ] : [n 4 : κ4 ]|n 5 : n 5 : [n 7 , l; κ7 ]).

(4.44)

222

4 Forward and Backward Discrete Systems

(i2b ) The condition of monotonic bifurcation or monotonic stability switching for the fixed point xk∗ in the sense of xk+1 = xk for r-directions (r ∈ {1, 2, . . . , n}) at a specific point p = pcr is    Df(x∗ ) = 0 k0

(4.45)

∗ where Df(xk0 ) = (∂f/∂xk )|xk0∗ is of an (n − r )-rank. In the vicinity of ∗ ∗ , if Df(xk∗ , p) is an (n − r1 )-rank (0 ≤ (xk0 , pcr ), for p = pcr and xk∗ = xk0 r1 < r ), then there are r1 -eigenvalues of λ j = 1 for j = j1 , j2 , . . . , jr1 ∈ {1, 2, . . . , n}

– If (xk∗ , p) is a unique solution without other solutions, then the fixed ∗ , pcr ) is a monotonic stability switching point of the fixed point at (xk0 point in the sense of xk+1 = xk . ∗ , pcr ) – If (xk∗ , p) is not a unique solution, then the fixed point at (xk0 is a monotonic bifurcation point of the fixed points with higher-order singularity in the sense of xk+1 = xk . (ii) For the ith component of g(xk , λ) given by xi,k+1 = gi (xk , λ) = xi,k + σi j x j,k ,

(4.46)

with j ∈ {i + 1, . . . , n}, σi j ∈ {0, ai j }, ai j = 0, there is a matrix for a constraint of xk+1 = Axk as ⎡

⎤ · · · σ1(n−1) σ1n · · · σ2(n−1) σ2n ⎥ ⎥ .. .. ⎥ .. = In×n + Nn×n , . . . ⎥ ⎥ ⎦ 0 · · · 1 σ(n−1)n 0 0 ··· 0 1 n×n ⎡ ⎤ 0 σ12 · · · σ1(n−1) σ1n ⎢ 0 0 · · · σ2(n−1) σ2n ⎥ ⎢ ⎥ ⎢ .. .. ⎥ = ⎢ ... ... . . . = 0n×n , . . ⎥ ⎢ ⎥ ⎣ 0 0 · · · 0 σ(n−1)n ⎦ 0 0 ··· 0 0 n×n

1 ⎢0 ⎢ ⎢ A = ⎢ ... ⎢ ⎣0

Nn×n

σ12 1 .. .

(4.47)

Nr = 0n×n for r ∈ {2, 3, · · · , n}. (ii1 )

The forward discrete system in Eq. (4.38) does not have any fixed points in the sense of xk+1 = (In×n + Nn×n )xk if f(xk∗ , p) = 0.

(4.48)

4.2 Forward Discrete Systems

(ii2 )

223

The fixed points xk∗ of the forward discrete system in Eq. (4.38) in the sense of xk+1 = (In×n + Nn×n )xk exist if f(xk∗ , p) = 0.

(4.49)

(ii2a ) The corresponding stability of the fixed point xk∗ in the sense of xk+1 = (In×n + Nn×n )xk is determined through |D P − λIn×n | = 0

(4.50)

where DP =

 ∂xk+1   x∗ = (In×n + Nn×n + Df(xk )) xk∗ . ∂xk k

(4.51)

Such stability of the fixed point xk∗ in the sense of xk+1 = (In×n +Nn×n )xk is classified by o m o ([n m 1 , n 1 ] : [n 2 , n 2 ] : [n 3 : κ3 ] : [n 4 : κ4 ]|n 5 : n 5 : [n 7 , l; κ7 ]). (4.52)

(ii2b ) The condition of bifurcation or stability switching for the fixed point xk∗ in the sense of xk+1 = (In×n + Nn×n )xk for r-directions (r ∈ {1, 2, . . . , n}) at a specific point p = pcr is   Nn×n + Df(x∗ ) = 0 k0

(4.53)

∗ ∗ where Nn×n + Df(xk0 ) is of an (n − r )-rank. In the vicinity of (xk0 , pcr ), ∗ ∗ ∗ for p = pcr and xk = xk0 , if Nn×n + Df(xk , p) is an (n − r1 )-rank (0 ≤ r1 < r ), then there are r1 -eigenvalues of λ j = 1 for j = j1 , j2 , . . . , jr1 ∈ {1, 2, . . . , n}

– If (xk∗ , p) is a unique solution without other solutions, then the fixed ∗ , pcr ) is a monotonic stability switching point of the fixed point at (xk0 point in the sense of xk+1 = (In×n + Nn×n )xk . ∗ , pcr ) is – If (xk∗ , p) is not a unique solution, then the fixed point at (xk0 a monotonic bifurcation point of the fixed points with higher-order singularity in the sense of xk+1 = (In×n + Nn×n )xk . Proof (i) From Eqs. (4.36) to (4.39), we have xk+1 = xk + f(xk , p). (i1 )

If

(4.54)

224

4 Forward and Backward Discrete Systems

f(xk , p) = 0 the foregoing equation from Eq. (4.54) gives f i1 (xk , p) > 0 ⇒ xi1 ,k+1 > xi1 ,k , i 1 ∈ Z 1 ⊆ {∅, 1, 2, . . . , n}, f i2 (xk , p) < 0 ⇒ xi2 ,k+1 < xi2 ,k , i 2 ∈ Z 3 ⊆ {∅, 1, 2, . . . , n}, f i1 (xk , p) = 0 ⇒ xi3 ,k+1 = xi3 ,k , i 3 ∈ Z 3 ⊂ {∅, 1, 2, . . . , n}, i i + i 2 + i 3 = n; Z 3 = {1, 2, . . . , n}. where f i1 (xk , p), f i2 (xk , p) and f i3 (xk , p) cannot exist for i 1 , i 2 , i 3 = ∅. Therefore, xk+1 = xk .

(i2 )

Thus a constraint of xk+1 = xk in Eq. (4.54) cannot hold. So under a constraint of xk+1 = xk , the fixed points of Eq. (4.54) cannot exist. If f(xk , p) = 0 the foregoing equation from Eq. (4.54) gives f i1 (xk , p) = 0 ⇒ xi1 ,k+1 = xi1 ,k , i 1 = 1, 2, . . . , n Therefore, xk+1 = xk . Thus a constraint of xk+1 = xk in Eq. (4.54) can hold. So under a constraint of xk+1 = xk , the fixed points xk∗ of Eq. (4.54) exist. In a neighborhood of xk∗ , the variational equation is xk+1 = xk + Df(xk∗ , p)xk and the corresponding matrix is DP =

∂xk+1 = I + Df(xk∗ , p) ∂xk

Therefore, the corresponding stability of such fixed point xk∗ is determined by   |D P − λI| =  Df(xk∗ , p) − (λ − 1)I = 0.

(4.55)

4.2 Forward Discrete Systems

225

(i2a ) The classification of eigenvalues for stability is given by o m o ([n m 1 , n 1 ] : [n 2 , n 2 ] : [n 3 : κ3 ] : [n 4 : κ4 ]|n 5 : n 5 : [n 7 , l; κ7 ]). ∗ , if there are r-eigenvalues in (i2b ) For a specific point of p = pcr and xk∗ = xk0 Eq. (4.55), satisfying

λi = 1 for i = i 1 , i 2 , . . . , ir ∈ {1, 2, . . . , n} and 1 ≤ r ≤ n then Eq. (4.55) gives ) (λ) = 0. (λ − 1)r (n−r n ) (λ) is an (n − r )-dimensional determinant of (Df(xk∗ , p) − where (n−r n (λ − 1)I). Thus the following determinant is zero, i.e.,

   Df(x∗ , pcr ) = 0 k0 ∗ ∗ for Df(xk0 , pcr ) with an (n − r )-rank. In other words, if Df(xk0 , pcr ) is an (n − r )-rank, there are r-eigenvalues of λi = 1 for i = i 1 , i 2 , . . . , ir ∈ {1, 2, . . . , n} in Eq. (4.55). ∗ ∗ In vicinity of (xk0 , pcr ), for p = pcr and xk∗ = xk0 , if Df(xk∗ , p) is an (n − r1 )-rank (0 ≤ r1 < r ), then there are r1 -eigenvalues of λ j = 1 for j = j1 , j2 , . . . , jr1 ∈ {1, 2, . . . , n}

– If (xk∗ , p) is a unique solution without other solutions, then the fixed ∗ , pcr ) is a monotonic stability switching of the fixed point. point at (xk0 ∗ ∗ , pcr ) is a – If (xk , p) is not a unique solution, then the fixed point at (xk0 monotonic bifurcation of the fixed point with higher-order singularity. (ii) From Eqs. (4.36) to (4.47), we have xk+1 = (In×n + Nn×n )xk + f(xk , p). (ii1 )

If f(xk , p) = 0 the foregoing equation from Eq. (4.56) gives f i1 (xk , p) > 0 ⇒ xi1 ,k+1 > xi1 ,k + σi1 j x j,k for i 1 ∈ Z 1 ⊆ {∅, 1, 2, . . . , n}, j = i 1 + 1, i 1 + 2, . . . , n; f i2 (xk , p) < 0 ⇒ xi2 ,k+1 < xi2 ,k + σi2 j x j,k , i 2 ∈ Z 3 ⊆ {∅, 1, 2, . . . , n}, j = i 2 + 1, i 2 + 2, . . . , n; f i1 (xk , p) = 0 ⇒ xi3 ,k+1 = xi3 ,k + σi3 j x j,k ,

(4.56)

226

4 Forward and Backward Discrete Systems

i 3 ∈ Z 3 ⊂ {∅, 1, 2, . . . , n}, j = i 3 + 1, i 3 + 2, . . . , n; i i + i 2 + i 3 = n; Z 3 = {1, 2, . . . , n}; where f i1 (xk , p), f i2 (xk , p) and f i3 (xk , p) cannot exist for i 1 , i 2 , i 3 = ∅. Therefore, xk+1 = (In×n + Nn×n )xk .

(ii2 )

Thus a constraint of xk+1 = (In×n + Nn×n )xk in Eq. (4.56) cannot hold. So under a constraint of xk+1 = (In×n + Nn×n )xk , the fixed points of Eq. (4.56) cannot exist. If f(xk , p) = 0 the foregoing equation from Eq. (4.56) gives f i (xk , p) = 0 ⇒ xi,k+1 = xi,k + σi j x j,k , i, j = 1, 2, · · · , n; j > i Therefore, xk+1 = (In×n + Nn×n )xk . Thus a constraint of xk+1 = (In×n + Nn×n )xk in Eq. (4.56) can hold. So under a constraint of xk+1 = (In×n + Nn×n )xk , the fixed points xk∗ of Eq. (4.56) exist. In neighborhood of xk∗ , the variational equation is xk+1 = (I + N)xk + Df(xk∗ , p)xk and the corresponding matrix is DP =

∂xk+1 = (I + N) + Df(xk∗ , p) ∂xk

Therefore, the corresponding stability of such fixed point xk∗ is determined by   |D P − λI| = (Df(xk∗ , p) + N) − (λ − 1)I = 0.

(4.57)

(ii2a ) The classification of eigenvalues for stability is given by o m o ([n m 1 , n 1 ] : [n 2 , n 2 ] : [n 3 : κ3 ] : [n 4 : κ4 ]|n 5 : n 5 : [n 7 , l; κ7 ]). ∗ , if there are r-eigenvalues (ii2b ) For a specific point of p = pcr and xk∗ = xk0 in Eq. (4.57), satisfying

4.2 Forward Discrete Systems

227

λi = 1 for i = i 1 , i 2 , · · · , ir , {1, 2, · · · , n} and 1 ≤ r ≤ n then Eq. (4.57) gives (λ) = 0 (λ − 1)r (n−1) n Thus the following determinant is zero, i.e.,    Df(x∗ , pcr ) + N = 0 k0 ∗ for Df(xk0 , pcr ) + N with an (n − r )-rank. In other words, if ∗ Df(xk0 , pcr ) + N is an (n − r )-rank, there are r-eigenvalues of λi = 1 for i = i 1 , i 2 , · · · , ir , {1, 2, · · · , n} in Eq. (4.57). ∗ ∗ In the vicinity of (xk0 , pcr ), for p = pcr and xk∗ = xk0 , if Df(xk∗ , p) + N is an (n − r1 )-rank (0 ≤ r1 < r ), then there are r1 -eigenvalues of λ j = 1 for j = j1 , j2 , · · · , jr {1, 2, · · · , n}

– If (xk∗ , p) is a unique solution without other solutions, then the fixed ∗ , pcr ) is a monotonic stability switching of the fixed point point at (xk0 in the sense of xk+1 = (In×n + Nn×n )xk . ∗ , pcr ) is a – If (xk∗ , p) is not a unique solution, then the fixed point at (xk0 monotonic bifurcation of the fixed point with higher-order singularity in the sense of xk+1 = (In×n + Nn×n )xk . 

Theorem 4.4 is approved.

Theorem 4.5 Consider a forward discrete system with an explicit mapping P:xk → xk+1 as in Eqs. (4.36) and (4.37). (i) For the ith component of g(xk , λ) given by xi,k+1 = gi (xk , λ) = −xi,k ,

(4.58)

there is a matrix for a constraint of xk+1 = Axk as ⎡

−1 ⎢ 0 ⎢ ⎢ A = ⎢ ... ⎢ ⎣ 0 0 (i1 )

0 −1 .. .

··· ··· .. .

0 0 .. .

0 0 .. .

⎤

⎥ ⎥ ⎥ = −In×n . ⎥ ⎥ ⎦ 0 · · · −1 0 0 · · · 0 −1 n×n

(4.59)

The forward discrete system in Eq. (4.36) does not have any fixed points in the sense of xk+1 = −In×n xk = −xk if f(xk∗ , p) = 0.

(4.60)

228

4 Forward and Backward Discrete Systems

(i2 )

The fixed points xk∗ of the forward discrete system in Eq. (4.36) in the sense of xk+1 = −In×n xk = −xk exist if f(xk∗ , p) = 0.

(4.61)

The stability of the fixed point xk∗ in the sense of xk+1 = −In×n xk = −xk is determined through |D P − λIn×n | = 0

(4.62)

where DP =

 ∂xk+1   x∗ = (−In×n + Df(xk )) xk∗ . ∂xk k

(4.63)

(i2a ) Such stability of the fixed point xk∗ in the sense of xk+1 = −xk is classified by o m o ([n m 1 , n 1 ] : [n 2 , n 2 ] : [n 3 : κ3 ] : [n 4 : κ4 ]|n 5 : n 5 : [n 7 , l; κ7 ]).

(4.64)

(i2b ) The condition of oscillatory bifurcation or oscillatory stability switching for the fixed point xk∗ in the sense of xk+1 = −xk for r-directions (r {1, 2, · · · , n}) at a specific point p = pcr is    Df(x∗ ) = 0 k0

(4.65)

 ∗ where Df(xk0 ) = (∂f/∂xk )xk0∗ is of an (n − r )-rank. In the vicinity ∗ ∗ , pcr ), for p = pcr and xk∗ = xk0 , if Df(xk∗ , p) is an (n − r1 )of (xk0 rank (0 ≤ r1 < r ), then there are r1 -eigenvalues of λ j = −1 for j = j1 , j2 , · · · , jr1 {1, 2, · · · , n}. – If (xk∗ , p) is a unique solution without other solutions, then the fixed ∗ , pcr ) is an oscillatory stability switching point of the fixed point at (xk0 point in the sense of xk+1 = −xk . ∗ , pcr ) is – If (xk∗ , p) is not a unique solution, then the fixed point at (xk0 an oscillatory bifurcation point of the fixed points with higher-order singularity in the sense of xk+1 = −xk . (ii) For the ith component of g(xk , λ) given by xi,k+1 = gi (xk , λ) = −xi,k + σi j x j,k ,

(4.66)

with j {i + 1, · · · , n}, σi j {0, ai j }, ai j = 0, there is a matrix for a constraint of xk+1 = Axk as

4.2 Forward Discrete Systems

⎡

⎤ · · · σ1(n−1) σ1n · · · σ2(n−1) σ2n ⎥ ⎥ .. .. ⎥ .. = −In×n + Nn×n , . . . ⎥ ⎥ 0 · · · −1 σ(n−1)n ⎦ 0 0 ··· 0 −1 n×n ⎡ ⎤ 0 σ12 · · · σ1(n−1) σ1n ⎢ 0 0 · · · σ2(n−1) σ2n ⎥ ⎢ ⎥ ⎢ .. .. ⎥ = ⎢ ... ... . . . = 0n×n , . . ⎥ ⎢ ⎥ ⎣ 0 0 · · · 0 σ(n−1)n ⎦ 0 0 ··· 0 0 n×n

−1 ⎢ 0 ⎢ ⎢ A = ⎢ ... ⎢ ⎣ 0

Nn×n

229

σ12 −1 .. .

(4.67)

Ns = 0n×n for s ∈ {2, 3, . . . , n}. (ii1 )

The forward discrete system in Eq. (4.38) does not have any fixed points in the sense of xk+1 = (−In×n + Nn×n )xk if f(xk∗ , p) = 0.

(ii2 )

(4.68)

The fixed points xk∗ of the forward discrete system in Eq. (4.38) in the sense of xk+1 = (−In×n + Nn×n )xk exist if f(xk∗ , p) = 0.

(4.69)

(ii2a ) The corresponding stability of the fixed point xk∗ in the sense of xk+1 = (−In×n + Nn×n )xk is determined through |D P − λIn×n | = 0

(4.70)

where DP =

 ∂xk+1   xk∗ = (−In×n + Nn×n + Df(xk )) xk∗ . ∂xk

(4.71)

Such stability of the fixed point xk∗ in the sense of xk+1 = (−In×n + Nn×n )xk is classified by o m o ([n m 1 , n 1 ] : [n 2 , n 2 ] : [n 3 : κ3 ] : [n 4 : κ4 ]|n 5 : n 5 : [n 7 , l; κ7 ]). (4.72)

(ii2b ) The condition of bifurcation or stability switching for the fixed point xk∗ in the sense of xk+1 = (−In×n +Nn×n )xk for r-directions (r {1, 2, · · · , n}) at a specific point p = pcr is

230

4 Forward and Backward Discrete Systems

  Nn×n + Df(x∗ ) = 0 k0

(4.73)

∗ ∗ where Nn×n + Df(xk0 ) is of an (n − r )-rank. In the vicinity of (xk0 , pcr ), ∗ ∗ ∗ for p = pcr and xk = xk0 , if Nn×n + Df(xk , p) is an (n − r1 )-rank (0 ≤ r1 < r ), then there are r1 -eigenvalues of λ j = −1 for j = j1 , j2 , · · · , jr1 {1, 2, · · · , n}.

– If (xk∗ , p) is a unique solution without other solutions, then the fixed ∗ , pcr ) is an oscillatory stability switching point of the point at (xk0 fixed point in the sense of xk+1 = (−In×n + Nn×n )xk . ∗ , pcr ) is – If (xk∗ , p) is not a unique solution, then the fixed point at (xk0 an oscillatory bifurcation point of the fixed points with higher-order singularity in the sense of xk+1 = (−In×n + Nn×n )xk . Proof The proof of this theorem is similar to Theorem 4.3.



