Bifurcation and stability in nonlinear dynamical systems 9783030229092, 9783030229108

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Nonlinear Systems and Complexity Series Editor: Albert C. J. Luo

Albert C. J. Luo

Bifurcation and Stability in Nonlinear Dynamical Systems

Nonlinear Systems and Complexity Volume 28

Series editor Albert C. J. Luo Southern Illinois University Edwardsville, IL, USA

Nonlinear Systems and Complexity provides a place to systematically summarize recent developments, applications, and overall advance in all aspects of nonlinearity, chaos, and complexity as part of the established research literature, beyond the novel and recent findings published in primary journals. The aims of the book series are to publish theories and techniques in nonlinear systems and complexity; stimulate more research interest on nonlinearity, synchronization, and complexity in nonlinear science; and fast-scatter the new knowledge to scientists, engineers, and students in the corresponding fields. Books in this series will focus on the recent developments, findings and progress on theories, principles, methodology, computational techniques in nonlinear systems and mathematics with engineering applications. The Series establishes highly relevant monographs on wide ranging topics covering fundamental advances and new applications in the field. Topical areas include, but are not limited to: Nonlinear dynamics; Complexity, nonlinearity, and chaos; Computational methods for nonlinear systems; Stability, bifurcation, chaos and fractals in engineering; Nonlinear chemical and biological phenomena; Fractional dynamics and applications; Discontinuity, synchronization and control.

More information about this series at http://www.springer.com/series/11433

Albert C. J. Luo

Bifurcation and Stability in Nonlinear Dynamical Systems

Albert C. J. Luo Southern Illinois University Edwardsville, IL, USA

ISSN 2195-9994 ISSN 2196-0003 (electronic) Nonlinear Systems and Complexity ISBN 978-3-030-22909-2 ISBN 978-3-030-22910-8 (eBook) https://doi.org/10.1007/978-3-030-22910-8 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

This book systematically presents a fundamental theory for the local analysis of bifurcation and stability of equilibriums in nonlinear dynamical systems. Until now, one does not have any efficient way to investigate stability and bifurcation of dynamical systems with higher-order singularity equilibriums. For instance, infinite-equilibrium dynamical systems have higher order singularity, which dramatically changes dynamical behaviors and possesses similar characteristics of discontinuous dynamical systems. The stability and bifurcation of equilibriums on the specific eigenvector are presented, and the spiral stability and Hopf bifurcation of equilibriums in nonlinear systems are presented through the Fourier series transformation. The bifurcation and stability of higher-order singularity equilibriums are presented through the (2m)th and (2m + 1)th-degree polynomial systems. From local analysis, dynamics of infinite-equilibrium systems is discussed. The research on infinite-equilibrium systems will bring us to the new era of dynamical systems and control. This book consists of eight chapters. The first chapter discusses the local theory of stability of equilibriums in nonlinear dynamical systems. The spiral stability of equilibriums on the eigenvector space is discussed. In addition, based on the Fourier series transformation, the spiral stability of equilibrium is presented. The extended Lyapunov stability theory is also presented. In Chap. 2, the local theory of bifurcations of equilibriums in nonlinear systems is presented on the specific eigenvector, and the Hopf bifurcation of equilibrium is discussed. In Chap. 3, the local analysis of stability and bifurcation for 1-dimensional and 2-dimensional dynamical systems is presented. In Chap. 4, equilibrium higher-order singularity in 1-dimensional systems is discussed globally. In Chap. 5, the global bifurcation theory for low-degree polynomial systems is presented. For a global view of bifurcation and stability of equilibriums in nonlinear dynamical systems, the bifurcation and stability of higher order singularity equilibriums are presented through the (2m)th and (2m + 1)th-degree polynomial systems in Chaps. 6 and 7, respectively. In Chap. 8, infinite-equilibrium systems are discussed through the local analysis of higher-order, singular dynamical

vii

viii

Preface

systems. The equilibrium computations and normal forms of equilibriums for nonlinear dynamical systems are presented. Finally, I would like to thank my wife for her support for this research work. The author hopes that the materials presented herein can last long for science and engineering. Such contributions will benefit human beings on their progress and development. Edwardsville, IL, USA

Albert C. J. Luo

Contents

1

Stability of Equilibriums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Continuous Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Equilibriums and Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Stability and Singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Hyperbolic Stability on Eigenvectors . . . . . . . . . . . . . . . 1.3.2 Spiral Stability on an Invariant Eigenplane . . . . . . . . . . 1.3.3 Spiral Stability Based on the Fourier Series Base . . . . . . 1.4 Spiral Stability in Second-Order Nonlinear Systems . . . . . . . . . 1.5 Lyapunov Functions and Stability . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . .

1 1 4 18 18 30 40 44 48 57

2

Bifurcations of Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Hyperbolic Bifurcations on Eigenvectors . . . . . . . . . . . . . . . . . 2.3 Hopf Bifurcation on an Eigenvector Plane . . . . . . . . . . . . . . . . 2.4 Hopf Bifurcation Based on the Fourier Series Base . . . . . . . . . . 2.5 Hopf Bifurcations in Second-Order Nonlinear Systems . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

59 59 60 69 75 80 85

3

Low-Dimensional Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . 3.1 1-Dimensional Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Stability and Singularity . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Sampled Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 2-Dimensional Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Stability and Singularity . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Hopf Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . .

87 87 87 97 104 109 110 117 122

ix

x

Contents

4

Equilibrium Stability in 1-Dimensional Systems . . . . . . . . . . . . . . . 4.1 System Classifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Equilibrium Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 One-Equilibrium Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Two-Equilibrium Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Three-Equilibrium Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

123 123 127 130 131 139 148

5

Low-Degree Polynomial Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Quadratic Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Cubic Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Quartic Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .

149 149 151 164 186 229

6

(2m)th-Degree Polynomial Systems . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Global Stability and Bifurcations . . . . . . . . . . . . . . . . . . . . . . . 6.2 Simple Equilibrium Bifurcations . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Appearing Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Switching Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Switching and Appearing Bifurcations . . . . . . . . . . . . . . 6.3 Higher Order Equilibrium Bifurcations . . . . . . . . . . . . . . . . . . . 6.3.1 Appearing Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Switching Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Appearing and Switching Bifurcations . . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . .

231 231 248 248 255 260 265 265 276 281 288

7

(2m+1)th-Degree Polynomial Systems . . . . . . . . . . . . . . . . . . . . . . . 7.1 Global Stability and Bifurcations . . . . . . . . . . . . . . . . . . . . . . . 7.2 Simple Equilibrium Bifurcations . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Appearing Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Switching Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Switching and Appearing Bifurcations . . . . . . . . . . . . . . 7.3 Higher Order Equilibrium Bifurcations . . . . . . . . . . . . . . . . . . . 7.3.1 Higher Order Equilibrium Bifurcations . . . . . . . . . . . . . 7.3.2 Switching Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Switching and Appearing Bifurcations . . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . .

289 289 306 306 319 324 331 331 351 356 363

8

Infinite-Equilibrium Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Equilibrium Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Normal Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Infinite-Equilibrium Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 One-Infinite-Equilibrium Systems . . . . . . . . . . . . . . . . . 8.3.2 Two-Infinite-Equilibrium Systems . . . . . . . . . . . . . . . . . 8.3.3 Higher Order Infinite-Equilibrium Systems . . . . . . . . . .

. . . . . . .

365 365 373 385 386 388 393

Contents

8.4

Network-Infinite-Equilibrium Systems . . . . . . . . . . . . . . . . . . . 8.4.1 A Network-Infinite-Equilibrium System . . . . . . . . . . . . 8.4.2 Circular Infinite-Equilibrium Systems . . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

. . . .

396 398 400 408

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409

Chapter 1

Stability of Equilibriums

In this chapter, basic concepts of nonlinear dynamical systems are introduced. A local theory of equilibrium stability for nonlinear dynamical systems is discussed. The spiral stability of equilibriums in nonlinear dynamical systems is presented through the Fourier series base. The higher order singularity and stability for nonlinear systems on the specific eigenvectors are developed. The Lyapunov function stability is briefly discussed, and the extended Lyapunov theory for equilibrium stability is also presented.

1.1

Continuous Dynamical Systems

Definition 1.1 For I
R , Ω
R n , and Λ
R m , consider a vector function f : Ω  I  Λ ! R n which is Cr (r  1)-continuous, and there is an ordinary differential equation in a form of x_ ¼ fðx, t, pÞ for t 2 I, x 2 Ω and p 2 Λ

ð1:1Þ

where x_ ¼ dx=dt is differentiation with respect to time t, which is simply called the velocity vector of the state variables x. With an initial condition of x(t0) ¼ x0, the solution of Eq. (1.1) is given by xðtÞ ¼ Φðx0 , t  t 0 , pÞ:

ð1:2Þ

(i) The ordinary differential equation with the initial condition is called a dynamical system. (ii) The vector function f(x, t, p) is called a vector field on domain Ω. (iii) The solution Φ(x0, t  t0, p) is called the flow of dynamical systems.

© Springer Nature Switzerland AG 2019 A. C. J. Luo, Bifurcation and Stability in Nonlinear Dynamical Systems, Nonlinear Systems and Complexity 28, https://doi.org/10.1007/978-3-030-22910-8_1

1

2

1 Stability of Equilibriums

(iv) The projection of the solution Φ(x0, t  t0, p) on domain Ω is called the trajectory, phase curve, or orbit of dynamical system, which is defined as Γ ¼ fxðtÞ 2 ΩjxðtÞ ¼ Φðx0 , t  t 0 , pÞ for t 2 Ig  Ω:

ð1:3Þ

Definition 1.2 If the vector field of the dynamical system in Eq. (1.1) is independent of time, such a system is called an autonomous dynamical system. Thus, equation (1.1) becomes x_ ¼ fðx, pÞ for t 2 I
R , x 2 Ω
R n and p 2 Λ
R m

ð1:4Þ

Otherwise, such a system is called nonautonomous dynamical systems if the vector field of the dynamical system in Eq. (1.1) is dependent on time and state variables. Definition 1.3 For a vector function f 2 R n with x 2 R n , the operator norm of f is defined by kfk ¼

Xn

max

i¼1 kxk1, t2I

j f i ðx, tÞ j:

ð1:5Þ

For an n  n matrix f(x, p) ¼ Ax with A ¼ (aij)nn, the corresponding norm is defined by kAk ¼

Xn j aij j: i, j¼1

ð1:6Þ

Definition 1.4 For a vector function xðtÞ ¼ ðx1 , x2 , . . . , xn ÞT 2 R n , the derivative and integral of x(t) are defined by  T dxðtÞ dx1 ðtÞ dx2 ðtÞ dx ðtÞ ¼ , ,..., n , dt dt dt dt ð ð ð ð xðtÞdt ¼ ð x1 ðtÞdt, x2 ðtÞdt, . . . , xn ðtÞdtÞT :

ð1:7Þ

For an nn matrix A ¼ (aij)nn, the corresponding derivative and integral are defined by dAðtÞ ¼ dt



daij ðtÞ dt

 nn

ð ð and AðtÞdt ¼ ð aij ðtÞdtÞ

nn

:

ð1:8Þ

Definition 1.5 For I
R , Ω
R n , and Λ
R m , the vector function f(x, t, p) with f : Ω  I  Λ ! R n is differentiable at x0 2 Ω if

1.1 Continuous Dynamical Systems

3

∂fðx, t, pÞ fðx0 þ Δx, t, pÞ  fðx0 , t, pÞ jðx0 , t, pÞ ¼ lim : Δx Δx!0 ∂x

ð1:9Þ

∂f/∂x is called the spatial derivative of f(x, t, p) at x0, and the derivative is given by the Jacobian matrix: ∂fðx, t, pÞ ¼ ð∂f i =∂xj Þnn : ∂x

ð1:10Þ

Definition 1.6 For I
R , Ω
R n , and Λ
R m , consider a vector function f(x, t, p) with f : Ω  I  Λ ! R n , t 2 I, and x 2 Ω and p 2 Λ. The vector function f(x, t, p) satisfies the Lipschitz condition with respect to x for I  Ω  Λ, kfðx2 , t, pÞ  fðx1 , t, pÞk  Lkx2  x1 k

ð1:11Þ

with x1, x2 2 Ω and L being a constant. The constant L is called the Lipschitz constant. Theorem 1.1 Consider a dynamical system as x_ ¼ fðx; t; pÞ with xðt 0 Þ ¼ x0

ð1:12Þ

with t0, t 2 I ¼ [t1, t2], x 2 Ω ¼ {xjkx  x0k  d}, and p 2 Λ. If the vector function f(x, t, p) is Cr-continuous (r  1) in G ¼ Ω  I  Λ, then the dynamical system in Eq. (1.12) has one and only one solution Φ(x0, t  t0, p) for j t  t 0 j min ðt 2  t 1 , d=MÞ with M ¼ max kfk: G

ð1:13Þ

Proof The proof of this theorem can be referred to the book by Coddington and Levinson (1955). ∎ Theorem 1.2 (Gronwall) Suppose there is a continuous real valued function g (t)  0 to satisfy gðtÞ  δ1

ðt

gðτÞdτ þ δ2

ð1:14Þ

t0

for all t 2 [t0, t1] and δ1 and δ2 are positive constants. For t 2 [t0, t1], one obtains gðtÞ  δ2 eδ1 ðtt0 Þ : Proof The proof can be referred in Luo (2011).

ð1:15Þ ∎

4

1 Stability of Equilibriums

Theorem 1.3 Consider a dynamical system as x_ ¼ fðx, t, pÞ with x(t0) ¼ x0 in Eq. (1.12) with t0, t 2 I ¼ [t1, t2], x 2 Ω ¼ {xjkx  x0k  d}, and p 2 Λ. The vector function f(x, t, p) is Cr-continuous (r  1) in G ¼ Ω  I  Λ. The solution of x_ ¼ fðx, t, pÞ with x(t0) ¼ x0 is x(t) on G and the solution of y_ ¼ fðy, t, pÞ with y(t0) ¼ y0 is y(t) on G. For a given ε > 0, if kx0  y0k  ε, then kxðtÞ  yðtÞk  εeLðtt0 Þ on I  Λ

ð1:16Þ ∎

Proof The proof can be referred in Luo (2011).

1.2

Equilibriums and Stability

Definition 1.7 Consider a metric space Ω and Ωα
Ω (α ¼ 1, 2, . . .). (i) A map h is called a homeomorphism of Ωα onto Ωβ (α, β ¼ 1, 2,   ) if the map h : Ωα ! Ωβ is continuous and one to one, and h1 : Ωβ ! Ωα is continuous. (ii) Two sets Ωα and Ωβ are homeomorphic or topologically equivalent if there is a homeomorphism of Ωα onto Ωβ. Definition 1.8 A connected, metric space Ω with an open cover {Ωα} (i.e., Ω ¼ [αΩα) is called an n-dimensional, Cr (r  1) differentiable manifold if the following properties exist. (i) There is an open unit ball B ¼ fx 2 R n jkxk < 1g. (ii) For all α, there is a homeomorphism hα : Ωα ! B. (iii) If hα : Ωα ! B and hβ : Ωβ ! B are homeomorphisms for Ωα \ Ωβ 6¼ ∅, then there is a Cr-differentiable map h ¼ hα ∘h1 for hα ðΩα \ Ωβ Þ  R n and β n hβ ðΩα \ Ωβ Þ  R with h : hβ ðΩα \ Ωβ Þ ! hα ðΩα \ Ωβ Þ,

ð1:17Þ

and for all x 2 hβ(Ωα \ Ωβ), the Jacobian determinant det Dh(x) 6¼ 0. The manifold Ω is called to be analytic if the maps h ¼ hα ∘h1 β are analytic. Definition 1.9 Consider an autonomous, nonlinear dynamical system x_ ¼ fðx, pÞ in Eq. (1.4). A point x 2 Ω is called an equilibrium point or critical point of a nonlinear system x_ ¼ fðx, pÞ if fðx ; pÞ ¼ 0:

ð1:18Þ

The linearized system of the nonlinear system x_ ¼ fðx, pÞ in Eq. (1.4) at the equilibrium point x is given by

1.2 Equilibriums and Stability

5

y_ ¼ Dfðx , pÞy where y ¼ x  x :

ð1:19Þ

Definition 1.10 Consider an n-dimensional, autonomous, nonlinear dynamical system x_ ¼ fðx, pÞ in Eq. (1.4) with an equilibrium point x . The linearized system of the nonlinear system at the equilibrium point x is y_ ¼ Dfðx , pÞy (y ¼ x  x ) in Eq. (1.19). The matrix Df(x , p) possesses n eigenvalues λk (k ¼ 1, 2, . . ., n). Set N ¼ {1, 2, . . . ,n} and N i ¼ fi1 , i2 , . . . , ini g [ ∅ with ij 2 N ( j ¼ 1, 2, . . ., ni; i ¼ 1, 2, 3) and Σ3i¼1 ni ¼ n. [3i¼1 N i ¼ N and Ni \ Nl ¼ ∅ (l 6¼ i). Ni ¼ ∅ if ni ¼ 0. The corresponding vectors for the negative, positive, and zero eigenvalues of Df(x , p) are {vk} (k 2 Ni, i ¼ 1, 2, 3), respectively. The stable, unstable, and invariant subspaces of the linearized nonlinear system in Eq. (1.19) are defined as E s ¼ spanfvk jðDfðx , pÞ  λk IÞvk ¼ 0, λk < 0, k 2 N 1
N [ ∅g; E

u

¼ spanfvk jðDfðx , pÞ  λk IÞvk ¼ 0, λk > 0, k 2 N 2
N [ ∅g;

ð1:20Þ

E i ¼ spanfvk jðDfðx , pÞ  λk IÞvk ¼ 0, λk ¼ 0, k 2 N 3
N [ ∅g: Definition 1.11 Consider a 2n-dimensional, autonomous dynamical system x_ ¼ f(x, p) in Eq. (1.4) with an equilibrium point x . The linearized system of the nonlinear system at the equilibrium point x is y_ ¼ Dfðx , pÞy (y ¼ x  x ) in Eq. (1.19). The matrix Df(x , p) has complex eigenvalues αk iβk with eigenvectors pffiffiffiffiffiffiffi uk ivk (k 2 {1, 2, . . . ,n}) and i ¼ 1, and the base of vector is B ¼ fu1 , v1 , . . . , uk , vk , . . . , un , vn g:

ð1:21Þ

The stable, unstable, and center subspaces of Eq. (1.19) are linear subspaces spanned by {uk, vk} (k 2 Ni, i ¼ 1, 2, 3), respectively. N i ¼ fi1 , i2 , . . . , ini g [ ∅
N [ ∅ and N ¼ {1, 2, . . . ,n} with ij 2 N ( j ¼ 1, 2, . . ., ni) and Σ3i¼1 ni ¼ n. [3i¼1 N i ¼ N and Ni \ Nl ¼ ∅ (l 6¼ i). Ni ¼ ∅ if ni ¼ 0. The stable, unstable, and center subspaces of the linearized nonlinear system in Eq. (1.19) are defined as

E ¼ span s

E

u

f j f j

¼ span

αk < 0, βk 6¼ 0,

g g

ðuk , vk Þ ðDfðx , pÞ  ðαk iβk ÞIÞðuk ivk Þ ¼ 0, ; k 2 N 1
f1, 2, . . . , ng [ ∅ αk > 0, βk 6¼ 0,

ðuk , vk Þ ðDfðx , pÞ  ðαk iβk ÞIÞðuk ivk Þ ¼ 0, ; k 2 N 2
f1, 2, . . . , ng [ ∅

6

1 Stability of Equilibriums

E ¼ span c

f j

g

αk ¼ 0, βk 6¼ 0,

ðuk , vk Þ ðDfðx , pÞ  ðαk iβk ÞIÞðuk ivk Þ ¼ 0, : k 2 N 3
f1, 2, . . . , ng [ ∅

ð1:22Þ

Theorem 1.4 Consider an n-dimensional, autonomous, nonlinear dynamical system x_ ¼ fðx, pÞ in Eq. (1.4) with an equilibrium point x . The linearized system of the nonlinear system at the equilibrium point x is y_ ¼ Dfðx , pÞy (y ¼ x  x ) in Eq. (1.19). The eigenspace of Df(x , p) (i.e., E
R n ) in the linearized dynamical system is expressed by direct sum of three subspaces: E ¼E sE uE

ð1:23Þ

c

where E s , E u , and E c are the stable, unstable, and center spaces, respectively. ∎

Proof This proof can be referred in Luo (2011).

Definition 1.12 Consider an n-dimensional, autonomous, nonlinear dynamical system x_ ¼ fðx, pÞ in Eq. (1.4) with an equilibrium point x and f(x, p) is Cr-continuous (r  1) in a neighborhood of the equilibrium x . The corresponding solution is x(t) ¼ Φ(x0, t  t0, p) ¼ Φt(x0). The linearized system of the nonlinear system at the equilibrium point x is y_ ¼ Dfðx , pÞy (y ¼ x  x ) in Eq. (1.19). Suppose there is a neighborhood of the equilibrium x as U(x )  Ω, and in the neighborhood kfðx þ y, pÞ  Dfðx , pÞyk ¼ 0: kyk kyk!0 lim

ð1:24Þ

(i) A Cr invariant manifold S loc ðx; x Þ ¼ fx 2 Uðx Þj lim xðtÞ ¼ x ; xðtÞ 2 Uðx Þ for all t  0g t!1

ð1:25Þ

is called the local stable manifold of x , and the corresponding global, stable manifold is defined as S ðx, x Þ ¼ [t0 Φt ðS

loc ðx, x



ÞÞ:

ð1:26Þ

(ii) A Cr invariant manifold U

loc ðx, x



Þ ¼ fx 2 Uðx Þj lim xðtÞ ¼ x , xðtÞ 2 Uðx Þ for all t  0g t!1

ð1:27Þ

is called the unstable manifold of x , and the corresponding global, unstable manifold is defined as

1.2 Equilibriums and Stability

7

U ðx, x Þ ¼ [t0 Φt ðU

loc ðx, x



ÞÞ:

ð1:28Þ

(iii) A Cr  1 invariant manifold C loc ðx, x Þ is called the center manifold of x if C loc ðx, x Þ possesses the same dimension of E c for x 2 S ðx, x Þ, and the tangential space of C loc ðx, x Þ is identical to E c . The stable and unstable manifolds are unique, but the center manifold is not unique. If the nonlinear vector field f is C1-continuous, then a Cr center manifold can be found for any r < 1. Theorem 1.5 Consider an n-dimensional, autonomous, nonlinear dynamical system x_ ¼ fðx, pÞ in Eq. (1.4) with a hyperbolic equilibrium point x and f(x, p) is Cr-continuous (r  1) in a neighborhood of the equilibrium x . The corresponding solution is x(t) ¼ Φ(x0, t  t0, p) ¼ Φt(x0). The linearized system of the nonlinear system at the equilibrium point x is y_ ¼ Dfðx , pÞy (y ¼ x  x ) in Eq. (1.19). Suppose there is a neighborhood of the hyperbolic equilibrium x as U(x )  Ω. If the homeomorphism between the local invariant subspace E(x, x )  U(x ) under the flow Φ(x0, t  t0, p) of x_ ¼ fðx, pÞ in Eq. (1.4) and the eigenspace E of the linearized system exists with the condition in Eq. (1.24), the local invariant subspace is decomposed by Eðx, x Þ ¼ S

loc ðx, x



ÞU

(i) The local stable invariant manifold S properties:

loc ðx, x

loc ðx, x





Þ:

ð1:29Þ

Þ possesses the following

(a) For x 2 S loc ðx, x Þ, S loc ðx, x Þ possesses the same dimension of E s and the tangential space of S loc ðx, x Þ is identical to E s . (b) For x0 2 S loc ðx, x Þ, xðtÞ 2 S loc ðx, x Þ for all time t  t0 and lim xðtÞ ¼ x . t!1

=S (c) For x0 2

loc ðx, x



Þ, kx  x k  δ for δ > 0 with t  t1  t0.

(ii) The local unstable invariant manifold U properties:

loc ðx, x



Þ possesses the following

(a) For x 2 U loc ðx, x Þ, U loc ðx, x Þ possesses the same dimension of E u and the tangential space of U loc ðx, x Þ is identical to E u . (b) For x0 2 U loc ðx, x Þ, xðtÞ 2 U loc ðx, x Þ for all time t  t0 and lim xðtÞ ¼ x . t!1

=U (c) For x0 2

loc ðx, x



Þ, kx  x k  δ for δ > 0 with t  t1  t0 .

Proof The proof for stable and unstable manifold can be referred in Hartman (1964). The proof for center manifold can be referenced in Marsden and McCracken (1976) or Carr (1981). ∎ Theorem 1.6 Consider an n-dimensional, autonomous, nonlinear dynamical system x_ ¼ fðx, pÞ in Eq. (1.4) with an equilibrium point x . Suppose there is a

8

1 Stability of Equilibriums

neighborhood of the equilibrium x as U(x )  Ω; then f(x, p) is Cr-continuous (r  1) in a neighborhood of the equilibrium x . The corresponding solution is x(t) ¼ Φ(x0, t  t0, p). The linearized system of the nonlinear system at the equilibrium point x is y_ ¼ Dfðx , pÞy (y ¼ x  x ) in Eq. (1.19). If the homeomorphism between the local invariant subspace E(x, x )  U(x ) under the flow Φ(x0, t  t0, p) of x_ ¼ fðx, pÞ in Eq. (1.4) and the eigenspace E of the linearized system exists with the condition in Eq. (1.24), in addition to the local stable and unstable invariant manifolds, there is a Cr1 center manifold C loc ðx, x Þ. The center manifold possesses the same dimension of E c for x 2 C loc ðx, x Þ, and the tangential space of C loc ðx, x Þ is identical to E c . Thus, the local invariant subspace is decomposed by Eðx, x Þ ¼ S

loc ðx, x



ÞU

loc ðx, x



ÞC

loc ðx, x



Þ:

ð1:30Þ

Proof The proof for stable and unstable manifolds can be referred in Hartman (1964). The proof for center manifolds can be referenced in Marsden and McCracken (1976) or Carr (1981). ∎ Definition 1.13 Consider an n-dimensional, autonomous, nonlinear dynamical system x_ ¼ fðx, pÞ in Eq. (1.4) with an equilibrium point x and f(x, p) is Cr-continuous (r  1) in a neighborhood of the equilibrium x . (i) The equilibrium x is stable if for all ε > 0 there is a δ > 0 such that for all x0 2 Uδ(x ) at t = t0 where Uδ(x ) = {xjkx  x k < δ} and t  t0, xðtÞ ¼ Φðx0 , t  t 0 , pÞ 2 U ε ðx Þ ¼ fxðtÞj kxðtÞ  x k: < ε, t 2 ðt 0 , 1Þg: ð1:31Þ (ii) The equilibrium x is unstable if it is not stable or if for all ε > 0 there is a δ > 0 such that for all x0 2 Uδ(x ) at t = t0 where Uδ(x ) = {xjkx  x k < δ} and t  t0 > t1, xðtÞ ¼ Φðx0 , t  t 0 , pÞ 2 = U ε ðx Þ ¼ fxðtÞj kxðtÞ  x k: < ε, t 2 ðt 0 , 1Þg: ð1:32Þ (iii) The equilibrium x is asymptotically stable if for all ε > 0 there is a δ > 0 such that for all x0 2 Uδ(x ) at t = t0 where Uδ(x ) = {xjkx  x k < δ} and t  t0, lim t!1 Φðx0 , t  t 0 , pÞ ¼ x :

ð1:33Þ

(iv) The equilibrium x is asymptotically unstable if for all ε > 0 there is a δ > 0 such that for all x0 2 Uδ(x ) at t = t0 where Uδ(x ) = {xjkx  x k < δ} and t  t0, lim t!1 Φðx0 , t  t 0 , pÞ ¼ x :

ð1:34Þ

Definition 1.14 Consider an n-dimensional, autonomous, nonlinear dynamical system x_ ¼ fðx, pÞ in Eq. (1.4) with an equilibrium point x and f(x, p) is

1.2 Equilibriums and Stability

9

Cr-continuous (r  1) in a neighborhood of the equilibrium x . The corresponding solution is x(t) ¼ Φ(x0, t  t0, p). Suppose U(x )  Ω is a neighborhood of equilibrium x , and there are n linearly independent vectors vk (k ¼ 1, 2, . . ., n). The independent eigenvector vk is called the covariant eigenvector. The corresponding contravariant vector of the independent eigenvector vk is defined by vk. For a perturbation of equilibrium y ¼ x  x , let y ¼ ck(t)vk ¼ ck(t)vk (tensor notation convention). P ¼ ðv1 , v2 , . . . , vn Þ

ð1:35Þ

is called the eigenvector covariant matrix. Q ¼ ðv1 , v2 , . . . , vn Þ

ð1:36Þ

is called the eigenvector contravariant matrix. The eigenvector contravariant vector is defined as vk ¼ ðak1 , ak2 , . . . , akn ÞT with QT ¼ P1 ¼ ðaij Þ:

ð1:37Þ

The component ck(t) is called the contravariant component. The corresponding vector c ¼ ðc1 , c2 , . . . , cn ÞT

ð1:38Þ

is called a contravariant component vector. The component ck is called the covariant component. The corresponding vector is c ¼ ðc1 , c2 , . . . , cn ÞT

ð1:39Þ

which 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 dynamical system as y ¼ ck(t)vk ¼ ck(t)vk. The two classes of vk and vk are the covariant and contravariant eigenvectors. Definition 1.15 The metric tensors of the covariant and contravariant eigenvectors are defined as for i, j ¼ 1, 2, . . . , n:

10

1 Stability of Equilibriums

gij ¼ ðvi ÞT  vj , gij ¼ ðvi ÞT  vj , gij ¼ ðvi ÞT  v j ¼ δij ;

ð1:40Þ

and the unit vectors of the covariant and contravariant eigenvectors are defined as pffiffiffiffiffi pffiffiffiffiffi ei ¼ vi = gii , ei ¼ vi = gii :

ð1:41Þ

Two transformations based on the covariant and contravariant eigenvectors are defined as y ¼ c i vi ¼ c j v j ,

ð1:42Þ

pffiffiffiffiffi pffiffiffiffiffi y ¼ ci gii ei ¼ cj gjj e j

ð1:43Þ

and

with T

T

T

ðv j Þ  y ¼ ðv j Þ  ci vi ¼ ci ðv j Þ  vi ¼ ci δi j ¼ c j ; ðv j ÞT  y ¼ ðv j ÞT  ci vi ¼ ci ðv j ÞT  vi ¼ ci δ ij ¼ c j :

ð1:44Þ

For the orthogonal eigenvectors, vi ¼ vi , gii ¼ gii ¼ 1, gij ¼ gij ¼ 0:

ð1:45Þ

Consider a solution in a 2-dimensional system for i, j ¼ 1, 2 as yðtÞ ¼ ci vi ¼ c1 v1 þ c2 v2 , yðtÞ ¼ cj v j ¼ c1 v1 þ c2 v2 :

ð1:46Þ

Based on unit covariant and contravariant eigenvectors, pffiffiffiffiffiffi pffiffiffiffiffiffi yðtÞ ¼ c1 g11 e1 þ c2 g22 e2 , pffiffiffiffiffiffi pffiffiffiffiffiffi yðtÞ ¼ c1 g11 e1 þ c2 g22 e2 :

ð1:47Þ

1.2 Equilibriums and Stability

11

The projected quantities on the covariant and contravariant eigenvectors are cj cj ci ðei ÞT  y ¼ ðei ÞT  cj vj ¼ pffiffiffiffiffi ðvi ÞT  vj ¼ pffiffiffiffiffi δij ¼ pffiffiffiffiffi , gii gii gii cj cj ci ffi: ðei ÞT  y ¼ ðei ÞT  cj vj ¼ pffiffiffiffiffi ðvi ÞT  vj ¼ pffiffiffiffiffi δji ¼ pffiffiffiffi gii gii gii Thus,

pffiffiffiffiffi T gii ðei Þ  y; pffiffiffiffiffi cj ¼ ðvj ÞT  y ¼ gjj ðej ÞT  y:

ci ¼ ðvi ÞT  y ¼

ð1:48Þ

ð1:49Þ

The covariant and contravariant eigenvectors as the coordinates to express a solution vector of dynamical system are presented in Fig. 1.1. The corresponding geometric interpretation of the relations of components is described. A vector y is depicted by the covariant and contravariant eigenvectors. The unit vectors and corresponding quantities based on the two eigenvectors are presented. Definition 1.16 Consider an n-dimensional, autonomous, nonlinear dynamical system x_ ¼ fðx, pÞ in Eq. (1.4) with an equilibrium point x . Suppose there is a neighborhood of the equilibrium x as U(x )  Ω; then f(x, p) is Cr-continuous (r  1) and Eq. (1.24) holds in U(x )  Ω. The corresponding solution is x(t) ¼ Φ(x0, t  t0, p). For a linearized dynamical system in Eq. (1.19), consider a real eigenvalue λk of matrix Df(x , p) (k 2 N ¼ {1, 2, . . . ,n}) with an eigenvector vk. For y ¼ ckvk (summation of k ¼ 1, 2,   , n ) and ck ¼ (vk)T  y, c_k ¼ ðvk ÞT  y_ with y_ ¼ Dfðx , pÞy ¼ Dfðx , pÞvj cj ¼ λj cj vj ; thus c_k ¼ λj cj δkj ¼ λk ck . (i) x at the equilibrium x on the direction vk is stable if lim ck ¼ lim ck0 eλk t ¼ 0 for λk < 0:

t!1

t!1

ð1:50Þ

(ii) x at the equilibrium x on the direction vk is unstable if lim j ck j¼ lim j ck0 eλk t j¼ 1for λk > 0:

t!1

t!1

ð1:51Þ

(iii) x at the equilibrium x on the direction vk is uncertain (critical) if lim ck ¼ lim eλk t ck0 ¼ ck0 for λk ¼ 0:

t!1

t!1

ð1:52Þ

Definition 1.17 Consider a 2n-dimensional, autonomous, nonlinear dynamical system x_ ¼ fðx, pÞ in Eq. (1.4) with an equilibrium point x . Suppose there is a neighborhood of the equilibrium x as U(x )  Ω; then f(x, p) is Cr-continuous (r  1) and Eq. (1.24) holds in U(x )  Ω. The corresponding solution in

12

1 Stability of Equilibriums

Fig. 1.1 A solution vector based on the covariant and contravariant eigenvectors in a 2-dimensional dynamical system. The thick black and blue colors are used for covariant and contravariant eigenvectors, respectively

x(t) ¼ Φ(x0, t  t0, p). For a linearized dynamical system in Eq. (1.19), consider a pffiffiffiffiffiffiffi pair of complex eigenvalue αk iβk (2k  1, 2k 2 N ¼ {1, 2, . . . ,n}, i ¼ 1) of matrix Df(x , p) with a pair of eigenvectors v2k  1 iv2k. On the invariant plane of (v2k  1, v2k), for y_ ¼ Dfðx , pÞy ¼ Dfðx , pÞvj cj with j ¼ 2k1, 2k, consider

1.2 Equilibriums and Stability

13

c2k1 ¼ ðv2k1 ÞT  y, c2k ¼ ðv2k ÞT  y

ð1:53Þ

with Dx f  v2k1 ¼ αk v2k1  βk v2k ,

ð1:54Þ

Dx f  v2k ¼ βk v2k1 þ αk v2k : Thus, T

T

c_ 2k1 ¼ ðv2k1 Þ  y_ ¼ ðv2k1 Þ  Dfðx ; pÞv j c j ¼ αk c2k1 þ βk c2k , T

T

c_ 2k ¼ ðv2k Þ  y_ ¼ ðv2k Þ  Dfðx ; pÞv j c j ¼ βk c2k1 þ αk c2k :

ð1:55Þ

For ck ¼ (c2k  1, c2k)T, c_ k ¼ Ek ck ) ck ¼ eαk t Bk ck0

ð1:56Þ

where  Ek ¼

αk

βk

βk

αk



 and Bk ¼

cos βk t

sin βk t

 sin βk t

cos βk t

 :

ð1:57Þ

(i) y at the equilibrium x on the plane of (v2k  1, v2k) is spirally stable if lim kck k ¼ lim eαk t kBk k  kck0 k ¼ 0 for Re λk ¼ αk < 0:

t!1

t!1

ð1:58Þ

(ii) y at the equilibrium x on the plane of (v2k1, v2k) is spirally unstable if lim kck k ¼ lim eαk t kBk k  kck0 k ¼ 1 for Re λk ¼ αk > 0:

t!1

t!1

ð1:59Þ

(iii) y at the equilibrium x on the plane of (v2k1, v2k) is on the invariant circle if lim kck k ¼ lim eαk t kBk k  kck0 k ¼ kck0 k for Re λk ¼ αk ¼ 0:

t!1

t!1

ð1:60Þ

(iv) y at the equilibrium x on the plane of (v2k1, v2k) is degenerate in the direction of v2k1 if Imλk ¼ 0. Definition 1.18 Consider an n-dimensional, autonomous, nonlinear dynamical system x_ ¼ fðx, pÞ in Eq. (1.4) with an equilibrium point x . Suppose there is a neighborhood of the equilibrium x as U(x )  Ω, and in the neighborhood f(x, p) is Cr-continuous (r  1) and Eq. (1.24) holds. The corresponding solution is

14

1 Stability of Equilibriums

x(t) ¼ Φ(x0, t  t0, p). The linearized system of the nonlinear system at the equilibrium point x is y_ ¼ Dfðx , pÞy (y ¼ x  x ) in Eq. (1.19). (i) The equilibrium x is called a hyperbolic equilibrium if none of the eigenvalues of Df(x , p) is zero real part (i.e., Reλk 6¼ 0 (k ¼ 1, 2, . . . , n)). (ii) The equilibrium x is called a sink if all of the eigenvalues of Df(x , p) have negative real parts (i.e., Re λk < 0 (k ¼ 1, 2, . . . , n)). (iii) The equilibrium x is called a source if all of the eigenvalues of Df(x , p) have positive real parts (i.e., Reλk > 0 (k ¼ 1, 2, . . . , n)). (iv) The equilibrium x is called a saddle if it is a hyperbolic equilibrium and Df(x , p) has at least one eigenvalue with a positive real part (i.e., Reλj > 0 ( j 2 {1, 2, . . . ,n}) and at least one with a negative real part (i.e., Reλk < 0 (k 2 {1, 2, . . . ,n}). (v) The equilibrium x is called a center if all of eigenvalues of Df(x , p) have zero real parts (i.e., Reλj ¼ 0 ( j ¼ 1, 2, . . . , n)) with distinct eigenvalues. (vi) The equilibrium x is called a stable node if all of eigenvalues of Df(x , p) are real λk < 0 (k ¼ 1, 2, . . . , n). (vii) The equilibrium x is called an unstable node if all of eigenvalues of Df(x , p) are real λk > 0 (k ¼ 1, 2, . . . , n). (viii) The equilibrium x is called a degenerate case if one of eigenvalues of Df(x , p) is zero λk ¼ 0 (k 2 {1, 2, . . . , n}). The generalized stability and bifurcation of flows in linearized, nonlinear dynamical systems in Eq. (1.4) will be discussed as follows. Definition 1.19 Consider an n-dimensional, autonomous, nonlinear dynamical system x_ ¼ fðx, pÞ in Eq. (1.4) with an equilibrium point x . Suppose there is a neighborhood of the equilibrium x as U(x )  Ω, and in the neighborhood f(x, p) is Cr-continuous (r  1) and Eq. (1.24) holds. The corresponding solution is x(t) ¼ Φ(x0, t  t0, p). The matrix Df(x , p) in Eq. (1.19) possesses n eigenvalues λk (k ¼ 1, 2, . . . , n). Set N ¼ {1, 2, . . . ,m, m+1, . . . ,(n+m)/2} and N i ¼ fi1 , i2 , . . . , ini g [ ∅ with ij 2 N ( j ¼ 1, 2, . . . , ni; i ¼ 1, 2, . . . , 6), Σ3i¼1 ni ¼ m and 2Σ6i¼4 ni ¼ n  m. [6i¼1 N i ¼ N with Ni \ Nl ¼ ∅ (l 6¼ i). Ni ¼ ∅ if ni ¼ 0. The matrix Df(x , p) possesses n1-stable, n2-unstable, and n3-invariant real eigenvectors plus n4-stable, n5-unstable, and n6-center pairs of complex eigenvectors. Without repeated complex eigenvalues of Reλk ¼ 0 (k 2 N3 [ N6), the flow Φ(t) of the nonlinear system x_ ¼ fðx, pÞ is an (n1 : n2 : [n3; m3]j n4 : n5 : n6) flow in the neighborhood of x . However, with repeated complex eigenvalues of Reλk ¼ 0 (k 2 N3 [ N6), the flow Φ(t) of the nonlinear system x_ ¼ fðx, pÞ is an (n1 : n2 : [n3; π 3]j n4 : n5 : [n6, r; s]) flow with Nm3 ¼ Am3 ¼ 0 (π 3 2 {∅, m3}), and r ¼ (r1, r2, . . . ,rl), s ¼ (s1, s2, . . . ,sl) for l  n/2 and qi 2 {∅, m6i} (i ¼ 1, 2, . . . , l). The meanings of notations in the aforementioned structures are defined as follows: (i) n1 represents exponential sinks on n1-directions of vk if λk < 0 (k 2 N1 and 1  n1  n) with distinct or repeated eigenvalues. (ii) n2 represents exponential sources on n2-directions of vk if λk > 0 (k 2 N2 and 1  n2  n) with distinct or repeated eigenvalues.

1.2 Equilibriums and Stability

15

(iii) n3 ¼ 1 represents an invariant center on 1-direction of vk if λk ¼ 0 (k 2 N3 and n3 ¼ 1). (iv) n4 represents spiral sinks on n4-pairs of (uk, vk) if Reλk < 0 and Imλk 6¼ 0 (k 2 N4 and 1  n4  n) with distinct or repeated eigenvalues. (v) n5 represents spiral sources on n5-pairs of (uk, vk) if Reλk > 0 and Imλk 6¼ 0 (k 2 N5 and 1  n5  n) with distinct or repeated eigenvalues. (vi) n6 represents invariant centers on n6-pairs of (uk, vk) if Reλk ¼ 0 and Imλk 6¼ 0 (k 2 N6 and 1  n6  n) with distinct eigenvalues. (vii) ∅ represents empty or none if ni ¼ 0 (i 2 {1, 2, . . . ,6}). (viii) [n3; m3] represents invariant centers on (n3  m3)-directions of vk3 (k3 2 N3) and sources in m3-directions of vj3 ( j3 2 N3 and j3 6¼ k3) if λk ¼ 0 (k 2 N3 and 3 þ1 n3  n) with the (m3 + 1)th-order nilpotent matrix Nm ¼ 0 (0 < m3  3 n2  1). (ix) [n3; ∅] represents invariant centers on n3-directions of vk if λk ¼ 0 (k 2 N3 and 1 < n3  n) with a nilpotent matrix N3 ¼ 0. (x) [n6, r; s] represents invariant centers on (ri  m6i) pairs of ðuk6i , vk6i Þ (k6i 2 N6i), and sources in m6i pairs of ðuj6 , vj6 Þ ( j6 2 N6i and j6 6¼ k6i) if Reλk ¼ 0 and Imλk 6¼ 0 (k 2 N6i and n6  n/2) for (l + 1) pairs of repeated eigenvalues with m6i þ1 the ðm6i þ 1Þth-order nilpotent matrix N6i ¼ 0 (0 < m6i  ri) and (i ¼ 1, 2,   , l). (xi) [n6, r; ∅] represents invariant centers on n6 pairs of (uk, vk) if Reλk ¼ 0 and Imλk 6¼ 0 (k 2 N6 and 1  n6  n) for ( pi + 1) pairs of repeated eigenvalues with a nilpotent matrix N6i ¼ 0 (i ¼ 1, 2,   , l ). Definition 1.20 Consider an n-dimensional, autonomous, nonlinear dynamical system x_ ¼ fðx, pÞ in Eq. (1.4) with an equilibrium point x . Suppose there is a neighborhood of the equilibrium x as U(x )  Ω, and in the neighborhood f(x, p) is Cr-continuous (r  1) and Eq. (1.24) holds. The corresponding solution is x(t) ¼ Φ(x0, t  t0, p). The matrix Df(x , p) in Eq. (1.19) possesses n eigenvalues λk (k ¼ 1, 2, . . . , n). Set N ¼ {1, 2, . . . ,m, m + 1, . . . ,(n + m)/2} and N i ¼ fi1 , i2 , . . . , ini g [ ∅ with ij 2 N ( j ¼ 1, 2, . . . , ni; i ¼ 1, 2, . . . , 6), Σ3i¼1 ni ¼ m and 2Σ6i¼4 ni ¼ n  m. [6i¼1 N i ¼ N with Ni \ Nl ¼ ∅ (l 6¼ i). Ni ¼ ∅ if ni ¼ 0. The matrix Df(x , p) possesses n1-stable, n2-unstable, and n3-invariant real eigenvectors plus n4-stable, n5-unstable, and n6-center pairs of complex eigenvectors. I. Nondegenerate cases (i) The equilibrium point x is an (n1 : n2 : ∅j n4 : n5 : ∅) hyperbolic point (or saddle) for the nonlinear system. (ii) The equilibrium point x is an (n1 : ∅ : ∅j n4 : ∅ : ∅) sink for the nonlinear system. (iii) The equilibrium point x is an (∅ : n2 : ∅j ∅ : n5 : ∅) source for the nonlinear system. (iv) The equilibrium point x is an (∅ : ∅ : ∅j ∅ : ∅ : n/2) center for the nonlinear system.

16

1 Stability of Equilibriums

(v) The equilibrium point x is an (∅ : ∅ : ∅j ∅ : ∅ : [n/2, r; ∅]) center for the nonlinear system. (vi) The equilibrium point x is an (∅ : ∅ : ∅j ∅ : ∅ : [n/2, r; s]) point for the nonlinear system. (vii) The equilibrium point x is an (n1 : ∅ : ∅j n4 : ∅ : n6) point for the nonlinear system. (viii) The equilibrium point x is an (∅ : n2 : ∅j ∅ : n5 : n6) point for the nonlinear system. (ix) The equilibrium point x is an (n1 : n2 : ∅j n4 : n5 : n6) point for the nonlinear system. II. Simple degenerate cases (i) The equilibrium point x is an (∅ : ∅ : [n; ∅]j ∅ : ∅ : ∅)-invariant (or static) center for the nonlinear system. (ii) The equilibrium point x is an (∅ : ∅ : [n; m3]j ∅ : ∅ : ∅) point for the nonlinear system. (iii) The equilibrium point x is an (∅ : ∅ : [n3; ∅]j ∅ : ∅ : n6) point for the nonlinear system. (iv) The equilibrium point x is an (∅ : ∅ : [n3; m3]j ∅ : ∅ : n6) point for the nonlinear system. (v) The equilibrium point x is an (∅ : ∅ : [n3; ∅]j ∅ : ∅ : [n6, l; ∅]) point for the nonlinear system. (vi) The equilibrium point x is an (∅ : ∅ : [n3; m3]j ∅ : ∅ : [n6, l; ∅]) point for the nonlinear system. (vii) The equilibrium point x is an (∅ : ∅ : [n3; ∅]j ∅ : ∅ : [n6, l; m6]) point for the nonlinear system. (viii) The equilibrium point x is an (∅ : ∅ : [n3; m3]j ∅ : ∅ : [n6, l; m6]) point for the nonlinear system. III. Complex degenerate cases (i) The equilibrium point x is an (n1 : ∅ : [n3; ∅]j n4 : ∅ : ∅) point for the nonlinear system. (ii) The equilibrium point x is an (n1 : ∅ : [n3; m3]j n4 : ∅ : ∅) point for the nonlinear system. (iii) The equilibrium point x is an (∅ : n2 : [n3; ∅]j ∅ : n5 : ∅) point for the nonlinear system. (iv) The equilibrium point x is an (∅ : n2 : [n3; m3]j ∅ : n5 : ∅) point for the nonlinear system. (v) The equilibrium point x is an (n1 : ∅ : [n3; ∅]j n4 : ∅ : n6) point for the nonlinear system. (vi) The equilibrium point x is an (n1 : ∅ : [n3; m3]j n4 : ∅ : n6) point for the nonlinear system. (vii) The equilibrium point x is an (∅ : n2 : [n3; ∅]j ∅ : n5 : n6) point for the nonlinear system. (viii) The equilibrium point x is an (∅ : n2 : [n3; m3]j ∅ : n5 : n6) point for the nonlinear system.