Theorem 4.6 Consider a forward discrete system with an explicit mapping P:xk → xk+1 as in Eqs. (4.36) and (4.37). (i) For the ith component of g(xk , λ) given by ) xi(rr ,k+1 = gir (xk , λ) = B(r ) xi(rr ,k) ,

(4.74)

and gir = (g2ir −1 , g2ir )T , xi(rr ,k) = (x2ir −1,k , x2ir ,k )T ,    αr βr B(r ) = , ρr = (αr )2 + (βr )2 = 1; −βr αr ir ∈ Z r = {qr −1 + 1, qr −1 + 2, . . . , qr −1 + lr } ⊂ {∅, 1, 2, . . . , n/2}, qr −1 =

r −1 

ls , r = 1, 2, . . . , l; l ≤ n/2, n = 2

s=1

r −1 

(4.75)

ls ,

s=1

there is a matrix for the constraint of xk+1 = Axk as ⎤ 02l1 ×2l2 · · · 02l1 ×2ll−1 02l1 ×2ll ⎢ 02l2 ×2l1 A2 · · · 02l2 ×2ll−1 02l2 ×2ll ⎥ ⎥ ⎢ ⎥ ⎢ . .. .. .. .. .. A=⎢ ⎥ , . . . . ⎥ ⎢ ⎣ 02l ×2l 02l ×2l · · · Al−1 02l ×2l ⎦ l−1 1 l−1 2 l−1 l 02ll ×2l1 02ll ×2l2 · · · 02ll ×2ll−1 Al n×n ⎤ ⎡ (r ) B 02×2 · · · 02×2 02×2 ⎢ 02×2 B(r ) · · · 02×2 02×2 ⎥ ⎥ ⎢ ⎢ .. . . .. .. ⎥ Ar = ⎢ ... (r = 1, 2, · · · , l). . . . . ⎥ ⎥ ⎢ ⎣ 02×2 02×2 · · · B(r ) 02×2 ⎦ ⎡

A1

02×2 02×2 · · · 02×2 B(r )

2lr ×2lr

(4.76)

4.2 Forward Discrete Systems

(i1 )

231

The forward discrete system in Eq. (4.36) does not have any fixed points in  the sense of xk+1 = Axk with ρr = (αr )2 + (βr )2 = 1 (r = 1, 2, · · · , l) if f(xk∗ , p) = 0.

(i2 )

(4.77)

The fixed points xk∗ of the forward  discrete system in Eq. (4.36) in the sense of xk+1 = Axk with ρr = (αr )2 + (βr )2 = 1 (r = 1, 2, · · · , l) exist if f(xk∗ , p) = 0.

(4.78)

The stability of the fixed point xk∗ in the sense of xk+1 = Axk with ρr =  (αr )2 + (βr )2 = 1 (r = 1, 2, · · · , l) is determined through |D P − λIn×n | = 0

(4.79)

where DP =

 ∂xk+1   xk∗ = (A + Df(xk )) xk∗ . ∂xk

(4.80)

(i2a ) Such spiral stability of the fixed point xk∗ in the sense of xk+1 = Axk with  ρr = (αr )2 + (βr )2 = 1 (r = 1, 2, · · · , l) is classified by o m o ([n m 1 , n 1 ] : [n 2 , n 2 ] : [n 3 : κ3 ] : [n 4 : κ4 ]|n 5 : n 5 : [n 7 , l; κ7 ]).

(4.81)

(i2b ) The condition of spiral bifurcation or spiral stability  switching for the fixed point xk∗ in the sense of xk+1 = Axk with ρr = (αr )2 + (βr )2 = 1 (r = 1, 2, · · · , l) for 2q-directions (q {1, 2, . . . , n/2}) at a specific point p = pcr is    Df(x∗ ) = 0 k0

(4.82)

 ∗ Df(xk0 ) = (∂f/∂xk )xk0∗ = (ci1 j1 )n/2×n/2 ,   c(2i1 −1)(2 j1 −1) c(2i1 −1)(2 j1 ) ci1 j1 = c(2i1 )(2 j1 −1) c(2i1 )(2 j1 ) 2×2

(4.83)

where

∗ , pcr ), for p = pcr and is of a 2(n/2 − q)-rank. In the vicinity of (xk0 ∗ ∗ ∗ xk = xk0 , if Df(xk , p) is an 2(n/2 − q1 )-rank (0 ≤ q1 < q), then there are

232

4 Forward and Backward Discrete Systems

r1 -pairs of eigenvalues of λ j = α j ± iβ j with ρ j = for j = j1 , j2 · · · , jq1 {1, 2, · · · , n/2} .



(α j )2 + (β j )2 = 1

– If (xk∗ , p) is a unique solution without other solutions, then the fixed ∗ , pcr ) is a spiral stability switching point at (xk0  point of the fixed point in the sense of xk+1 = Axk with ρr = (αr )2 + (βr )2 = 1 (r = 1, 2, · · · , l). ∗ , pcr ) is – If (xk∗ , p) is not a unique solution, then the fixed point at (xk0 a spiral bifurcation point of the fixed points withhigher-order singularity in the sense of xk+1 = Axk with ρr = (αr )2 + (βr )2 = 1 (r = 1, 2, · · · , l). (ii) For the ith component of g(xk , λ) given by xir ,k+1 = gir (xk , λ) = B(r ) xir ,k + σir j x j,k

(4.84)

gir = (g2ir −1 , g2ir )T , xir ,k = (x2ir −1,k , x2ir ,k )T , x j,k = (x2 j−1,k , x2 j,k )T for j = ir + 1, ir + 2, . . . , n/2    αr βr , ρr = (αr )2 + (βr )2 = 1; B(r ) = −βr αr   a(2ir −1)(2 j−1) a(2ir −1)(2 j) = 02×2 σir j ∈ {02×2 , air j }, air j = a(2ir )(2 j−1) a(2ir )(2 j)

(4.85)

ir ∈ Z r = {qr −1 + 1, qr −1 + 2, . . . , qr −1 + lr } ⊂ {∅, 1, 2, . . . , n/2}, qr −1 =

r −1 

ls , r = 1, 2, . . . , l; l ≤ n/2, n = 2

s=1

r −1 

ls ,

s=1

there is a matrix for the constraint of xk+1 = (A + N)xk as ⎡

A1 + N1 02l1 ×2l2 ⎢ 02l2 ×2l1 A2 + N2 ⎢ ⎢ .. .. A+N=⎢ . . ⎢ ⎣ 02l ×2l 02l ×2l l−1 1 l−1 2 02ll ×2l1 02ll ×2l2

··· ··· .. .

02l1 ×2ll−1 02l2 ×2ll−1 .. .

02l1 ×2ll 02l2 ×2ll .. .

· · · Al−1 + Nl−1 02ll−1 ×2ll · · · 02ll ×2ll−1 Al + Nl

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

, n×n

(4.86)

4.2 Forward Discrete Systems

233

⎤ B(r ) 02×2 · · · 02×2 02×2 ⎢ 02×2 B(r ) · · · 02×2 02×2 ⎥ ⎥ ⎢ ⎢ .. . . .. .. ⎥ Ar = ⎢ ... , . . . . ⎥ ⎥ ⎢ ⎣ 02×2 02×2 · · · B(r ) 02×2 ⎦ 02×2 02×2 · · · 02×2 B(r ) 2lr ×2lr ⎡ ⎤ 0 σ(qr −1 +1)(qr −1 +2) · · · σ(qr −1 +1)(qr +lr −1) σ(qr −1 +1)(qr −1 +lr ) ⎢0 0 · · · σ(qr −1 +2)(qr +lr −1) σ(qr −1 +2)(qr −1 +lr ) ⎥ ⎢ ⎥ ⎢ .. ⎥ . .. .. .. .. Nr = ⎢ . ⎥ . . . ⎢ ⎥ ⎣0 0 ··· 0 σ(qr −1 +lr −1)(qr −1 +lr ) ⎦ 0 0 ··· 0 0 (2lr )×(2lr ) ⎡

= 0n×n , Nsr = 0n×n for sr ∈ {2, 3, . . . , lr }, (r = 1, 2, . . . , l). (4.87) (ii1 )

The forward discrete system in Eq. (4.38) does not have any fixed points in the sense of xk+1 = (A + N)xk with ρr = (αr )2 + (βr )2 = 1 (r = 1, 2, · · · , l) if f(xk∗ , p) = 0.

(ii2 )

(4.88)

The fixed points xk∗ of the forward discrete system  in Eq. (4.38) in the sense of xk+1 = (A + N)xk with ρr = (αr )2 + (βr )2 = 1 (r = 1, 2, · · · , l) exist if f(xk∗ , p) = 0.

(4.89)

(ii2a ) The corresponding stability ofthe fixed point xk∗ in the sense of xk+1 = (A + N)xk with ρr = (αr )2 + (βr )2 = 1 (r = 1, 2, · · · , l) is determined through |D P − λIn×n | = 0

(4.90)

where DP =

 ∂xk+1   x∗ = (A + N + Df(xk )) xk∗ . ∂xk k

(4.91)

∗ Such stability  of the fixed point xk in the sense of xk+1 = (A + N)xk 2 2 with ρr = (αr ) + (βr ) = 1 (r = 1, 2, · · · , l) is classified by o m o ([n m 1 , n 1 ] : [n 2 , n 2 ] : [n 3 : κ3 ] : [n 4 : κ4 ]|n 5 : n 5 : [n 7 , l; κ7 ]). (4.92)

234

4 Forward and Backward Discrete Systems

(ii2b ) The condition of bifurcation or stability switchingfor the fixed point xk∗ in the sense of xk+1 = (A + N)xk with ρr = (αr )2 + (βr )2 = 1 (r = 1, 2, · · · , l) for r-directions (r = 1, 2, · · · , l) at a specific point p = pcr is   N + Df(x∗ ) = 0 k0

(4.93)

∗ ∗ where N+ Df(xk0 ) is of an 2(n/2−q1 )-rank. In the vicinity of (xk0 , pcr ), ∗ ∗ ∗ for p = pcr and xk = xk0 , if N + Df(xk ) is an 2(n/2 − q1 )-rank (0 ≤ q1 ε,

(5.17)

( j)

then the point xk is not an approximate solution of fixed-point for f(xk∗ , p0 ) ≈ 0 ( j) in the sense of ε. Thus, let xk,0 = xk and repeat the same procedure until Eq. (5.16) is satisfied. Definition 5.3 Consider an n-dimensional, discrete system in Eq. (5.1). For a given point x0 with parameter p0 , there is a neighborhood U (x0 ). Suppose for parameter p0 , the discrete system in Eq. (5.1) has a fixed-point xk∗ . The Jacobian matrix of the discrete vector field f(xk , p) in U (xk,0 ) have distinct eigenvalues λk (k = 1, 2, . . . , n) and the corresponding covariant and contravariant eigenvectors are vk and vk (k = 1, 2, . . . , n), respectively. The covariant component of the discrete vector field on the covariant direction of vk is in Eq. (5.11). (i) For a specific p = p0 , if λs = 0 (s = 1, 2, . . . , n but s = i) (2) i 1 2 |G i( j1 j2 ) (x0 , p0 )z k z k |  |λi z k δ j | j

j

j

(5.18)

( j, j1 , j2 = 1, 2, . . . , n), then the covariant component of the discrete vector field is expanded through the Taylor series as j

f s ≈ (vs )T · f(xk,0 , p0 ) + (vs )T · Df(xk,0 , p0 )v j z k = 0.

(5.19)

Thus, with Df(xk,0 , p0 )v j = λ j v j , there is an approximate solution of fixedpoint for a specific p = p0 , z ks ≈ −(λs )−1 (vs )T · f(xk,0 , p0 ), for s = 1, 2, . . . , n but s = i.

(5.20)

(ii) For a specific p = p0 , on the covariant direction of vi λs = 0 (s = 1, 2, . . . , n but s = i) (si ) 1 2 i |G i( j1 j2 ... js ) (xk,0 , p0 )z k z k . . . z k | j

js

j

i

(si +1) 1 2 i  |G i( |≤ε j1 j2 ... js +1 ) (xk,0 , p0 )z k z k . . . z k j

j

i

( j1 , j2 , . . . jsi +1 = 1, 2, . . . , n),

js

+1

(5.21)

5.1 Fixed-Point Computations

255

then the covariant component of the discrete vector field is expanded through the Taylor series as f i ≈ (vi )T · f(xk,0 , p0 ) + λi z ki +

sk  1 (q) j j j G i( j1 j2 ... jq ) (z k1 z k2 . . . z kq ) = 0. q! q=2

(5.22) (ii1 ) Equation (5.22) is equivalent to f i ≈ A0 (z ki )2m i + A1 (z ki )2m i −1 + · · · + A2m i −1 z ki + A2m i = 0 Al = Al (z k1 , z k2 , . . . , z ki−1 , z ki+1 , . . . , z kn ) (l = 0, 1, 2, . . . , 2m i ),

(5.23)

and the standard form for simple fixed-point solution is f i ≈ a0

mi 

[(z ki )2 + B j z ki + C j ] = 0.

(5.24)

j=1

(a) If  j = B 2j − 4C j < 0 for j = 1, 2, . . . , m i ,

(5.25)

then Eq. (5.22) does not have any solution. Thus, the fixed-point does not exist in the neighborhood U (xk,0 ). (b) If  j = B 2j − 4C j > 0 for j ∈ { j1 , . . . , jr1 } ⊂ {1, 2, . . . , m i }, l = Bl2 − 4Cl < 0 for l ∈ {l1 , . . . , lr2 } ⊂ {1, 2, . . . , m i }, r1 + r2 = m i ,

(5.26)

then Eq. (5.22) has r1 -pairs of solutions in the neighborhood U (xk,0 ). Thus, there are 2r1 -approximate solutions of fixed-points for a specific p = p0 , i( j)

z k,1,2 = − 21 B j ±



 j for j ∈ { j1 , . . . , jr1 } ⊂ {1, 2, . . . , m i },

i i i {z k,1 , z k,2 , . . . , z k,2r } 1 (i) (i) ≤ z k,s+1 , with z k,s

i( j )

i( j )

k( jr )

k(ir )

= sort{z k,11 , z k,21 , . . . , z k,1 1 , z k,2 1 } (5.27)

256

5 Infinite-Fixed-Point Discrete Systems

and (l) (l−1) (l−1) (l−1) zk, j , (l = 1, 2, . . . ; j = 1, 2, . . . , 2r1 ), xk, j = xk, j + Q j (0) (0) (0) with xk, = Q for l = 1, j ≡ xk, j0 , zk, j ≡ zk, j0 , Q (l−1) (l−1) Q(l−1) = (v(l−1) j j1 , v j2 , . . . , v jn ) for

(Df(x(l−1) , p0 ) − λ(l−1) I)v(l−1) =0 j ji ji (i = 1, 2, . . . , n). (5.28) If (l) ||f(xk, j , p0 )|| ≤ ε,

(5.29)

(l) ∗ then xk, j ≈ xk, j ( j = 1, 2, . . . , 2r 1 ) is called a set of approximate solutions of fixed-point for f(xk∗ , p0 ) ≈ 0 in the sense of ε. (c) If

 j = B 2j − 4C j = 0 for j ∈ { j1 , . . . , jr1 } ⊂ {1, 2, . . . , m i }, l = Bl2 − 4Cl > 0 for l ∈ {l1 , . . . , lr2 } ⊂ {1, 2, . . . , m i }, s = Bs2 − 4Cs < 0 for s ∈ {s1 , . . . , sr3 } ⊂ {1, 2, . . . , m i },

(5.30)

r1 + r2 + r3 = m i , then Eq. (5.22) has r1 -pairs of repeated solutions and r2 -pairs of simple solutions in the neighborhood U (xk,0 ).Thus, there are (r1 + 2r2 )-approximate solutions of fixed-points for a specific p = p0 , i( j)

z k,1,2 = − 21 B j ,  j = 0 for j ∈ { j1 , . . . , jr1 } ⊂ {1, 2, . . . , m i }  i(l) z k,1,2 = − 21 Bl ± l for l ∈ {l1 , . . . , lr2 } ⊂ {1, 2, . . . , m i },   i i i z k,1 , z k,2 , . . . , z k,2(r (5.31) 1 +r2 )   i( jr ) i(l1 ) i(l1 ) i(lr ) i( j ) i( j ) = sort z k,11 , z k,21 , . . . , z k,2 1 ; z k,1 , z k,2 , . . . , z k,22 (i) (i) ≤ z k,s+1 , with z k,s

5.1 Fixed-Point Computations

257

and (l) (l−1) (l−1) (l−1) = xk, zk, j , (l = 1, 2, . . . ; j = 1, 2, . . . , r1 + 2r2 ) xk,l j +Q (0) (0) (0) with xk, = Q for l = 1, j ≡ xk, j0 , zk, j ≡ zk, j0 , Q (l−1) (l−1) Q(l−1) = (v(l−1) j j1 , v j2 , . . . , v jn ) for (l−1) (l−1) (Df(xk, I)v(l−1) =0 j , p0 ) − λ ji ji

(i = 1, 2, . . . , n). (5.32) If (l) ||f(xk, j , p0 )|| ≤ ε,

(5.33)

(l) ∗ then xk, j ≈ xk, j ( j = 1, 2, . . . , r 1 + 2r 2 ) is called a set of approximate solutions of fixed-points for f(xk∗ , p0 ) ≈ 0 in the sense of ε. (ii2 ) Equation (5.22) is equivalent to

f i ≈ A0 (z ki )2m i +1 + A1 (z ki )2m i + · · · + A2m i z ki + A2m i +1 = 0 A j = A j (z k1 , z k2 , . . . , z ki−1 , z ki+1 , . . . , z kn ) ( j = 0, 1, 2, . . . , 2m i + 1), (5.34) and the standard form for simple fixed-point solutions is f i ≈ a0 (z ki − a)

mk 

[(z ki )2 + B j z ki + C j ] = 0.

(5.35)

 j = B 2j − 4C j < 0 for j = 1, 2, . . . , m i ,

(5.36)

j=1

(a) If

then Eq. (5.22) has a solution. Thus, the fixed-point has one solution in the neighborhood U (xk,0 ). (b) If  j = B 2j − 4C j > 0 for j ∈ { j1 , . . . , jr1 } ⊂ {1, 2, . . . , m i }, l = Bl2 − 4Cl < 0 for l ∈ {l1 , . . . , lr2 } ⊂ {1, 2, . . . , m i }, r1 + r2 = m k ,

(5.37)

258

5 Infinite-Fixed-Point Discrete Systems

then Eq. (5.22) has (2r1 + 1)-simple solutions in the neighborhood U (xk,0 ).Thus, there exist (2r1 + 1)-approximate solutions of fixed-points for a specific p = p0 as i( j)

z k,1,2 = − 21 B j ±



 j for j ∈ { j1 , . . . , jr1 } ⊂ {1, 2, . . . , m i }, i( j )

i( j )

i( jr )

i( jr )

i i i {z k,1 , z k,2 , . . . , z k,2r } = sort{a, z k,11 , z k,21 , . . . , z k,1 1 , z k,2 1 } 1 +1 (i) (i) ≤ z k,s+1 , with z k,s

(5.38) and (l) (l−1) (l−1) (l−1) xk, zk, j , (l = 1, 2, . . . ; j = 1, 2, . . . , 2r1 + 1) j = xk, j + Q (0) (0) with xk, j ≡ xk, j0 , zk, j ≡ zk, j0 for l = 1.