1.2 Equilibriums and Stability

17

Definition 1.21 Consider an n-dimensional, autonomous, nonlinear dynamical system x_ ¼ fðx, pÞ in Eq. (1.4) with an equilibrium point x . Suppose there is a neighborhood of the equilibrium x as U(x )  Ω, and in the neighborhood f(x, p) is Cr-continuous (r  1) and Eq. (1.24) holds. The corresponding solution is x(t) ¼ Φ(x0, t  t0, p). The matrix Df(x , p) in Eq. (1.19) possesses n eigenvalues λk (k ¼ 1, 2, . . . , n). Set N ¼ {1, 2, . . . ,n} and N i ¼ fi1 , i2 , . . . , ini g [ ∅, with ij 2 N ( j ¼ 1, 2, . . . , ni, i ¼ 1, 2, 3) and Σ3i¼1 ni ¼ n. [3i¼1 N i ¼ N and Ni \ Nl ¼ ∅ (l 6¼ i). Ni ¼ ∅ if ni ¼ 0. The matrix Df(x , p) possesses n1-stable, n2-unstable, and n3-invariant real eigenvectors. Without repeated eigenvalues of λk ¼ 0 (k 2 N3), the flow Φ(t) of the nonlinear system x_ ¼ fðx, pÞ in Eq. (1.4) is a (n1 : n2 : ∅ j or (n1 : n2 : 1j local flow in the neighborhood of equilibrium point x . However, with repeated eigenvalues of λk ¼ 0 (k 2 N3), the flow Φ(t) of the nonlinear system x_ ¼ fðx, pÞ in Eq. (1.4) is a (n1 : n2 : [n3; m3]j local flow in the neighborhood of equilibrium point x . I. Nondegenerate cases (i) The equilibrium point x is an (n : ∅ : ∅ j-stable node for the nonlinear system. (ii) The equilibrium point x is an (∅ : n : ∅ j-unstable node for the nonlinear system. (iii) The equilibrium point x is an (n1 : n2 : ∅ j saddle for the nonlinear system. II. Degenerate cases (i) The equilibrium point x is an (n1 : n2 : 1j-critical state for the nonlinear system. (ii) The equilibrium point x is an (n1 : n2 : [n3; ∅]j point for the nonlinear system. (iii) The equilibrium point x is an (n1 : n2 : [n3; m3]j point for the nonlinear system. Definition 1.22 Consider a 2n-dimensional, autonomous, nonlinear dynamical system x_ ¼ fðx, pÞ in Eq. (1.4) with an equilibrium point x . Suppose there is a neighborhood of the equilibrium x as U(x )  Ω, and in the neighborhood f(x, p) is Cr-continuous (r  1) and Eq. (1.24) holds. The corresponding solution is x(t) ¼ Φ(x0, t  t0, p). The matrix Df(x , p) in Eq. (1.19) possesses n pairs of complex eigenvalues (k ¼ 1, 2, . . . , n). Set N ¼ {1, 2, . . . ,n} and N i ¼ fi1 , i2 , . . . , ini g [ ∅ with ij 2 N ( j ¼ 1, 2, . . . , ni; i ¼ 4, 5, 6) and Σ6i¼4 ni ¼ n. [6i¼4 N i ¼ N and Ni \ Nl ¼∅ (l 6¼ i). Ni ¼ ∅ if ni ¼ 0. The matrix Df(x , p) possesses n4-stable, n5-unstable, and n6-center pairs of complex eigenvectors. Without repeated eigenvalues of Reλk ¼ 0 (k 2 N6), the flow Φ(t) of the nonlinear system x_ ¼ fðx, pÞ in Eq. (1.4) is a jn4 : n5 : n6) local flow in the neighborhood of equilibrium point x . However, with repeated eigenvalues of Reλk ¼ 0 (k 2 N6), the flow Φ(t) of the nonlinear system x_ ¼ fðx, pÞ in Eq. (1.4) is a jn4 : n5 : [n6, r; s]) local flow in the neighborhood of equilibrium point x .

18

1 Stability of Equilibriums

I. Nondegenerate cases (i) The equilibrium point x is an jn : ∅ : ∅ )-spiral sink for the nonlinear system. (ii) The equilibrium point x is an j ∅ : n : ∅ )-spiral source for the nonlinear system. (iii) The equilibrium point x is an j ∅ : ∅ : n) center for the nonlinear system. (iv) The equilibrium point x is an jn4 : n5 : ∅ )-spiral saddle for the nonlinear system. II. Quasi-degenerate cases (i) The equilibrium point x is an jn4 : ∅ : n6) point for the nonlinear system. (ii) The equilibrium point x is an j ∅ : n5 : n6) point for the nonlinear system. (iii) The equilibrium point x is an jn4 : ∅ : [n6, r; ∅]) point for the nonlinear system. (iv) The equilibrium point x is an jn4 : ∅ : [n6, r; s]) point for the nonlinear system. (v) The equilibrium point x is an j ∅ : n5 : [n6, r; ∅]) point for the nonlinear system. (vi) The equilibrium point x is an j ∅ : n5 : [n6, r; s]) point for the nonlinear system.

1.3

Stability and Singularity

To extend the idea of Definitions 1.16 and 1.17, a new function will be defined to determine the stability and the stability-state switching as in Luo (2019).

1.3.1

Hyperbolic Stability on Eigenvectors

Definition 1.24 Consider an n-dimensional, autonomous, nonlinear dynamical system x_ ¼ fðx, pÞ in Eq. (1.4) with an equilibrium point x and f(x, p) is Cr-continuous (r  1) in a neighborhood of the equilibrium x . The corresponding solution is x(t) ¼ Φ(x0, t  t0, p). Suppose U(x )  Ω is a neighborhood of equilibrium x , and there are n linearly independent vectors vk (k ¼ 1, 2, . . . , n). For a perturbation of equilibrium (y ¼ x  x ), let y ¼ ckvk ¼ ckvk and y_ ¼ c_k vk (summation with k ¼ 1, 2, . . . , n): ck ¼ ðvk ÞT  y ¼ ðvk ÞT  ðx  x Þ:

ð1:61Þ

1.3 Stability and Singularity

19

Thus c_k ¼ Gk ðx, pÞ ¼ ðvk ÞT  x_ ¼ ðvk ÞT  fðx, pÞ ¼

1 X 1 ðmÞ ðx , pÞcj1 cj2   cjrk G r ! kðj1 j2 jm Þ r ¼1 k

ð1:62Þ

k

ðsummation of jr ¼ 1, 2,   , n and r k ¼ 1, 2,   , m Þ, where ð1Þ

Gkðj Þ ðx , pÞ ¼ ðvk ÞT  Dc j1 fðx ðc1 , . . . , cn Þ, pÞ 1

¼ ðvk ÞT  Dx fðx ðc1 , . . . , cn Þ, pÞ∂c j1 x

k T

ð1:63Þ

¼ ðv Þ  Dx fðx ðc , . . . , c Þ, pÞv j1 1

n

ðj1 ¼ 1, 2, . . . , nÞ, ðmÞ

Gkðj

ðmÞ

1 j2 jm Þ

ðx , pÞ ¼ ðvk ÞT  Dc j1 c j2 c jm fðx ðc1 , . . . , cn Þ, pÞ 1 n ¼ ðvk ÞT  DðmÞ x fðx ðc , . . . , c Þ, pÞðvj1 vj2 . . . vjm Þ

ð1:64Þ

ðj1 , j2 , . . . , jm ¼ 1, 2, . . . , nÞ, ð0Þ

ðmÞ

with Gk ðx, pÞ ¼ Gk ðx, pÞ if m ¼ 0, and Dx f ðxðc1 ,   , c2 Þ is the (m + 1)th-order tensor. Definition 1.25 Consider an n-dimensional, autonomous, nonlinear dynamical system x_ ¼ fðx, pÞ in Eq. (1.4) with an equilibrium point x and f(x, p) is Cr-continuous (r  1) in a neighborhood of the equilibrium x . The corresponding solution is x(t) ¼ Φ(x0, t  t0, p). Suppose U(x )  Ω is a neighborhood of equilibrium x , and there are n linearly independent vectors vk (k ¼ 1, 2, . . . , n). For a perturbation of equilibrium y ¼ x  x , let y ¼ ckvk and y_ ¼ c_k vk (summation of k). (i) x at the equilibrium x on the direction vk is stable if T

T

T

T

ðvk Þ  ðxðt þ εÞ  xðtÞÞ < 0 for ðvk Þ  ðxðtÞ  x Þ > 0, ðvk Þ  ðxðt þ εÞ  xðtÞÞ > 0 for ðvk Þ  ðxðtÞ  x Þ < 0

ð1:65Þ

for all x 2 U(x )  Ω and all t 2 [t0, 1). The equilibrium x is called a sink (or a stable node) on the direction vk.

20

1 Stability of Equilibriums

(ii) x at the equilibrium x on the direction vk is unstable if ðvk ÞT  ðxðt þ εÞ  xðtÞÞ > 0 for ðvk ÞT  ðxðtÞ  x Þ > 0, ðvk ÞT  ðxðt þ εÞ  xðtÞÞ < 0 for ðvk ÞT  ðxðtÞ  x Þ < 0

ð1:66Þ

for all x 2 U(x )  Ω and all t 2 [t0, 1). The equilibrium x is called a source (or an unstable node) on the direction vk. (iii) x at the equilibrium x on the direction vk is increasingly unstable if ðvk ÞT  ðxðt þ εÞ  xðtÞÞ > 0 for ðvk ÞT  ðxðtÞ  x Þ > 0, ðvk ÞT  ðxðt þ εÞ  xðtÞÞ > 0 for ðvk ÞT  ðxðtÞ  x Þ < 0

ð1:67Þ

for all x 2 U(x )  Ω and all t 2 [t0, 1). The equilibrium x is called an increasing saddle (or an upper saddle) on the direction vk. (iv) x at the equilibrium x on the direction vk is decreasingly unstable if ðvk ÞT  ðxðt þ εÞ  xðtÞÞ < 0 for ðvk ÞT  ðxðtÞ  x Þ > 0, ðvk ÞT  ðxðt þ εÞ  xðtÞÞ < 0 for ðvk ÞT  ðxðtÞ  x Þ < 0

ð1:68Þ

for all x 2 U(x )  Ω and all t 2 [t0, 1). The equilibrium x is called a decreasing saddle (or a lower saddle) on the direction vk. (v) x at the equilibrium x on the direction vk is invariant if ðvk ÞT  ðxðt þ εÞ  xðtÞÞ ¼ 0 for ðvk ÞT  ðxðtÞ  x Þ 6¼ 0

ð1:69Þ

for all x 2 U(x )  Ω and all t 2 [t0, 1). The equilibrium x is called to be degenerate on the direction vk. Theorem 1.7 Consider an n-dimensional, autonomous, nonlinear dynamical system x_ ¼ fðx, pÞ in Eq. (1.4) with an equilibrium point x and f(x, p) is Cr (r  1)continuous in a neighborhood of the equilibrium x (i.e., U(x )  Ω). The corresponding solution is x(t) ¼ Φ(x0, t  t0, p). Suppose Eq. (1.24) holds in U(x )  Ω. For a linearized dynamical system in Eq. (1.19), consider a real eigenvalue λk of matrix Df(x , p) (k 2 N ¼ {1, 2, . . . ,n}) with an eigenvector vk. Let y ¼ ckvk and y_ ¼ c_k vk (summation of k ¼ 1, 2, . . . , n). Define c_k ¼ ðvk ÞT  y_ ¼ ðvk ÞT  x_ ¼ ðvk ÞT  fðx, pÞ:

ð1:70Þ

1.3 Stability and Singularity

21

(i) x at the equilibrium x on the direction vk is stable if and only if Gk ðx, pÞ ¼ ðvk ÞT  fðx, pÞ < 0 for ck ¼ ðvk ÞT  ðxðtÞ  x Þ > 0, Gk ðx, pÞ ¼ ðvk ÞT  fðx, pÞ > 0 for ck ¼ ðvk ÞT  ðxðtÞ  x Þ < 0

ð1:71Þ

for all x 2 U(x )  Ω and all t 2 [t0, 1). (ii) x at the equilibrium x on the direction vk is unstable if and only if Gk ðx, pÞ ¼ ðvk ÞT  fðx, pÞ > 0 for ck ¼ ðvk ÞT  ðxðtÞ  x Þ > 0, Gk ðx, pÞ ¼ ðvk ÞT  fðx, pÞ < 0 for ck ¼ ðvk ÞT  ðxðtÞ  x Þ < 0

ð1:72Þ

for all x 2 U(x )  Ω and all t 2 [t0, 1). (iii) x at the equilibrium x on the direction vk is increasingly unstable if and only if Gk ðx, pÞ ¼ ðvk ÞT  fðx, pÞ > 0 for sk ¼ ðvk ÞT  ðxðtÞ  x Þ > 0, Gk ðx, pÞ ¼ ðvk ÞT  fðx, pÞ > 0 for sk ¼ ðvk ÞT  ðxðtÞ  x Þ < 0

ð1:73Þ

for all x 2 U(x )  Ω and all t 2 [t0, 1). (iv) x at the equilibrium x on the direction vk is decreasingly unstable if and only if Gk ðx, pÞ ¼ ðvk ÞT  fðx, pÞ < 0 for ck ¼ ðvk ÞT  ðxðtÞ  x Þ > 0, Gk ðx, pÞ ¼ ðvk ÞT  fðx, pÞ < 0 for ck ¼ ðvk ÞT  ðxðtÞ  x Þ < 0

ð1:74Þ

for all x 2 U(x )  Ω and all t 2 [t0, 1). (v) x at the equilibrium x on the direction vk is invariant if Gk ðx, pÞ ¼ ðvk ÞT  fðx, pÞ ¼ 0 for all x 2 U(x )  Ω and all t 2 [t0, 1). Proof Because _ ðvk ÞT  ðxðt þ εÞ  xðtÞÞ ¼ ðvk ÞT  ðxðtÞ þ xðtÞε þ oðεÞ  xðtÞÞ _ ¼ ðvk ÞT  xðtÞε þ oðεÞ, and x_ ¼ fðx, pÞ, we have ðvk ÞT  ðxðt þ εÞ  xðtÞÞ ¼ ðvk ÞT  fðx, pÞε þ oðεÞ ¼ Gk ðx, pÞε þ oðεÞ:

ð1:75Þ

22

1 Stability of Equilibriums

(i) Due to any selection of ε > 0, for ck ¼ ðvk ÞT  ðxðtÞ  x Þ < 0, ðvk ÞT  ðxðt þ εÞ  xðtÞÞ < 0 if Gk ðx, pÞ > 0, and vice versa; and for ck ¼ (vk)T  (x(t)  x ) > 0, ðvk ÞT  ðxðt þ εÞ  xðtÞÞ > 0 if Gk ðx, pÞ < 0, and vice versa. (ii) For ck ¼ (vk)T  (x(t)  x ) > 0, ðvk ÞT  ðxðt þ εÞ  xðtÞÞ > 0 if Gk ðx, pÞ > 0, and vice versa; and for ck ¼ (vk)T  (x(t)  x ) < 0, ðvk ÞT  ðxðt þ εÞ  xðtÞÞ < 0 if Gk ðx, pÞ < 0, and vice versa. (iii) For ck ¼ (vk)T  (x(t)  x ) > 0, ðvk ÞT  ðxðt þ εÞ  xðtÞÞ > 0 if Gk ðx, pÞ > 0, and vice versa; and for sk ¼ ðvk ÞT  ðxðtÞ  x Þ < 0, ðvk ÞT  ðxðt þ εÞ  xðtÞÞ < 0 if Gk ðx, pÞ > 0, and vice versa. (iv) For ck ¼ (vk)T  (x(t)  x ) > 0, ðvk ÞT  ðxðt þ εÞ  xðtÞÞ > 0 if Gk ðx, pÞ < 0, and vice versa; and for ck ¼ (vk)T  (x(t)  x ) < 0, ðvk ÞT  ðxðt þ εÞ  xðtÞÞ < 0 if Gk ðx, pÞ < 0, and vice versa. (v) For ck ¼ (vk)T  (x(t)  x ) > 0, ðvk ÞT  ðxðt þ εÞ  xðtÞÞ ¼ 0 if Gk ðx, pÞ ¼ 0, and vice versa. Similarly, for ck ¼ (vk)T  (x(t)  x ) > 0,

1.3 Stability and Singularity

23

ðvk ÞT  ðxðt þ εÞ  xðtÞÞ ¼ 0 if Gk ðx, pÞ ¼ 0, and vice versa. The theorem is proved.

∎

Theorem 1.8 Consider an n-dimensional, autonomous, nonlinear dynamical system x_ ¼ fðx, pÞ in Eq. (1.4) with an equilibrium point x and f(x, p) is Cr-continuous (r  1) in a neighborhood of the equilibrium x (i.e., U(x )  Ω). The corresponding solution is x(t) ¼ Φ(x0, t  t0, p). Suppose Eq. (1.24) holds in U(x )  Ω. For a linearized dynamical system in Eq. (1.19), consider a real eigenvalue λk of matrix Df(x , p) (k 2 N ¼ {1, 2, . . . ,n}) with an eigenvector vk. Let y ¼ ckvk and y_ ¼ c_k vk (summation with k ¼ 1, 2, . . . , n ). ck ¼ (vk)T  y and c_k ¼ ðvk ÞT  y_ ¼ ðvk ÞT  fðx, pÞ. ð2Þ Suppose kGk ðx , pÞk < 1 and Df(x , p)vj ¼ λjvj (j ¼ 1, 2, . . . , n). (i) x at the equilibrium x on the direction vk is stable if and only if λk < 0

ð1:76Þ

for all x 2 U(x )  Ω and all t 2 [t0, 1). (ii) x at the equilibrium x on the direction vk is unstable if and only if λk > 0

ð1:77Þ

for all x 2 U(x )  Ω and all t 2 [t0, 1). (iii) x at the equilibrium x on the direction vk is increasingly unstable if and only if ð2Þ

λk ¼ 0, and Gkðj

1 j2 Þ

ðx , pÞcj1 cj2 > 0

ð1:78Þ

for all x 2 U(x )  Ω and all t 2 [t0, 1). (iv) x at the equilibrium x on the direction vk is decreasingly unstable if and only if ð2Þ

λk ¼ 0, and Gkðj

1 j2 Þ

ðx , pÞcj1 cj2 < 0

ð1:79Þ

for all x 2 U(x )  Ω and all t 2 [t0, 1). (v) x at the equilibrium x on the direction vk is invariant if and only if ðmÞ

Gkðj

1 j2 jm Þ

ðx , pÞ ¼ 0 ðm ¼ 0, 1, 2, . . .Þ

for all x 2 U(x )  Ω and all t 2 [t0, 1).

ð1:80Þ

24

1 Stability of Equilibriums

Proof For x ¼ x , ck ¼ 0. With Df(x , p)vj ¼ λjvj, the Taylor series expansion gives c_k ¼ ðvk ÞT  fðx, pÞ ¼ ðvk ÞT  ðfðx , pÞ þ Dx fðx , pÞvi ci Þ þ oðkckÞ ¼ ðvk ÞT  ðDx fðx , pÞvi ci Þ þ oðkckÞ ð1Þ

¼ GkðiÞ ðx , pÞci þ oðkckÞ ¼ ðvk ÞT  λi vi ci þ oðkckÞ ¼ ðλi δki Þci þ oðkckÞ ¼ λk ck þ oðkckÞ ð1Þ

¼ GkðkÞ ðx , pÞck þ oðkckÞ, and ð1Þ

GkðkÞ ðx , pÞ ¼ ðvk ÞT  ðDx fðx , pÞvk Þ ¼ λk : (i) For ck > 0, c_k ¼ λk ck < 0, and for ck < 0, c_k ¼ λk ck > 0: ð1Þ

Thus, GkðkÞ ðx , pÞ ¼ λk < 0. (ii) For ck > 0, Gk ðx, pÞ ¼ c_k ¼ λk ck > 0, and for ck < 0, Gk ðx, pÞ ¼ c_k ¼ λk ck < 0: ð1Þ

Thus, GkðkÞ ðx , pÞ ¼ λk > 0.

1.3 Stability and Singularity

25

(iii) For ck > 0, Gk ðx, pÞ ¼ c_k ¼ λk ck > 0, and for ck < 0, Gk ðx, pÞ ¼ c_k ¼ λk ck > 0: ð1Þ

Thus, GkðkÞ ðx , pÞ ¼ λk ¼ 0 and the higher order should be considered. The higher order Taylor series expansion gives c_k ¼ ðvk ÞT  fðx, pÞ ¼ ðvk ÞT  ð½fðx , pÞ þ Dcj1 fðx , pÞcj1 : þ

1 ð2Þ D ðx , pÞcj1 cj2 þ oðkck2 Þ 2! cj1 cj2

1 k T ð2Þ ½ðv Þ  Dx ðx , pÞðvj1 vj2 Þcj1 cj2 þ oðkck2 Þ 2! 1 ð2Þ ¼ Gkðj j Þ ðx , pÞcj1 cj2 þ oðkck2 Þ: 1 2 2!

¼

For ck > 0 with summation of j1, j2 ¼ 1, 2, . . . , n, Gk ðx, pÞ ¼ c_k ¼

1 ð2Þ G ðx , pÞcj1 cj2 > 0, 2! kðj1 j2 Þ

Gk ðx, pÞ ¼ c_k ¼

1 ð2Þ ðx , pÞcj1 cj2 > 0: G 2! kðj1 j2 Þ

and for ck < 0,

So we have ð2Þ

Gkðj

1 j2 Þ

ðx , pÞcj1 cj2 > 0:

ð1Þ

(iv) Similar to (iii), we have Gk ðx , pÞ ¼ λk ¼ 0. In addition, for ck > 0, Gk ðx, pÞ ¼ c_k ¼ and for sk < 0,

1 ð2Þ ðx , pÞcj1 cj2 < 0, G 2! kðj1 j2 Þ

26

1 Stability of Equilibriums

Gk ðx, pÞ ¼ c_k ¼

1 ð2Þ G ðx , pÞcj1 cj2 < 0: 2! kðj1 j2 Þ

So ð2Þ

Gkðj

1 j2 Þ

ðx , pÞcj1 cj2 < 0:

(v) Using the Taylor series expansion yields c_k ¼ ðvk ÞT  fðx, pÞ XN 1 ðmÞ G ðx , pÞcj1 cj2 . . . cjm þ oðkckN Þ ¼ m¼1 m! kðj1 j2 jm Þ ¼0 ðN ¼ 1, 2, . . .Þ Thus only if ðmÞ

Gkðj

1 j2 jm Þ

ðx , pÞ ¼ 0 ðm ¼ 1, 2, . . .Þ,

the above equation of c_k ¼ 0 holds, and vice versa. The theorem is proved.

∎

Definition 1.26 Consider an n-dimensional, autonomous, nonlinear dynamical system x_ ¼ fðx, pÞ in Eq. (1.4) with an equilibrium point x and f(x, p) is Cr (r  1)continuous in a neighborhood of the equilibrium x (i.e., U(x )  Ω). The corresponding solution is x(t) ¼ Φ(x0, t  t0, p). Suppose Eq. (1.24) holds in U(x )  Ω. For a linearized dynamical system in Eq. (1.19), consider a real eigenvalue λk of matrix Df(x , p) (k 2 N ¼ {1, 2, . . . ,n}) with an eigenvector vk T and let y ¼ ckvk. ck ¼ (vk)T  y and c_k ¼ ðvk Þ  y_ ¼ ðvk ÞT  fðx, pÞ. (i) x at the equilibrium x on the direction vk is stable of the mkth order if ðr Þ ðx , pÞ 1 j2 jrk Þ

Gkðjk

¼ 0, r k ¼ 0, 1, 2, . . . , mk  1;

ðvk ÞT  ðxðt þ εÞ  xðtÞÞ < 0 for ðvk ÞT  ðxðtÞ  x Þ > 0,

ð1:81Þ

ðvk ÞT  ðxðt þ εÞ  xðtÞÞ > 0 for ðvk ÞT  ðxðtÞ  x Þ < 0 for all x 2 U(x )  Ω and all t 2 [t0, 1). The equilibrium x is called a sink (or a stable node) of the mkth order on the direction vk.

1.3 Stability and Singularity

27

(ii) x at the equilibrium x on the direction vk is unstable of the mkth order if ðr Þ ðx , pÞ 1 j2 jrk Þ

Gkðjk

¼ 0, r k ¼ 0, 1, 2, . . . , mk  1;

ðvk ÞT  ðxðt þ εÞ  xðtÞÞ > 0 for ðvk ÞT  ðxðtÞ  x Þ > 0,

ð1:82Þ

ðvk ÞT  ðxðt þ εÞ  xðtÞÞ < 0 for ðvk ÞT  ðxðtÞ  x Þ < 0 for all x 2 U(x )  Ω and all t 2 [t0, 1). The equilibrium x is called a source (or an unstable node) of the mkth order on the direction vk. (iii) x at the equilibrium x on the direction vk is increasingly unstable of the mkth order if ðr Þ ðx , pÞ 1 j2 jrk Þ

Gkðjk

¼ 0, r k ¼ 0, 1, 2, . . . , mk  1;

ðvk ÞT  ðxðt þ εÞ  xðtÞÞ > 0 for ðvk ÞT  ðxðtÞ  x Þ > 0,

ð1:83Þ

ðvk ÞT  ðxðt þ εÞ  xðtÞÞ > 0 for ðvk ÞT  ðxðtÞ  x Þ < 0 for all x 2 U(x )  Ω and all t 2 [t0, 1). The equilibrium x is called an increasing saddle (or an upper saddle) of the mkth order on the direction vk. (iv) x at the equilibrium x on the direction vk is decreasingly unstable of the mkth order if ðr Þ ðx , pÞ 1 j2 jrk Þ

Gkðjk

¼ 0, r k ¼ 0, 1, 2, . . . , mk  1;

ðvk ÞT  ðxðt þ εÞ  xðtÞÞ < 0 for ðvk ÞT  ðxðtÞ  x Þ > 0,

ð1:84Þ

ðvk ÞT ðxðt þ εÞ  xðtÞÞ < 0 for ðvk ÞT  ðxðtÞ  x Þ < 0 for all x 2 U(x )  Ω and all t 2 [t0, 1). The equilibrium x is called a decreasing saddle (or a lower saddle) of the mkth order on the direction vk. Theorem 1.9 Consider an n-dimensional, autonomous, nonlinear dynamical system x_ ¼ fðx, pÞ in Eq. (1.4) with an equilibrium point x and f(x, p) is Cr (r  1)continuous in a neighborhood of the equilibrium x (i.e., U(x )  Ω). The corresponding solution is x(t) ¼ Φ(x0, t  t0, p). Suppose Eq. (1.24) holds in U(x )  Ω. For a linearized dynamical system in Eq. (1.19), consider a real eigenvalue λk of matrix Df(x , p) (k 2 N ¼ {1, 2, . . . ,n}) with an eigenvector vk and let y ¼ ckvk. ck ¼ (vk)T  y and c_k ¼ ðvk ÞT  y_ ¼ ðvk ÞT  fðx, pÞ.

28

1 Stability of Equilibriums

(i) x at the equilibrium x on the direction vk is stable of the mkth order if and only if ðr Þ ðx , pÞ 1 j2...jr k Þ

Gkðjk

ðm þ1Þ

ck  Gkðjk j ...j 1 2

2mk þ1

¼ 0, r k ¼ 0, 1, 2, . . . , mk  1; jmk j1 j2 Þ ðx , pÞðc c   c Þ < 0

ð1:85Þ

for all x 2 U(x )  Ω and all t 2 [t0, 1). (ii) x at the equilibrium x on the direction vk is unstable of the mkth order if and only if ðr Þ ðx , pÞ 1 j2...jr k Þ

Gkðjk

c  k

¼ 0, r k ¼ 0, 1, 2, . . . , mk  1;

ðm Þ Gkðjk j ...j Þ ðx , pÞðcj1 cj2   cjmk Þ 1 2 mk

ð1:86Þ >0

for all x 2 U(x )  Ω and all t 2 [t0, 1). (iii) x at the equilibrium x on the direction vk is increasingly unstable of the mkth order if and only if ðr Þ ðx , pÞ 1 j2...jr k Þ

Gkðjk

ðm Þ

Gkðjk j ...j 1 2

mk

¼ 0, r k ¼ 0, 1, 2, . . . , mk  1;

jmk j1 j2 Þ ðx , pÞðc c   c Þ > 0

ð1:87Þ

for all x 2 U(x )  Ω and all t 2 [t0, 1). (iv) x at the equilibrium x on the direction vk is decreasingly unstable of the mkth order if and only if ðr Þ ðx , pÞ 1 j2...jr k Þ

Gkðjk

¼ 0, r k ¼ 0, 1, 2, . . . , mk  1;

ðm Þ Gkðjk j ...j Þ ðx , pÞðcj1 cj2   cjmk Þ 1 2 mk

ð1:88Þ 0, Gk ðx, pÞ ¼ c_k ¼

1 ðmk Þ ðx , pÞðcj1 cj2   cjmk Þ < 0, G mk ! kðj1 j2...jmk Þ

and for ck < 0, Gk ðx, pÞ ¼ c_k ¼

1 ðmk Þ G ðx , pÞðcj1 cj2   cjmk Þ > 0: mk ! kðj1 j2...jmk Þ

Thus ðm Þ

ck  Gkðjk j ...j 1 2

mk Þ

ðx , pÞðcj1 cj2   cjmk Þ < 0:

(ii) For ck > 0, Gk ðx, pÞ ¼ c_k ¼

1 ðmk Þ ðx , pÞðcj1 cj2   cjmk Þ > 0, G mk ! kðj1 j2...jmk Þ

and for ck < 0, Gk ðx, pÞ ¼ c_k ¼

1 ðmk Þ ðx , pÞðcj1 cj2   cjmk Þ < 0: G mk ! kðj1 j2...jmk Þ

Thus ðm Þ

ck  Gkðjk j ...j 1 2

mk Þ

ðx , pÞðcj1 cj2   cjmk Þ > 0:

(iii) For ck > 0, Gk ðx, pÞ ¼ c_k ¼

1 ðmk Þ ðx , pÞðcj1 cj2   cjmk Þ > 0, G mk ! kðj1 j2...jmk Þ

30

1 Stability of Equilibriums

and for ck < 0, Gk ðx, pÞ ¼ c_k ¼

1 ðmk Þ ðx , pÞðcj1 cj2   cjmk Þ > 0: G mk ! kðj1 j2...jmk Þ

So we have ðm Þ

Gkðjk j ...j

mk Þ

1 2

ðx , pÞðcj1 cj2   cjmk Þ > 0:

(iv) Similar to (iii), for ck > 0, Gk ðx, pÞ ¼ c_k ¼

1 ðmk Þ ðx , pÞðcj1 cj2   cjmk Þ < 0, G mk ! kðj1 j2...jmk Þ

and for ck < 0, Gk ðx, pÞ ¼ c_k ¼

1 ðmk Þ G ðx , pÞðcj1 cj2   cjmk Þ < 0: mk ! kðj1 j2...jmk Þ

So ðm Þ

Gkðjk j ...j 1 2

mk Þ

ðx , pÞðcj1 cj2   cjmk Þ < 0:

The theorem is proved.

1.3.2

∎

Spiral Stability on an Invariant Eigenplane

Definition 1.27 Consider an n-dimensional, autonomous, nonlinear dynamical system x_ ¼ fðx, pÞ in Eq. (1.4) with an equilibrium point x and f(x, p) is Cr (r  1)continuous in a neighborhood of the equilibrium x (i.e., U(x )  Ω). The corresponding solution is x(t) ¼ Φ(x0, t  t0, p). Suppose Eq. (1.24) holds in U(x )  Ω. For a linearized dynamical system in Eq. (1.19), consider a pair of pffiffiffiffiffiffiffi complex eigenvalue αk iβk (2k  1, 2k 2 N ¼ {1, 2, . . . ,n}, i ¼ 1) of matrix Df(x , p) with a pair of eigenvectors v2k  1 iv2k. On the invariant plane of (v2k  1, v2k), the contravariants are defined as c2k1 ¼ ðv2k1 ÞT  y, c2k ¼ ðv2k ÞT  y

ð1:89Þ

1.3 Stability and Singularity

31

with Dx f  v2k1 ¼ αk v2k1  βk v2k ,

ð1:90Þ

Dx f  v2k ¼ βk v2k1 þ αk v2k : The time change rates of the contravariants are c_2k1 ¼ G2k1 ðx, pÞ ¼ ðv2k1 ÞT  fðx, pÞ ¼

1 X 1 ðrk Þ ðx , pÞcj1 cj2   cjrk , G r ! ð2k1Þðj1 j2 jrk Þ r ¼1 k k

c_2k ¼ G2k ðx, pÞ ¼ ðv2k ÞT  fðx, pÞ ¼

ð1:91Þ

1 X 1 ðrk Þ Gð2kÞðj j j Þ ðx , pÞcj1 cj2   cjrk , 1 2 rk r ! k r ¼1 k

where for jrk 2 f1, 2, . . . , ng, r k ¼ 1, 2, . . . ðr Þ

k Gð2k1Þðj

1 j2 jr k Þ

ðx , pÞ ¼ ðv2k1 ÞT  D

ðrk Þ j cj1 cj2 c rk

fðx , pÞ

¼ ðv2k1 ÞT  Dxðrk Þ fðx , pÞvj1 vj2   vjrk , ðr Þ

k Gð2kÞðj

1 j2 jr k

2k T Þ ðx , pÞ ¼ ðv Þ  D

ðrk Þ j cj1 cj2 c rk

fðx , pÞ

ð1:92Þ

¼ ðv2k ÞT  Dxðrk Þ fðx , pÞvj1 vj2   vjrk : Thus, the time change rates of the contravariants in Eq. (1.91) become (

c_2k1 c_2k

)

" ¼

αk

βk

#( c2k1 )

βk

αk

c2k

1 X 1 þ r ! r ¼2 k k

f

ðr Þ

k Gð2k1Þðj

1 j2 jrk Þ

ðx , pÞ

ðrk Þ ðx , pÞ Gð2kÞðj 1 j2 jr k Þ

g

ð1:93Þ cj1 cj2   cjrk :

A radial variable is defined as ρk ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðc2k1 Þ2 þ ðc2k Þ2 ,

ð1:94Þ

32

1 Stability of Equilibriums

with c2k1 ¼ ρk cos θk , c2k ¼ ρk sin θk ,

ð1:95Þ

c_2k1 ¼ ρ_ k cos θk  θ_ k ρk sin θk , c_2k ¼ θ_ k ρk cos θk þ ρ_ k sin θk :

ð1:96Þ

ρ_ k ¼ c_2k1 cos θk þ c_2k sin θk , θ_ k ρk ¼ c_2k1 sin θk  c_2k cos θk

ð1:97Þ

where we have

Thus

and ρ_ k ¼

1 1 X X 1 ðrk Þ 1 ðrk Þ Gρk and θ_ k ¼ G r r ! ! θk k r ¼1 r ¼1 k k

ð1:98Þ

k

where ðr Þ

k Gρðrkk Þ ¼ ½Gð2k1Þðj

1 j2 jr k Þ

ðr Þ

k þ Gð2kÞðj

1 j2 jr k Þ

ðx , pÞcj1 cj2   cjrk cos θk

ðx , pÞcj1 cj2   cjrk sin θk ,

1 ðrk Þ ½G ðx , pÞcj1 cj2   cjrk sin θk ρk ð2k1Þðj1 j2 jrk Þ

ðr Þ

Gθkk ¼

ðr Þ

k  Gð2kÞðj

1 j2 jr k Þ

ð1:99Þ

ðx , pÞcj1 cj2   cjrk cos θk

ðsummation of j1 , j2 ,   , jrk Þ: For ρk ! 0, the linearized equation is (

c_2k1 c_2k

)

" ¼

αk

βk

βk

αk

#(

c2k1 c2k

)

( þ

oðρk Þ oðρk Þ

) ð1:100Þ

Equation (1.98) becomes ρ_ k ¼ αk ρk þ oðρk Þ, θ_ k ρk ¼ βk ρk þ oðρk Þ:

ð1:101Þ

1.3 Stability and Singularity

33

For ρk ! 0, we have ρ_ k ¼ αk ρk , θ_ k ¼ βk :

ð1:102Þ

ρk ¼ ρk0 eαk , θk ¼ βk t þ θ0 :

ð1:103Þ

Thus

So c2k1 ¼ ρk0 eαk t cos ðβk t  θ0 Þ ¼ eαk t ðc2k1 cos βk t þ c2k 0 0 sin βk tÞ,

ð1:104Þ

c2k ¼ ρk0 eαk t sin ðβk t  θ0 Þ ¼ eαk t ðc2k1 sin βk t þ c2k 0 0 cos β k tÞ and (

c2k1

)

c2k

" ¼e

αk t

cos βk t

sin βk t

 sin βk t

cos βk t

#(

c2k1 0

)

c2k 0

:

ð1:105Þ

Letting ck ¼ (c2k1, c2k)T, we have c_ k ¼ Ek ck ) ck ¼ eαk t Bk ck0

ð1:106Þ

where " Ek ¼

αk

βk

βk

αk

"

# and Bk ¼

cos βk t

sin βk t

 sin βk t

cos βk t

# :

ð1:107Þ

Definition 1.28 Consider an n-dimensional, autonomous, nonlinear dynamical system x_ ¼ fðx, pÞ in Eq. (1.4) with an equilibrium point x and f(x, p) is Cr-continuous (r  1) in a neighborhood of the equilibrium x . The corresponding solution is x(t) ¼ Φ(x0, t  t0, p). Suppose U(x )  Ω is a neighborhood of equilibrium x . For a linearized dynamical system in Eq. (1.19), consider a pair of pffiffiffiffiffiffiffi complex eigenvalue αk iβk (2k  1, 2k 2 N ¼ {1, 2, . . . ,n},i ¼ 1) of matrix Df (x , p) with a pair of eigenvectors v2k  1 iv2k. On the invariant plane of (v2k  1, v2k), c2k1 ¼ ðv2k1 ÞT  y, c2k ¼ ðv2k ÞT  y with

ð1:108Þ

34

1 Stability of Equilibriums

Dx f  v2k1 ¼ αk v2k1  βk v2k , Dx f  v2k ¼ βk v2k1 þ αk v2k

ð1:109Þ

and ρk ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðc2k1 Þ2 þ ðc2k Þ2 ,

ð1:110Þ

c2k1 ¼ ρk cos θk , c2k ¼ ρk sin θk : For any arbitrarily small ε > 0, the stability of the equilibrium x on the invariant plane of (v2k  1, v2k) can be determined. (i) x at the equilibrium x on the plane of (v2k  1, v2k) is spirally stable if ρk ðt þ εÞ  ρk ðtÞ < 0:

ð1:111Þ

(ii) x at the equilibrium x on the plane of (v2k  1, v2k) is spirally unstable if ρk ðt þ εÞ  ρk ðtÞ > 0: ð1:112Þ (iii) x at the equilibrium x on the plane of (v2k  1, v2k) is stable with the mkth-order singularity if for θk 2 [0, 2π] Gρðrkk Þ ðx , pÞ ¼ 0 for r k ¼ 1, 2, . . . , mk  1 ρk ðt þ εÞ  ρk ðtÞ < 0:

ð1:113Þ

(iv) x at the equilibrium x on the plane of (v2k  1, v2k) is spirally unstable with the mkth-order singularity if for θk 2 [0, 2π] Grðsk k Þ ðθk Þ ¼ 0 for sk ¼ 0, 1, 2, . . . , mk  1 ρk ðt þ εÞ  ρk ðtÞ > 0:

ð1:114Þ

(v) x at the equilibrium x on the plane of (v2k  1, v2k) is circular if for θk 2 [0, 2π] r k ðt þ εÞ  r k ðtÞ ¼ 0:

ð1:115Þ

(vi) x at the equilibrium x on the plane of (uk, vk) is degenerate in the direction of uk if βk ¼ 0 and θk ðt þ εÞ  θk ðtÞ ¼ 0:

ð1:116Þ

Theorem 1.10 Consider an n-dimensional, autonomous, nonlinear dynamical system x_ ¼ fðx, pÞ in Eq. (1.4) with an equilibrium point x and f(x, p) is Cr-continuous (r  1) in a neighborhood of the equilibrium x . The corresponding solution is

1.3 Stability and Singularity

35

x(t) ¼ Φ(x0, t  t0, p). Suppose U(x )  Ω is a neighborhood of equilibrium x . For a linearized dynamical system in Eq. (1.19), consider a pair of complex eigenvalue pffiffiffiffiffiffiffi αk iβk (2k  1, 2k 2 N ¼ {1, 2, . . . ,n}, i ¼ 1) of matrix Df(x , p) with a pair of eigenvectors v2k  1 iv2k. On the invariant plane of (v2k  1, v2k), c2k1 ¼ ðv2k1 ÞT  y, c2k ¼ ðv2k ÞT  y

ð1:117Þ

with Dx f  v2k1 ¼ αk v2k1  βk v2k , Dx f  v2k ¼ βk v2k1 þ αk v2k

ð1:118Þ

and ρk ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðc2k1 Þ2 þ ðc2k Þ2 ,

ð1:119Þ

c2k1 ¼ ρk cos θk , c2k ¼ ρk sin θk : For any arbitrarily small ε > 0, the stability of the equilibrium x on the invariant plane of (v2k  1, v2k) can be determined. (i) x at the equilibrium x on the plane of (v2k  1, v2k) is spirally stable if and only if Gð1Þ ρk ðx , pÞ ¼ αk < 0:

ð1:120Þ

(ii) x at the equilibrium x on the plane of (v2k  1, v2k) is spirally unstable if and only if Gð1Þ ρk ðx , pÞ ¼ αk > 0:

ð1:121Þ

(iii) x at the equilibrium x on the plane of (v2k  1, v2k) is spirally stable with the mkth-order singularity if and only if for θk 2 [0, 2π] Gρðrkk Þ ðx , pÞ ¼ 0 for r k ¼ 1, 2, . . . , mk  1 and Gρðmk k Þ ðx , pÞ < 0:

ð1:122Þ

(iv) x at the equilibrium x on the plane of (v2k  1, v2k) is spirally unstable with the mkth-order singularity if and only if for θk 2 [0, 2π] Gρðrkk Þ ðx , pÞ ¼ 0 for r k ¼ 1, 2, . . . , mk  1 and Gρðmk k Þ ðx , pÞ > 0:

ð1:123Þ

36

1 Stability of Equilibriums

(v) x at the equilibrium x on the plane of (v2k  1, v2k) is circular if and only if for θk 2 [0, 2π] Gρðrkk Þ ðx , pÞ ¼ 0 for r k ¼ 1, 2, . . . : (vi) x at the equilibrium x on the plane of (v2k direction of v2k  1 if and only if

 1, v2k)

ð1:124Þ is degenerate in the

ðs Þ

Imλk ¼ βk ¼ 0 and Gθkk ðθk Þ ¼ 0 for sk ¼ 2, 3, . . . : Proof For x ¼ x , c2k1 ! 0, and c2k ! 0, consider ρk ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðc2k1 Þ2 þ ðc2k Þ2 :

Assuming c2k1 ¼ ρk cos θk , c2k ¼ ρk sin θk , we have c_2k1 ¼ ρ_ k cos θk  θ_ k ρk sin θk , c_2k ¼ θ_ k ρk cos θk þ ρ_ k sin θk : Thus ρ_ k ¼ c_2k1 cos θk þ c_2k sin θk , θ_ k ρk ¼ c_2k1 sin θk  c_2k cos θk , where c_2k1 ¼ G2k1 ðx, pÞ ¼ ðv2k1 ÞT  fðx, pÞ ¼

1 X 1 ðrk Þ Gð2k1Þðj j j Þ ðx , pÞcj1 cj2   cjrk , 1 2 rk r ! k r ¼1 k

c_2k ¼ G2k ðx, pÞ ¼ ðv2k ÞT  fðx, pÞ ¼

1 X 1 ðrk Þ Gð2kÞðj j j Þ ðx , pÞcj1 cj2   cjrk , 1 2 rk r ! k r ¼1 k

ðjrk 2 f1, 2,   , ng, r k ¼ 1, 2,   Þ:

ð1:125Þ

1.3 Stability and Singularity

37

Therefore, 1 1 X X 1 ðrk Þ 1 ðrk Þ Gρk and θ_ k ¼ G , r r ! ! θk r ¼1 k r ¼1 k

ρ_ k ¼

k

k

where ðr Þ

k Gρðrkk Þ ðx , pÞ ¼ ½Gð2k1Þðj

1 j2 jrk Þ

ðr Þ

k þ Gð2kÞðj

1 j2 jrk Þ

ðr Þ

Gθkk ðx , pÞ ¼

ðx , pÞcj1 cj2   cjrk cos θk

ðx , pÞcj1 cj2   cjrk sin θk ,

1 ðrk Þ ½G ðx , pÞcj1 cj2   cjrk sin θk ρk ð2k1Þðj1 j2 jrk Þ ðr Þ

k  Gð2kÞðj

1 j2 jrk Þ

ðx , pÞcj1 cj2   cjrk cos θk ,

ðsummation of j1 , j2 , . . . , jrk Þ: To the first-order approximation of c_2k1 and c_2k , c_2k1 ¼ αk c2k1 þ βk c2k þ oðρk Þ, c_2k ¼ βk c2k1 þ αk c2k þ oðρk Þ or 

c_ 2k1 c_ 2k





αk ¼ βk

βk αk



   oðρk Þ c2k1 : þ oðρk Þ c2k

Using ρ_ k ¼ c_2k1 cos θk þ c_2k sin θk , 2k 2k1 θ_ k ¼ ρ1 sin θk Þ, k ðc_ cos θ k  c_

we obtain ρ_ k ¼ ðαk c2k1 þ βk c2k1 Þ cos θk þ ðβk c2k1 þ αk c2k Þ sin θk ¼ αk ρk þ oðρk Þ, θ_ k ρk ¼ ðβk c2k1 þ αk c2k Þ cos θk þ ðαk c2k1 þ βk c2k Þ sin θk ¼ βk ρk þ oðρk Þ:

38

1 Stability of Equilibriums

Further ρ_ k ¼ αk ρk þ oðρk Þ, θ_ k ρk ¼ βk ρk þ oðρk Þ: So ρ_ k ¼ αk ρk θ_ k ρk ¼ βk ρk ) θ_ k ¼ βk : Therefore, ð1Þ

Gð1Þ ρk ðx , pÞ ¼ αk ρk and Gθk ðx , pÞ ¼ β k θ k :

The corresponding higher order expression is given by ρ_ k ¼

Xmk 1 1 1 ðmk Þ k Gðsk Þ ðx , pÞ þ G ðx , pÞ þ oðρm k Þ: rk ¼1 r k ! ρk mk ! ρk

Because for ε > 0 and ε ! 0, ρk ðt þ εÞ  ρk ðtÞ ¼ ρ_ k ε Xmk 1 1 1 ðmk Þ k ¼ ε r ¼1 Gðsk Þ ðx , pÞ þ ε G ðx , pÞ þ oðερm k Þ: k r k ! ρk mk ! ρk (i) For equilibrium stability, ρk > 0 and ρk ! 0. If αk 6¼ 0, we have ρ_ k ¼ Gð1Þ ρk ðx , pÞ ¼ αk ρk þ oðρk Þ:

Due to ρk > 0, if αk < 0, then ρ_ k < 0. Therefore, ρk ðt þ εÞ  ρk ðtÞ ¼ ρ_ k ε < 0 which implies that x at the equilibrium x on the plane of (v2k1, v2k) is spirally stable, and vice versa. (ii) Due to ρk > 0, if αk > 0, then ρ_ k > 0. Thus, ρk ðt þ εÞ  ρk ðtÞ ¼ ρ_ k ε > 0, which implies that x at the equilibrium x on the plane of (v2k1, v2k) is spirally unstable, and vice versa.

1.3 Stability and Singularity

39

(iii) If for θk 2 [0, 2π] the following conditions exist Gρðrkk Þ ðx , pÞ ¼ 0 for rk ¼ 1, 2, . . . , mk  1; Gρðmk k Þ ðx , pÞ 6¼ 0, and j Gρðrkk Þ ðx , pÞ j< 1 for r k ¼ mk þ 1, mk þ 2, . . . , then the higher order terms can be ignored, i.e., ρ_ k ¼

1 ðmk Þ k G ðx , pÞ þ oðρm k Þ: mk ! ρk

Due to ρk > 0, if Gρðmk k Þ ðx , pÞ < 0, then ρ_ k < 0. Therefore, ρk ðt þ εÞ  ρk ðtÞ ¼ ρ_ k ε < 0: In other words, x at the equilibrium x on the plane of (v2k1, v2k) is spirally stable with the mkth-order singularity, and vice versa. (iv) Due to ρk > 0, if Gρðmk k Þ ðx , pÞ > 0, then ρ_ k > 0. Therefore, ρk ðt þ εÞ  ρk ðtÞ ¼ ρ_ k ε > 0: In other words, x at the equilibrium x on the plane of (v2k1, v2k) is spirally unstable with the (mk1)th-order singularity, and vice versa. (v) If for θk 2 [0, 2π] the following conditions exist Gρðrkk Þ ðx , pÞ ¼ 0 for r k ¼ 1, 2, . . . , then ρk ðt þ εÞ  ρk ðtÞ ¼ ρ_ k ε ¼ 0 and vice versa. Therefore ρk(t) is constant. x at the equilibrium x on the plane of (v2k1, v2k) is circular. (vi) Consider θk ðt þ εÞ  θk ðtÞ ¼ θ_ k ε 1 Xmk 1 1 ðrk Þ 1 ðmk Þ k ¼ εfβk þ ½ r ¼2 G ðx , pÞ þ G ðx , pÞ þ oðρm k Þ g: k ρk r k ! θk mk ! θk If for θk 2 [0, 2π] the following conditions exist ðr Þ

βk ¼ 0 and Gθkk ðx , pÞ ¼ 0 for r k ¼ 2, 3, . . .