(5.39) If (l) ||f(xk, j , p0 )|| ≤ ε,

(5.40)

(l) ∗ then xk, j ≈ xk, j ( j = 1, 2, . . . , 2r 1 + 1) is called a set of approximate solutions of fixed-point for f(xk∗ , p0 ) ≈ 0 in the sense of ε. (c) If

 j = B 2j − 4C j = 0 for j ∈ { j1 , . . . , jr1 } ⊂ {1, 2, . . . , m i }, l = Bl2 − 4Cl > 0 for l ∈ {l1 , . . . , lr2 } ⊂ {1, 2, . . . , m i }, s = Bs2 − 4Cs < 0 for s ∈ {s1 , . . . , sr3 } ⊂ {1, 2, . . . , m i }, r1 + r2 + r3 = m i ,

(5.41)

then Eq. (5.22) has r1 -pairs of repeated solutions and r2 -pairs of simple solutions in the neighborhood U (xk,0 ). Thus, there are (r1 + 2r2 + 1)-approximate solutions of fixed-points for a specific p = p0 , i( j)

z k,1,2 = − 21 B j ,  j = 0 for j ∈ { j1 , . . . , jr1 } ⊂ {1, 2, . . . , m i }  i(l) z k,1,2 = − 21 Bl ± l for l ∈ {l1 , . . . , lr2 } ⊂ {1, 2, . . . , m i }, i i i {z k,1 , z k,2 , . . . , z k,2(r } 1 +r2 )+1 i( j )

i( j )

i( jr )

i(lr )

i(l1 ) i(l1 ) , z k,2 , . . . , z k,22 } = sort{a, z k,11 , z k,21 , . . . , z k,2 1 ; z k,1 i i with z k,s ≤ z k,s+1

(5.42)

5.1 Fixed-Point Computations

259

and (l) (l−1) (l−1) (l−1) zk, j , (l = 1, 2, . . . ; j = 1, 2, . . . , r1 + 2r2 + 1) xk, j = xk, j + Q (0) (0) (0) with xk, = Q for l = 1 j ≡ xk, j0 , zk, j ≡ zk, j0 , Q (l−1) (l−1) Q (l−1) = (v(l−1) j j1 , v j2 , . . . , v jn ) for (l−1) (l−1) (Df(xk, I)v(l−1) = 0. j , p0 ) − λ ji ji

(i = 1, 2, . . . , n). (5.43) If (l) ||f(xk, j , p0 )|| ≤ ε,

(5.44)

(l) ∗ then xk, j ≈ xk, j ( j = 1, 2, . . . , r 1 + 2r 2 + 1) is called a set of approximate solutions of fixed-point for f(xk∗ , p0 ) ≈ 0 in the sense of ε.

The afore-presented method is an extension of the Newton-Raphson method. For such an extension, the solutions of fixed-points of nonlinear discrete systems are determined through the eigenvector space of the linearized systems at the initial guessed solutions rather than the state space of original nonlinear systems. In fact, such a new method is completely different from the Newton-Raphson method. The singularity of nonlinear discrete systems in eigenvector space is employed to determine solution existence and multiplicity.

5.2 Normal Forms of Discrete Systems In this section, a general normal form of discrete systems is presented through the eigenvector space of fixed-points. As in Luo (2019, 2020), the eigenvector space is developed, and the corresponding normal forms are developed by the Taylor series expansion. Nonlinear discrete systems experiencing the simple eigenvalues are discussed first, and the eigenvector space is developed through the following definition. Definition 5.4 Consider an n-dimensional, discrete system xk+1 = xk + f(xk , p).

(5.45)

There is a fixed-point xk∗ with parameter p0 , f(xk∗ , p0 ) = 0.

(5.46)

260

5 Infinite-Fixed-Point Discrete Systems

In a neighborhood U (xk∗ ) at parameter p0 , the discrete vector field in Eq. (5.45) is linearized by f(xk , p0 ) = Df(xk∗ , p0 )(xk − xk∗ ).

(5.47)

For the Jacobian matrix, the corresponding eigenvalues are determined by |I + Df(xk∗ , p0 ) − λ¯ I| = 0 or |Df(xk∗ , p0 )

(5.48)

− λI| = 0 with λ = λ¯ − 1.

Thus, the foregoing equation gives f (λ) = (−1)n λn + a1 λn−1 + · · · + an−1 λ + an = 0.

(5.49)

(i) For n = 2m, consider simple eigenvalues of Eq. (5.48) as f (λ) = λ2m + a1 λ2n−1 + · · · + a2m−1 λ + a2m = 0 equivalent to f (λ) =

m 

(5.50) (λ + Bi λ + Ci ) = 0. 2

i=1

(i1 ) If i1 = Bi21 − 4Ci1 > 0 (i 1 ∈ {s1(1) , s2(1) , . . . , sn(1) } ⊂ {1, 2, . . . , 2m}), 1

(5.51)

then (i 1 ) λ1,2 = − 21 Bi1 ±

1 2



Bi21 − 4Ci1 ,

(Df(xk∗ , p0 ) − λ(ij 1 ) I)v(ij 1 ) = 0 ( j = 1, 2),

(5.52)

(i 1 ∈ {s1(1) , s2(1) , . . . , sn(1) } ⊂ {1, 2, . . . , m}). 1 (i2 ) If i2 = Bi22 − 4Ci2 < 0 (i 2 ∈ {s1(2) , s2(2) , . . . , sn(2) } ⊂ {1, 2, . . . , m}), 2 then

(5.53)

5.2 Normal Forms of Discrete Systems

λ(i1,22 ) = − 21 Bi2 ±

261 1 2

 4Ci2 − Bi22 i ≡ α (i2 ) ± β (i2 ) i,

Df(xk∗ , p0 )u1(i2 ) = α (i2 ) u1(i2 ) + β (i2 ) v1(i2 ) , Df(xk∗ , p0 )v1(i2 ) = −β (i2 ) u1(i2 ) + α (i2 ) v1(i2 ) , √ } ⊂ {1, 2, . . . , m}). i = −1, (i 2 ∈ {s1(2) , s2(2) , . . . , sn(2) 2

(5.54)

(i3 ) If i3 = Bi23 − 4Ci3 = 0, Bi3 = B j3 (i 3 , j3 ∈ {s1(3) , s2(3) , . . . , sn(3) } ⊂ {1, 2, . . . , m}), 3

(5.55)

then (i 3 ) λ1,2 = − 21 Bi3 = λ(i3 ) ,

Df(xk∗ , p0 )v1(i3 ) = λ(i3 ) v1(i3 ) Df(xk∗ , p0 )v2(i3 ) = λ(i3 ) v2(i3 ) + δ (i3 ) v1(i3 ) , (Df(xk∗ , p0 ) − λ(i3 ) I)v(ij 3 ) = 0 for j = 1, 2 and δ (i3 ) = 0;

(5.56)

(Df(xk∗ , p0 ) − λ(i3 ) I)2 v(ij 3 ) = 0 for j ∈ {1, 2} and δ (i3 ) = 1 (i 3 ∈ {s1(3) , s2(3) , . . . , sn(3) } ⊂ {1, 2, . . . , m}). 3 (ii) For n = 2m + 1, consider simple eigenvalues of Eq. (5.48) as f (λ) = −λ2m+1 + a1 λ2m + · · · + a2m λ + a2m+1 = 0 equivalent to f (λ) = −(λ − λ1 )

m 

(5.57) (λ + Bi λ + Ci ) = 0. 2

i=1

(ii1 ) For λ = λ1 (Df(xk∗ , p0 ) − λ1 I)v1 = 0.

(5.58)

(ii2 ) If i1 = Bi21 − 4Ci1 > 0 (i 1 ∈ {s1(1) , s2(1) , . . . , sn(1) } ⊂ {1, 2, . . . , m}), 1

(5.59)

262

5 Infinite-Fixed-Point Discrete Systems

then (i 1 ) = − 21 Bi1 ± λ1,2

1 2



Bi21 − 4Ci1 ,

(Df(xk∗ , p0 ) − λ(ij 1 ) I)v(ij 1 ) = 0 ( j = 1, 2),

(5.60)

(i 1 ∈ {s1(1) , s2(1) , . . . , sn(1) } ⊂ {1, 2, . . . , m}). 1 (ii3 ) If i2 = Bi22 − 4Ci2 < 0 (i 2 ∈ {s1(2) , s2(2) , . . . , sn(2) } ⊂ {1, 2, . . . , m}), 2

(5.61)

then λ(i1,22 ) = − 21 Bi2 ±

1 2



4Ci2 − Bi22 i ≡ α (i2 ) ± β (i2 ) i,

Df(xk∗ , p0 )u1(i2 ) = α (i2 ) u1(i2 ) + β (i2 ) v1(i2 ) , Df(xk∗ , p0 )v1(i2 ) = −β (i2 ) u1(i2 ) + α (i2 ) v1(i2 ) , √ } ⊂ {1, 2, . . . , 2m}). i = −1, (i 2 ∈ {s1(2) , s2(2) , . . . , sn(2) 2

(5.62)

(ii4 ) If i3 = Bi23 − 4Ci3 = 0, Bi3 = B j3 (i 3 , j3 ∈ {s1(3) , s2(3) , . . . , sn(3) } ⊂ {1, 2, . . . , m}), 3

(5.63)

then (i 3 ) λ1,2 = − 21 Bi3 = λ(i3 ) ,

Df(xk∗ , p0 )v1(i3 ) = λ(i3 ) v1(i3 ) Df(xk∗ , p0 )v2(i3 ) = λ(i3 ) v2(i3 ) + δ (i3 ) v1(i3 ) , (Df(xk∗ , p0 ) − λ(i3 ) I)v(ij 3 ) = 0 for j = 1, 2 and δ

(i 3 )

(5.64)

= 0;

(Df(xk∗ , p0 )) − λ(i3 ) I)2 v(ij 3 ) = 0 for j ∈ {1, 2} and δ (i3 ) = 1 (i 3 ∈ {s1(3) , s2(3) , . . . , sn(3) } ⊂ {1, 2, . . . , m}). 3 Nonlinear discrete systems experiencing the repeated eigenvalues is discussed first. The eigenvector space is developed through the following definition. Definition 5.5 Consider an n-dimensional discrete system of xk+1 = xk +f(xk , p) in Eq. (5.45). A fixed-point x∗ with parameter p0 is given by f(xk∗ , p0 ) = 0 in Eq. (5.46).

5.2 Normal Forms of Discrete Systems

263

In a neighborhood U (xk∗ ) at parameter p0 , the discrete vector field in Eq. (5.45) is linearized, i.e., f(xk , p0 ) = Df(xk∗ , p0 )(xk −xk∗ ) in Eq. (5.47). For the Jacobian matrix of Df(xk∗ , p0 ), the corresponding eigenvalues are determined by Eq. (5.48). Thus, the eigenvalue equation is given in Eq. (5.49). Consider a general case for simple and repeated eigenvalues of f (λ) = (−1)n λn + a1 λn−1 + · · · + an−1 λ + an = 0 with λ = λ¯ − 1 equivalent to f (λ) = (−1)n

li 

(λ − λi(1) ) 1

i 1 =1

×

l3 

l2 

(λ − αi(2) − βi(2) i)(λ − αi(2) + βi(2) i) 2 2 2 2

i 2 =1

(λ − λi(3) )qi3 3

i 3 =1

l4 

(5.65)

((λ − αi(4) − βi(4) i)(λ − αi(4) + βi(4) i))qi4 4 4 4 4

i 4 =1

= 0. (i)

For simple real eigenvalues, if λ = λi(1) 1 (i 1 ∈ {s1(1) , s2(1) , . . . , sl(1) } ⊂ {1, 2, . . . , n}), 1

(5.66)

then I)v(i1 ) = 0 (Df(xk∗ , p0 ) − λi(1) 1 } ⊂ {1, 2, . . . , n}). (i 1 ∈ {s1(1) , s2(1) , . . . , sl(1) 1

(5.67)

(ii) For complex eigenvalues with simple pairs, if λ1 = αi(2) + βi(2) i, λ2 = αi(2) − βi(2) i, 2 2 2 2 √ } ⊂ {1, 2, . . . , n}), i = −1, (i 2 ∈ {s1(2) , s2(2) , . . . , sl(2) 2

(5.68)

then v1(i2 ) + βi(2) v2(i2 ) , Df(xk∗ , p0 )v1(i2 ) = αi(2) 2 2 v1(i2 ) + αi(2) v2(i2 ) , Df(xk∗ , p0 )v1(i2 ) = −βi(2) 2 2 (i 2 ∈

{s1(2) , s2(2) , . . . , sl(2) } 2

(5.69)

⊂ {1, 2, . . . , n}).

(iii) For repeated real eigenvalues, if with qi3 -repeated λ = λi(3) 3 } ⊂ {1, 2, . . . , n}), (i 3 ∈ {s1(3) , s2(3) , . . . , sl(3) 3

(5.70)

264

5 Infinite-Fixed-Point Discrete Systems

then v1(i3 ) Df(xk∗ , p0 )v1(i3 ) = λi(3) 3 3) Df(xk∗ , p0 )v(ij 3 ) = λi(3) v(ij 3 ) + δ (ij 3 ) v(ij−1 , 3

(5.71)

( j = 2, 3, . . . , Ni3 ), δ (ij 3 ) ∈ {0, 1} (i 3 ∈ {s1(3) , s2(3) , . . . , sl(3) } ⊂ {1, 2, . . . , n}). 3 (iv) For complex eigenvalues with repeated pairs, if λ1 = αi(4) + βi(4) i, λ2 = αi(4) − βi(4) i, with qi4 -repeated. 4 4 4 4 √ (4) (4) (4) i = −1, (i 4 ∈ {s1 , s2 , . . . , sl4 } ⊂ {1, 2, . . . , n}),

(5.72)

then Df(xk∗ , p0 )u1(i4 ) Df(xk∗ , p0 )v1(i4 )

⎫

= =

αi(4) u1(i4 ) + βi(4) v1(i4 ) , ⎬ 4 4 −βi(4) u1(i4 ) + αi(4) v1(i4 ) , ⎭ 4 4

⎫ (i 4 ) (4) (i 4 ) (i 4 ) (i 4 ) ⎬ Df(xk∗ , p0 )u(ij 4 ) = αi(4) u + β v + δ u , j i4 j j j−1 4 Df(x∗ , p0 )v(i4 ) = −β (4) u(i4 ) + α (4) v(i4 ) + δ (i4 ) v(i4 ) , ⎭ j

k

i4

j

i4

j

j

(5.73)

j−1

δ (ij 3 ) ∈ {0, 1}, ( j = 2, 3, . . . , qi4 ) (i 4 ∈ {s1(4) , s2(4) , . . . , sl(4) } ⊂ {1, 2, . . . , n}). 4 The covariant matrix for the Jacobian matrix Df(xk∗ , p0 ) is Q = (v1 , v2 , . . . , vn ) = (ai j )n×n , (sl(1) )

(s (1) )

{v1 , v2 , . . . , vn } = inorder{v1 1 , . . . , vl1 1 ; (s (2) )

(s (2) )

(sl(2) )

(sl(2) )

u1 1 , v1 1 , . . . , ul2 2 , vl2 2 ; ); v1(i3 ) , . . . , vq(ii3 ) ; (i 3 = s1(3) , . . . , sl(3) 3 3

 

(5.74)

l3 -group

u1(i4 ) , v1(i4 ) , . . . , uq(ii4 ) , vq(ii4 ) ; (i 4 4 4





= s1(4) , . . . , sl(4) )}, 3 

l4 -group

and the contravariant matrix of the Jacobian matrix Df(xk∗ , p0 ) is P = (v1 , v2 , . . . , vn ) = (a i j )n×n = Q−1 .

(5.75)

5.2 Normal Forms of Discrete Systems

265

Thus PDf(xk∗ , p0 )Q = Q−1 Df(xk∗ , p0 )Q = diag(A(1) , A(2) , A(3) , A(4) ),

(5.76)

where (1) (1) A(1) = diag(λ(1) 1 , λ2 , . . . , λl1 );

(5.77)

(2) (2) A(2) = diag(A(2) 1 , A2 , . . . , Al2 ), ⎡ ⎤ α (2) β (2) 2 ⎦ Ar(2) = ⎣ i2(2) i(2) (i 2 = sr(2) ; r = 1, 2, . . . , l2 ); −βi2 αi2

(5.78)

(3) (3) A(3) = diag(B(3) 1 , B2 , . . . , Bl3 ), ⎡ (3) (i ) ⎤ λi3 δ1 3 · · · 0 0 ⎢ ⎥ ⎢ 0 λ(3) · · · 0 0 ⎥ ⎢ ⎥ i3 ⎢ . ⎥ . . . . (3) ⎢ .. . . .. .. ⎥ Br = ⎢ .. ⎥ ⎢ ⎥ ⎢ 0 0 · · · λ(3) δ (i3 ) ⎥ ⎣ qi3 −1 ⎦ i3 0 0 · · · 0 λi(3) 3

(5.79)

qi3 ×qi3

(i 3 = δs(i3 )

sr(3) ; r

= 1, 2, . . . , l3 ),

∈ {0, 1} (s = 1, 2, . . . , qi3 − 1);

(4) (4) A(4) = diag(B(4) 1 , B2 , . . . , Bl4 ), ⎡ (4) (i ) ⎤ Ai4 δ1 4 · · · 0 0 ⎢ ⎥ ⎢ 0 A(4) · · · 0 0 ⎥ ⎢ ⎥ i4 ⎢ . .. . . .. .. ⎥ .. Br(4) = ⎢ . . . . ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 0 · · · A(4) δ(i4 ) ⎥ ⎣ qi4 −1 ⎦ i4 0 0 · · · 0 Ai(4) 4

(5.80) (2qi4 ×2qi4 )

(i 4 = sr(4) ; r = 1, 2, . . . , l4 ), ⎤ ⎤ ⎡ ⎡ (4) (i 4 ) αi(4) δ β 0 (4) s i (i ) 4 ⎦ ⎦ Ai4 = ⎣ 4(4) (4) , δs 4 = ⎣ 0 δs(i4 ) −βi4 αi4 2×2

δs(i4 ) ∈ {0, 1} (s = 1, 2, . . . , qi4 − 1).

and 2×2

266

5 Infinite-Fixed-Point Discrete Systems

From the eigenvector space of the linearized Jacobian matrix of fixed-points, the corresponding normal forms are developed through the following definition. Definition 5.6 Consider an n-dimensional, discrete system of xk+1 = xk +f(xk , p) in Eq. (5.45). A fixed-point xk∗ with parameter p0 is given by f(xk∗ , p0 ) = 0 in Eq. (5.46). In a neighborhood U (xk∗ ) at parameter p0 , the discrete vector field in Eq. (5.45) is linearized, i.e., f(xk , p0 ) = Df(xk∗ , p0 )(xk − xk∗ ) in Eq. (5.47). For the Jacobian matrix of Df(xk∗ , p0 ), the corresponding eigenvalues are determined by Eq. (5.48). Thus, the eigenvalue equation is given in Eq. (5.49). The covariant matrix for the Jacobian matrix Df(xk∗ , p0 ) is Q = (v1 , v2 , . . . , vn ) = (ai j )n×n ,

(5.81)

and the contravariant matrix of the Jacobian matrix Df(xk∗ , p0 ) is P = (v1 , v2 , . . . , vn ) = (a αβ )n×n = Q−1 .

(5.82)

Suppose a new transform is x = cα vα , x∗ = c∗α vα ,

(5.83)

and the covariant component is cα = (vα )T · x = (vα )T · cβ vβ = cβ δβα .

(5.84)

Thus, the component of the discrete vector field on the covariant direction of vk is expanded by f α = (vα )T · f(xk , p) = (vα )T · f(xk∗ , p0 ) + G k( j1 ) z k1 j

where

+

mα  1 (q) j j j G α( j1 j2 ... jq ) (z k1 z k2 . . . z kq ) q! q=2

+

1 j j j G (m α +1) (θ j z 1 )(θ j1 z k1 ) . . . (θ jmα +1 z kmα +1 ), (m α + 1)! k( j1 j2 ... jmα +1 ) 1 k

(5.85)

5.2 Normal Forms of Discrete Systems

267

j

∗j

j

z k = (v j )T · y = ck − ck ( j = 1, 2, . . . , n),   G α( j1 ) = (vα )T · ∂c j1 f(xk , p) ∗ k (x ,p0 )  k = (vα )T · ∂x f(xk , p)(x∗ ,p0 ) v j1 , k   (r ) (r ) α T G α( j1 j2 ... jr ) = (v ) · ∂c j1 c j2 ...c jr f(xk , p) ∗ (xk ,p0 )  α T (r ) = (v ) · ∂x f(xk , p)(x∗ ,p0 ) v j1 v j2 . . . v jr

(5.86)

k

) ∗ = G (r α( j1 j2 ... jr ) (xk , p0 ).

(i)

For simple real eigenvalues, consider λ = λi(1) 1 (i 1 ∈ {s1(1) , s2(1) , . . . , sl(1) } ⊂ {1, 2, . . . , n}) 1

(5.87)

for a specific p = p0 on the covariant direction of vi1 . If (1) (1) (1) λ(1) α  = 0 (α = i 1 ∈ {s1 , s2 , . . . , sl1 } ⊂ {1, 2, . . . , n}) (si )

j

j

j

|G α(1j1 j2 ... jsα ) (x0 , p0 )z k1 z k2 . . . z ksα | (si +1)

j

j

j

 |G α(1j1 j2 ... jsα +1 ) (x0 , p0 )z k1 z k2 . . . z ksα +1 | ≤ ε

(5.88)

( j1 , j2 , . . . jsα +1 = 1, 2, . . . , n), then the normal form on the covariant direction of vi1 is i1 i1 z k+1 ≈ (λ(1) k + 1)z k +

(i 1 ∈

si1  1 (q) j j j G i1 ( j1 j2 ... jq ) (z k1 z k2 . . . z kq ) q! q=2

{s1(1) , s2(1) , . . . , sl(1) } 1

(5.89)

⊂ {1, 2, . . . , n}).

(ii) For complex eigenvalues with simple pairs, consider λ(i1 2 ) = αi(2) + βi(2) i, λ(i2 2 ) = αi(2) − βi(2) i, 2 2 2 2 } ⊂ {1, 2, . . . , n}) (i 2 ∈ {s1(2) , s2(2) , . . . , sl(2) 2

(5.90)

for a specific p = p0 on the covariant eigenvector plane of u1(i2 ) and v1(i2 ) . If

268

5 Infinite-Fixed-Point Discrete Systems (s(α) )

j

js(α)

j

|G α( j11j2 ... jsα ) (x0 , p0 )z k1 z k2 . . . z k

1

1

(s(α) +1)

 |G α( j11j2 ... jsα

1 +1

j1 j2 ) (x0 , p0 )z k z k

(s(α) )

j

js(α)

|G α( j12j2 ... jsα ) (x0 , p0 )z k1 z k2 . . . z k

2

2



1 +1

. . . zk

js(α)

j

|

(s(α) +1) j j |G α2 ( j21 j2 ... jsα +1 ) (x0 , p0 )z k1 z k2 2

| = ε,

|

(5.91) js(α)

. . . zk

2 +1

| = ε,

( j1 , j2 , . . . jq = 1, 2, . . . , n), } ⊂ {1, 2, . . . , n}), (α = i 2 ∈ {s1(2) , s2(2) , . . . , sn(2) 2 then the normal form on the covariant eigenvector plane of u1(i2 ) and v1(i2 ) is (i 2 )1 z k+1 ≈ z k(i2 )1 + αi(2) z k(i2 )1 − βi(2) z k(i2 )2 + 2 2

(i 2 )2 z k+1

≈

z k(i2 )2

+

βi(2) z k(i2 )1 2

+

αi(2) z k(i2 )2 2

s(i2 )1

 1 (q) j j j G (i2 )1 ( j1 j2 ... jq ) (z k1 z k2 . . . z kq ), q! q=2

s(i2 )2

+

 1 (q) j j j G (i2 )2 ( j1 j2 ... jq ) (z k1 z k2 . . . z kq ), q! q=2

with (i 2 ∈ {s1(2) , s2(2) , . . . , sl(2) } ⊂ {1, 2, . . . , n}), ( j1 , j2 , . . . jq = 1, 2, . . . , n). 2 (5.92) (iii) For repeated real eigenvalues, consider with qi3 -repeated λ = λi(3) 3 } ⊂ {1, 2, . . . , n}) (i 3 ∈ {s1(3) , s2(3) , . . . , sl(3) 3

(5.93)

  for a specific p = p0 on the covariant space of v(ij 3 ) j = 2, 3, . . . , qi3 . If = 0(k = i 3 ∈ {s1(3) , s2(3) , . . . , sl(3) } ⊂ {1, 2, . . . , n}) λi(3) 3 3 (s(α) )

|G (α) j (j j1 j2 ... js

(α) j

) (x0 , p0 )z

(s(α) +1)

 |G (α) j (j j1 j2 ... js

(α) j

z ...z

j1 j2

(x0 , p0 )z +1 )

js(α)

j

|

z ...z

j1 j2

js(α)

j

+1

|≤ε

(5.94)

( j1 , j2 , . . . js(α) j +1 = 1, 2, . . . , n),   then the normal form on the covariant space of v(ij 3 ) j = 1, 2, 3, . . . , qi3 is

5.2 Normal Forms of Discrete Systems

269

s(α)1  1 (q) j j j G (α)1 ( j1 j2 ... jq ) (z k1 z k2 . . . z kq ), q! q=2 ⎫ (α) (α) (α) (α) j ⎪ z z k+1j ≈ z k j + δ (ij 3 ) z k j−1 + λi(3) ⎬ k 3 s(α) j ( j = 2, 3, . . . , qi3 ),  1 (q) j j j + G (α) j ( j1 j2 ... jq ) (z k1 z k2 . . . z kq ) ⎪ ⎭ q=2 q!

(α)1 z k+1 ≈ z k(α)1 + λi(3) z k(α)1 + 3

(5.95)

} ⊂ {1, 2, . . . , n}). δ (ij 3 ) ∈ {0, 1}(α = i 3 ∈ {s1(3) , s2(3) , . . . , sl(3) 3 (iv) For complex eigenvalues with repeated pairs, consider λ1 = αi(4) + βi(4) i, λ2 = αi(4) − βi(4) i, with qi4 -repeated 4 4 4 4 } ⊂ {1, 2, . . . , n}) (i 4 ∈ {s1(4) , s2(4) , . . . , sl(4) 4

(5.96)

  on the covariant eigenvector space of u(ij 2 ) and v(ij 2 ) j = 1, 2, 3, . . . , qi4 . If (s(α) )

|G (α) j1j1( j1 j2 ... js

(k) j1

j1 j2 ) (x0 , p0 )z k z k

(s(α) +1)

 |G (α) j1j1( j1 j2 ... js

(α) j1

(s(α) )

|G (α) j2j2( j1 j2 ... js

(α) j2

+1 )

(s(k) +1)

(α) j2

j

j1

| js(α)

j

(x0 , p0 )z k1 z k2 . . . z k

j1 j2 ) (x0 , p0 )z k z k

 |G (α) j2j2( j1 j2 ... js

js(α)

. . . zk

js(α)

. . . zk j

j

j2 +1

| ≤ ε;

j2 +1

| ≤ ε;

|

js(α)

(x0 , p0 )z k1 z k2 . . . z k +1 )

j1 +1

(5.97)

( j1 , j2 , . . . jq = 1, 2, . . . , n), } ⊂ {1, 2, . . . , n}), (α = i 4 ∈ {s1(4) , s2(4) , . . . , sl(4) 2 ( j = 1, 2, . . . , qi4 ) then ⎫ (α)11 (α)11 (4) (α)12 ⎪ z k+1 ≈ z k(α)11 + αi(4) z − β z ⎪ i4 k k 4 ⎪ ⎪ s(α)11 ⎪  1 (q) jq ⎪ j1 j2 ⎪ G (α)11 ( j1 j2 ... jq ) (z k z k . . . z k ), ⎪ + ⎬ q=2 q! (α)12 ⎪ ⎪ z k+1 ≈ z k(α)12 + βi(4) z k(α)11 + αi(4) z k(α)12 ⎪ 4 4 ⎪ ⎪ s(α)12  1 (q) jq ⎪ ⎪ j1 j2 G (α)12 ( j1 j2 ... jq ) (z k z k . . . z k ); ⎪ + ⎭ q=2 q!

(5.98a)

270

5 Infinite-Fixed-Point Discrete Systems (α)

(α)

(α)

(α)

(α) j2

(α)

(α)

(α)

(α)

(α) j2

z k+1j1 ≈ z k j1 + δ (ij 4 ) z k ( j−1)1 + αi(4) z k j1 − βi(4) zk 4 4 s(α) j1  1 (q) j j j G (α) j1 ( j1 j2 ... jq ) (z k1 z k2 . . . z kq ), + q=2 q!

z k+1j2 ≈ z k j2 + δ (ij 4 ) z k ( j−1)2 + βi(4) z k j1 + αi(4) zk 4 4 s(α) j2  1 (q) j j j G (α) j2 ( j1 j2 ... jq ) (z k1 z k2 . . . z kq ); + q=2 q!

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(5.98b)

( j = 2, 3, . . . , qi4 ), } ⊂ {1, 2, . . . , n}) with (α = i 4 ∈ {s1(4) , s2(4) , . . . , sl(4) 4 ( j1 , j2 , . . . jq = 1, 2, . . . , n). The normal-forms of nonlinear discrete systems presented herein are based on the eigenvector space of the corresponding linearized systems at fixed-points in general. From simple and repeated eigenvalues of the linearized systems at fixed-points, the corresponding normal forms of the original nonlinear systems are developed through the differential geometry representation. The infinite-fixed-point systems in the following section can be introduced from the normal-forms of nonlinear discrete systems at bifurcation points via the local analysis.

5.3 Infinite-Fixed-Point Systems In this section, using singularity analysis, the stability of fixed-points in the infinitefixed-point discrete system will be discussed as in Luo (2019, 2020). Definition 5.7 Consider a discrete system as xk+1,i = xk,i + gi1 (xk , pi1 )gi2 (xk , pi2 ) . . . giri (xk , piri ) (i = 1, 2, . . . , n), ri ≥ 1

(5.99)

where gi (xk , p) ≡ gi1 (xk , pi1 )gi2 (xk , pi2 ) . . . giri (xk , piri ).

(5.100)

Such a discrete system is called an infinite-fixed-point discrete system if the following condition exists gi (xk , p) = 0, i ∈ {1, 2, . . . , n}; gi1 j1 (xk , pi1 j1 ) ≡ gi2 j2 (xk , pi2 j2 ) = ϕ(xk , p) = 0 for i 1 , i 2 ∈ {1, 2, . . . , n}, i 1 = i 2 ; j1 ∈ {1, 2, . . . , ri1 }, j2 ∈ {1, 2, . . . , ri2 }.

(5.101)

5.3 Infinite-Fixed-Point Systems

271

The corresponding surface of ϕ(xk , p) = 0 is called an infinite-fixed-point surface in such a discrete system. Definition 5.8 Consider a discrete system as xk+1,i = xk,i + ϕ(xk , p1 )gi (xk , p) (i = 1, 2, . . . , n) (i)

(5.102)

An (n − 1)-dimensional surface of ϕ(xk , p1 ) = 0 is called an (n − 1)dimensional infinite-fixed-point surface if fixed-point xk∗ in Eq. (5.102) satisfies the following condition ϕ(xk∗ , p1 ) = 0.

(5.103)

(ii) The fixed-point xk∗ is called a simple fixed-point if ∗ ∗ xk∗ = xkα for xk∗ , xkα ∈ E and α = 1, 2, . . . ∗ ∗ with E = {xk |gi (xk , p) = 0, i = 1, 2, . . . , n}.

(5.104)

(iii) The fixed-point xk∗ is called a r-repeated fixed-point (or r th -order fixed-point) if ∗ ∗ ∗ = xk2 = . . . = xkr xk∗ = xk1 ∗ for xkα ∈ E and α = 1, 2, . . . , r

with E =

{xk∗ |gi (xk∗ , p)

(5.105)

= 0, i = 1, 2, . . . , n}.

(iv) The fixed-point xk∗ is called a simple fixed-point on the infinite-fixed-point surface if ⎫ ⎬ ϕ(xk∗ , p1 ) = 0, (5.106) gi (xk∗ , p) = 0, i = i 1 , i 2 , . . . , i n−1 ⎭ i α ∈ {1, 2, . . . , n} and α = 1, 2, . . . , n − 1.

5.3.1 One-Infinite-Fixed-Point Systems Consider a simple discrete system with one infinite-fixed-point as xk+1,1 = xk,1 + xk,2 , xk+1,2 = xk,2 + xk,1 xk,2 . The fixed-points are

(5.107)

272

5 Infinite-Fixed-Point Discrete Systems ∗ ∗ xk,1 ∈ (−∞, ∞), xk,2 = 0;

(5.108)

∗ ∗ xk,1 = 0, xk,2 = 0.

From Eq. (5.107), the variational equations at the fixed-points are xk+1,1 = xk,1 + xk,2 = G (1) j1 z k1 , j

xk+1,2 = =

1(z k ) ∗ ∗ xk,2 + xk,2 xk,1 + xk,1 xk,2 j1 j1 j2 (1) 1 (2) G j1 z k + 2 G j1 j2 z k z k 2(z ) 2(z z ) k

k

+ xk,1 xk,2

(5.109)

k

where j

z k = xk, j ( j = 1, 2), (1) (1) (1) ∗ ∗ G (1) 1(1) = 1, G 1(2) = 1; G 2(1) = x k,2 , G 2(2) = 1 + x k,1 ,

G (2) 2(11)

=

G (2) 2(22)

=

0, G (2) 2(12)

=

G (2) 2(21)

(5.110)

= 1.

∗ ∗ For (xk,1 , xk,2 ) = (a, 0), the variational equation is

⎡ ⎣

xk+1,1 xk+1,2

⎤

⎡

⎦=⎣

⎤⎡ 1

1

0 1+a

⎦⎣

xk,1 xk,2

⎤ ⎦.

(5.111)

Consider ⎡ ⎣

xk+1,1 xk+1,2

⎤

⎡

⎦ = λ⎣

xk,1 xk,2

⎤ ⎦.

(5.112)

Thus ⎡ ⎣

⎤⎡

1−λ

1

0

1+a−λ

⎦⎣

xk,1 xk,2

⎤ ⎦=0

(5.113)

The eigenvalues of Eq. (5.116) are given by      1 − λ 1 =0     0 1 + a − λ

(5.114)

That is, λ1 = 1 and λ2 = 1 + a

(5.115)

5.3 Infinite-Fixed-Point Systems

273

and the corresponding vectors are v1 = (1, 0)T and v2 = (1, a)T

(5.116)

(i) For λ1 = 1, on the vector of v1 = (1, 0)T , the fixed point of (a, 0) is invariant. (ii) For λ2 = 1 + a, on the vector of v2 = (1, a)T , the fixed point of (a, 0) is • a monotonic source if a > 0; • a monotonic invariance if a = 0 where the higher-order singularity should be considered; • a monotonic sink if a ∈ (−1, 0); • a zero-invariant sink if a = −1; • an oscillatory sink if a ∈ (−2, −1); • a flipped invariance if a = −2 where the higher-order singularity should be considered; • an oscillatory source if a ∈ (−∞, −2). ∗ ∗ For the fixed-point of (xk,1 , xk,2 ) = (0, 0), the variational equation is

xk+1,1 = xk,1 + xk,2 and xk+1,2 = xk,2 + xk,1 xk,2 .

(5.117)

(i) On the xk,1 -direction, ⎫ xk+1,1 = (1 + xk,2 /xk,1 )xk,1 > xk,1 if xk,1 > 0 ⎬ for xk,2 > 0 xk+1,1 = (1 + xk,2 /xk,1 )xk,1 > xk,1 if xk,1 < 0 ⎭ (5.118) If xk,2 > 0, xk+1,1 − xk,1 = xk,2 > 0. So the fixed-point of (0, 0) is a first-order upper-saddle. However, ⎫ xk+1,1 = (1 + xk,2 /xk,1 )xk,1 < xk,1 if xk,1 > 0 ⎬ for xk,2 < 0 xk+1,1 = (1 + xk,2 /xk,1 )xk,1 > xk,1 if xk,1 < 0 ⎭ (5.119) xk+1,1 − xk,1 = xk,2 < 0 if xk,2 < 0. So the fixed-point of (0, 0) is a first-order lower-saddle. (ii) On the xk,2 -direction, ⎫ xk+1,2 − xk,2 = xk,1 xk,2 < 0 if xk,1 < 0 ⎬ for xk,2 > 0 (5.120) xk+1,2 − xk,2 = xk,1 xk,2 > 0 if xk,1 > 0 ⎭

274

5 Infinite-Fixed-Point Discrete Systems

Thus, the fixed-point of (0, 0) is a second-order upper-saddle for xk,2 > 0. However, ⎫ xk+1,2 − xk,2 = xk,1 xk,2 > 0 if xk,1 < 0 ⎬ for xk,2 < 0 (5.121) xk+1,2 − xk,2 = xk,1 xk,2 < 0 if xk,1 > 0 ⎭ Thus, the fixed-point of (0, 0) is a second-order lower-saddle for xk,2 < 0 on the xk,2 -direction. Therefore, the fixed-point of (0, 0) is a second-order uppersaddle. ∗ ∗ , xk,2 ) = (−2, 0), the variational equation is For the fixed-point of (xk,1

xk+1,1 = xk,1 + xk,2 and xk+1,2 = −xk,2 + xk,1 xk,2 .