40

1 Stability of Equilibriums

then θk ðt þ εÞ  θk ðtÞ ¼ θ_ k ε ¼ 0: Therefore, x at the equilibrium x on the plane of (v2k1, v2k) is degenerate in the direction of v2k1. This theorem is proved. ∎

1.3.3

Spiral Stability Based on the Fourier Series Base

From the previous method, the spiral stability of the equilibrium x is difficult, and such spiral stability is based on the circular assumption as in Eq. (1.119), which is difficult to separate ρk(t) and θk(t). For the linear case, it is very easy to do so. However, for the nonlinear case, only special cases can be discussed for the spiral stability. Thus, an alternative approach is presented to determine the spiral stability of the equilibrium x as in Luo (2019). Definition 1.29 Consider an n-dimensional, autonomous, nonlinear dynamical system x_ ¼ fðx, pÞ in Eq. (1.4) with an equilibrium point x and f(x, p) is Cr-continuous (r  1) in a neighborhood of the equilibrium x (i.e., U(x )  Ω). The corresponding solution is x(t) ¼ Φ(x0, tt0, p). Suppose Eq. (1.24) holds in U(x )  Ω. For a linearized dynamical system in Eq. (1.19), consider a pair of pffiffiffiffiffiffiffi complex eigenvalue αk iβk (2k1, 2k 2 N ¼ {1, 2,   , n},i ¼ 1) of matrix Df(x , p) with a pair of eigenvectors v2k1 iv2k. In the vicinity of x , for x ¼ x + y, there is a solution in the form of x ¼ x þ y ¼ x þ a0 ðtÞ þ

1 X

bj ðtÞ cos ðjβk tÞ þ cj ðtÞ sin ðjβk tÞ

ð1:126Þ

j¼1

with the condition of lim kAj k ¼ 0

j!1

ð1:127Þ

where a0 ¼ ða01 , a02 , . . . , a0n ÞT , bj ¼ ðbj1 , bj2 , . . . , bjn ÞT , cj ¼ ðcj1 , cj2 , . . . , cjn ÞT , Aj ¼ ðAj1 , Aj2 , . . . , Ajn ÞT , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ajl ¼ b2jl þ c2jl ðj ¼ 1, 2, . . . ;l ¼ 1, 2, . . . , nÞ:

ð1:128Þ

1.3 Stability and Singularity

41

Thus, the approximate solution of x in the vicinity of the equilibrium x is for a large number N ! 1 as x ¼ x þ y x þ a0 ðtÞ þ

N X

bj ðtÞ cos ðjβk tÞ þ cj ðtÞ sin ðjβk tÞ:

ð1:129Þ

j¼1

With the transformation in Eq. (1.129), Eq. (1.4) becomes a_ 0 ¼ F0 ða0 , b, cÞ, b_ ¼ βk kc þ Fc ða0 , b, cÞ,

ð1:130Þ

c_ ¼ βk kb þ F ða0 , b, cÞ s

where b ¼ ðb1 , b2 , . . . , bN ÞT , c ¼ ðc1 , c2 , . . . , cN ÞT , Fc ¼ ðFc1 , Fc2 , . . . , FcN ÞT , Fs ¼ ðFs1 , Fs2 , . . . , FsN ÞT , k ¼ diag ð1, 2, . . . , NÞ, A ¼ ðA1 , A2 , . . . , AN ÞT ; ð 1 T fðx, pÞdt, F0 ða0 , b, cÞ ¼ T 0 ð 2 T fðx, pÞ cos ðjβk tÞdt, Fcj ða0 , b, cÞ ¼ T 0 ð 2 T Fsj ða0 , b, cÞ ¼ fðx, pÞ sin ðjβk tÞdt, T 0 T¼

ð1:131Þ

ð1:132Þ

2π , ðj ¼ 1, 2, . . . , NÞ: βk

The equilibrium of Eq. (1.130) is given by 0 ¼ F0 ða 0 , b , c Þ, 0 ¼ βk kc þ Fc ða 0 , b , c Þ,

ð1:133Þ

0 ¼ βk kb þ Fs ða 0 , b , c Þ: If Eq. (1.133) has a zero solution, i.e., a 0 ¼ 0, b ¼ 0, c ¼ 0, and then the equilibrium x has the following stability.

ð1:134Þ

42

1 Stability of Equilibriums

(i) The equilibrium x is spirally stable if lim ka0 ðtÞk ¼ 0, lim kbðtÞk ¼ 0, lim kcðtÞk ¼ 0

t!1

t!1

t!1

ðor lim ka0 ðtÞk ¼ 0 lim kAðtÞk ¼ 0Þ: t!1

ð1:135Þ

t!1

(ii) The equilibrium x is spirally unstable if lim ka0 ðtÞk ¼ 1, or lim kbðtÞk ¼ 1, or lim kcðtÞk ¼ 1

t!1

t!1

t!1

ð or lim ka0 ðtÞk ¼ 1, or lim kAðtÞk ¼ 1Þ: t!1

ð1:136Þ

t!1

(iii) The equilibrium x is stable if lim ka0 ðtÞk ¼ 0, bðtÞ ¼ 0, cðtÞ ¼ 0

t!1

ðor lim ka0 ðtÞk ¼ 0, AðtÞ ¼ 0Þ:

ð1:137Þ

t!1

(iv) The equilibrium x is unstable if lim ka0 ðtÞk ¼ 1, bðtÞ ¼ 0, cðtÞ ¼ 0

t!1

ðor lim ka0 ðtÞk ¼ 1, AðtÞ ¼ 0Þ:

ð1:138Þ

t!1

Definition 1.30 Consider an n-dimensional, autonomous, nonlinear dynamical system x_ ¼ fðx, pÞ in Eq. (1.4) with an equilibrium point x and f(x, p) is Cr-continuous (r  1) in a neighborhood of the equilibrium x (i.e.,U(x )  Ω). The corresponding solution is x(t) ¼ Φ(x0, tt0, p). Suppose Eq. (1.24) holds in U(x )  Ω. For a linearized dynamical system in Eq. (1.19), consider a pair of pffiffiffiffiffiffiffi complex eigenvalue αk iβk (2k1, 2k 2 N ¼ {1, 2, . . . ,n},i ¼ 1) of matrix Df(x , p) with a pair of eigenvectors v2k1 iv2k. In the vicinity of x , there exists a transformation of x ¼ x þ y ¼ x þ a0 ðtÞ þ

N X

bj ðtÞ cos ðjβk tÞ þ cj ðtÞ sin ðjβk tÞ

ð1:139Þ

j¼1

under lim kAj k ¼ 0 with Eq. (1.128). The corresponding dynamical system of j!1

coefficients is

1.3 Stability and Singularity

43

a_ 0 ¼ F0 ða0 , b, cÞ, b_ ¼ βk kc þ Fc ða0 , b, cÞ,

ð1:140Þ

c_ ¼ βk kb þ F ða0 , b, cÞ s

with Eqs. (1.131) and (1.132). The equivalent equation of Eq. (1.140) is z_ ¼ gðzÞ

ð1:141Þ

z ¼ ða0 , b, cÞT , g ¼ ðF0 , Fc , Fs ÞT :

ð1:142Þ

where

If Eq. (1.141) has an equilibrium of z ¼ 0 (i.e., a 0 ¼ 0, b ¼ 0, c ¼ 0), and in a neighborhood of z ¼ 0, the linearized equation with Δz ¼ zz is Δ_z ¼ Dgðz ÞΔz,

ð1:143Þ

and then the eigenvalues of Dg(z )determined by j Dgðz Þ  λInð2Nþ1Þnð2Nþ1Þ j¼ 0

ð1:144Þ

yield the stability of the equilibrium x with a kind of ðn1 : n2 : n3 jn4 : n5 : n6 Þ

ð1:145Þ

with Σ6r¼1 nr ¼ nð2N þ 1Þ: (i) If Reλl < 0 (l ¼ 1, 2, . . . , n(2N + 1)) with at least b 6¼ 0 or c 6¼ 0, the equilibrium x is spirally stable. (ii) If Reλl > 0 (l 2 {1, 2, . . . ,n(2N + 1)}) with at least b 6¼ 0 or c 6¼ 0, the equilibrium x is spirally unstable. (iii) The boundary between the stable and unstable solutions of equilibrium x is determined by the higher singularity. Remark 1.1 (i) For A(t) ¼ 0, a0(t) 6¼ 0, Eq. (1.139) is recalled to Eq. (1.19) with n-eigenvalues, and the stability of the equilibrium x is determined identically by Eq. (1.19). (ii) For A(t) 6¼ 0, a0(t) ¼ 0, Eq. (1.144) gives the spiral stability of the equilibrium x without the shifted center. (iii) From eigenvectors of Eq. (1.144), the stability on each eigenvector direction for z in the vicinity of z ¼ 0 can be determined in the similar fashion.

44

1 Stability of Equilibriums

1.4

Spiral Stability in Second-Order Nonlinear Systems

The stability of equilibriums in the second-order dynamical systems can be discussed through the first-order dynamical system. Since second-order dynamical systems are used extensively in physics and engineering, the spiral stability of equilibrium is specially discussed as in Luo (2019). Definition 1.31 Consider a second-order, autonomous, nonlinear dynamical system with 2n-dimensions as € _ pÞ 2 Rn : x ¼ fðx, x,

ð1:146Þ

There is an equilibrium point (x , 0) determined by fðx , 0, pÞ ¼ 0,

ð1:147Þ

_ pÞ is Cr-continuous (r  1) in a neighborhood of the equilibrium (x , 0) and fðx, x, (i.e., U(x , 0)  Ω). The corresponding solution is x(t) ¼ Φ(x0, tt0, p). Suppose _ pÞ  Dx fðx , 0, pÞΔx  Dx_ fðx , 0, pÞΔxk _ kfðx, x, ¼0 _ kΔxk þ kΔxk _ kΔxkþkΔxk!0 lim

ð1:148Þ

in U(x , 0)  Ω. The linearization of Eq. (1.146) is _ Δ€ x ¼ Dx fðx , 0, pÞΔx þ Dx_ fðx , 0, pÞΔx:

ð1:149Þ

For the linearized system, consider a pair of complex eigenvalues αk iβk (2k1, pffiffiffiffiffiffiffi 2k 2 N ¼ {1, 2, . . . ,2n},i ¼ 1) with a pair of eigenvectors v2k1 iv2k. In the vicinity of (x , 0), for x ¼ x + y, there is a solution in the form of x ¼ x þ y ¼ x þ a0 ðtÞ þ

1 X

bj ðtÞ cos ðjβk tÞ þ cj ðtÞ sin ðjβk tÞ

ð1:150Þ

j¼1

with the condition of lim kAj k ¼ 0

j!1

ð1:151Þ

where a0 ¼ ða01 , a02 , . . . , a0n ÞT , bj ¼ ðbj1 , bj2 , . . . , bjn ÞT , cj ¼ ðcj1 , cj2 , . . . , cjn ÞT , Aj ¼ ðAj1 , Aj2 , . . . , Ajn ÞT , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ajl ¼ b2jl þ c2jl ðj ¼ 1, 2, . . . ;l ¼ 1, 2, . . . , nÞ:

ð1:152Þ

1.4 Spiral Stability in Second-Order Nonlinear Systems

45

Thus, the approximate solution of x in the vicinity of the equilibrium (x , 0) is for a large number N ! 1 as x ¼ x þ y x þ a0 ðtÞ þ

N X

bj ðtÞ cos ðjβk tÞ þ cj ðtÞ sin ðjβk tÞ:

ð1:153Þ

j¼1

With the transformation in Eq. (1.153), Eq. (1.146) becomes _ c_ Þ, € a0 ¼ F0 ða0 , b, c, a_ 0 , b, _ c_ Þ, € ¼ 2βk k1 c_ þ β2 k2 b þ Fc ða0 , b, c, a_ 0 , b, b k

ð1:154Þ

_ c_ Þ €c ¼ 2βk k1 b_ þ β2k k2 c þ Fs ða0 , b, c, a_ 0 , b, where b ¼ ðb1 , b2 , . . . , bN ÞT , c ¼ ðc1 , c2 , . . . , cN ÞT , Fc ¼ ðFc1 , Fc2 , . . . , FcN ÞT , Fs ¼ ðFs1 , Fs2 , . . . , FsN ÞT , k1 ¼ diag ð1, 2, . . . , NÞ, k2 ¼ diag ð12 , 22 , . . . , N 2 Þ, A ¼ ðA1 , A2 , . . . , AN ÞT ; _ c_ Þ ¼ F0 ða0 , b, c, a_ 0 , b,

1 T

_ c_ Þ ¼ 2 Fcj ða0 , b, c, a_ 0 , b, T _ c_ Þ ¼ 2 Fsj ða0 , b, c, a_ 0 , b, T T¼

ðT

ð1:155Þ

_ pÞdt, fðx, x,

0

ðT

_ pÞ cos ðjβk tÞdt, fðx, x, ð1:156Þ

0

ðT

_ pÞ sin ðjβk tÞdt, fðx, x,

0

2π , ðj ¼ 1, 2, . . . , NÞ: βk

The equilibrium of Eq. (1.154) is given by 0 ¼ F0 ða 0 , b , c , 0, 0, 0Þ, 0 ¼ β2k k2 b þ Fc ða 0 , b , c , 0, 0, 0Þ,

ð1:157Þ

0 ¼ β2k k2 c þ Fs ða 0 , b , c , 0, 0, 0Þ: If Eq. (1.157) has a zero solution, i.e., a 0 ¼ 0, b ¼ 0, c ¼ 0,

ð1:158Þ

46

1 Stability of Equilibriums

then the equilibrium x has the following stability: (i) The equilibrium x is spirally stable if lim ka0 ðtÞk ¼ 0, lim kbðtÞk ¼ 0, lim kcðtÞk ¼ 0

t!1

t!1

t!1

ðor lim ka0 ðtÞk ¼ 0, lim kAðtÞk ¼ 0Þ: t!1

ð1:159Þ

t!1

(ii) The equilibrium x is spirally unstable if lim ka0 ðtÞk ¼ 1, or lim kbðtÞk ¼ 1, or lim kcðtÞk ¼ 1

t!1

t!1

t!1

ðor lim ka0 ðtÞk ¼ 1, or lim kAðtÞk ¼ 1Þ: t!1

ð1:160Þ

t!1

(iii) The equilibrium x is asymptotically stable if lim ka0 ðtÞk ¼ 0, bðtÞ ¼ 0, cðtÞ ¼ 0

t!1

ðor lim ka0 ðtÞk ¼ 0, AðtÞ ¼ 0Þ:

ð1:161Þ

t!1

(iv) The equilibrium x is asymptotically unstable if lim ka0 ðtÞk ¼ 0, bðtÞ ¼ 0, cðtÞ ¼ 0

t!1

ðor lim ka0 ðtÞk ¼ 0, AðtÞ ¼ 0Þ:

ð1:162Þ

t!1

Definition 1.32 Consider a 2n-dimensional, autonomous, nonlinear dynamical _ pÞ in Eq. (1.146) with an equilibrium point (x , 0) and fðx, x, _ pÞ system € x ¼ fðx, x, r is C -continuous (r  1) in a neighborhood of the equilibrium (x , 0) (i.e., U(x , 0)  Ω). The corresponding solution is x(t) ¼ Φ(x0, tt0, p). Suppose Eq. (1.148) holds in U(x , 0)  Ω. For a linearized dynamical system in Eq. (1.149), consider a pair of complex eigenvalues αk iβk (2k1, pffiffiffiffiffiffiffi 2k 2 N ¼ {1, 2, . . . ,2n}, i ¼ 1) with a pair of eigenvectors v2k1 iv2k. In the vicinity of (x , 0), there exists a transformation of x ¼ x þ y ¼ x þ a0 ðtÞ þ

N X

bj ðtÞ cos ðjβk tÞ þ cj ðtÞ sin ðjβk tÞ

ð1:163Þ

j¼1

under lim kAj k ¼ 0 with Eq. (1.152). The corresponding dynamical system of j!1

coefficients is

1.4 Spiral Stability in Second-Order Nonlinear Systems

47

_ c_ Þ, € a0 ¼ F0 ða0 , b, c, a_ 0 , b, € ¼ 2βk k1 c_ þ β2 k2 b þ Fc ða0 , b, c, a_ 0 , b, _ c_ Þ, b k €c ¼ 2βk k1 b_ þ

β2k k2 c

ð1:164Þ

_ c_ Þ þ F ða0 , b, c, a_ 0 , b, s

with Eqs. (1.155) and (1.156). The equivalent equation of Eq. (1.164) is €z ¼ gðz, z_ Þ

ð1:165Þ

_ c_ ÞT , g ¼ ðF0 , Fc , Fs ÞT : z ¼ ða0 , b, cÞT , z_ ¼ ða_ 0 , b,

ð1:166Þ

where

The equivalent first-order equation of Eq. (1.165) is u_ ¼ fðuÞ

ð1:167Þ

u ¼ ðz, z_ ÞT ;f ¼ ð_z, gÞT

ð1:168Þ

where

If Eq. (1.165) has an equilibrium of (z , 0) (i.e., u ¼ 0), and in a neighborhood of u ¼ 0 the linearized equation with Δu ¼ uu is Δu_ ¼ Dfðu ÞΔu

ð1:169Þ

with  Þ ¼ Dfðu



0nð2Nþ1Þnð2Nþ1Þ Gnð2Nþ1Þnð2Nþ1Þ

Inð2Nþ1Þnð2Nþ1Þ Hnð2Nþ1Þnð2Nþ1Þ

 ð1:170Þ

and   ∂g  ∂g  G¼ ,H ¼ ∂z u ¼ 0 ∂_z u ¼0

ð1:171Þ

then the eigenvalues of Dfðu Þ determined by j Dfðu Þ  λI2nð2Nþ1Þ2nð2Nþ1Þ j¼ 0 yield the stability of the equilibrium x with a kind of

ð1:172Þ

48

1 Stability of Equilibriums

ðn1 : n2 : n3 jn4 : n5 : n6 Þ

ð1:173Þ

with Σ6r¼1 nr ¼ 2nð2N þ 1Þ. (i) If Reλl < 0 (l ¼ 1, 2, . . . , 2n(2N + 1)) with at least b 6¼ 0 and c 6¼ 0, the equilibrium x is spirally stable. (ii) If Reλl > 0 (l 2 {1, 2, . . . ,2n(2N + 1)}) with at least b 6¼ 0 and c 6¼ 0, the equilibrium x is spirally unstable. (iii) The boundary between the stable and unstable solutions of the equilibrium x is determined by the higher singularity.

1.5

Lyapunov Functions and Stability

Consider an n-dimensional, autonomous, nonlinear dynamical system x_ ¼ fðx, pÞ in Eq. (1.4). Let V : U ! R be a differentiable function defined in a neighborhood of equilibrium x on U/{x }. A function V_ : U ! R is defined by _ VðxÞ ¼ DVðxÞ ¼ nT  fðx, tÞ

ð1:175Þ

where nT ¼ ð∂x1 V, . . . , ∂xn VÞ and f(x, t) ¼ ( f1, . . . ,fn)T. Definition 1.33 Consider an n-dimensional, autonomous, nonlinear dynamical system x_ ¼ fðx, pÞ in Eq. (1.4) with an equilibrium point x and f(x, p) is Cr-continuous (r  1) in a neighborhood of the equilibrium x . Suppose U(x )  Ω is a neighborhood of equilibrium x . There is a continuous function V : U ! R which is differentiable on U/{x }, such that V(x ) ¼ 0 and V(x) > 0 if x 6¼ x . (i) If V_  0 in U/{x }, the function V is called a Lyapunov function for equilibrium x . (ii) If V_ < 0 in U/{x }, the function V is called a strict Lyapunov function for equilibrium x . Theorem 1.11 Consider an n-dimensional, autonomous, nonlinear dynamical system x_ ¼ fðx, pÞ in Eq. (1.4) with an equilibrium point x and f(x, p) is Cr-continuous (r  1) in a neighborhood of the equilibrium x . Suppose U(x )  Ω is a neighborhood of equilibrium x . There is a continuous function V : U ! R which is differentiable on U/{x }, such that V(x ) ¼ 0 and V(x) > 0 if x 6¼ x . (i) If V_  0 in U/{x }, the equilibrium x is stable. (ii) If V_ < 0 in U/{x }, the equilibrium x is asymptotically stable. Proof The proof can be referred in Hirsch et al. (2004).

∎

1.5 Lyapunov Functions and Stability

49

As in Luo (2012), to investigate the stability of the continuous systems in the vicinity of equilibrium x , a measuring function should be introduced through the relative position vector to equilibrium x . The relative position vector is given by r ¼ x  x :

ð1:176Þ

Definition 1.34 For an equilibrium point x , consider a flow x 2 U(x ) in dynamical system of Eq. (1.4). A relative distance function for the flow x to the equilibrium x is defined by dðx, x Þ ¼ kx  x k:

ð1:177Þ

If d(x, xp) ¼ C ¼ const, there is a surface given by kx  x k ¼ C

ð1:178Þ

which is called the equi-distance surface of equilibrium x . Further, if there is a monotonically increasing or decreasing function of the relative distance d(x, x ), E ¼ Vðx, x Þ  f ðdðx, x ÞÞ

ð1:179Þ

with the following property: Vðx, x Þ ¼ f ðdðx, x ÞÞ ¼ min ðor max Þ if dðx, x Þ ¼ 0:

ð1:180Þ

Such a monotonic function V(x, x ) is called a generalized measuring function of dynamical system in a neighborhood of the equilibrium x . If E ¼ C ¼ const, there is a surface given by Vðx, x Þ ¼ C

ð1:181Þ

which is called the equi-measuring function surface. To explain the concept of the measuring function, consider a domain U(x ) in phase space in the vicinity of the equilibrium x with big circular symbol, as shown in Fig. 1.2. Suppose a point x with a small circular symbol is the solution of the subsystem. The relative location vector to the equilibrium x is expressed by a vector r ¼ xx and the corresponding relative distance is expressed by d(x, x ). Such a point x is on the equi-measuring function surface of E ¼ V(d(x, x )). For different values of E ¼ Ei (i ¼ 1, 2,   ), a set of the equi-measuring function surfaces will fill the entire domain of U(x), which are depicted by the thick curves in Fig. 1.2. Based on the equi-measuring function surface, there is a dynamical system.

50

1 Stability of Equilibriums

Fig. 1.2 A relative position vector from xp to x and a measuring function with n1 + n2 ¼ n

Definition 1.35 For any equi-measuring function surface in Eq. (1.181), there is a dynamical system as x_ m ¼ f m ðxm Þ

ð1:182Þ

with the initial condition ðxm 0 , t 0 Þ and the equi-measuring function surface can be expressed by Vðxm , x Þ ¼ Vðxm 0 , x Þ ¼ E:

ð1:183Þ

The dynamical system given in Eq. (1.182) is invariant in the sense of the measuring function in Eq. (1.181). The subscript or superscript “m” represents the flow on the “equi-measuring function surface.” To measure dynamical behaviors of dynamical system in Eq. (1.4) to the equi-measuring function surface, from Luo (2008a,b), the following functions are introduced. Definition 1.36 Consider a flow x(t) of the dynamical system with a vector field f(x, t, p) in Eq. (1.4). At time t, if the flow x(t) arrives to the equi-measuring function surface in Eq. (1.181), the kth-order, G-functions at the constant measuring function level are defined as

1.5 Lyapunov Functions and Stability GðkÞ m ðxðtÞ, x Þ ¼ ðk þ 1Þ! lim

ε!0

1 f½nðxm ðt þ εÞ, x Þ T  xðt þ εÞ εkþ1

 ½nm ðxðtÞ, x Þ T  xðtÞ  ¼

Xkþ1 r¼1

51

k X 1

q! q¼1

q GðkÞ m ðxðtÞ, x Þε g

ð1:184Þ

Crkþ1 Dðkþ1rÞ ½nðxm ðtÞ, x Þ T  ½Dðr1Þ fðxðtÞ, pÞ

Dðr1Þ f m ðxm , x Þ jxm ¼x for k ¼ 0,1,2,. . . . The normal vector of the equi-measuring function surface is nðxm , x Þ ¼

∂Vðxm , x Þ ∂V ∂V ∂V ¼ ð m , m , . . . , m ÞT ∂xm ∂x1 ∂x2 ∂xn

ð1:185Þ

where the total differential operator is given by D¼

∂ðÞ ∂ðÞ , x_ þ ∂x ∂t

DðrÞ ðÞ ¼ D∘Dðr1Þ ðÞ ¼ DðDðr1Þ ðÞÞ,

ð1:186Þ

and Dð0Þ ðÞ ¼ I with C rkþ1 ¼

ðk þ 1Þ! and r! ¼ 1  2  3    r: r!ðk þ 1  rÞ!