(5.122)

(i) On the xk,1 -direction, ⎫ xk+1,1 = (1 + xk,2 /xk,1 )xk,1 > xk,1 if xk,1 > 0 ⎬ for xk,2 > 0 xk+1,1 = (1 + xk,2 /xk,1 )xk,1 > xk,1 if xk,1 < 0 ⎭ (5.123) If xk,2 > 0, xk+1,1 − xk,1 = xk,2 > 0. So the fixed-point of (−2, 0) is a first-order upper-saddle. However, ⎫ xk+1,1 = (1 + xk,2 /xk,1 )xk,1 < xk,1 if xk,1 > 0 ⎬ for xk,2 < 0 xk+1,1 = (1 + xk,2 /xk,1 )xk,1 > xk,1 if xk,1 < 0 ⎭ (5.124) xk+1,1 − xk,1 = xk,2 < 0 if xk,2 < 0. So the fixed-point of (−2, 0) is a first-order upper-saddle. (ii) On the xk,2 -direction, ⎫ xk+1,2 + xk,2 = xk,1 xk,2 < 0 if xk,1 < 0 ⎬ for xk,2 > 0 (5.125) xk+1,2 + xk,2 = xk,1 xk,2 > 0 if xk,1 > 0 ⎭ Thus, the fixed-point of (−2, 0) is an oscillatory lower-saddle for xk,2 > 0. However, ⎫ xk+1,2 + xk,2 = xk,1 xk,2 > 0 if xk,1 < 0 ⎬ for xk,2 < 0 (5.126) xk+1,2 + xk,2 = xk,1 xk,2 < 0 if xk,1 > 0 ⎭

5.3 Infinite-Fixed-Point Systems

275

Thus, the fixed-point of (−2, 0) is an oscillatory lower-saddle for xk,2 < 0 on the xk,2 -direction. Therefore, the fixed-point of (−2, 0) is an oscillatory lower-saddle. As shown in Fig. 5.1a, the infinite-fixed-point of (a, 0) is a monotonic sink (a ∈ (−1, 0)), zero-sink (a = −1), oscillatory sink (a ∈ (−1, 0)), flipped (a = −2) with lower-saddle, and oscillatory source (a ∈ (−∞, −2)). The infinite-fixed point of (a, 0) is a monotonic source (a ∈ (0, ∞)) and an invariance with an uppersaddle (a = 0). The solid and dashed thick lines are for stable and unstable infinite∗ = 0, respectively. The corresponding eigenvalues are presented fixed-points of xk,2 through the unit circles. In Fig. 5.1b, the mapping points for such an infinite-fixedpoint system are presented through black circular symbols connected by red lines for mapping. The blue curves are for the mapping iteration references, which is based on the first integrals of the similar continuous dynamical system.

5.3.2 Two-Infinite-Fixed-Point Systems Consider a nonlinear discrete system with two infinite-fixed-points xk+1,1 = xk,1 + xk,2 (xk,1 − a1 )(xk,2 − b1 ), xk+1,2 = xk,2 − xk,1 (xk,1 − a1 )(xk,2 − b1 ).

(5.127)

where a1 > 0, b1 > 0. The one simple fixed-point and two infinite-fixed-points are ⎫ ∗ ∗ xk,1 = a1 , xk,2 ∈ (−∞, ∞), ⎬ for two infinite fixed points, x ∗ ∈ (−∞, ∞), x ∗ = b1 ; ⎭ k,1

∗ xk,1 ∗ xk,1

k,2

= =

∗ a1 , xk,2 = b1 ; for intersection fixed points, ∗ 0, xk,2 = 0 for simple fixed point,

(5.128)

⎫ ∗ ∗ xk,1 = 0, xk,2 = b1 ; ⎬ for simple fixed points. x ∗ = a1 , x ∗ = 0 ⎭ k,1

k,2

The stability of fixed-points is determined by the local analysis. The corresponding G-functions are as follows.

276

5 Infinite-Fixed-Point Discrete Systems

xk ,2

mSI

oSO

(a)

mUS

mSO

xk ,1

oSI

xk ,2

xk ,1

(b) Fig. 5.1 a Local analysis of fixed-points with eigenvalue charts for an infinite-fixed-point system, b Discrete trajectories. The solid and dashed thick lines are for stable and unstable infinite-fixedpoints, respectively. The circular symbols are for the monotonic upper-saddle (mUS) with invariance, oscillatory lower-saddle (oUS) with flip, and zero-sink with invariance (iSI). The corresponding eigenvalues are presented through the unit circles. The mapping points for such an infinitefixed-point system are presented by the black circular symbols connected by red lines for mapping. The blue curves are for the mapping iteration references, which is based on the first integrals of the similar discrete system. mSO: monotonic source; mSI: monotonic sink; oSI: oscillatory sink; oSO: oscillatory source

5.3 Infinite-Fixed-Point Systems

277

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ (1) ∗ ∗ ∗ ∗ G 1(2) = (xk,1 − a1 )(xk,2 − b1 ) + xk,2 (xk,1 − a1 ); ⎬ for the first order, ∗ ∗ ∗ ∗ ⎪ G (1) ⎪ 2(1) = −(x k,1 − a1 )(x k,2 − b1 ) − x k,1 (x k,2 − b1 ), ⎪ ⎪ ⎪ ⎭ ∗ ∗ G (1) 2(2) = −x k,1 (x k,1 − a1 ); ⎫ ⎪ ⎪ G (2) ⎪ 1(11) = 0, (5.129) ⎪ ⎪ ⎪ (2) (2) ∗ ∗ G 1(12) = G 1(21) = 2[(xk,2 − b1 ) + xk,2 ], ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ∗ G (2) = 2(x − a ), 1 k,1 1(22) for the second order. ∗ ⎪ ⎪ G (2) ⎪ 2(11) = −2(x k,2 − b1 ), ⎪ ⎪ ⎪ (2) ∗ ∗ ⎪ G (2) ⎪ 2(12) = G 2(21) = −[(x k,1 − a1 ) + x k,1 ], ⎪ ⎪ ⎪ (2) ⎭ G = 0.

∗ ∗ G (1) 1(1) = x k,2 (x k,2 − b1 ),

2(22)

∗ ∗ Consider the infinite-fixed-points of xk,1 = a1 , xk,2 ∈ (−∞, ∞). The corresponding G-functions and variational equations are given by

⎫ ∗ ∗ ⎪ G (1) 1(1) = x k,2 (x k,2 − b1 ), ⎪ ⎪ ⎪ ⎪ ⎬ G (1) = 0; 1(2)

∗ ⎪ G (1) ⎪ 2(1) = −a1 (x k,2 − b1 ), ⎪ ⎪ ⎪ ⎭ G (1) = 0; 2(2)

(5.130)

∗ ∗ xk+1,1 = xk,1 + G (1) 1(1) x k,1 = x k,1 + x k,2 (x k,2 − b1 )x k,1 , ∗ xk+1,2 = xk,2 + G (1) 2(1) x k,1 = x k,2 − a1 (x k,2 − b1 )x k,1 .

Thus, 

 ⎡ ∗ ∗ xk+1,1 (xk,2 − b1 ) 1 + xk,2 =⎣ xk+1,2 −a1 (x ∗ − b1 ) k,2

0 1

⎤ ⎦

 xk,1 . xk,2

(5.131)

For such a linearized equation, the corresponding eigenvalues are ∗ ∗ λ1 = 1 + xk,2 (xk,2 − b1 ) and λ2 = 1.

(5.132)

∗ ∈ (b1 , ∞), there exist For xk,2 ∗ ∗ λ1 = 1 + xk,2 (xk,2 − b1 ) ∈ (1, ∞) and λ2 = 1.

(5.133)

∗ ∈ (b1 , ∞) is a monotonic source. Such an infinite-fixed-point interval of xk,2 ∗ For xk,2 ∈ (0, b1 ), such an infinite-fixed-point is

278

(i)

5 Infinite-Fixed-Point Discrete Systems

a monotonic sink if ∗ ∗ (xk,2 − b1 ) ∈ (0, 1) and λ2 = 1, λ1 = 1 + xk,2 ∗ xk,2 ∈ (0, b1 ) for b1 ∈ (0, 2); ∗ xk,2 ∈ (0, 11 ) ∪ (12 , b1 ) with   12 = 21 (b1 + b12 − 4) and 11 = 21 (b1 − b12 − 4)

(5.134)

for b1 ∈ [2, ∞); (ii) a zero-invariant sink if ∗ ∗ λ1 = 1 + xk,2 (xk,2 − b1 ) = 0 and λ2 = 1,  ∗ = 21 (b1 ± b12 − 4) and b1 ∈ [2, ∞); xk,2

(5.135)

(iii) an oscillatory sink if ∗ ∗ λ1 = 1 + xk,2 (xk,2 − b1 ) ∈ (−1, 0) and λ2 = 1, ∗ xk,2 ∈ (11 , 21 ) ∪ (22 , 12 ) with   1 1 2 12 = 2 (b1 + b1 − 4) and 11 = 2 (b1 − b12 − 4)

for b1 ∈ [2, ∞),   22 = 21 (b1 + b12 − 8) and 21 = 21 (b1 − b12 − 8) √ for b1 ∈ [2 2, ∞);

(5.136)

(iv) a flipped invariance if ∗ ∗ λ1 = 1 + xk,2 (xk,2 − b1 ) = −1 and λ2 = 1,  √ ∗ = 21 (b1 ± b12 − 8) for b1 ∈ [2 2, ∞); xk,2

(5.137)

(v) an oscillatory source if ∗ ∗ λ1 = 1 + xk,2 (xk,2 − b1 ) ∈ (−∞, −1) and λ2 = 1, ∗ xk,2 ∈ (21 , 22 ) with   22 = 21 (b1 + b12 − 8) and 21 = 21 (b1 − b12 − 8) √ for b1 ∈ (2 2, ∞);

(5.138)

∗ For xk,2 ∈ (−∞, 0), we have ∗ ∗ λ1 = 1 + xk,2 (xk,2 − b1 ) ∈ (1, ∞) and λ2 = 1.

(5.139)

5.3 Infinite-Fixed-Point Systems

279

∗ Such an infinite-fixed-point interval of xk,2 ∈ (−∞, 0) is a monotonic source. ∗ ∗ ∈ (−∞, ∞), xk,2 = b1 . The correConsider the infinite-fixed-points of xk,1 sponding G-functions and variational equations are

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

G (1) 1(1) = 0, ∗ G (1) 1(2) = b1 (x k,1 − a1 );

G (1) 2(1) = 0, G (1) 2(2)

∗ ∗ = −xk,1 (xk,1 − a1 );

⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(5.140)

∗ xk+1,1 = xk,1 + G (1) 1(2) x k,2 = x k,1 + b1 (x k,1 − a1 )x k,2 , ∗ ∗ xk+1,2 = xk,2 + G (1) 2(2) x k,2 = x k,2 − x k,1 (x k,1 − a1 )x k,2 .

Thus, ⎡ ⎣

xk+1,1 xk+1,2

⎤

⎡

⎦=⎣

1

∗ b1 (xk,1 − a1 )

∗ ∗ 0 1 − xk,1 (xk,1 − a1 )

⎤⎡ ⎦⎣

xk,1 xk,2

⎤ ⎦

(5.141)

For such a linearized equation, the corresponding eigenvalues are ∗ ∗ λ1 = 1 and λ2 = 1 − xk,1 (xk,1 − a1 ).

(5.142)

∗ ∈ (a1 , ∞), such an infinite-fixed-point is: For xk,1

(i)

a monotonic sink if ∗ ∗ λ1 = 1 and λ2 = 1 − xk,1 (xk,1 − a1 ) ∈ (0, 1),  ∗ ∈ (a1 , 1 ) with 1 = 21 (a1 + a12 + 4); xk,1

(5.143)

(ii) a zero-invariant sink if ∗ ∗ (xk,1 − a1 ) = 0, λ1 = 1 and λ2 = 1 − xk,1  ∗ = 21 (a1 + a12 + 4); xk,1

(5.144)

(iii) an oscillatory sink if ∗ ∗ λ1 = 1 and λ2 = 1 − xk,1 (xk,1 − a1 ) ∈ (−1, 0),  ∗ ∈ (1 , 2 ) with 1 = 21 (a1 + a12 + 4), xk,1  2 = 21 (a1 + a12 + 8);

(5.145)

280

5 Infinite-Fixed-Point Discrete Systems

(iv) a flipped invariance if ∗ ∗ λ1 = 1 and λ2 = 1 − xk,1 (xk,1 − a1 ) = −1,  ∗ = 21 (a1 + a12 + 8); xk,1

(5.146)

(v) an oscillatory source if ∗ ∗ λ1 = 1 and λ2 = 1 − xk,1 (xk,1 − a1 ) ∈ (−∞, −1).  ∗ ∈ (2 , ∞) with 2 = 21 (a1 + a12 + 8). xk,1

(5.147)

∗ For xk,1 ∈ (0, a1 ), such an infinite-fixed-point is a monotonic source if ∗ ∗ λ1 = 1 and λ2 = 1 − xk,1 (xk,1 − a1 ) ∈ (1, ∞).

(5.148)

∗ ∈ (−∞, 0), such an infinite-fixed-point is For xk,1

(i)

a monotonic sink if ∗ ∗ λ1 = 1 and λ2 = 1 − xk,1 (xk,1 − a1 ) ∈ (0, 1),  ∗ ∈ (−3 , 0) with 3 = 21 ( a12 + 4 − a1 ); xk,1

(5.149)

(ii) a zero-invariant sink if ∗ ∗ λ1 = 1 and λ2 = 1 − xk,1 (xk,1 − a1 ) = 0,  ∗ = − 21 ( a12 + 4 − a1 ); xk,1

(5.150)

(iii) an oscillatory sink if ∗ ∗ λ1 = 1 and λ2 = 1 − xk,1 (xk,1 − a1 ) ∈ (−1, 0),  ∗ ∈ (−4 , −3 ) with 3 = 21 ( a12 + 4 − a1 ), xk,1  4 = 21 ( a12 + 8 − a1 );

(5.151)

(iv) a flipped invariance if ∗ ∗ (xk,1 − a1 ) = −1, λ1 = 1 and λ2 = 1 − xk,1  ∗ = − 21 ( a12 + 8 − a1 ); xk,1

(v) an oscillatory source if

(5.152)

5.3 Infinite-Fixed-Point Systems

281

∗ ∗ λ1 = 1 and λ2 = 1 − xk,1 (xk,1 − a1 ) ∈ (−∞, −1).  ∗ ∈ (−∞, −4 ) with 4 = 21 ( a12 + 8 − a1 ). xk,1

(5.153)

∗ ∗ For the fixed-point point of (xk,1 , xk,2 ) = (0, b1 ), the variational equation is

xk+1,1 = xk,1 + G (1) 1(2) x k,2 = x k,1 − a1 b1 x k.2 , (2) 1 xk+1,2 = xk,2 + 2 [G (2) 2(12) + G 2(21) ]x k,1 x k,2 = xk,2 + a1 xk,1 xk,2 ; where (1) (1) (1) G (1) 1(1) = 0, G 1(2) = −a1 b1 ; G 2(1) = 0, G 2(2) = 0; (1) (1) (1) G 2(11) = 0, G 2(12) = 2a1 , G 2(22) = 0.

(5.154)

(i) On the xk,1 -direction, xk+1,1 − xk,1 = −a1 b1 xk.2 > 0 for xk,2

⎫ < 0, ⎬

xk+1,1 − xk,1 = −a1 b1 xk.2 < 0 for xk,2 > 0 ⎭

(5.155)

∗ ∗ Thus, (xk,1 , xk,2 ) = (0, b1 ) is a lower-saddle of the first-order for xk,2 > 0, and an upper-saddle of the first-order for xk,2 < 0. (ii) On the xk,2 -direction, for xk,1 > 0,

⎫ xk+1,2 − xk,2 = a1 xk,1 xk,2 < 0 for xk,2 < 0, ⎬ xk+1,2 − xk,2 = a1 xk,1 xk,2 > 0 for xk,2 > 0. ⎭

(5.156)

∗ ∗ Thus, (xk,1 , xk,2 ) = (0, b1 ) is a second-order monotonic source on the xk,2 direction. For xk,1 < 0,

⎫ xk+1,2 − xk,2 = a1 xk,1 xk,2 > 0 for xk,2 < 0, ⎬ xk+1,2 − xk,2 = a1 xk,1 xk,2 < 0 for xk,2 > 0. ⎭

(5.157)

∗ ∗ Thus, (xk,1 , xk,2 ) = (0, b1 ) is a second-order monotonic sink. Therefore, the ∗ ∗ , xk,2 ) = (0, b1 ) is a second-order upper-saddle. fixed-point point of (xk,1 ∗ ∗ , xk,2 ) = (a1 , 0), the variational equation is For the fixed-point point of (xk,1

282

5 Infinite-Fixed-Point Discrete Systems (1) xk+1,1 = xk,1 + 21 [G (1) 1(12) + G 1(21) ]x k,1 x k,2 = xk,1 − b1 xk,1 xk,2 , xk+1,2 = xk,2 + G (1) 2(1) x k,1 = x k,2 + a1 b1 x k,1 ; where (1) G (1) 1(1) = 0, G 1(2) = 0; (1) (1) (1) G 1(11) = 0, G (1) 1(12) = G 1(21) = −b1 , G 1(22) = 0; (1) G (1) 2(1) = a1 b1 , G 2(2) = 0.

(5.158)

(i) On the xk,1 -direction, for xk,2 > 0, ⎫ xk+1,1 − xk,1 = −b1 xk,1 xk,2 > 0 for xk,1 < 0, ⎬ xk+1,1 − xk,1 = −b1 xk,1 xk,2 < 0 for xk,1 > 0 ⎭

(5.159)

∗ ∗ Thus (xk,1 , xk,2 ) = (a1 , 0) is a second-order monotonic sink. For xk,2 < 0,

⎫ xk+1,1 − xk,1 = −b1 xk,1 xk,2 < 0 for xk,1 < 0, ⎬ xk+1,1 − xk,1 = −b1 xk,1 xk,2 > 0 for xk,1 > 0 ⎭

(5.160)

∗ ∗ Thus (xk,1 , xk,2 ) = (a1 , 0) is a second-order monotonic source. (ii) On the xk,2 -direction,

⎫ xk+1,2 − xk,2 = a1 b1 xk,1 < 0 for xk,1 < 0, ⎬ xk+1,2 − xk,2 = a1 b1 xk,1 > 0 for xk,1 > 0. ⎭

(5.161)

∗ ∗ Thus, (xk,1 , xk,2 ) = (a1 , 0) is an upper-saddle of the first-order for xk,1 > ∗ ∗ ) = (a1 , 0) is a lower-saddle of the first-order for xk,1 < 0, and (xk,1 , xk,2 ∗ ∗ , xk,2 ) = (a1 , 0) is a second-order lower-saddle in the two 0. Therefore, (xk,1 directions. ∗ ∗ , xk,2 ) = (a1 , b1 ), the variational equation is For the fixed-point point of (xk,1

xk+1,1 − xk,1 = xk+1,2 − xk,2 =

1 [G (2) 1(12) 2! 1 [G (2) 2(12) 2!

+ G (2) 1(21) ]x k,1 x k,2 = b1 x k,1 x k,2 , (2) + G a(21) ]xk,1 xk,2 = −a1 xk,1 xk,2 ;

where (1) (1) (1) G (1) 1(1) = 0, G 1(2) = 0; G 2(1) = 0, G 2(2) = 0; (2) (2) (2) G (2) 1(11) = 0, G 1(12) = G 1(21) = b1 , G 1(22) = 0; (2) (2) (2) G (2) 2(11) = 0, G 2(12) = G 2(21) = −a1 , G 2(22) = 0.

(i) On the xk,1 -direction, for xk,2 > 0,

(5.162)

5.3 Infinite-Fixed-Point Systems

283

⎫ xk+1,1 − xk,1 = −b1 xk,1 xk,2 < 0 for xk,1 < 0, ⎬ xk+1,1 − xk,1 = −b1 xk,1 xk,2 > 0 for xk,1 > 0. ⎭

(5.163)

∗ ∗ Thus, (xk,1 , xk,2 ) = (a1 , b1 ) is a monotonic source of the second order. For xk,2 < 0,

xk+1,1 − xk,1 = −b1 xk,1 xk,2 > 0 for xk,1

⎫ < 0, ⎬

xk+1,1 − xk,1 = −b1 xk,1 xk,2 < 0 for xk,1 > 0. ⎭

(5.164)

∗ ∗ Thus, (xk,1 , xk,2 ) = (a1 , b1 ) is a monotonic sink of the second-order. (ii) On the xk,2 -direction, for xk,1 > 0,

xk+1,1 − xk,1 = −b1 xk,1 xk,2 > 0 for xk,2

⎫ < 0, ⎬

xk+1,1 − xk,1 = −b1 xk,1 xk,2 < 0 for xk,2 > 0. ⎭

(5.165)

∗ ∗ Thus, (xk,1 , xk,2 ) = (a1 , b1 ) is a source of the second order. For xk,1 < 0,

xk+1,1 − xk,1 = −b1 xk,1 xk,2 < 0 for xk,2

⎫ < 0, ⎬

xk+1,1 − xk,1 = −b1 xk,1 xk,2 > 0 for xk,2 > 0. ⎭

(5.166)

∗ ∗ Thus, (xk,1 , xk,2 ) = (a1 , b1 ) is a sink of the second order. ∗ ∗ , xk,2 ) = (a1 , b1 ) is an upper-saddle of the secondTherefore, the fixed-point of (xk,1 order along the diagonal direction. ∗ ∗ = 0, xk,2 = 0, we have For fixed-point of xk,1

xk+1,1 − xk,1 = G (1) 1(2) x k,2 = a1 b1 x k,2 , xk+1,2 − xk,2 = G (1) 2(1) x k,1 = −a1 b1 x k,1 ; where G (1) 1(1) G (1) 2(1)

=

(5.167) 0, G (1) 1(2)

= a1 b1 ;

= −a1 b1 , G (1) 2(2) = 0.

The linearized equation is ⎡ ⎣

xk+1,1 xk+1,2

⎤

⎡

⎦=⎣

⎤⎡ 1

a1 b1

−a1 b1

1

⎦⎣

xk,1 xk,2

⎤ ⎦

For such a linearized equation, the corresponding eigenvalues are

(5.168)

284

5 Infinite-Fixed-Point Discrete Systems

Table 5.1 Summarization of fixed-point stability for Eq. (5.127) Infinite Infinite Simple

x1 ∈ (a1 , ∞), x2 = b1

x1 ∈ (0, a1 ), x2 = b1

x1 ∈ (−∞, 0), x2 = b1

mSI-oSO

mSO

mSI-oSO

x1 = a1 , x2 ∈ (b1 , ∞),

x1 = a1 , x2 ∈ (0, b1 )

x1 = a1 , x2 ∈ (−∞, 0)

mSO

mSI-oSO-mSI

mSO

x1 = a1 , x2 = 0,

x1 = 0, x2 = b1

x1 = a1 , x2 = b1

mLS

mUS

mUS

Infinite: Infinite-fixed-point. Simple: Simple fixed-point. mSI-oSO: monotonic sink to oscillatory source, mSO: monotonic source, mUS: monotonic upper-saddle, mLS: monotonic lower-saddle. mSI-oSO-mSI: monotonic sink to oscillatory source to monotonic sink

λ1,2 = 1 ± a1 b1 i.

(5.169)

Thus, this fixed-point is an unstable focus. From the foregoing local analysis of fixed-points, the stability of fixed-points are summarized in Table 5.1. The fixed-point of (0, 0) is an unstable focus, which is not included. The rich dynamics in the infinite-fixed-point discrete systems can be obtained. Such fixed-points possess higher-order singularity, which cannot be analyzed by the traditional eigenvalue analysis. From the discrete systems in Eq. (5.127), the different domains are separated by the infinite-fixed-points, as tabulated in Table 5.1, and the illustrations of the infinite-fixed point stability are presented in Fig. 5.2. From the local analysis of fixed-point stability, the discrete trajectories with of fixed-points are presented in Fig. 5.3. The solid and dashed thick lines are for monotonic sink and monotonic source infinite-fixed-points, respectively. The hollow circular symbols are for simple and intersection fixed-points. The discrete trajectories are presented through by black circular symbols and red lines. The blue circular curves are for the mapping iteration references, which is based on the first integrals of the similar continuous dynamical system. The filled circular symbol is for the monotonic upper-saddle of the second order in two directions. The hollow circular symbols are for lower-saddle and upper-saddles in one direction.

5.3.3 Higher-Order Infinite-Fixed-Point Systems Consider a discrete system with an infinite-fixed-point of the second-order singularity as 2 , xk+1,2 = xk,2 + xk,1 xk,2 sin xk,1 xk+1,1 = xk,1 − xk,1 xk,2

where a1 > 0, b1 > 0.

(5.170)

5.3 Infinite-Fixed-Point Systems

285

xk ,2 mSO mSI-oSO

mUS

mSO

xk ,2 = b1

mUS mSI-oSO

mSI-oSO-mSI UF

mLS

xk ,1

mSO

xk ,1 = a1 Fig. 5.2 Local analysis of fixed-points for the first integral manifolds. The solid and dashed thick lines are for sink and source infinite-fixed-points, respectively. The filled circular symbol is for the intersection fixed-point. The hollow circular symbols are for the simple fixed-points. mSI-oSO: monotonic sink to oscillatory source, mSO: monotonic source, mUS: monotonic upper-saddle, mLS: monotonic lower-saddle. mSI-oSO-mSI: monotonic sink to oscillatory source to monotonic sink. UF: unstable focus

The simple fixed-points and infinite-fixed-points of the nonlinear discrete system in Eq. (5.170) are ∗ ∗ xk,1 = 0, xk,2 ∈ (−∞, ∞); ∗ ∗ xk,1 ∈ (−∞, ∞), xk,2 =0

 for infintite-fixed-points,

∗ ∗ = 0, xk,2 = 0for intersection fixed-point, xk,1

(5.171)

∗ ∗ xk,1 = ±lπ (l = 1, 2, . . .), xk,2 = 0for simple fixed-points. ∗ ∗ For the infinite-fixed-points of xk,1 = 0, xk,2 = 0 the variational equation is ∗ xk+1,1 − xk,1 = −xk,1 xk,2 xk,2 , ∗ ∗ xk+1,2 − xk,2 = xk,1 sin xk,1 xk,2 .

(5.172)

∗ On the x1 -direction, xk+1,1 − xk,1 < 0 for xk,1 > 0 which is a monotonic lower∗ saddle, and xk+1,1 − xk,1 > 0 for xk,1 < 0 which is a monotonic upper-saddle. ∗ ∗ On the xk,2 -direction, the fixed-point is a monotonic source for xk,1 sin xk,1 > 0, and

286

5 Infinite-Fixed-Point Discrete Systems

xk ,2 xk ,2 = b1

xk ,1

xk ,1 = a1 Fig. 5.3 Discrete trajectories for an infinite-fixed-point system in Eq. (5.127). The solid and dashed thick lines are for sink and source infinite-fixed-points, respectively. The filled circular symbol is for the intersection fixed-point. The hollow circular symbols are for simple and intersection fixedpoints. The mapping points for such an infinite-fixed-point system are presented by the black circular symbols connected by red lines for mapping. The blue circular curves are for mapping iteration references, which is based on the first integrals of the similar continuous dynamical system. mSIoSO: monotonic sink to oscillatory source, mSO: monotonic source, mUS: monotonic upper-saddle, mLS: monotonic lower-saddle. mSI-oSO-mSI: monotonic sink to oscillatory source to monotonic sink ∗ ∗ ∗ a monotonic sink for xk,1 sin xk,1 < 0. Thus, the infinite-fixed-point for xk,2 = 0 and ∗ ∗ xk,1 sin xk,1 = 0 are named monotonic sink and monotonic sources. ∗ ∗ = 0, xk,2 = 0, the variational equation is For the infinite-fixed-point of xk,1 ∗ 2 ) xk,1 , xk+1,1 − xk,1 = −(xk,2 ∗ ∗ ∗ xk+1,2 − xk,2 = xk,2 cos xk,1 xk,1 xk,1 = xk,2 xk,1 xk,1 .

(5.173)

On the xk,1 -direction, the fixed-point is a monotonic sink. On the xk,2 -direction, xk+1,2 − xk,2 > 0 for x2∗ > 0 which is a monotonic upper-saddle, and xk+1,2 − xk,2 < 0 for x2∗ < 0 which is a monotonic lower-saddle. Thus, the infinite-fixed∗ ∗ = 0, xk,2 = 0 is named a monotonic sink with different type on the point of xk,1 xk,2 -direction. ∗ ∗ = 0, xk,1 = ±mπ the variational equation is For the simple fixed-points of xk,2 ∗ xk+1,1 − xk,1 = −xk,1 xk,2 xk,2 , ∗ ∗ xk+1,2 − xk,2 = xk,1 cos xk,1 xk,1 xk,2 .

(5.174)

5.3 Infinite-Fixed-Point Systems

287

∗ On the xk,1 -direction, xk+1,1 − xk,1 < 0 for xk,1 > 0 which is a monotonic ∗ lower-saddle, and xk+1,1 − xk,1 > 0 for xk,1 < 0 which is a monotonic uppersaddle. On the x2 -direction, we have the following four cases: ∗ ∗ > 0 for xk,1 cos xk,1 > 0 with xk,1 > 0, which is a monotonic

xk+1,2 −xk,2 source. (ii) xk+1,2 −xk,2 sink. (iii) xk+1,2 −xk,2 sink. (iv) xk+1,2 −xk,2 source. (i)

∗ ∗ < 0 for xk,1 cos xk,1 > 0 with xk,1 < 0, which is a monotonic ∗ ∗ < 0 for xk,1 cos xk,1 < 0 with xk,1 > 0, which is a monotonic ∗ ∗ > 0 for xk,1 cos xk,1 < 0 with xk,1 < 0, which is a monotonic

Thus, the fixed-point based on the xk,1 -direction is called a monotonic lower-saddle or upper-saddle of the second-order. ∗ ∗ , xk,2 ) = (0, 0), the variational equation is For the fixed-point of (xk,1 xk+1,1 − xk,1 = −xk,1 xk,2 xk,2 ,

(5.175)

xk+1,2 − xk,2 = xk,1 xk,1 xk,2 .

∗ ∗ On the xk,1 -direction, (xk,1 , xk,2 ) = (0, 0) is a monotonic sink of the third-order. ∗ ∗ On the xk,2 -direction, (xk,1 , xk,2 ) = (0, 0) is a monotonic source of the third-order. ∗ ∗ , xk,2 ) = (0, 0) is named a monotonic center of the third-order herein. (xk,1 From the foregoing local analysis, the stability of fixed-points is tabulated in Table 5.2. From the local analysis, the stability of fixed-points are presented in Fig. 5.4a. The solid and dashed thick lines are for sink and source infinite-fixed-points, respectively. The hollow circular symbols are for switching saddles. The filled circular symbol is

Table 5.2 Summarization of fixed-point stability for Eq. (5.170) ∗ =0 xk,2

x2∗ = 0 x1∗

=0

x1∗ ∈ (4π, 3π )

x1∗ ∈ (3π, 2π )

x1∗ ∈ (2π, π )

x1∗ ∈ (π, 0)

mSI-oSO-mSI

mSO

mSI-oSO-mSI

mSO

x1∗ ∈ (−π, 0)

x1∗ ∈ (−2π, −π )

x1∗ ∈ (−3π, −2π )

x1∗ ∈ (−4π, −3π )

mSO

mSI-oSO-mSI

mSO

mSI-oSO-mSI

x2∗

x2∗

∈ (0, ∞)

mSI-oSO x2∗ = 0 x2∗ = 0

∈ (−∞, 0)

mSI-oSO

x1∗ = 2mπ

x1∗ = (2m − 1)π

x1∗ = −2mπ

x1∗ = −(2m − 1)π

mLS

mLS

mUS

mUS

x1∗ = 0 Center with 3rd order mSI and mSO

mSI-oSO-mSI: monotonic sink to oscillatory source to monotonic sink, mSO: monotonic source, mUS: monotonic upper-saddle, mLS: monotonic lower-saddle. mSI-oSO: monotonic sink to oscillatory source

288

5 Infinite-Fixed-Point Discrete Systems

xk ,2 @: mSI-oSO-mSI mSI-oSO

@

mUS

mUS

mUS mSO

@

mLS mSO

mSO

mLS @

mLS mSO

@

xk ,1

mSI-oSO

(a)

xk ,2

xk ,1

(b) Fig. 5.4 a Fixed-point stability for an infinite-fixed-point system, b discrete trajectories. The solid and dashed thick lines are for sink and source infinite-fixed-points, respectively. The filled circular symbol is for the center with third-order sink and source. The hollow circular symbols are for monotonic upper-saddle (mUS) and monotonic lower-saddle (mLS) on the infinite-fixed-point line. mSIoSO-mSI: monotonic sink to oscillatory source to monotonic sink, mSO: monotonic source, mUS: monotonic upper-saddle, mLS: monotonic lower-saddle. mSI-oSO: monotonic sink to oscillatory source

for the center with third-order sink and source. In Fig. 5.4b, the discrete trajectories are presented through by black circular symbols and red lines. The blue circular curves are for the mapping iteration references, which is based on the first integrals of the similar continuous dynamical system.

5.4 Infinite-Fixed-Point-Network Systems

289

5.4 Infinite-Fixed-Point-Network Systems In this section, nonlinear discrete systems with a network of infinite-fixed-points are discussed. Definition 5.9 Consider a discrete system as xk+1,i = xk,i + ϕ1 (xk , p1 )ϕ2 (xk , p2 ) . . . ϕl (xk , pl )gi (xk , p) (i = 1, 2, . . . , n), ri ≥ 1.

(5.176)

An (n − 1)-dimensional surface network of ϕ j (xk , p j ) = 0 ( j = 1, 2, . . . , l) in phase space is called an infinite-fixed-point network of discrete system in Eq. (5.176). (ii) An (n − 1)-dimensional surface of ϕ j (xk , p j ) = 0 ( j ∈ {1, 2, . . . , l}) is called an (n − 1)-dimensional infinite-fixed-point surface if the fixed-point xk∗ of Eq. (5.176) satisfies the following condition, i.e., (i)

ϕ j (xk∗ , p j ) = 0 for j ∈ {1, 2, . . . , l}.

(5.177)

(iii) An (n−2)-dimensional intersection edge of ϕ j1 (xk , p j1 ) = 0 and ϕ j2 (xk , p j2 ) = 0 is called an (n − 2)-dimensional infinite-fixed-point edge if the fixed-point xk∗ of Eq. (5.176) satisfies the following conditions, i.e., ϕ j1 (xk∗ , p j1 ) = 0, ϕ j2 (xk∗ , p j2 ) = 0 ϕ j1 (xk , p j1 ) = ϕ j2 (xk , p j2 ) and Dϕ j1 (xk , p j1 ) = Dϕ j2 (xk , p j2 ) for j1 , j2 ∈ {1, 2, . . . , l}.

(5.178)

(iv) An (n −r )-dimensional intersection edge of ϕ jα (xk , p jα ) = 0 (α = 1, 2, . . . , r ) is called an (n − r )-dimensional infinite-fixed-point edge if the fixed-point xk∗ of Eq. (5.176) satisfies the following condition, i.e., ϕ j1 (xk∗ , p j1 ) = 0, ϕ j2 (xk∗ , p j2 ) = 0, . . . , ϕ jr (xk∗ , p jr ) = 0, ϕ jα (xk , p jα ) = ϕ jβ (xk , p jβ ) and Dϕ jα (xk , p jα ) = Dϕ jβ (xk , p jβ )

(5.179)

for j1 , j2 , . . . , jr ∈ {1, 2, . . . , l}; α, β ∈ {1, 2, . . . , r }, α = β. (v) An intersection vertex of ϕ jα (xk , p jα ) = 0 (α = 1, 2, . . . , n) is called an infinite-fixed-point vertex if the fixed-point xk∗ of Eq. (5.176) satisfies the following condition, i.e., ϕ j1 (xk∗ , p j1 ) = 0, ϕ j2 (xk∗ , p j2 ) = 0, . . . , ϕ jn (xk∗ , p jn ) = 0, ϕ jα (xk , p jα ) = ϕ jβ (xk , p jβ ) and Dϕ jα (xk , p jα ) = Dϕ jβ (xk , p jβ ) for j1 , j2 , . . . , jn ∈ {1, 2, . . . , l}, α, β ∈ {1, 2, . . . , n}, α = β.

(5.180)

290

5 Infinite-Fixed-Point Discrete Systems

(vi) An (n − 1)-dimensional surface of ϕ j (xk , p j ) = 0 ( j ∈ {1, 2, . . . , l}) in phase space is called a r th -order, (n − 1)-dimensional, infinite-fixed-point if ϕ j (xk , p j ) ≡ ϕ j1 (xk , p j1 ) = . . . = ϕ jr (xk , p jr ) for j1 , j2 , . . . , jr ∈ {1, 2, . . . , l}.

(5.181)

5.4.1 An Infinite-Fixed-Point-Network System To demonstrate the infinite-fixed-point network discrete systems, consider a discrete system of four infinite-fixed-points, i.e., xk+1,1 = xk,1 + xk,2 (xk,1 − a1 )(xk,2 − b1 )(xk,1 + a1 )(xk,2 + b1 ), xk+1,2 = xk,2 − xk,1 (xk,1 − a1 )(xk,2 − b1 )(xk,1 + a1 )(xk,2 + b1 ).

(5.182)

where a1 > 0, b1 > 0. Such four infinite-fixed-points form a network of the infinite-fixed-point in phase space. The discrete behaviors of the discrete systems are separated in different subdomains. The simple fixed-points and infinite-fixed-points are ⎫ ∗ ∗ = a1 , xk,2 ∈ (−∞, ∞); ⎪ xk,1 ⎪ ⎪ ⎪ ⎪ ∗ ∗ x = −a , x ∈ (−∞, ∞); ⎬ k,1 ∗ xk,1 ∗ xk,1

1

k,2

∗ ⎪ ∈ (−∞, ∞), xk,2 = b1 ; ⎪ ⎪ ⎪ ⎪ ⎭ ∗ ∈ (−∞, ∞), xk,2 = −b1

for infintite-fixed-points, ∗ xk,1 ∗ xk,1

= =

∗ a1 , xk,2 ∗ a1 , xk,2

= =

⎫

∗ ∗ b1 ; xk,1 = −a1 , xk,2 = b1 ; ⎬ ∗ ∗ −b1 ; xk,1 = −a1 , xk,2 = −b1 ⎭

(5.183)

for intersection fixed-points, ∗ ∗ xk,1 = 0, xk,2 = 0, for a simple fixed-point,

⎫ ∗ ∗ ∗ ∗ xk,1 = 0, xk,2 = b1 ; xk,1 = 0, xk,2 = −b1 ; ⎬ x ∗ = a1 , x ∗ = 0; x ∗ = −a1 , x ∗ = 0 ⎭ k,1

k,2

k,1

k,2

for simple fixed-points. ∗ ∗ = ±a1 , xk,2 ∈ (−∞, ∞), we have For xk,1

⎡ ⎣

xk+1,1 xk+1,2

⎤

⎡

⎦=⎣

1

∗ ∗2 ± 2a1 xk,2 (xk,2 − b12 ) ∗2 ∓2a12 (xk,2 − b12 )

⎤⎡ 0 1

⎦⎣

xk,1 xk,2

⎤ ⎦

(5.184)

5.4 Infinite-Fixed-Point-Network Systems

291

The corresponding eigenvalues are ∗ ∗2 (xk,2 − b12 ), λ2 = 1. λ1 = 1 ± 2a1 xk,2

(5.185)

∗ ∗ = ±a1 , xk,2 ∈ (−∞, ∞) is Therefore, the infinite-fixed point of xk,1

• • • • • •

∗ ∗2 a monotonic source for ±2a1 xk,2 (xk,2 − b12 ) ∈ (1, ∞), ∗ ∗2 a monotonic sink for ±2a1 xk,2 (xk,2 − b12 ) ∈ (−1, 0), ∗ ∗2 a zero-invariant sink for ±2a1 xk,2 (xk,2 − b12 ) = −1, ∗ ∗2 an oscillatory sink for ±2a1 xk,2 (xk,2 − b12 ) ∈ (−2, −1), ∗ ∗2 a flipped invariance for ±2a1 xk,2 (xk,2 − b12 ) = −2, ∗ ∗2 an oscillatory source for ±2a1 xk,2 (xk,2 − b12 ) ∈ (−∞, −2). ∗ ∗ For xk,1 ∈ (−∞, ∞), xk,2 = ±b1 , we have

⎡ ⎣

xk+1,1 xk+1,2

⎤

⎡

⎦=⎣

⎤⎡

∗2 − a12 ) ±2b12 (xk,1

1

∗ ∗2 0 1 ∓ 2b1 xk,1 (xk,1 − a12 )

⎦⎣

xk,1 xk,2

⎤ ⎦

(5.186)

The corresponding eigenvalues are ∗ ∗2 λ1 = 1, λ2 = 1 ∓ 2b1 xk,1 (xk,1 − a12 ).