ð1:187Þ

From Eq. (1.181), the following relation holds: 0 ¼ x_ T 

∂V ¼ ½nðxm , x Þ T  f m ðxm , x Þ: ∂x

ð1:188Þ

For a zero-order G-function (k ¼ 0), one obtains m m m T T Gð0Þ m ðx, x Þ ¼ ½nðx , x Þ  ½fðx, pÞ  f ðx , x Þ jxm ¼x ¼ ½nðx, x Þ  fðx, pÞ:

ð1:189Þ The zero-order G-function is the dot product of the vector field f(x, p) for the dynamical system and the normal vector n(xm, x ). Consider an instantaneous value of the equi-measuring function at time t. In other words, letting xm ¼ x, Eq. (1.179) gives

52

1 Stability of Equilibriums

EðtÞ ¼ Vðx, x Þ:

ð1:190Þ

The corresponding time change ratio of the measuring function is dEðtÞ ∂Vðx, x Þ ¼  x_ dt ∂x ¼ ½nðx, x Þ T  fðx, pÞ

ð1:191Þ

¼ Gð0Þ m ðx, x Þ:

From the foregoing equation, the change of the equi-measuring function for the dynamical system for time t 2 [tk, tk+1] can be defined as in Luo (2008a,b). Definition 1.37 For a flow x(t) of the dynamical system with a vector field f(x, p) in Eq. (1.4), consider the equi-measuring function V(x, x ) in Eq. (1.179) to monotonically increase to a metric function d(x, xp) in Eq. (1.177). The total change of the equi-measuring function for the time interval [tk, t] is defined as ðt dEðtÞ ¼ Gð0Þ Lðx , t k , tÞ ¼ m ðx, x Þdt t k dt tk ðt ¼ ½nðx, x Þ T  fðx, pÞdt ðt



ð1:192Þ

tk

¼ VðxðtÞ, x Þ  Vðxk , x Þ where xk ¼ x(tk). For a given t ¼ tk+1>tk, the increment of the equi-measuring function to dynamical system in Eq. (1.4) for t 2 [tk, tk+1] is Lðx , t k , t kþ1 Þ ¼

ð tkþ1 tk

¼

ð tkþ1

dEðtÞ ¼ dt

ð tkþ1 tk

Gð0Þ m ðx, x Þdt

½nðx, x Þ T  fðx, pÞdt

ð1:193Þ

tk

¼ Vðxkþ1 , x Þ  Vðxk , x Þ: From Eq. (1.193), the equi-measuring function quantity is used to measure changes of the dynamical system. Thus such a function can be used to investigate the stability of dynamical systems. Definition 1.38 For a dynamical system in Eq. (1.4), consider the equi-measuring function V(x, x ) in Eq. (1.179) to monotonically increase to a metric function d(x, xp) in Eq. (1.177) in Uðx Þ  R n . A flow x(t) at xk for t¼tk is:

1.5 Lyapunov Functions and Stability

53

(i) locally decreasing to the equi-measuring function surface in Uðx Þ  R n if Vðxk , x Þ  Vðxkε , x Þ < 0, Vðxkþε , x Þ  Vðxk , x Þ < 0;

ð1:194Þ

(ii) locally increasing to the equi-measuring function surface in Uðx Þ  R n if Vðxk , x Þ  Vðxkε , x Þ > 0, Vðxkþε , x Þ  Vðxk , x Þ > 0;

ð1:995Þ

(iii) locally tangential to the equi-measuring function surface in Uðx Þ  R n if Vðxk , x Þ  Vðxkε , x Þ < 0,  either

Vðxkþε , x Þ  Vðxk , x Þ > 0 Vðxk , x Þ  Vðxkε , x Þ > 0,

or

Vðxkþε , x Þ  Vðxk , x Þ < 0

for the external surface; ð1:196Þ gfor the interior surface:

From the previous definitions, the locally increasing and decreasing of a flow x(t) at xk to the measuring function surface are depicted in Fig. 1.3 in the vicinity of the point x . A flow x(t) at xk, locally tangential to the equi-measuring function surface, can be similarly sketched. Theorem 1.12 For a dynamical system in Eq. (1.4), consider the equi-measuring function V(x, x )in Eq. (1.179) to be monotonically increased to a metric function d (x, x ) in Eq. (1.177). A flow x(t) at xk for t¼tk in the domain Uðx Þ  R n is: (i) locally decreasing to the equi-measuring function surface in Uðx Þ  R n if and only if T Gð0Þ m ðxk , x Þ ¼ ½nðxk , x Þ  fðxk , pÞ < 0;

ð1:197Þ

(ii) locally increasing to the equi-measuring function surface in Uðx Þ  R n if and only if T Gð0Þ m ðxk , x Þ ¼ ½nðxk , x Þ  fðxk , pÞ < 0;

ð1:198Þ

(iii) locally tangential to the equi-measuring function surface in Uðx Þ  R n if and only if T Gð0Þ m ðxk , x , t k Þ ¼ ½nðxk , x Þ  fðxk , pÞ ¼ 0 Gð1Þ m ðxk , x , t k Þ > 0 for the external surface; Gð1Þ m ðxk , x , t k Þ < 0 for the interior surface:

ð1:199Þ

54

1 Stability of Equilibriums

a

b Fig. 1.3 (a) A locally increasing flow and (b) a locally decreasing flow to a measuring function with n1 + n2 ¼ n

1.5 Lyapunov Functions and Stability

55

Proof Using G-function and the Taylor series expansion, the theorem can be proved directly. ∎ Definition 1.39 For a dynamical system in Eq. (1.4), consider the equi-measuring function V(x, x ) in Eq. (1.179) to monotonically increase to a metric function d(x, xp) in Eq. (1.177) in Uðx Þ  R n . A flow x(t) for xk to xk+1 in U(x ) for t ¼ ts 2 (tk, tk+1) is: (i) uniformly decreasing to the equi-measuring function surface in Uðx Þ  R n if Vðxs , x Þ  Vðxsε , x Þ < 0, Vðxsþε , x Þ  Vðxs , x Þ < 0;

ð1:200Þ

(ii) uniformly increasing to the equi-measuring function surface in Uðx Þ  R n if Vðxs , x Þ  Vðxsε , x Þ < 0, Vðxsþε , x Þ  Vðxs , x Þ < 0;

ð1:201Þ

(iii) uniformly invariant to the equi-measuring function surface in Uðx Þ  R n if Vðxsε , x Þ ¼ Vðxs , x Þ ¼ Vðxsþε , x Þ:

ð1:202Þ

Theorem 1.13 For a dynamical system in Eq. (1.4), consider the equi-measuring function V(x, x ) in Eq. (1.179) to monotonically increase to a metric function d(x, xp) in Eq. (1.177) in Uðx Þ  R n . A flow x(t) for xk to xk + 1 for t 2 (tk, tk + 1) in Uðx Þ  R n is: (i) uniformly decreasing to the equi-measuring function surface in Uðx Þ  R n if and only if all points x(t) for t 2 [tk, tk+1] on the flow γ satisfy the following condition: T Gð0Þ m ðx, x Þ ¼ ½nðx, x Þ  fðx, pÞ < 0;

ð1:203Þ

(ii) uniformly increasing to the equi-measuring function surface in Uðx Þ  R n if and only if all points x(t) for t 2 [tk, tk+1] on the flow γ satisfy the following condition: T Gð0Þ m ðx, x Þ ¼ ½nðx, x Þ  fðx, pÞ > 0;

ð1:204Þ

(iii) uniformly invariant to the equi-measuring function surface in Uðx Þ  R n if and only if all points x(i)(t) for t 2 [tk, tk+1] on the flow γ satisfy the following condition: GðkÞ m ðx, x Þ ¼ 0 k ¼ 0, 1, 2, . . . :

ð1:205Þ

56

1 Stability of Equilibriums

Proof Using G-function and the Taylor series expansion, the theorem can be proved directly. ∎ Definition 1.40 For a dynamical system in Eq. (1.4), consider the equi-measuring function V(x, x ) in Eq. (1.179) to monotonically increase to a metric function d(x, xp) in Eq. (1.177) in Uðx Þ  R n . A flow x(t) at xk for t¼tk is: (i) locally decreasing with the (2s)th order to the equi-measuring function surface in Uðx Þ  R n if GðrÞ m ðxk ; x Þ ¼ 0,for r ¼ 0,1,2,. . . ,2s  1;

Vðxk ; x Þ  Vðxkε ; x Þ < 0,



Vðxkþε ; x Þ  Vðxk ; x Þ < 0;

g

ð1:206Þ

(ii) locally increasing with the (2s)th order to the equi-measuring function surface in Uðx Þ  R n if GðrÞ m ðxk ; x Þ ¼ 0,for r ¼ 0,1,2,. . . ,2s  1

Vðxk ; x Þ  Vðxkε ; x Þ > 0, Vðxkþε ; x Þ  Vðxk ; x Þ > 0;

g

ð1:207Þ

(iii) locally tangential with the (2s + 1)th order to the equi-measuring function surface in Uðx Þ  R n if GðrÞ m ðxk ; x Þ ¼ 0,for r ¼ 0,1,2,. . . ,2s;

Vðxk ; x Þ  Vðxkε ; x Þ < 0, Vðxkþε ; x Þ  Vðxk ; x Þ > 0 Vðxk ; x Þ  Vðxkε ; x Þ > 0, Vðxkþε ; x Þ  Vðxk ; x Þ < 0

g for the external surface,

ð1:208Þ

g for the interior surface:

Theorem 1.14 For a dynamical system in Eq. (1.4), consider the equi-measuring function V(x, x ) in Eq. (1.179) to monotonically increase to a metric function d(x, xp) in Eq. (1.177) in Uðx Þ  R n . A flow x(t) at xk for t¼tk is: (i) locally decreasing with the (2s)th order to the equi-measuring function surface in Uðx Þ  R n if and only if GðrÞ m ðxk , x Þ ¼ 0, for r ¼ 0, 1, 2, . . . , 2s  1; Gð2sÞ m ðxk , x Þ < 0;

ð1:209Þ

References

57

(ii) locally increasing with the (2s)th order to the equi-measuring function surface in Uðx Þ  R n if and only if GðrÞ m ðxk , x Þ ¼ 0, for r ¼ 0, 1, 2, . . . , 2s  1; Gð2sÞ m ðxk , x Þ > 0;

ð1:210Þ

(iii) locally tangential with the (2s + 1)th order to the equi-measuring function surface in Uðx Þ  R n if and only if GðrÞ m ðxk , x Þ ¼ 0, for r ¼ 0, 1, 2, . . . , 2s;

Gð2sþ1Þ ðxk , x Þ > 0 for the external surface, m

ð1:211Þ

Gð2sþ1Þ ðxk , x Þ < 0 for the interior surface: m Proof Using G-function and the Taylor series expansion, the theorem can be proved directly. ∎

References Carr, J.,1981, Applications of Center Manifold Theory, Applied Mathematical Science 35, Springer-Verlag, New York. Coddington, E.A. and Levinson, N., 1955, Theory of Ordinary Differential Equations, New York: McGraw-Hill. Hartman, P., Ordinary Differential Equations, Wiley, New York. (2nd ed. Birkhauser, Boston Basel Stuttgart, 1964). Hirsch, M.W., Smale, S. and Devaney, R.L., 2004, Differential Equations, Dynamical Systems, and An Introduction to Chaos, Amsterdam: Elsevier. Luo, A.C.J., 2008a, A theory for flow switchability in discontinuous dynamical systems, Nonlinear Analysis: Hybrid Systems, 2(4), pp. 1030–1061. Luo, A.C.J., 2008b, Global Transversality, Resonance and Chaotic Dynamics, Singapore: World Scientific. Luo, A.C.J., 2011, Regularity and Complexity in Dynamical Systems, New York: Springer. Luo, A.C.J., 2012, Continuous Dynamical Systems, HEP/L&H Scientific, Beijing/Glen Carbon. Luo, A.C.J., 2019, On stability and bifurcation of equilibriums in nonlinear systems, Journal of Vibration Testing and System Dynamics, 3(2), pp. 147–232. Marsden, J.E. and McCracken, M.F., 1976, The Hopf Bifurcation and Its Applications, Applied Mathematical Science 19, Springer-Verlag, New York.

Chapter 2

Bifurcations of Equilibrium

In this chapter, the hyperbolic bifurcations of equilibriums on the eigenvectors in nonlinear dynamical systems are discussed, and the Hopf bifurcation of an equilibrium on a specific eigenvector plane is presented. Based on the Fourier series base, the transformation for the spiral stability is introduced for the Hopf bifurcation of equilibriums. The Hopf bifurcation of equilibriums in the second-order nonlinear dynamical systems is discussed from the Fourier series transformation.

2.1

Bifurcations

The dynamical characteristics of equilibriums in nonlinear dynamical systems in Eq. (1.4) are based on the given parameters. With varying parameters in dynamical systems, the corresponding dynamical behaviors will change qualitatively. The qualitative switching of dynamical behaviors in dynamical systems is called bifurcation and the corresponding parameter values are called bifurcation values. To understand the qualitative changes of dynamical behaviors of nonlinear systems with parameters in the neighborhood of equilibriums, the bifurcation theory for equilibrium of nonlinear dynamical system in Eq. (1.4) will be presented. Dx(
) ¼ ∂(
)/∂x and Dp(
) ¼ ∂(
)/∂p will be adopted from now on. For no specific notice, D  Dx. Definition 2.1 Consider an n-dimensional, autonomous, nonlinear dynamical system x_ ¼ fðx, pÞ in Eq. (1.4) with an equilibrium point (x, p). Suppose there is a neighborhood of the equilibrium x as U(x)  Ω, and in the neighborhood Eq. (1.24) holds. The linearized system of the nonlinear system at the equilibrium point (x, p) is y_ ¼ Dx fðx , pÞy (y ¼ x  x) in Eq. (1.19). (i) The equilibrium point ðx0 , p0 Þ is called the switching point of equilibrium solutions if Dxf(x, p) at ðx0 , p0 Þ possesses at least one real eigenvalue (or one pair of complex eigenvalues) with zero real part. © Springer Nature Switzerland AG 2019 A. C. J. Luo, Bifurcation and Stability in Nonlinear Dynamical Systems, Nonlinear Systems and Complexity 28, https://doi.org/10.1007/978-3-030-22910-8_2

59

60

2 Bifurcations of Equilibrium

(ii) The value p0 in Eq. (1.4) is called a switching value of p if the dynamical characteristics at point ðx0 , p0 Þ change from one state into another state. (iii) The equilibrium point ðx0 , p0 Þ is called the bifurcation point of equilibrium solutions if Dxf(x, p) at ðx0 , p0 Þ possesses at least one real eigenvalue (or one pair of complex eigenvalues) with zero real part, and more than one branches of equilibrium solutions appear or disappear. (iv) The value p0 in Eq. (1.4) is called a bifurcation value of p if the dynamical characteristics at point ðx0 , p0 Þ change from one stable state into another unstable state.

2.2

Hyperbolic Bifurcations on Eigenvectors

Definition 2.2 Consider an n-dimensional, autonomous, nonlinear dynamical system x_ ¼ fðx, pÞ in Eq. (1.4) with an equilibrium point x and f(x, p) is Cr-continuous (r  1) in a neighborhood of the equilibrium x (i.e., U(x)  Ω). The corresponding solution is x(t) ¼ Φ(x0, t  t0, p). Suppose Eq. (1.24) holds in U(x)  Ω. For a linearized dynamical system in Eq. (1.19), consider a real eigenvalue λk of matrix Df (x, p) (k 2 N ¼ {1, 2, . . . ,n}) with an eigenvector vk. Suppose one of n independent solutions y ¼ ckvk and y_ ¼ c_k vk , ck ¼ ðvk ÞT
y ¼ ðvk ÞT
ðx  x Þ

ð2:1Þ

c_k ¼ ðvk ÞT
y_ ¼ ðvk ÞT
x_ ¼ ðvk ÞT
fðx, pÞ:

ð2:2Þ

and

In the vicinity of point ðx0 , p0 Þ, (vk)T
f(x, p) can be expanded for (0 < θi < 1) and (0 < γ j < 1) with γ ¼ (γ 1, γ 2, . . . ,γ m)T as ðvk ÞT
fðx, pÞ ¼ ðvk ÞT
fðx0 þ y , p0 þ ΔpÞ ¼ akðkÞ zk þ bTk
ðp  p0 Þ þ

þ

q mk X X 1 r ðqr, rÞ C q akðj j


j Þ zj1  zj2 


zjqr  ðp  p0 Þr 1 2 qr q! q¼2 r¼0 m k þ1 X r¼0

1 ðm þ1r , rÞ ðθ zj1  Þðθ2 zj2  Þ Cr a k ðmk þ 1Þ! mk þ1 kðj1 j2


jmk þ1r Þ 1




ðθmk þ1r zjmk þ1r  ÞðγT
ðp  p0 ÞÞr ,

ð2:3Þ

2.2 Hyperbolic Bifurcations on Eigenvectors

61

where zs ¼ ðvs ÞT
y ¼ cs  cs ðs ¼ 1; 2; . . . ; nÞ, 0 Δp ¼p  p0 ; T

bkT ¼ ðvk Þ
∂p fðx; pÞjðx ;p0 Þ , 0

k T

akðj1 Þ ¼ ðv Þ
∂cj1 fðx; pÞjðx ;p0 Þ 0

k T

¼ ðv Þ
∂x fðx; pÞjðx ;p0 Þ vj1 0

¼ ðr;sÞ akðj j


j Þ 1 2 r

ð1Þ Gkðj Þ ðx0 ; p0 Þ, 1 k T

¼ ðv Þ
T

ð2:4Þ

ðrÞ ðsÞ ∂cj1 cj2


cjr ∂p fðx; pÞjðx ;p Þ 0 0 ðrÞ ðsÞ

¼ ðvk Þ
∂x ∂p fðx; pÞjðx ;p Þ vj1 vj2


vjr ; 0

ðr;0Þ akðj j


j Þ 1 2 r

k T

¼ ðv Þ
T

0

ðrÞ ∂cj1 cj2


cjr fðx; pÞjðx ;p Þ 0 0 ðrÞ

¼ ðvk Þ
∂x fðx; pÞjðx ;p0 Þ vj1 vj2


vjr 0

¼

ðrÞ Gkðj j ...j Þ ðx0 ; p0 Þ: 1 2 r

Definition 2.3 Consider an n-dimensional, autonomous, nonlinear dynamical system x_ ¼ fðx, pÞ in Eq. (1.4) with an equilibrium point x and f(x, p) is Cr-continuous (r  1) in a neighborhood of the equilibrium x (i.e., U(x)  Ω). The corresponding solution is x(t) ¼ Φ(x0, t  t0, p). Suppose Eq. (1.24) holds in U(x)  Ω. For a linearized dynamical system in Eq. (1.19), consider a real eigenvalue λk of matrix Df (x, p) (k 2 N ¼ {1, 2, . . . ,n}) with an eigenvector vk. Suppose one of n independent solutions y ¼ ckvk and y_ ¼ c_k vk . For a specific j 6¼ k, equilibrium equation in the vicinity of (x0, p0) is ðvj ÞT
fðx0 þ y , p0 þ ΔpÞ ¼ 0 ðor λ0j zj þ bTj
ðp  p0 Þ  0Þ, ðj ¼ 1, 2, . . . , n but j 6¼ kÞ, λ0j 6¼ 0:

ð2:5Þ

62

2 Bifurcations of Equilibrium

For a specific k, if ak(k) ¼ λ0k ¼ 0 at (x0, p0), equilibrium equation is ðvk ÞT
fðx0 þ y , p0 þ ΔpÞ ¼ 0 with λ0k ¼ 0; ð1Þ

ðor Gkðj Þ ðx0 , p0 Þzj1  þ bTk
ðp  p0 Þ 1

q sk X X 1 r ðqr, rÞ þ C q akðj j


j Þ zj1  zj2  . . . zjqr  ðp  p0 Þr  0Þ 1 2 qr q! q¼2 r¼0

ð2:6Þ

equivalent to z_k ¼ Ak0 ðzk Þsk þ Ak1 ðzk Þsk 1 þ


þ Aksk ¼ 0 Aki ¼ Aki ðz1 , z2 ,


, zk1 , zkþ1 , . . . , zn , pÞ, ði ¼ 0, 1, 2, . . . , sk Þ: Equations (2.5) and (2.6) give lk-equilibrium zk (0 < lk sk) with the eigenvector vk direction.

Plk

i¼1 αi

¼ mk on

(i) If ðr Þ ðx0 , p0 Þ 1 j2 ...jrkÞ

λ0k ¼ 0, Gkðjk ðm Þ

ðx0 , p0 Þzj1 zj2 2 ...jmkÞ

Gkðjk j 1

¼ 0 ðr k ¼ 1, 2, 3, . . . , mk  1Þ,

. . . zjmk > 0,

g

ð2:7Þ

the bifurcation of equilibrium x at point ðx0 , p0 Þ is called an increasing saddlenode (or an upper-saddle-node) bifurcation of the (mk)th order on the eigenvector vk. The bifurcation point ðx0 , p0 Þ on the eigenvector vk is an increasing saddle (or an upper saddle) of the (mk)th order. (ii) If ðr Þ ðx0 , p0 Þ 1 j2 ...jrkÞ

λ0k ¼ 0, Gkðjk ðm Þ

Gkðjk j

1 2 ...jmk

¼ 0 ðr k ¼ 1, 2, 3, . . . , mk  1Þ,

jmk  j1 j2 < 0, Þ ðx0 , p0 Þz z . . . z

g

ð2:8Þ

the bifurcation of equilibrium x at point ðx0 , p0 Þ is called a decreasing saddlenode (or a lower-saddle-node) bifurcation of the mkth order on the eigenvector vk. The bifurcation point ðx0 , p0 Þ on the eigenvector vk is a decreasing saddle (or a lower saddle) of the mkth order. (iii) If ðr Þ ðx0 , p0 Þ 1 j2 ...jrkÞ

λ0k ¼ 0, Gkðjk ðm Þ

zk Gkðjk j

¼ 0 ðr k ¼ 1, 2, 3, . . . , mk  1Þ,

jmk  j1 j2 < 0, Þ ðx0 , p0 Þz z . . . z

1 2 ...jmk

g

ð2:9Þ

2.2 Hyperbolic Bifurcations on Eigenvectors

63

the bifurcation of equilibrium x at point ðx0 , p0 Þ is called an (mk)th-order sink bifurcation on the eigenvector vk. The bifurcation point ðx0 , p0 Þ on the eigenvector vk is an (mk)th-order sink. (iv) If ðr Þ ðx0 , p0 Þ 1 j2 ...jr kÞ

λ0k ¼ 0, Gkðjk ðm Þ

zk Gkðjk j

¼ 0 ðr k ¼ 1, 2, 3, . . . , mk  1Þ,

jmk  j1 j2 > 0, Þ ðx0 , p0 Þz z . . . z

1 2 ...jmk

g

ð2:10Þ

the bifurcation of equilibrium x at point ðx0 , p0 Þ is called an (mk)th-order source bifurcation of the eigenvector vk. The bifurcation point ðx0 , p0 Þ on the eigenvector vk is an (mk)th-order source. Definition 2.5 Consider an n-dimensional, autonomous, nonlinear dynamical system x_ ¼ fðx, pÞ in Eq. (1.4) with an equilibrium point x and f(x, p) is Cr (r  1)continuous in a neighborhood of the equilibrium x (i.e., U(x)  Ω). The corresponding solution is x(t) ¼ Φ(x0, t  t0, p). Suppose Eq. (1.24) holds in U(x)  Ω. For a linearized dynamical system in Eq. (1.19), consider a real eigenvalue λk of matrix Df(x, p) (k 2 N ¼ {1, 2, . . . ,n}) with an eigenvector vk. Suppose one of n independent solutions y ¼ ckvk and y_ ¼ c_k vk . Consider alα ðlα Þ ¼ λlα 6¼ 0 (l 2 S ¼ {l1, l2, . . . ,lp} and S  N ¼ {1, 2, . . . ,n}); the higher order term is ignorable in Eq. (2.3). The p equations for the corresponding directions are ðvlα ÞT
fðx0 þ y , p0 þ ΔpÞ ¼ 0, ðor λ0lα zlα  þ bTlα
ðp  p0 Þ  0Þ

ð2:10Þ

ðlα ¼ l1 , l2 , . . . , lp Þ, λ0lα 6¼ 0 and at p ¼ p0, for a specific kα, if akα ðkα Þ ¼ λkα ¼ 0 (kα 2 N\S ¼ {lp + 1, lp + 2, . . . ,ln} N ), the s equations (s ¼ n  p) on the corresponding directions are given by ðvkα ÞT
fðx0 þ y , p0 þ ΔpÞ ¼ 0 with λ0kα ¼ 0 ð1Þ

ðor Gkα ðj Þ zj1  þ bTk
ðp  p0 Þ 1

þ

sk X q¼2

q X

1 r ðqr, rÞ C q akα ðj j


j Þ zj1  zj2  . . . zjqr  ðp  p0 Þr  0Þ; 1 2 qr q! r¼0

equivalent to z_kα ¼ Akα 0 ðzkα Þskα þ Akα 1 ðzkα Þsk 1 þ


þ Akα skα ¼ 0, Akε i ¼ Akα i ðz1 , z2 , . . . , zk1 , zkþ1 , . . . , zn , pÞ, ði ¼ 0, 1, 2, . . . , skα ;kα ¼ lpþ1 , lpþ2 , . . . , ln Þ:

ð2:11Þ

64

2 Bifurcations of Equilibrium

Pl k α k α Equations (2.10) and (2.11) give lkα -equilibrium zkα  with i¼1 αi ¼ mk α ð0 < mkα skα Þ on the eigenvector vkα direction. Such a bifurcation at equilibrium point ðx0 , p0 Þ is called the hyperbolic bifurcation of ðβlpþ1 : βlpþ2 :


: βln Þ order on the eigenvectors of vkα (kα ¼ lp + 1, lp + 2, . . ., ln) with βα 2 {2mα, 2mα + 1}. (i) If ðr Þ

λ0kα ¼ 0, Gkαkðjα

1 j2 ...jrk α Þ

ðm Þ

ðx0 , p0 Þ ¼ 0 ðr kα ¼ 1, 2, 3, . . . , mkα  1Þ,

ðx0 , p0 Þzj1 zj2


z mkα > 0 j

Gkα ðjkα j

1 2 ...j2mkα Þ

ðkα 2 flpþ1 , lpþ2 , . . . , ln gÞ,

g

ð2:12Þ

the bifurcation of equilibrium x at point ðx0 , p0 Þ is called an increasing saddlenode (or an upper-saddle-node) bifurcation of the ðmkα Þth order on the eigenvector vkα . The bifurcation point ðx0 , p0 Þ on the eigenvector vkα is an increasing saddle (or an upper saddle) of the (mk)th order. (ii) If ðr Þ

λ0kα ¼ 0, Gkαkðjα

1 j2 ...jrk α Þ

ðm Þ

ðx0 , p0 Þ ¼ 0 ðr kα ¼ 1, 2, 3, . . . , mkα  1Þ,

ðx0 , p0 Þzj1 zj2


z mkα < 0 j

Gkα ðjkα j

1 2 ...jmkα Þ

ðkα 2 flpþ1 , lpþ2 , . . . , ln gÞ,

g

ð2:13Þ

the bifurcation of equilibrium x at point ðx0 , p0 Þ is called a decreasing saddlenode (or a lower-saddle-node) bifurcation of the (mk)th order on the eigenvector vkα . The bifurcation point ðx0 , p0 Þ on the eigenvector vkα is a decreasing saddle (or a lower saddle) of the (mk)th order (iii) If ðr Þ

λ0kα ¼ 0, Gkαkðjα

1 j2 ...jrk α Þ

ðm Þ

zkα Gkα ðjkα j

1 2 ...jmkα Þ

ðx0 , p0 Þ ¼ 0 ðr kα ¼ 1, 2, 3, . . . , mkα  1Þ,

ðx0 , p0 Þzj1 zj2 . . . z mkα < 0

ðkα 2 flpþ1 , lpþ2 , . . . , ln gÞ,

j

g

ð2:14Þ

the bifurcation of equilibrium x at point ðx0 , p0 Þ is called an ðmkα Þth-order sink bifurcation on the eigenvector vkα . The bifurcation point ðx0 , p0 Þ on the eigenvector vkα is an ðmkα Þth -order sink.