(5.187)

∗ ∗ ∈ (−∞, ∞), xk,2 = ±b1 is Therefore, the infinite-fixed point of xk,1

• • • • • •

∗ ∗2 a monotonic source for ∓2b1 xk,1 (xk,1 − a12 ) ∈ (1, ∞), ∗ ∗2 a monotonic sink for ∓2b1 xk,1 (xk,1 − a12 ) ∈ (−1, 0), ∗ ∗2 (xk,1 − a12 ) = −1, a zero-invariant sink for ∓2b1 xk,1 ∗ ∗2 an oscillatory sink for ∓2b1 xk,1 (xk,1 − a12 ) ∈ (−2, −1), ∗ ∗2 a flipped invariance for ∓2b1 xk,1 (xk,1 − a12 ) = −2, ∗ ∗2 an oscillatory source for ∓2b1 xk,1 (xk,1 − a12 ) ∈ (−∞, −2). ∗ ∗ For xk,1 = 0 and xk,2 = 0, we have

⎡ ⎣

xk+1,1 xk+1,2

⎤

⎡

⎦=⎣

⎤⎡ 1

a12 b12

−a12 b12

1

⎦⎣

xk,1 xk,2

⎤ ⎦.

(5.188)

The corresponding eigenvalues are λ1 = 1 − a12 b12 i, λ2 = 1 + a12 b12 i with i =

√

−1.

(5.189)

∗ ∗ The simple fixed point of (xk,1 , xk,2 ) = (0, 0) is a monotonic unstable focus. ∗ ∗ , xk,2 ) = (α1 , β1 ), we have For the intersection fixed-point of (xk,1

292

5 Infinite-Fixed-Point Discrete Systems

xk+1,1 = (1 + β1 (α1 − α2 )(β1 − β2 )xk,2 )xk,1 , xk+2,1 = (1 + α1 (α1 − α2 )(β1 − β2 )xk,1 )xk,2

(5.190)

where xk,1 = xk,1 − α1 , xk+1,1 = xk+1,1 − α1 ; xk,2 = xk,1 − β1 , xk+1,2 = xk+1,2 − β1 ; α1 , α2 ∈ {a1 , −a1 }; β1 , β2 ∈ {b1 , −b1 }; α1 = α2 , β1 = β2 . (i)

(5.191)

On the xk,1 -direction, if β1 (α1 − α2 )(β1 − β2 )xk,2 > 0,

(5.192)

then xk+1,1 > xk,1 > 0, and xk+1,1 < xk,1 < 0.

(5.193)

(ii) On the xk,1 -direction, if β1 (α1 − α2 )(β1 − β2 )xk,2 < 0,

(5.194)

then 0 < xk+1,1 < xk,1 , and 0 > xk+1,1 > xk,1

(5.195)

(iii) On the xk,2 -direction, if α1 (α1 − α2 )(β1 − β2 )xk,1 > 0,

(5.196)

then xk+1,2 > xk,2 > 0, and xk+1,2 < xk,2 < 0.

(5.197)

(iv) On the xk,2 -direction, if α1 (α1 − α2 )(β1 − β2 )xk,1 < 0,

(5.198)

then 0 < xk+1,2 < xk,2 , and 0 > xk+1,2 > xk,2 .

(5.199)

5.4 Infinite-Fixed-Point-Network Systems

293

∗ ∗ Therefore, the intersection fixed point (xk,1 , xk,2 ) = (α1 , β1 ) is

• a monotonic upper-saddle of the second order if xk+1,2 > xk,2 , • a monotonic lower-saddle of the second order if xk+1,2 < xk,2 . ∗ ∗ For the intersection fixed-point of (xk,1 , xk,2 ) = (0, β1 ), we have

⎡ ⎣

xk+1,1 xk+1,2

⎤

⎡

⎦=⎣

⎤⎡

1 01

⎤

−β1 a12 (β1 − β2 ) x ⎦⎣ k,1 ⎦ 2 + a1 (β1 − β2 )xk,1 xk,2

(5.200)

where xk,1 = xk,1 − 0, xk+1,1 = xk+1,1 − 0; xk,2 = xk,1 − β1 , xk+1,2 = xk+1,2 − β1 ; β1 , β2 ∈ {b1 , −b1 }; β1 = β2

(5.201)

and the corresponding eigenvalues are λ1 = 1, λ2 = 1 + a12 (β1 − β2 )xk,1 .

(5.202)

(i) On the xk,2 -direction, if a12 (β1 − β2 ) > 0,

(5.203)

∗ ∗ , xk,2 ) = (0, β1 ) is a monotonic upperthen the intersection fixed-point of (xk,1 saddle of the second order. (ii) On the xk,2 -direction, if

a12 (β1 − β2 ) < 0,

(5.204)

∗ ∗ , xk,2 ) = (0, β1 ) is a monotonic lowerthen the intersection fixed-point of (xk,1 saddle of the second order. ∗ ∗ , xk,2 ) = (α1 , 0), we have For the intersection fixed-point of (xk,1

⎡ ⎣

where

xk+1,1 xk+1,2

⎤

⎡

⎦=⎣

1 − b12 (α1 − α2 )xk,2 0 α1 b12 (α1 − α2 )

1

⎤⎡ ⎦⎣

xk,1 xk,2

⎤ ⎦

(5.205)

294

5 Infinite-Fixed-Point Discrete Systems

xk,1 = xk,1 − α1 , xk+1,1 = xk+1,1 − α1 ; xk,2 = xk,1 − 0, xk+1,2 = xk+1,2 − 0;

(5.206)

α1 , α2 ∈ {a1 , −a1 }; a1 = a2 . and the corresponding eigenvalues are λ1 = 1 − b12 (α1 − α2 )xk,2 , λ2 = 1.

(5.207)

(i) On the xk,1 -direction, if −b12 (α1 − α2 ) > 0

(5.208)

∗ ∗ , xk,2 ) = (α1 , 0) is a monotonic upperthen the intersection fixed-point of (xk,1 saddle of the second order. (ii) On the xk,1 -direction, if

−b12 (α1 − α2 ) < 0

(5.209)

∗ ∗ , xk,2 ) = (α1 , 0) is a monotonic lowerthen the intersection fixed-point of (xk,1 saddle of the second order.

The local analysis of fixed-points is completed, and the stability of fixed-points is summarized in Table 5.3. From the local analysis, the fixed-point stability is presented in Fig. 5.5a. The solid and dashed thick lines are for monotonic sink and monotonic source infinite-fixedpoints, respectively. The filled circular symbol is for the monotonic upper-saddle of the second order in two directions. The hollow circular symbols are for monotonic lower-saddle and monotonic upper-saddles in one direction. The network of four infinite-fixed-points has nine (9) subdomains separated by the four infinite-fixedpoints. In Fig. 5.5b, discrete trajectories are presented through by black circular symbols and red lines. The blue circular curves are for the mapping iteration references, which is based on the first integrals of the similar continuous dynamical system.

5.4.2 A Spiral Infinite-Fixed-Point System Consider a discrete system of a circular infinite-fixed-point with spiral mapping trajectories, i.e., 2 2 xk+1,1 = xk,1 + (xk,1 + xk,2 − R 2 )m xk,2 , 2 2 xk+1,2 = xk,2 − (xk,1 + xk,2 − R 2 )m xk,1 .

(5.210)

5.4 Infinite-Fixed-Point-Network Systems

295

Table 5.3 Summarization of fixed-point stability for Eq. (5.182) ∗ =b xk,2 1 ∗ xk,2 ∗ xk,1 ∗ xk,1 ∗ xk,1

= −b1 = a1 = −a1 = a1

∗ =0 xk,1 ∗ xk,1

= −a1

∗ ∈ (a , ∞), xk,1 1

∗ ∈ (0, a ) xk,1 1

∗ ∈ (−a , 0), xk,1 1

∗ ∈ (−∞, −a ) xk,1 1

mSI-oSO

mSO

mSI-oSO-mSI

mSO

∗ xk,1

∗ xk,1

∗ xk,1

∗ ∈ (−∞, −a ) xk,1 1

∈ (a1 , ∞),

∈ (0, a1 )

∈ (−a1 , 0),

mSO

mSI-oSO-mSI

mSO

mSI-oSO

∗ xk,2

∗ xk,2

∗ xk,2

∗ ∈ (−∞, −b ) xk,2 1

∈ (b1 , ∞),

∈ (0, b1 )

∈ (−b1 , 0)

mSO

mSI-oSO-mSI

mSO

mSI-oSO

∗ xk,2

∗ xk,2

∗ xk,2

∗ ∈ (−∞, −b ) xk,2 1

∈ (b1 , ∞),

∈ (0, b1 )

∈ (−b1 , 0)

mSI

mSO

mSI-oSO-mSI

∗ xk,2

∗ xk,2

∗ xk,2

= b1 ,

=0

= −b1

mUS

mLS

mUS

∗ =b , xk,2 1

∗ =0 xk,2

∗ = −b xk,2 1

mUS

UF

mLS

∗ xk,2

∗ xk,2

= b1 ,

mLS

=0

mUS

mSO

∗ = −b xk,2 1

mLS

mSI-oSO-mSI: monotonic sink to oscillatory source to monotonic sink, mSO: monotonic source, mUS: upper-saddle, mLS: monotonic lower-saddle, UF: unstable focus. mSI-oSO: monotonic sink to oscillatory source

where R > 0. The simple fixed-points and infinite-fixed-points are ∗2 ∗2 xk,1 + xk,2 = R 2 for a infintite-fixed-point circle; ⎫ ∗ ∗ xk,1 = ±R, xk,2 = 0; ⎬ for intersection fixed-points; x ∗ = 0, x ∗ = ±R ⎭ k,1

(5.211)

k,2

∗ ∗ xk,1 = 0, xk,2 = 0 for a simple fixed-point.

Let xk,1 = ρk cos θk , xk,2 = ρk sin θk ; xk+1,1 = ρk+1 cos θk+1 , xk+1,2 = ρk+1 sin θk+1 .

(5.212)

Equation (5.210) becomes ρk+1 cos θk+1 = ρk cos θk + (ρk2 − R 2 )m ρk sin θk , ρk+1 sin θk+1 = ρk sin θk − (ρk2 − R 2 )m ρk cos θk .

(5.213)

296

5 Infinite-Fixed-Point Discrete Systems

xk ,2

@:mSI-oSO-mSI

mSI-oSO

mSO mUS

mLS

mSO

@

mUS mSO

mSO

mSI-oSO

@

mUS

mLS

UF

xk ,1 mSO

@ mSO

mSI-oSO

mLS

mLS mSO

(a)

@ mUS

mSO

mSI-oSO

xk ,2 xk ,1 = − a1

xk ,1 = a1

xk ,2 = b1

xk ,1

xk ,2 = −b1

(b) Fig. 5.5 a Fixed-point stability for an infinite-fixed-point-network discrete system, b discrete trajectories. The solid and dashed thick lines are for sink and source infinite-fixed-points, respectively. The circles are for the first integral manifolds. The filled circular symbol is for the intersection fixed-point. The hollow circular symbols are for simple fixed-points. mSI: monotonic sink. mSO: monotonic source. mUS: monotonic upper-saddle. mLS: monotonic lower-saddle

Thus yk+1 = yk + yk2m+1 + R 2 yk2m with 2 yk+1 = ρk+1 − R 2 = (ρk+1 + R)(ρk+1 − R)

and yk =

ρk2

− R = (ρk + R)(ρk − R); 2

(5.214)

5.4 Infinite-Fixed-Point-Network Systems

297

and tan θk+1 =

tan θk − (ρk2 − R 2 )m . 1 + (ρk2 − R 2 )m tan θk

(5.215)

Therefore for ρk∗ = R, we have θk∗ ∈ [0, 2π ). From Eq. (5.213), for ρk∗ = 0, we have θk∗ ∈ [0, 2π ). The infinite-fixed-point is at ρk∗ = R and θk∗ ∈ [0, 2π ). From Eq. (5.214), we have yk+1 = (1 + R 2 yk2m−1 )yk .

(5.216)

If yk < 0, 0 > yk+1 > yk . If yk > 0, 0 < yk < yk+1 . Thus the infinite-fixed-point of ρk∗ = R and θk∗ ∈ [0, 2π ) is the monotonic upper-saddle of the ∗2 ∗2 + xk,2 = R 2 is a monotonic (2m)th -order. Therefore, the infinite-fixed-point of xk,1 th upper-saddle of the (2m) -order. The fixed-point is at ρk∗ = 0. From Eq. (5.214), we have yk+1 = (1 + R 2m )yk ,

(5.217)

Thus, the fixed-point of ρk∗ = 0 is a monotonic source in the ρk -direction. Therefore, with θk ∈ [0, 2π ), the fixed-point at xk∗ = 0 is a monotonic unstable focus. For the infinite-fixed-point of ρk∗ = R and θk∗ ∈ [0, 2π ), the local analysis gives xk+1,1 = xk,1 + (yk )m R sin θk , xk+1,2 = xk,2 − (yk )m R cos θk

(5.218)

yk = ρk2 − R 2 − (ρk∗2 − R 2 ) = ρk2 − R 2 .

(5.219)

where

(i) For m = 2l − 1, when yk < 0, θk to θk+1 is counter-clockwise, and when yk > 0, θk to θk+1 is clockwise. (ii) For m = 2l, when yk < 0, θk to θk+1 is clockwise, and when yk > 0, θk to θk+1 is clockwise. For the infinite-fixed-point of ρk∗ = 0 and θk∗ ∈ [0, 2π ), the local analysis gives xk+1,1 = xk,1 + (−R)m ρk sin θk , xk+1,2 = xk,2 − (−R)m ρk cos θk

(5.220)

ρk = ρk − ρk∗ = ρk > 0.

(5.221)

where

298

5 Infinite-Fixed-Point Discrete Systems

(i) For m = 2l − 1, θk to θk+1 is counter-clockwise. (ii) For m = 2l, θk to θk+1 is clockwise. From the local analysis, the discrete trajectories for two upper-saddle infinitefixed-points are presented in Fig. 5.6a, b. The hollow circular symbols are for monotonic lower-saddle and upper-saddles in one direction. The discrete trajectories are presented through by black circular symbols and red lines. The blue circular curves are for the mapping iteration references, which is based on the first integrals of the similar continuous dynamical system.

5.4.3 A Linear Infinite-Fixed-Point System Consider a discrete system of a circular infinite-fixed-point with linear mapping trajectories, i.e., 2 2 xk+1,1 = xk,1 + (xk,1 + xk,2 − R 2 )m xk,1 , 2 2 xk+1,2 = xk,2 + (xk,1 + xk,2 − R 2 )m xk,2

(5.222)

where R > 0. The simple fixed-points and infinite-fixed-points are ∗2 ∗2 xk,1 + xk,2 = R 2 for the infintite-fixed-point; ⎫ ∗ ∗ xk,1 = ±R, xk,2 = 0; ⎬ for intersection fixed point; x ∗ = 0, x ∗ = ±R ⎭ k,1

(5.223)

k,2

∗ ∗ xk,1 = 0, xk,2 = 0 for simple-fixed point.

Consider xk,1 = ρk cos θk , xk,2 = ρk sin θk ; xk+1,1 = ρk+1 cos θk+1 , xk+1,2 = ρk+1 sin θk+1 .

(5.224)

Thus, equation (5.222) becomes ρk+1 = ρk + (ρk2 − R 2 )m ρk = [1 + (ρk − R)m (ρk + R)m ]ρk , θk+1 = θk + 2lπ, l = 0, 1, 2, . . .

(5.225)

The fixed-points are ρk∗ = 0 and ρk∗ = R for a specific θk∗ ∈ [0, 2π ). We have

5.4 Infinite-Fixed-Point-Network Systems

299

xk ,2 [2(2l − 1)]th mUS

xk ,1 UF

(a) xk ,2

[2(2l )]th mUS

xk ,1 UF

(b) Fig. 5.6 Discrete trajectories for an infinite-fixed-point system with focus mapping. a The opposite directions of mapping rotation and b the same directions of mapping rotation on both sides of the infinite-fixed-point. The solid thick circle is for monotonic lower-saddle infinite-fixed-points. The thin circles are for iteration reference. The hollow circular symbols are for a monotonic unstable focus and the intersection fixed-points between the infinite-fixed-point circle with coordinates. The red lines are discrete trajectories inside and outside the infinite-fixed-point circle. mUS: monotonic upper-saddle. UF: unstable focus

300

5 Infinite-Fixed-Point Discrete Systems

ρk+1 = ρk for ρk∗ = R,

⎫ ρk+1 = [1 + (ρk − R)m (ρk + R)m ]ρk < ρk for ρk < R, ⎬ ρk+1 = [1 + (ρk − R)m (ρk + R)m ]ρk > ρk for ρk > R ⎭

for m = 2l − 1 with l = 1, 2, . . . ; ρk+1 ρk+1

⎫ = [1 + (ρk − R)m (ρk + R)m ]ρk > ρk for ρk < R, ⎬ = [1 + (ρk − R)m (ρk + R)m ]ρk > ρk for ρk > R ⎭

(5.226)

for m = 2l with l = 1, 2, . . . ; The iterations of ρk > R and ρk < R for m = 2l − 1 in the radial direction are outwards and inwards the center, respectively. With ρk = R, the discrete system is invariant, which is an infinite-fixed-point. Thus, the infinite-fixed-point of ρk∗ = R is the (2l − 1)th -order monotonic source (mSO) in the radial direction. The motions of ρk > R and ρk < R for m = 2l in the radial direction are outwards the center. With ρk = R, the discrete system is invariant, which is an infinite-fixedpoint. Thus, the infinite-fixed-point of ρk∗ = R is the (2l)th -order upper-saddle (US) in the radial direction. For a specific θk∗ ∈ [0, 2π ), we have ρk+1 = ρk = 0 for ρk∗ = 0, ρk+1 = (1 + (−R 2 )m )ρk < ρk for ρk > 0,

(5.227)

for m = 2l − 1 with l = 1, 2, . . . . • • • • •

For For For For For

R R R R R

∈ (0, 1), the fixed point of xk∗ = 0 is a monotonic sink. ∗ = 1, the √ fixed point of xk = 0 is∗ a zero-invariant sink. 2m ∈ (1, xk = 0 is an oscillatory sink. √ 2), the fixed point of 2m ∗ = √2, the fixed point of xk = 0 is a flipped invariance. ∈ ( 2m 2, ∞), the fixed point of xk∗ = 0 is an oscillatory source.