2.2 Hyperbolic Bifurcations on Eigenvectors

65

(iv) If ðr Þ

λ0kα ¼ 0, Gkαkðjα

1 j2 ...jr kα Þ

ðm Þ

ðx0 , p0 Þ ¼ 0 ðr kα ¼ 2, 3, . . . , mkα  1Þ,

ðx0 , p0 Þzj1 zj2 . . . z mkα < 0 j

zkα Gkα ðjkα j

1 2 ...jmkα Þ

ðk α 2 flpþ1 , lpþ2 , . . . , ln gÞ,

g

ð2:15Þ

the bifurcation of equilibrium x at point ðx0 , p0 Þ is called an (mk)th-order source bifurcation on the eigenvector vkα . The bifurcation point ðx0 , p0 Þ on the eigenvector vkα is an (mk)th-order source. Definition 2.6 Consider an n-dimensional, autonomous, nonlinear dynamical system x_ ¼ fðx, pÞ in Eq. (1.4) with an equilibrium point x and f(x, p) is Cr-continuous (r  1) in a neighborhood of the equilibrium x (i.e., U(x)  Ω). The corresponding solution is x(t) ¼ Φ(x0, t  t0, p). Suppose Eq. (1.24) holds in U(x)  Ω. For a linearized dynamical system in Eq. (1.19), consider a real eigenvalue λk of matrix Df (x, p) (k 2 N ¼ {1, 2, . . . ,n}) with an eigenvector vk. Suppose one of n independent solutions y ¼ ckvk and y_ ¼ c_k vk . Three special cases of bifurcations are defined as follows. (i) Consider λs 6¼ 0 ðs ¼ 1, 2, . . . , n but s 6¼ kÞ

ð2:16Þ

with λs zs þ bTs
ðp  p0 Þ ¼ 0,



s ¼ 1, 2, . . . , n but s 6¼ k; bTk
ðp  p0 Þ þ

1 ð0, 2Þ ð1, 1Þ a ðp  p0 Þ2 þ akðj Þ ðp  p0 Þzj1  1 2! k

1 ð2, 0Þ j1  j2  a z z ¼ 0, 2! kðj1 j2 Þ equivalent to

þ

k 2

z_ ¼ Ak0 ðz Þ þ Ak1 z þ Ak2 ¼ 0; k

k

Aki ¼ Aki ðz1 , z2 , . . . , zk1 , zkþ1 , . . . , zn , pÞ ði ¼ 0, 1, 2Þ; Δk ¼ B2k1  4C k1  0 with Bk1 ¼

Ak1 A , C ¼ k2 , Ak0 k1 Ak0

ð2, 0Þ Ak1 ¼ 0, Ak0 ¼ akðkkÞ 6¼ 0, C1k 0,

g

ð2:17Þ

66

2 Bifurcations of Equilibrium

where ð2;0Þ 1 j2 Þ

akðj

T

ð2Þ

ð0Þ

T

ð2Þ

¼ ðvk Þ
∂cj1 cj2 ∂p fðx; pÞjðx ;p Þ ¼ ðvk Þ
∂cj1 cj2 fðx; pÞjðx ;p 0

k T

¼ ðv Þ
ð1;1Þ

T

0

0

ð2Þ ∂x fðx; pÞðvj1 vj1 Þjðx ;p Þ 0 0

¼

ð1Þ ð1Þ

0Þ

ð2Þ Gkðj j Þ ðx0 ; p0 Þ, 1 2 T

akðj Þ ¼ ðvk Þ
∂cj1 ∂p fðx; pÞjðx ;p Þ ¼ ðvk Þ
∂x ∂p fðx; pÞvj1 jðx ;p Þ , 1

ð0;2Þ

T

bkT ¼ ðvk Þ
∂p fðx; pÞjðx ;p0 Þ , ak 0

ð2;0Þ

0

0

0

T

0

ð2:18Þ

ð2Þ

¼ ðvk Þ
∂p fðx, pÞ,

ð2Þ

akðkkÞ ¼ GkðkkÞ ðx0 ; p0 Þ 6¼ 0: If Eq. (2.17) has two equilibrium solutions, such a bifurcation at point ðx0 , p0 Þ is called the saddle-node bifurcation on the eigenvector vk. (i1) If  ð2, 0Þ ð2Þ λk ¼ 0, akðj j Þ zj1 zj2 ¼ Gkðj j Þ ðx0 , p0 Þzj1 zj2 > 0 1 2

1 2

for j1 , j2 2 f1, 2, . . . , ng

ð2:19Þ

such a bifurcation at point ðx0 , p0 Þ is called an increasing saddle-node (or an upper-saddle-node) switching/appearing bifurcation on the eigenvector vk. The bifurcation point at ðx0 , p0 Þ is an increasing saddle (or an upper saddle) on the eigenvector vk. (i2) If  ð2, 0Þ ð2Þ λk ¼ 0, akðj j Þ zj1 zj2 ¼ Gkðj j Þ ðx0 , p0 Þzj1 zj2 < 0 1 2

1 2

for j1 , j2 2 f1, 2, . . . , ng;

ð2:20Þ

such a bifurcation at point ðx0 , p0 Þ is called a decreasing saddle-node (or a lower-saddle-node) switching/appearing bifurcation of the second order on the eigenvector vk. The bifurcation point at ðx0 , p0 Þ is a decreasing saddle (or a lower saddle) of the second order on the eigenvector vk. (ii) Consider λs 6¼ 0 ðs ¼ 1, 2, . . . , n but s 6¼ kÞ

ð2:21Þ

with λs zs þ bTs
ðp  p0 Þ ¼ 0, s ¼ 1, 2, . . . , n but s 6¼ k;

 ð2:22aÞ

2.2 Hyperbolic Bifurcations on Eigenvectors

bTk
ðp  p0 Þ þ

1 ð0, 2Þ ð1, 1Þ a ðp  p0 Þ2 þ akðj Þ ðp  p0 Þzj1  1 2! k

1 ð2, 0Þ j1  j2  a z z ¼ 0, 2! kðj1 j2 Þ equivalent to

þ

z_k ¼ Ak0 ðzk Þ2 þ Ak1 zk þ Ak2 ¼ 0; Aki ¼ Aki ðz1 , z2 , . . . , zk1 , zkþ1 , . . . , zn , pÞ ði ¼ 0, 1, 2Þ; Δk ¼ B2k1  4C k1  0 with Bk1 ¼

67

g

ð2:22bÞ

Ak1 A , C ¼ k2 , Ak0 k1 Ak0

ð2, 0Þ Ak2 ¼ 0, Ak0 ¼ akðkkÞ 6¼ 0, Ak1 2 ð1, 1Þ,

where ð2;0Þ 1 j2 Þ

akðj

T

ð2Þ

ð0Þ

ð2Þ

T

¼ ðvk Þ
∂cj1 cj2 ∂p fðx; pÞjðx ;p Þ ¼ ðvk Þ
∂cj1 cj2 fðx; pÞjðx ;p 0

0

T

0

ð2Þ

ð2Þ

¼ ðvk Þ
∂x fðx; pÞðvj1 vj2 Þjðx ;p Þ ¼ Gkðj 0

ð1;1Þ

T

0

ð1Þ ð1Þ

1 j2 Þ

0Þ

ðx0 ; p0 Þ,

T

akðj Þ ¼ ðvk Þ
∂cj1 ∂p fðx; pÞjðx ;p Þ ¼ ðvk Þ
∂cj1 ∂p fðx; pÞjðx ;p0 Þ 1

0

ð2:23Þ

0

0

T

¼ ðvk Þ
∂x ∂p fðx; pÞvj1 jðx ;p Þ 6¼ 0, 0

ð2;0Þ akðkkÞ

¼

ð2Þ GkðkkÞ ðx0 ; p0 Þ

0

6¼ 0:

If Eq. (2.20) has two solutions of equilibriums, such a bifurcation at point ðx0 , p0 Þ is called a saddle-node transcritical (or switching) bifurcation of the second order on the eigenvector of vk. (ii1) If  ð2, 0Þ ð2Þ λk ¼ 0, akðj j Þ zj1 zj2 ¼ Gkðj j Þ ðx0 , p0 Þzj1 zj2 < 0 1 2

for j1 , j2 2 f1, 2,


, ng,

1 2

ð2:24Þ

such a bifurcation at point ðx0 , p0 Þ is called an increasing transcritical (or an upper-saddle-node) switching bifurcation of the second order on the eigenvector vk. The bifurcation point at ðx0 , p0 Þ is an increasing saddle (or an upper saddle) of the second order on the eigenvector vk.

68

2 Bifurcations of Equilibrium

(ii2) If ð2, 0Þ ð2Þ λk ¼ 0, akðj j Þ zj1 zj2 ¼ Gkðj j Þ ðx0 , p0 Þzj1 zj2 > 0  1 2

ð2:25Þ

1 2

for j1 , j2 2 f1, 2, . . . , ng; such a bifurcation at point ðx0 , p0 Þ is called a decreasing transcritical (or a lower-saddle-node) switching bifurcation of the second order on the eigenvector vk. The bifurcation point at ðx0 , p0 Þ is a decreasing saddle (or a lower saddle) of the second order on the eigenvector vk. (iii) Consider λs 6¼ 0 ðs ¼ 1, 2, . . . , n but s 6¼ kÞ

ð2:26Þ

with λs zs þ bTs
ðp  p0 Þ ¼ 0,  s ¼ 1, 2, . . . , n but s 6¼ k; ðvk ÞT
fðx0 þ y , p0 þ ΔpÞ ¼ 0 with λ0k ¼ 0; q 3 P P 1 r ðqr, rÞ Cq akðj j


j Þ zj1  zj2  . . . zjqr  ðp  p0 Þr þ bTk
ðp  p0 Þ  0 1 2 qr q! q¼2 r¼0

equivalent to z_k ¼ Ak0 ðzk Þ3 þ Ak1 ðzk Þ2 þ Ak2 zk þ Ak3 ¼ 0, Aki ¼ Aki ðz1 , z2 , . . . , zk1 , zkþ1 , . . . , zn , pÞ ði ¼ 0, 1, 2, 3Þ; ð3, 0Þ Ak3 ¼ Ak1 ¼ 0, Ak0 ¼ akðkkkÞ 6¼ 0, Ak2 Ak0 < 0,

g

ð2:27Þ

where ð3;0Þ 1 j2 j3 Þ

akðj

T

ð3Þ

ð0Þ

T

ð3Þ

¼ ðvk Þ
∂cj1 cj2 cj3 ∂p fðx; pÞjðx ;p Þ ¼ ðvk Þ
∂cj1 cj2 cj3 fðx; pÞjðx ;p 0

k T

¼ ðv Þ
ð1;1Þ

T

0

ð3Þ ∂x fðx; pÞðvj1 vj2 vj3 Þjðx ;p Þ 0 0 ð1Þ ð1Þ

0

¼ T

akðj Þ ¼ ðvk Þ
∂cj1 ∂p fðx; pÞjðx ;p Þ ¼ ðvk Þ
∂cj1 ∂p fðx; pÞjðx ;p0 Þ 1

0

0

T

¼ ðvk Þ
∂x ∂p fðx; pÞvj1 jðx ;p Þ , 0

ð3;0Þ akðkkkÞ

¼

ð3Þ GkðkkkÞ ðx0 ; p0 Þ

6¼ 0:

0

0Þ

ð3Þ Gkðj j j Þ ðx0 ; p0 Þ, 1 2 3

0

ð2:28Þ

2.3 Hopf Bifurcation on an Eigenvector Plane

69

If Eq. (2.27) has three solutions, such a bifurcation at point ðx0 , p0 Þ is called the pitchfork-appearing bifurcation on the eigenvector of vk. (iii1) If ð3, 0Þ ð3Þ λk ¼ 0, zk akðj j j Þ zj1 zj2 zj3 ¼ zk Gkðj j

1 2 j3 Þ

1 2 3

ðx0 , p0 Þzj1 zj2 zj3 < 0

 ð2:29Þ

for j1 , j2 , j3 2 f1, 2, . . . , ng,

such a bifurcation at point ðx0 , p0 Þ is called a stable pitchfork bifurcation of the third order (or a third-order sink) on the eigenvector vk. The bifurcation point at ðx0 , p0 Þ is a third-order sink on the eigenvector vk. (iii2) If ð3, 0Þ ð3Þ λk ¼ 0, zk akðj j j Þ zj1 zj2 zj3 ¼ zk Gkðj j

1 2 j3 Þ

1 2 3

ðx0 , p0 Þzj1 zj2 zj3 > 0

for j1 , j2 , j3 2 f1, 2, . . . , ng,

 ð2:30Þ

such a bifurcation at point ðx0 , p0 Þ is called an unstable pitchfork bifurcation of the third order (or a third-order source) on the eigenvector vk. The bifurcation point at ðx0 , p0 Þ is a third-order source on the eigenvector vk.

2.3

Hopf Bifurcation on an Eigenvector Plane

Definition 2.7 Consider an n-dimensional, autonomous, nonlinear dynamical system x_ ¼ fðx, pÞ in Eq. (1.4) with an equilibrium point x and f(x, p) is Cr (r  1)continuous in a neighborhood of the equilibrium x. The corresponding solution is x (t) ¼ Φ(x0, t  t0, p). Suppose U(x)  Ω is a neighborhood of equilibrium x. For a linearized dynamical system in Eq. (1.19), consider a pair of complex eigenvalue pffiffiffiffiffiffiffi αk iβk (2k  1, 2k 2 N ¼ {1, 2, . . . ,n}, i ¼ 1) of matrix Df(x, p) with a pair of eigenvectors v2k  1 iv2k. On the invariant plane of (v2k  1, v2k), T

c2k1 ¼ ðv2k1 Þ
y, c2k ¼ ðv2k ÞT
y,

ð2:31Þ

with Dx f
v2k1 ¼ αk v2k1  βk v2k , Dx f
v2k ¼ βk v2k1 þ αk v2k , and

ð2:32Þ

70

2 Bifurcations of Equilibrium

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρk ¼ ðc2k1 Þ2 þ ðc2k Þ2 , c2k1 ¼ ρk cos θk , c2k ¼ ρk sin θk ,

ð2:33Þ

and c_2k1 ¼ Gð2k1Þ ðx, pÞ ¼ ðv2k1 ÞT
fðx, pÞ, c_2k ¼ Gð2kÞ ðx, pÞ ¼ ðv2k ÞT
fðx, pÞ,

ð2:34Þ

where Gð2k1Þ ðx, pÞ ¼ ðv2k1 ÞT
fðx, pÞ ¼ aTð2k1Þ
ðp  p0 Þ þ að2k1Þð2k1Þ z2k1 þ að2k1Þð2kÞ z2k þ

q m X X 1 r ðqr, rÞ C q að2k1Þðj j


j Þ ðx , p0 Þzj1 zj2 . . . zjqr ðp  p0 Þr 1 2 qr q! q¼2 r¼0

þ

mþ1 X 1 ðmþ1r, rÞ Cr a ðx , p0 Þzj1 zj2 . . . zjmþ1r ðp  p0 Þr , ðm þ 1Þ! r¼0 mþ1 ð2k1Þðj1 j2


jmþ1r Þ

Gð2kÞ ðx, pÞ ¼ ðv2k ÞT
fðx, pÞ ¼ aTð2kÞ
ðp  p0 Þ þ að2kÞð2k1Þ z2k1 þ að2kÞð2kÞ z2k þ

q m X X 1 r ðqr, rÞ C q að2kÞðj j


j Þ ðx , p0 Þzj1 zj2 . . . zjqr ðp  p0 Þr 1 2 qr q! q¼2 r¼0

þ

mþ1 X 1 ðmþ1r, rÞ Cr a ðx , p0 Þzj1 zj2 . . . zjmþ1r ðp  p0 Þr , ðm þ 1Þ! r¼0 mþ1 ð2kÞðj1 j2


jmþ1r Þ

ð2:35Þ

and ðrk , sk Þ að2k1Þðj

1 j2


jr k Þ

ðx , pÞ ¼ ðv2k1 ÞT
∂

ðrk Þ j cj1 cj2


c rk

ðs Þ

∂p k fðx , pÞ

ðr Þ ðs Þ

¼ ðv2k1 ÞT
∂x k ∂p k fðx , pÞvj1 vj2 . . . vjrk , ðr k , sk Þ að2kÞðj j

1 2


jr k Þ

ðx , pÞ ¼ ðv2k ÞT
∂

ðrk Þ j cj1 cj2


c rk

ðs Þ

∂p k fðx , pÞ

ðr Þ ðs Þ

¼ ðv2k ÞT
∂x k ∂p k fðx , pÞvj1 vj2 . . . vjrk ; aTð2k1Þ ¼ ðv2k1 ÞT
∂p fðx, pÞ, aTð2kÞ ¼ ðv2k ÞT
∂p fðx, pÞ;

ð2:36Þ

2.3 Hopf Bifurcation on an Eigenvector Plane

71

að2k1Þð2k1Þ ¼ ðv2k1 ÞT
∂x fðx, pÞv2k1 , að2k1Þð2kÞ ¼ ðv2k1 ÞT
∂x fðx, pÞv2k ;

ð2:37Þ

að2kÞð2k1Þ ¼ ðv2k ÞT
∂x fðx, pÞv2k1 , að2kÞð2kÞ ¼ ðv2k ÞT
∂x fðx, pÞv2k : Thus ρ_ k ¼ c_

2k1

qk 1 X X 1 ðqk rk , rk Þ cos θk þ c_ sin θk ¼ G q ! ρk q ¼1 r ¼0 k 2k

k

k

ð1, 1Þ ¼ ðαk þ aρ ðkÞ ðθk , p0 Þ
ðp  p0 ÞÞρk k q 1 X X 1 ðqr, rÞ þ aρ ðj


j Þ ðθk , p0 Þzj1 . . . zjqr ðp  p0 Þr , k 1 qr q! q¼2 r¼0 q 1 X X 1 ðqk rk , rk Þ 2k 2k1 _ _ ð c cos θ  c sin θ Þ ¼ G θ_ k ¼ ρ1 k k k q ! θk qk ¼1 rk ¼0 k

ð2:38Þ

ð1, 1Þ ¼ βk þ aθk ðkÞ ðθk , p0 Þ
ðp  p0 Þ

þ

q 3 1 X X 1 ðqr, rÞ a ðθ , p Þzj1 . . . zjqr ðp  p0 Þr , ρk q¼2 r¼0 q! θk ðj1


jqr Þ k 0

where ðs, rÞ Gρðsk, rÞ ¼ aρ ðj j k

1 2


js Þ

zj1 zj2 . . . zjs ðp  p0 Þr ,

1 ðs, rÞ ðs, rÞ Gθk ¼ aθk ðj j


j Þ zj1 zj2 . . . zjs ðp  p0 Þr 1 2 s ρk

ð2:39Þ

ðsummation of j1 , j2 , . . . , js Þ, and ðs, rÞ aρ ðj j

ðs, rÞ ¼ cos θk að2k1Þðj

ðs, rÞ aθk ðj j

ðs, rÞ ¼  sin θk að2k1Þðj

k

1 2


js Þ 1 2


js Þ

1 j2


js Þ

1 j2

ðs, rÞ ðx , pÞ þ sin θk ðað2kÞðj

1 j2


js Þ

ðs, rÞ 


j Þ ðx , pÞ þ cos θ k ðað2kÞðj s

1 j2

ðx , pÞ,




j Þ ðx , pÞ:

ð2:40Þ

s

Definition 2.8 Consider an n-dimensional, autonomous, nonlinear dynamical system x_ ¼ fðx, pÞ in Eq. (1.4) with an equilibrium point x and f(x, p) is Cr-continuous (r  1) in a neighborhood of the equilibrium x (i.e., U(x)  Ω). The corresponding solution is x(t) ¼ Φ(x0, t  t0, p). Suppose Eq. (1.24) holds in U(x)  Ω. For a linearized dynamical system in Eq. (1.19), consider a real eigenvalue λk of matrix

72

2 Bifurcations of Equilibrium

Df(x, p) (k 2 N ¼ {1, 2, . . . ,n}) with an eigenvector vk. Suppose one of n independent solutions y ¼ ckvk and y_ ¼ c_k vk . Consider as(s) ¼ λs 6¼ 0 (s 2 S ¼ {1, 2, . . . ,p} and S  N ¼ {1, 2, . . . ,n}); the higher order term is ignorable in Eq. (2.3). The p equations for the corresponding directions are λsα zsα þ bTsα
ðp  p0 Þ ¼ 0

ð2:41Þ

λsα 6¼ 0 ðsα ¼ 1, 2, . . . , pÞ:

For p ffiffiffiffiffiffiaffi pair of complex eigenvalues αkα iβkα (2kα  1, 2kα 2 N ¼ {1, 2, . . . ,n}, i ¼ 1 ) of matrix Df(x , p) with a pair of eigenvectors v2kα 1 iv2kα . On the invariant plane of ðv2kα 1 , v2kα Þ and at p ¼ p0, for a specific kα, if akα ðkα Þ ¼ αkα ¼ 0 (kα 2 N=S ¼ f12 p þ 1, 12 p þ 2, . . . , 12 ng  N), the s-paired equations (s ¼ (n  p)/2) on the corresponding directions are given by ð1, 1Þ ðθkα , p0 Þ kα ðk α Þ

0 ¼ ðαkα þ aρ þ


ðp  p0 ÞÞρkα

mkα X q X 1 ðqr, rÞ aρ ðj


j Þ ðθkα , p0 Þzj1 . . . zjqr ðp  p0 Þr , kα 1 qr q! q¼2 r¼0

equivalent to ρ_ kα ¼ Akα 0 ðρkα Þmkα þ Akα 1 ðρkα Þmkα 1 þ


þ Akα mα ¼ 0, Akα i ¼ Akα i ðz1 , z2 , . . . , z2kα 2 , z2kα þ1 , . . . , zn , θkα , p0 Þ ði ¼ 0, 1, 2, . . . , mkα Þ, ð1, 1Þ θ_ kα ¼ βkα þ aθk ðkα Þ ðθkα , p0 Þ
ðp  p0 Þ

g

mk α X q 1 X 1 ðqr, rÞ þ ðθ , p Þzj1 . . . zjqr ðp  p0 Þr , a ρkα q¼2 r¼0 q! θkα ðj1


jqr Þ kα 0

equivalent to 1 θ_ kα ¼ ½B ðρ Þmkα þ Bkα 1 ðρkα Þmkα þ


þ Bkα mα , ρk α k α 0 k α Bkα i ¼ Bkα i ðz , z , . . . , z 1

2

2k α 2

,z

2k α þ1

ði ¼ 0, 1, 2, . . . , mkα Þ, 1 1 1 ðk α ¼ p þ 1, p þ 2, . . . , nÞ 2 2 2

, . . . , z , θ k α , p0 Þ n

g

ð2:42Þ

2.3 Hopf Bifurcation on an Eigenvector Plane

73

with ðmkα , 0Þ ðθkα , p0 Þzj1 kα ðj1


jmk Þ

kαÞ Gðm ¼ aρ ρk α

α

j

. . . z mkα 6¼ 0:

ð2:43Þ

Equations (2.41)–(2.43) possess lkα -branch spiral equilibriums x with Plkα kα i¼1 αi ¼ mkα ð0 < lkα mk αÞ. Such lk α -branch solutions are called the bifurcation solutions of equilibrium x on the eigenvector of vk in the neighborhood of ðx0 , p0 Þ. Such a bifurcation at point ðx0 , p0 Þ is called the Hopf bifurcation of ðm12pþ1 : m12pþ2 :


: m12n Þ order on the eigenvector of ðv2kα 1 , v2kαÞ. (i) If kαÞ αkα ¼ 0, Gðr ¼ 0 ðr kα ¼ 2, 3, . . . , mkα  1Þ, ρk α

Gρðmk kαÞ α

¼

ðm , 0Þ j aρ kðjα


j Þ ðθkα , p0 Þzj1


z mkα kα 1 mα

>0

1 1 1 ðk α 2 f p þ 1, p þ 2, . . . , ngÞ, 2 2 2

g

ð2:44Þ

(i1) The bifurcation of equilibrium x at point ðx0 , p0 Þ is called the source Hopf bifurcation of the ð2lkα þ 1Þth order on the eigenvector plane of ðv2kα 1 , v2kα Þ if mkα ¼ 2lkα þ 1. The bifurcation point ðx0 , p0 Þ on the eigenvector plane of ðv2kα 1 , v2kα Þ is the ð2lkα þ 1Þth-order source. (i2) The bifurcation of equilibrium x at point ðx0 , p0 Þ is called the upper-saddle Hopf bifurcation of the ð2lkα Þth order on the eigenvector plane of ðv2kα 1 , v2kα Þ if mkα ¼ 2lkα : The bifurcation point ðx0 , p0 Þ on the eigenvector plane of ðv2kα 1 , v2kα Þ is the ð2lkα Þth-order upper saddle. (ii) If αkα ¼ 0, GρðrkkαÞ ¼ 0 ðr kα ¼ 2, 3, . . . , mkα  1Þ α

Gρðmk kαÞ α

¼

ðm , 0Þ j aρ kðjα


j Þ ðθkα , p0 Þzj1


z mkα kα 1 mα

1 1 1 ðk α 2 f p þ 1, p þ 2, . . . , ngÞ, 2 2 2

0, k

1 2 3

ð2:51Þ

the bifurcation of equilibrium x at point ðx0 , p0 Þ is called an unstable Hopf bifurcation of the third order on the eigenvector plane of (v2k  1, v2k). The bifurcation point ðx0 , p0 Þ on the eigenvector plane of (v2k  1, v2k) is a thirdorder source. (ii) If ð3, 0Þ j1 j2 j3 Gð3Þ ρk ¼ aρ ðj j j Þ ðθ k , p0 Þz z z < 0, k

1 2 3

ð2:52Þ

the bifurcation of equilibrium x at point ðx0 , p0 Þ is called a stable Hopf bifurcation of the third order on the eigenvector plane of (v2k  1, v2k). The bifurcation point ðx0 , p0 Þ on the eigenvector plane of (v2k  1, v2k) is a thirdorder sink.