The fixed-point of xk∗ = 0 is a monotonic source if ρk+1 = (1 + (−R 2 )m )ρk > ρk for ρk > 0, for m = 2l with l = 1, 2, . . . .

(5.228)

From the local analysis, the center point for mSI, oSI, oSO and mSO are presented in Fig. 5.7a–d, respectively. The solid thick curve is for infinite-fixed-points. The hollow circular symbols are for monotonic upper-saddles in one direction. The mapping points for such an infinite-fixed-point discrete system are presented on the specific line with a constant angle. The blue straight lines for a specific rotation angle are for the mapping iteration references, which is based on the first integrals of the similar continuous dynamical system.

5.4 Infinite-Fixed-Point-Network Systems

301

x2

(2l − 1)th mSO

mSI

x1

(a)

xk ,2

(2l − 1)th mSO

oSI

xk ,1

(b) Fig. 5.7 Fixed-point stability for an infinite-fixed-point system in Eq. (5.222). The infinite-fixedpoint of the (2l − 1)th order monotonic source: a the monotonic sink fixed point (0, 0), b the oscillatory sink fixed-point (0, 0), c the oscillatory source fixed point (0, 0); d the monotonic source fixed point (0, 0) and the infinite-fixed-point of the (2l)th order monotonic upper-saddle. The solid circular curve is for infinite-fixed-points. The straight lines for specific rotation angle are for iteration references. The hollow circular symbols are for simple fixed-point and the intersection fixed-points. mSI: monotonic sink. mSO: monotonic source. mUS: monotonic upper-saddle. oSI: oscillatory sink. oSO: oscillatory source

302

5 Infinite-Fixed-Point Discrete Systems

xk ,2

(2l − 1)th mSO

xk ,1

oSO

(c)

xk ,2

(2l )th mUS

mSO

(d) Fig. 5.7 (continued)

xk ,1

5.4 Infinite-Fixed-Point-Network Systems

303

5.4.4 A Hyperbolic Infinite-Fixed-Point System Consider a discrete system of a circular infinite-fixed-point with hyperbolic mapping trajectories as 2 2 xk+1,1 = xk,1 + (xk,1 + xk,2 − R 2 )m xk,2 , 2 2 xk+1,2 = xk,2 + (xk,1 + xk,2 − R 2 )m xk,1

(5.229)

where R > 0. The simple fixed-points and infinite-fixed-points are ∗2 ∗2 xk,1 + xk,2 = R 2 for the infintite-fixed-point; ⎫ ∗ ∗ xk,1 = ±R, xk,2 = 0; ⎬ for intersection fixed-point; x ∗ = 0, x ∗ = ±R ⎭ k,1

(5.230)

k,2

∗ ∗ = 0, xk,2 = 0 for simple fixed-point. xk,1

Consider xk,1 = ρk cos θk , xk,2 = ρk sin θk ; xk+1,1 = ρk+1 cos θk+1 , xk+1,2 = ρk+1 sin θk+1 .

(5.231)

Equation (5.229) becomes 2 ρk+1 = ρk2 + ρk2 (ρk2 − R 2 )m sin 2θk + ρk2 (ρk2 − R 2 )2m ,

tan(θk+1 − θk ) =

(ρk2 − R 2 )m . 1 + (ρk2 − R 2 )m tan 2θk

(5.232)

The corresponding fixed-points are ρk∗ = 0, for simple fixed point, ρk∗ = R, for infinite-fixed-point, θk∗ θk∗

= =

0, 21 π, π, 23 π ; ρk∗ = R, 1 π, 43 π, 43 π, 54 π ; ρk∗ = 4

⎫ ⎬

R⎭

(5.233) for intersection fixed-points.

Let 2 yk = ρk2 − R 2 , yk+1 = ρk+1 − R2;

yk = ρk2 − R 2 − (ρk∗2 − R 2 ) = ρk2 − R 2 .

(5.234)

304

5 Infinite-Fixed-Point Discrete Systems

The local analysis of equation (5.232) in vicinity of infinite-fixed-point gives yk+1 = (1 + R 2 (yk )m−1 sin 2θk∗ )yk , tan(θk+1 − θk ) = (i)

(yk2 )m . 1 + (yk )m tan 2θk∗

(5.235)

For θk∗ ∈ (0, 21 π ) and θk∗ ∈ (π, 23 π ), equation (5.232) gives ⎫ yk+1 < yk < 0 if yk < 0 ⎬ for m = 2l − 1. yk+1 > yk > 0 if yk > 0 ⎭

(5.236)

Thus, the infinite-fixed-point is a monotonic source of the (2l − 1)th -order for θk∗ ∈ (0, 21 π ) and θk∗ ∈ (π, 23 π ). (ii) For θk∗ ∈ ( 21 π, π ) and θk∗ ∈ ( 23 π, 2π ), equation (5.232) gives ⎫ 0 > yk+1 > yk if yk < 0 ⎬ for m = 2l − 1. 0 < yk+1 < yk if yk > 0 ⎭

(5.237)

Thus, the infinite-fixed-point is a monotonic sink of the (2l − 1)th -order for θk∗ ∈ ( 21 π, π ) and θk∗ ∈ ( 23 π, 2π ). (iii) For θk∗ ∈ (0, 21 π ) and θk∗ ∈ (π, 23 π ), equation (5.232) gives ⎫ 0 > yk+1 > yk < 0 if yk < 0 ⎬ for m = 2l. ⎭ yk+1 > yk > 0 if yk > 0

(5.238)

Thus, the infinite-fixed-point is a monotonic upper-saddle of the (2l)th -order for θk∗ ∈ (0, 21 π ) and θk∗ ∈ (π, 23 π ). (iv) For θk∗ ∈ ( 21 π, π ) and θk∗ ∈ ( 23 π, 2π ), equation (5.232) gives ⎫ yk+1 < yk < 0 if yk < 0 ⎬ for m = 2l. 0 < yk+1 < yk if yk > 0 ⎭

(5.239)

Thus, the infinite-fixed-point is a monotonic lower-saddle of the (2l)th -order for θk∗ ∈ ( 21 π, π ) and θk∗ ∈ ( 23 π, 2π ). For the intersection fixed points, the local analysis of Eq. (5.232) in vicinity of infinite-fixed-point gives

5.4 Infinite-Fixed-Point-Network Systems

yk+1 = (1 + 2 cos 2θk∗ R 2 (yk )m−1 θk )yk , θk+1 ≈ θk + (yk )m (i)

305

(5.240)

For θk∗ = 0, we have yk+1 = (1 + 2R 2 (yk )m−1 θk )yk .

(5.241)

Thus ⎫ yk+1 > yk > 0 for θk > 0 ⎬ if yk > 0. yk > yk+1 > 0 for θk < 0 ⎭

(5.242)

With Eq. (5.231), therefore, the intersection fixed point of (θk∗ , ρk∗ ) = (0, R) is a monotonic upper-saddle of the (m + 1)th -order. (ii) For θk∗ = 21 π , we have yk+1 = (1 − 2R 2 (yk )2l−2 θk )yk .

(5.243)

Thus ⎫ 0 < yk+1 < yk for θk > 0 ⎬ if yk > 0 0 < yk < yk+1 for θk < 0 ⎭

(5.244)

Therefore, with Eq. (5.231), the intersection fixed point of (θk∗ , ρk∗ ) = ( 21 π, R) is a monotonic upper-saddle of the (m + 1)th -order. (iii) For θk∗ = π , we have yk+1 = (1 + 2R 2 (yk )2l−2 θk )yk

(5.245)

Thus ⎫ yk+1 > yk > 0 for θk > 0 ⎬ if yk > 0 yk > yk+1 > 0 for θk < 0 ⎭

(5.246)

Therefore, with Eq. (5.231), the intersection fixed point of (θk∗ , ρk∗ ) = (π, R) is a monotonic lower-saddle of the (m + 1)th -order. (iv) For θk∗ = 23 π , we have yk+1 = (1 − 2R 2 (yk )2l−2 θk )yk Thus

(5.247)

306

5 Infinite-Fixed-Point Discrete Systems

⎫ 0 < yk+1 < yk for θk > 0 ⎬ if yk > 0 0 < yk < yk+1 for θk < 0 ⎭

(5.248)

Therefore, with Eq. (5.231), the intersection fixed point of (θk∗ , ρk∗ ) = ( 23 π, R) is a monotonic lower-saddle of the (m + 1)th -order. For the simple fixed-point of xk∗ = 0, the linearized equation is ⎡ ⎣

xk+1,1 xk+1,2

⎤

⎡

⎦=⎣

1

(−R 2 )m

(−R 2 )m

1

⎤⎡ ⎦⎣

xk,1 xk,2

⎤ ⎦

(5.249)

with the corresponding eigenvalues are λ1 = 1 − R 2m , λ2 = 1 + R 2m .

(5.250)

Therefore, the fixed-point of xk∗ = 0 is a monotonic saddle-nodes. (i) On the eigenvector relative to λ2 = 1 + R 2m , the fixed-point of xk∗ = 0 is a monotonic source. (ii) On the eigenvector relative to λ1 = 1 − R 2m , the fixed-point of xk∗ = 0 is • • • • •

a monotonic sink for R ∈ (0, 1), an invariant sink for R = 1, √ an oscillatory sink for R ∈ (1,√2m 2), a flipped invariance for R = 2m √ 2, an oscillatory source for R ∈ ( 2m 2, ∞).

where θk = θk − θk∗ and m ∈ {2l − 1, 2l}. From the local analysis of fixed-point stability, fixed-points with iteration references are presented in Fig. 5.8. In Fig. 5.8a, c, the (2l − 1)th order monotonic sink and monotonic source on the infinite-fixed-point circle are presented for the monotonic saddle and monotonic-oscillatory saddle-node of the fixed point of xk∗ = 0, respectively. However, in Fig. 5.8b, d the (2l)th order monotonic upper-saddle and monotonic lower-saddle infinite-fixed-points on the infinite-fixed-point circle are for the monotonic saddle and monotonic-oscillatory saddle-node of the fixed point of xk∗ = 0, respectively. The hollow circular symbols are for intersection points between the infinite-fixed-point circle and coordinates. The reference curves are based on the first integrals of the similar continuous dynamical system. The mapping points for such an infinite-fixed-point system are presented at θk = 14 π, 43 π, 45 π, 74 π . From the

5.4 Infinite-Fixed-Point-Network Systems

307

xk ,2

(2l )th mUS (2l − 1)th mSO

(2l − 1)th mSI

(2l )th mLS

(2l )th mUS xk ,1

mSN

(2l − 1)th mSI

(2l − 1) th mSO (2l )th mLS

(a)

xk ,2

(2l + 1)th mUS

(2l )th mLS

(2l ) th mUS

(2l + 1)th mLS

(2l + 1)th mUS

mSN

xk ,1 (2l )th mLS

(2l )th mUS

(b)

(2l + 1)th mLS

Fig. 5.8 Fixed-point stability for an infinite-fixed-point system in Eq. (5.229). a The monotonic saddle-node of xk∗ = 0 and (2l − 1)th order monotonic sources and monotonic sinks, and (b) the monotonic saddle-node of xk∗ = 0 and (2l)th order upper-saddles and lower-saddles. (c) The monotonic source with oscillatory sink of xk∗ = 0 and (2l − 1)th order monotonic sources and monotonic sinks, and (d) the monotonic source with oscillatory sink of xk∗ = 0 and (2l)th order upper-saddles and lower-saddles. The thick circle is for the infinite-fixed-point. The hollow circular symbols are for simple fixed-point and intersection fixed-points. mSI: monotonic sink. mSO: monotonic source. mUS: monotonic upper-saddle. mLS: monotonic lower-saddle. mSN: monotonic saddle-node

308

5 Infinite-Fixed-Point Discrete Systems

xk ,2

(2l )th mUS

(2l − 1)th mSO

(2l − 1)th mSI

(2l )th mLS

(2l )th mUS

xk ,1

mSN

(2l − 1)th mSI

(2l − 1)th mSO (2l )th mLS

(c)

xk ,2

(2l + 1)th mUS

(2l )th mLS

(2l ) th mUS

(2l + 1)th mLS

(2l + 1)th mUS

mSN

xk ,1 (2l )th mLS

(2l )th mUS

(d) Fig. 5.8 (continued)

(2l + 1)th mLS

5.4 Infinite-Fixed-Point-Network Systems

309

Table 5.4 Summarization of fixed-point stability for Eq. (5.229) ρ∗ = R

θ ∗ ∈ (0, 21 π ),

θ ∗ ∈ ( 21 π, π )

θ ∗ ∈ (π, 23 π ),

− 1)th mSI

− 1)th mSO

(2l

m = 2l

(2l)th mUS

(2l)th mLS

(2l)th mUS

(2l)th mLS

ρ∗

θ∗

θ∗

θ∗

θ ∗ = 23 π

m = 2l − 1

=0

(2l)th mUS + 1)th mUS

(2l

=

1 2π

(2l)th mUS (2l

+ 1)th mUS

(2l

θ ∗ ∈ ( 23 π, 2π )

m = 2l − 1 =R

− 1)th mSO

=π

(2l)th mLS (2l

+ 1)th mLS

(2l − 1)th mSI

(2l)th mUS

m = 2l

(2l

ρ∗ = R m = 2l − 1

θ ∗ = 41 π (2l − 1)th mSO

θ ∗ = 43 π (2l − 1)th mSI

θ ∗ = 45 π (2l − 1)th mSO

(2l + 1)th mLS (2l − 1)th mSI

m = 2l

(2l)th mUS

(2l)th mLS

(2l)th mUS

(2l)th mLS

θ ∗ = 74 π

mSI: monotonic sink, mSO: monotonic source, mUS: monotonic upper-saddle, mLS: monotonic lower-saddle

local analysis, the behaviors of fixed-points are summarized in Table 5.4. Other cases can be similarly sketched. In other words, on the eigenvector of λ2 , the oscillatory source, invariant sink and an flipped invariance can be presented.

References Luo ACJ (2019) On dynamics of infinite-equilibrium systems, Int J Dyn Control Luo ACJ (2020) Bifurcation and stability in nonlinear dynamical system. Springer, New York

Index

A Appearing bifurcation, 56, 62, 68, 71

B Backward discrete system, 235–237 Backward mapping, 208 Backward solution, 208 Bifurcation, 52, 117

C Center, 14, 15 Center manifold, 9 Center subspace, 6 Contraction map, 9 Contravariant component vector, 10 Covariant vector, 10

D Degenerate case, 15 Degenerate fixed-point, 49 Discrete flow, 1 Discrete map, 2 Discrete system, 1 Discrete system partially with fixed-points, 173 Discrete system without fixed-points, 173 Discrete trajectory, 1 Discrete vector field, 1

E Eigenvector contravariant matrix, 10 Eigenvector covariant matrix, 10

F Finite-fixed-point, 173 Fixed-point, 3 Flipped degenate case, 20, 79 Flip subspace, 5 Forward discrete system, 218–220, 234 Forward mapping, 208 Forward solution, 208

G Global stability, 175 Global stable manifold, 8 Global unstable manifold, 8

H Hyperbolic fixed-point, 8 Hyperbolic infinite-fixed-point system, 303

I Infinite-fixed-point, 173 Infinite-fixed-point network system, 290 Infinite-fixed-point surface, 271 Infinite-fixed-point system, 174, 251, 270, 290 Invariant degenerate case, 78 Invariant discrete system, 171 Invariant manifold, 8 Invariant subspace, 8, 9

J Jacobian matrix, 3

© Higher Education Press 2020 A. C. J. Luo, Bifurcation and Stability in Nonlinear Discrete Systems, Nonlinear Physical Science, https://doi.org/10.1007/978-981-15-5212-0

311

312 L Linear infinite-fixed-point system, 298 Lipschitz condition, 3 Local bifurcation, 52 Local stable manifold, 8 Local unstable manifolds, 8

M Metric tensor, 10 Monotonically stable node, 164 Monotonically unstable node, 164 Monotonic decreasing-saddle-node bifurcation, 58 Monotonic increasing-saddle-node bifurcation, 58 Monotonic lower-saddle, 79, 95, 165, 166, 177 Monotonic lower-saddle-node bifurcation, 118, 120 Monotonic sink, 78, 98, 166, 167, 177 Monotonic sink bifurcation, 119, 120 Monotonic sink pitchfork bifurcation, 140 Monotonic source, 78, 98, 165, 167, 176, 177 Monotonic source bifurcation, 118, 119, 121 Monotonic source pitchfork bifurcation, 152 Monotonic stability, 78, 181, 222, 223 Monotonic stability switching, 240 Monotonic three-fixed-point system, 194 Monotonic two-fixed-point system, 185 Monotonic upper-saddle, 79, 95, 165, 166, 176–177 Monotonic upper-saddle-node bifurcation, 118, 120

N Negative mapping, 208 Negative set, 207 Neimark bifurcation, 160 Non-uniform discrete system, 2 Normal form, 259

O One-dimensional discrete system, 77 One-fixed-point system, 180 One-infinite-fixed-point system, 271 Operator norm, 2 Oscillartorilly stable node, 162 Oscillatorilly unstable node, 164 Oscillatory bifurcation, 228, 242 Oscillatory-decreasing-saddle-node bifurcation, 63

Index Oscillatory-increasing-saddle-node bifurcation, 60 Oscillatory lower-saddle, 80, 102, 178 Oscillatory lower-saddle bifurcation, 121– 124, 166–168 Oscillatory pitchfork bifurcation, 64 Oscillatory sink, 79, 99, 179 Oscillatory sink bifurcation, 121–124, 126, 127, 167 Oscillatory source, 79, 99, 179, 180, 197 Oscillatory source bifurcation, 121–124, 126, 127, 167 Oscillatory stability, 78, 183 Oscillatory stability switching, 228, 243, 244, 246 Oscillatory three-fixed-point system, 198 Oscillatory two-fixed-point system, 189 Oscillatory unstable pitchfork bifurcation, 65 Oscillatory upper-saddle, 79, 100, 178, 179 Oscillatory-upper-saddle bifurcation, 121– 123, 165, 167

P Period-doubling bifurcation, 127 Period-1 solution, 3 Pitchfork bifurcation, 59, 65 Positive mapping, 208, 212 POSITIVE SET, 207

S Saddle, 160 Saddle-node appearing bifurcation, 126 Saddle-node switching bifurcation, 132 Saddle-stable node bifurcation, 160 Saddle-unstable node bifurcation, 160 Sink, 14 Source, 14 Spiral infinite-fixed-point system, 294 Spirally stable fixed-point, 48–49 Spirally unstable fixed-point, 48–49 Spiral sink, 15 Spiral source, 15 Spiral stability, 45 Spiral stability switching, 223, 231, 246, 248 Stability switching, 17 Stable focus, 160 Stable node, 160 Stable subspace, 5–6 Switching bifurcation, 57, 63–64

Index T Tensor, 10 Two-fixed-point system, 185 Two-infinite-fixed-point system, 275

U Uniform discrete system, 2 Unstable focus, 160 Unstable node, 160 Unstable subspace, 5–7

313 Y Yang chaos, 218 Yang solution, 208 Yin chaos, 218 Yin-Yang Chaos, 218 Yin-Yang solution, 208, 213 Yin-yang theory, 207

Z Zero-sink, 78 Zero-stable node, 78, 164