2.4

Hopf Bifurcation Based on the Fourier Series Base

In the traditional Hopf bifurcation analysis, it is very difficult to determine the limit cycle and stability. As in Luo (2019), an alternative method was presented herein.

76

2 Bifurcations of Equilibrium

Definition 2.10 Consider an n-dimensional, autonomous, nonlinear dynamical system x_ ¼ fðx, pÞ in Eq. (1.4) with an equilibrium point x and f(x, p) is Cr-continuous (r  1) in a neighborhood of the equilibrium x (i.e., U(x)  Ω). The corresponding solution is x(t) ¼ Φ(x0, t  t0, p). Suppose Eq. (1.24) holds in U(x)  Ω. For a linearized dynamical system in Eq. (1.19), consider a pair of pffiffiffiffiffiffiffi complex eigenvalues αk iβk (2k  1, 2k 2 N ¼ {1, 2, . . . ,n},i ¼ 1) of matrix Df(x, p) with a pair of eigenvectors v2k  1 iv2k. In the vicinity of x, there exists a transformation of x ¼ x þ y ¼ x þ a0 ðtÞ þ

N X

bj ðtÞ cos ðjβk tÞ þ cj ðtÞ sin ðjβk tÞ

ð2:53Þ

j¼1

under lim kAj k ¼ 0 with Eq. (1.128). The corresponding dynamical system of j!1

coefficients is a_ 0 ¼ F0 ða0 , b, cÞ, b_ ¼ βk kc þ Fc ða0 , b, cÞ,

ð2:54Þ

c_ ¼ βk kb þ F ða0 , b, cÞ s

with Eqs. (1.131) and (1.132). If Eq. (2.54) has a nonzero solution, i.e., ða0 , b , c Þ 6¼ 0,

ð2:55Þ

then in the vicinity of equilibrium x, there is a bifurcation solution of equilibriumx. (i) The point ðx0 , p0 Þ is called a hyperbolic bifurcation of the equilibrium (x, p) if a0 ðx , pÞ 6¼ 0 and

lim

ðx , pÞ!ðx0 , p0 Þ,

a0 ðx , pÞ ¼ 0;

A ðx , pÞ ¼ 0 ðor b ðx , pÞ ¼ 0, c ðx , pÞ ¼ 0Þ

ð2:56Þ

or βk ¼ 0: The new equilibrium x ¼ x þ a0 ðx , pÞ

ð2:57Þ

is called the bifurcation branch of the equilibrium (x, p). If there are l new branches of equilibrium points x , such a bifurcation is called the l-branch bifurcation of the equilibrium x. (i1) For l ¼ 1, the bifurcation at the point ðx0 , p0 Þ is called a transcritical bifurcation of the equilibrium x if x is linear with parameter p.

2.4 Hopf Bifurcation Based on the Fourier Series Base

77

(i2) For l ¼ 1, the bifurcation at the point ðx0 , p0 Þ is called a saddle-node bifurcation of the equilibrium x if x and x are parabolic with parameter p. (i3) For l ¼ 2, the bifurcation at the point ðx0 , p0 Þ is called a pitchfork bifurcation of the equilibrium x if two new equilibriums x and x form a pitchfork. (i4) For l ¼ 2m  1, the bifurcation at the point ðx0 , p0 Þ is called a (2m)th-order transcritical bifurcation of the equilibrium x if x is a (2m)th-order inflexion point at ðx0 , p0 Þ with parameter p. (i5) For l ¼ 2m  1, the bifurcation at the point ðx0 , p0 Þ is called a (2m)th-order saddle-node bifurcation of the equilibrium x if the (2m  1) branches of x and x are parabolic-alike of the (2m)th order with parameter p. (i6) For l ¼ 2m, the bifurcation at the point ðx0 , p0 Þ is called a (2m + 1)th-order pitchfork bifurcation of the equilibrium x if the (2m) branch of x and x forms a (2m + 1)th-order pitchfork with (2m + 1) branches. (ii) The point ðx0 , p0 Þ is called a Hopf bifurcation of the equilibrium (x, p) if a0 ðx , pÞ 6¼ 0 and

lim

a0 ðx , pÞ ¼ 0;

lim

A ðx , pÞ ¼ 0:

ðx , pÞ!ðx0 , p0 Þ,

A ðx , pÞ 6¼ 0 and

ðx , pÞ!ðx0 , p0 Þ,

ð2:58Þ

The periodic motion near the equilibrium (x, p), xðtÞ ¼ x þ a0 ðx , pÞ þ

N X

bj ðx , pÞ cos ðjβk tÞ þ cj ðx , pÞ sin ðjβk tÞ

ð2:59Þ

j¼1

is called the limit cycle (or periodic motion) near the equilibrium (x, p). Definition 2.11 Consider an n-dimensional, autonomous, nonlinear dynamical system x_ ¼ fðx, pÞ in Eq. (1.4) with an equilibrium point x and f(x, p) is Cr-continuous (r  1) in a neighborhood of the equilibrium x (i.e., U(x)  Ω). The corresponding solution is x(t) ¼ Φ(x0, t  t0, p). Suppose Eq. (1.24) holds in U(x)  Ω. For a linearized dynamical system in Eq. (1.19), consider a pair of pffiffiffiffiffiffiffi complex eigenvalues αk iβk (2k  1, 2k 2 N ¼ {1, 2, . . . ,n},i ¼ 1) of matrix Df(x, p) with a pair of eigenvectors v2k  1 iv2k. In the vicinity of x, there exists a transformation of x ¼ x þ y ¼ x þ a0 ðtÞ þ

N X

bj ðtÞ cos ðjβk tÞ þ cj ðtÞ sin ðjβk tÞ

ð2:60Þ

j¼1

under lim kAj k ¼ 0 with Eq. (1.128). The corresponding dynamical system of j!1

coefficients is

78

2 Bifurcations of Equilibrium

a_ 0 ¼ F0 ða0 , b, cÞ, b_ ¼ βk kc þ Fc ða0 , b, cÞ, c_ ¼ βk kb þ Fs ða0 , b, cÞ

ð2:61Þ

with Eqs. (1.131) and (1.132). If the equilibrium of Eq. (2.61) has a nonzero solution, i.e., ða0 , b , c Þ 6¼ 0,

ð2:62Þ

the new steady-state motion or equilibrium is x ¼ x þ y ¼ x þ a0 þ

N X

bj cos ðjβk tÞ þ cj sin ðjβk tÞ:

ð2:63Þ

j¼1

1. The new solution x is stable if lim ka0 ðtÞ  a0 k ¼ 0, lim kbðtÞ  b k ¼ 0, lim kcðtÞ  c k ¼ 0

t!1

t!1

t!1

ðor lim ka0 ðtÞ  a0 k ¼ 0, lim kAðtÞ  A k ¼ 0Þ: t!1

ð2:64Þ

t!1

2. The new solution x is unstable if lim ka0 ðtÞ  a0 k ¼ 1, or lim kbðtÞ  b k ¼ 1, or lim kcðtÞ  c k ¼ 1

t!1

t!1

t!1

ðor lim ka0 ðtÞ  a0 k ¼ 1, or lim kAðtÞ  A k ¼ 1Þ: t!1

t!1

ð2:65Þ 3. The new equilibrium x ¼ x þ a0 is stable if lim ka0 ðtÞ  a0 k ¼ 0, bðtÞ ¼ b ¼ 0, cðtÞ ¼ c ¼ 0:

t!1

ð2:66Þ

4. The new equilibrium x is unstable if lim ka0 ðtÞ  a0 k ¼ 0, bðtÞ ¼ b ¼ 0, cðtÞ ¼ c ¼ 0:

t!1

ð2:67Þ

Definition 2.12 Consider an n-dimensional, autonomous, nonlinear dynamical system x_ ¼ fðx, pÞ in Eq. (1.4) with an equilibrium point x and f(x, p) is Cr-continuous (r  1) in a neighborhood of the equilibrium x (i.e., U(x)  Ω). The corresponding solution is x(t) ¼ Φ(x0, t  t0, p). Suppose Eq. (1.24) holds in U(x)  Ω. For a linearized dynamical system in Eq. (1.19), consider a pair of pffiffiffiffiffiffiffi complex eigenvalues αk iβk (2k  1, 2k 2 N ¼ {1, 2, . . . ,n}, i ¼ 1) of matrix

2.4 Hopf Bifurcation Based on the Fourier Series Base

79

Df(x, p) with a pair of eigenvectors v2k  1 iv2k. In the vicinity of x, there exists a transformation of x ¼ x þ y ¼ x þ a0 ðtÞ þ

N X

bj ðtÞ cos ðjβk tÞ þ cj ðtÞ sin ðjβk tÞ

ð2:68Þ

j¼1

under lim kAj k ¼ 0 with Eq. (1.128). The corresponding dynamical system of j!1

coefficients is a_ 0 ¼ F0 ða0 , b, cÞ, b_ ¼ βk kc þ Fc ða0 , b, cÞ,

ð2:69Þ

c_ ¼ βk kb þ Fs ða0 , b, cÞ with Eqs. (1.131) and (1.132). The equivalent equation of Eq. (2.69) is z_ ¼ gðzÞ

ð2:70Þ

z ¼ ða0 , b, cÞT , g ¼ ðF0 , Fc , Fs ÞT :

ð2:71Þ

where

If the equilibrium of Eq. (2.70) has a nonzero solution, i.e., z 6¼ 0 ) ða0 , b , c Þ 6¼ 0,

ð2:72Þ

and in a neighborhood of z the linearized equation with Δz ¼ z  z is Δ_z ¼ Dgðz ÞΔz,

ð2:73Þ

then the eigenvalues of Dg(z) determined by j Dgðz Þ  λInð2Nþ1Þ nð2Nþ1Þ j¼ 0

ð2:74Þ

yield the stability of the steady-state motion xðtÞ and equilibrium x of ðn1 : n2 : n3 jn4 : n5 : n6 Þ

ð2:75Þ

accordingly for Σ6r¼1 nr ¼ nð2N þ 1Þ and Σ6r¼1 nr ¼ n. The periodic motion xðtÞ generated by the Hopf bifurcation of the equilibrium x is determined as follows:

80

2 Bifurcations of Equilibrium

(i) If Reλl < 0 (l ¼ 1, 2, . . ., n(2N + 1)), the periodic motion xðtÞ near the equilibrium x is stable. (ii) If Reλl > 0 (l 2 {1, 2, . . . ,n(2N + 1)}), the periodic motion xðtÞ near the equilibrium x is unstable. (iii) The boundary between the stable and unstable periodic motions xðtÞ is determined by the higher singularity. The equilibrium x generated by the hyperbolic bifurcation of the equilibrium x is determined as follows: (iv) If Reλl < 0 (l ¼ 1, 2, . . ., n), the equilibrium x ¼ x þ a0 is stable. (v) If Reλl > 0 (l 2 {1, 2, . . . ,n}), the equilibrium x ¼ x þ a0 is unstable. (vi) The boundary between the stable and unstable equilibrium x ¼ x þ a0 is determined by the higher singularity.

2.5

Hopf Bifurcations in Second-Order Nonlinear Systems

In this section, the Hopf bifurcation of equilibrium for the second-order dynamical systems will be discussed through the finite Fourier series transformation as in Luo (2019). Definition 2.13 Consider a 2n-dimensional, autonomous, nonlinear dynamical _ pÞ in Eq. (1.146) with an equilibrium point (x, 0) and fðx, x, _ pÞ system € x ¼ fðx, x, r is C -continuous (r  1) in a neighborhood of the equilibrium (x, 0) (i.e., U(x, 0)  Ω). The corresponding solution is x(t) ¼ Φ(x0, t  t0, p). Suppose Eq. (1.148) holds in U(x, 0)  Ω. For a linearized dynamical system in Eq. (1.149), consider a pair of complex eigenvalues αk iβk (2k  1, pffiffiffiffiffiffiffi 2k 2 N ¼ {1, 2, . . . ,2n}, i ¼ 1) with a pair of eigenvectors v2k  1 iv2k. In the vicinity of (x, 0), there exists a transformation of x ¼ x þ y ¼ x þ a0 ðtÞ þ

N X

bj ðtÞ cos ðjβk tÞ þ cj ðtÞ sin ðjβk tÞ

ð2:76Þ

j¼1

under lim kAj k ¼ 0 with Eq. (1.152). The corresponding dynamical system of j!1

coefficients is _ c_ Þ, € a0 ¼ F0 ða0 , b, c, a_ 0 , b, € ¼ 2βk k1 c_ þ β2 k2 b þ Fc ða0 , b, c, a_ 0 , b, _ c_ Þ, b k

_ c_ Þ €c ¼ 2βk k1 b_ þ β2k k2 c þ Fs ða0 , b, c, a_ 0 , b,

with Eqs. (1.155) and (1.156). The equilibrium of Eq. (2.77) is given by

ð2:77Þ

2.5 Hopf Bifurcations in Second-Order Nonlinear Systems

81

0 ¼ F0 ða0 , b , c , 0, 0, 0Þ, 0 ¼ β2k k2 b þ Fc ða0 , b , c , 0, 0, 0Þ, 0¼

β2k k2 c

þF

s

ð2:78Þ

ða0 , b , c , 0, 0, 0Þ:

If Eq. (2.78) has a nonzero solution, i.e., ða0 , b , c Þ 6¼ 0,

ð2:79Þ

then in the vicinity of equilibrium x, there is a bifurcation solution of equilibrium x. (i) The point ðx0 , p0 Þ is called a hyperbolic bifurcation of the equilibrium (x, p) if a0 ðx , pÞ 6¼ 0 and

lim

ðx , pÞ!ðx0 , p0 Þ,

a0 ðx , pÞ ¼ 0;

A ðx , pÞ ¼ 0 ðor b ðx , pÞ ¼ 0, c ðx , pÞ ¼ 0Þ

ð2:80Þ

or βk ¼ 0: The new equilibrium x ¼ x þ a0 ðx , pÞ

ð2:81Þ

is called a bifurcation branch of the equilibrium (x, p). If there are l new branches of equilibrium points x , such a bifurcation is called an l-branch bifurcation of the equilibrium x. (i1) For l ¼ 1, the bifurcation at the point ðx0 , p0 Þ is called a transcritical bifurcation of the equilibrium x if x is linear with parameter p. (i2) For l ¼ 1, the bifurcation at the point ðx0 , p0 Þ is called a saddle-node bifurcation of the equilibrium x if x and x are parabolic with parameter p. (i3) For l ¼ 2, the bifurcation at the point ðx0 , p0 Þ is called a pitchfork bifurcation of the equilibrium x if two new equilibriums x and x form a pitchfork. (i4) For l ¼ 2m  1, the bifurcation at the point ðx0 , p0 Þ is called the (2m)th-order transcritical bifurcation of the equilibrium x if x is a (2m)th-order inflexion point at ðx0 , p0 Þ with parameter p. (i5) For l ¼ 2m  1, the bifurcation at the point ðx0 , p0 Þ is called a (2m)th-order saddle-node bifurcation of the equilibrium x if the (2m  1) branches of x and x are parabolic-alike of the (2m)th order with parameter p. (i6) For l ¼ 2m, the bifurcation at the point ðx0 , p0 Þ is called a (2m + 1)th-order pitchfork bifurcation of the equilibrium x if the (2m) branch of x and x forms a (2m + 1)th-order pitchfork with (2m + 1) branches.

82

2 Bifurcations of Equilibrium

(ii) The point ðx0 , p0 Þ is called a Hopf bifurcation of the equilibrium (x, p) if a0 ðx , pÞ 6¼ 0 and

lim

a0 ðx , pÞ ¼ 0;

lim

A ðx , pÞ ¼ 0:

ðx , pÞ!ðx0 , p0 Þ,

A ðx , pÞ 6¼ 0 and

ðx , pÞ!ðx0 , p0 Þ,

ð2:82Þ

The periodic motion near the equilibrium (x, p), xðtÞ ¼ x þ a0 ðx , pÞ þ

N X

bj ðx , pÞ cos ðjβk tÞ þ cj ðx , pÞ sin ðjβk tÞ,

ð2:83Þ

j¼1

is called the limit cycle (or periodic motion) near the equilibrium (x, p). Definition 2.14 Consider a 2n-dimensional, autonomous, nonlinear dynamical _ pÞ in Eq. (1.146) with an equilibrium point (x, 0) and fðx, x, _ pÞ system € x ¼ fðx, x, r is C -continuous (r  1) in a neighborhood of the equilibrium (x, 0) (i.e., U(x, 0)  Ω). The corresponding solution is x(t) ¼ Φ(x0, t  t0, p). Suppose Eq. (1.148) holds in U(x, 0)  Ω. For a linearized dynamical system in Eq. (1.149), consider a pair of complex eigenvalues αk iβk (2k  1, pffiffiffiffiffiffiffi 2k 2 N ¼ {1, 2, . . . ,2n}, i ¼ 1) with a pair of eigenvectors v2k  1 iv2k. In the vicinity of (x, 0), there exists a transformation of x ¼ x þ y ¼ x þ a0 ðtÞ þ

N X

bj ðtÞ cos ðjβk tÞ þ cj ðtÞ sin ðjβk tÞ

ð2:84Þ

j¼1

under lim kAj k ¼ 0 with Eq. (1.152). The corresponding dynamical system of j!1

coefficients is _ c_ Þ, € a0 ¼ F0 ða0 , b, c, a_ 0 , b, € ¼ 2βk k1 c_ þ β2 k2 b þ Fc ða0 , b, c, a_ 0 , b, _ c_ Þ, b k

ð2:85Þ

_ c_ Þ €c ¼ 2βk k1 b_ þ β2k k2 c þ Fs ða0 , b, c, a_ 0 , b, with Eqs. (1.155) and (1.156). If the equilibrium of Eq. (2.85) has a nonzero solution, i.e., ða0 , b , c Þ 6¼ 0,

ð2:86Þ

the new steady-state motion or equilibrium is x ¼ x þ y ¼ x þ a0 þ

N X j¼1

bj cos ðjβk tÞ þ cj sin ðjβk tÞ:

ð2:87Þ

2.5 Hopf Bifurcations in Second-Order Nonlinear Systems

83

(i) The new solution x is stable if lim ka0 ðtÞ  a0 k ¼ 0, lim kbðtÞ  b k ¼ 0, lim kcðtÞ  c k ¼ 0

t!1

t!1

ðor lim ka0 ðtÞ  t!1

a0 k

t!1



¼ 0, lim kAðtÞ  A k ¼ 0Þ:

ð2:88Þ

t!1

(ii) The new solution x is unstable if lim ka0 ðtÞ  a0 k ¼ 1, or lim kbðtÞ  b k ¼ 1, or lim kcðtÞ  c k ¼ 1

t!1

t!1

ðor lim ka0 ðtÞ  t!1

a0 k

t!1



¼ 1, or lim kAðtÞ  A k ¼ 1Þ: t!1

ð2:89Þ (iii) The new equilibrium x ¼ x þ a0 is stable if lim ka0 ðtÞ  a0 k ¼ 0, bðtÞ ¼ b ¼ 0, cðtÞ ¼ c ¼ 0:

t!1

ð2:90Þ

(iv) The new equilibrium x ¼ x þ a0 is unstable if lim ka0 ðtÞ  a0 k ¼ 1, bðtÞ ¼ b ¼ 0, cðtÞ ¼ c ¼ 0:

t!1

ð2:91Þ

Definition 2.15 Consider a 2n-dimensional, autonomous, nonlinear dynamical _ pÞ in Eq. (1.146) with an equilibrium point (x, 0) and fðx, x, _ pÞ system € x ¼ fðx, x, r is C (r  1)-continuous in a neighborhood of the equilibrium (x, 0) (i.e., U(x, 0)  Ω). The corresponding solution is x(t) ¼ Φ(x0, t  t0, p). Suppose Eq. (1.148) holds in U(x, 0)  Ω. For a linearized dynamical system in Eq. (1.149), consider a pair of complex eigenvalues αk iβk (2k  1, pffiffiffiffiffiffiffi 2k 2 N ¼ {1, 2, . . . ,2n}, i ¼ 1) with a pair of eigenvectors v2k  1 iv2k. In the vicinity of (x, 0), there exists a transformation of x ¼ x þ y ¼ x þ a0 ðtÞ þ

N X

bj ðtÞ cos ðjβk tÞ þ cj ðtÞ sin ðjβk tÞ

ð2:92Þ

j¼1

under lim kAj k ¼ 0 with Eq. (1.152). The corresponding dynamical system of j!1

coefficients is _ c_ Þ, € a0 ¼ F0 ða0 , b, c, a_ 0 , b, € ¼ 2βk k1 c_ þ β2 k2 b þ Fc ða0 , b, c, a_ 0 , b, _ c_ Þ, b k _ c_ Þ €c ¼ 2βk k1 b_ þ β2k k2 c þ Fs ða0 , b, c, a_ 0 , b,

ð2:93Þ

84

2 Bifurcations of Equilibrium

with Eqs. (1.155) and (1.156). The equivalent equation of Eq. (2.93) is €z ¼ gðz, z_ Þ

ð2:94Þ

_ c_ ÞT , g ¼ ðF0 , Fc , Fs ÞT : z ¼ ða0 , b, cÞT , z_ ¼ ða_ 0 , b,

ð2:95Þ

where

The equivalent first-order equation of Eq. (2.94) is u_ ¼ fðuÞ

ð2:96Þ

u ¼ ðz, z_ ÞT ;f ¼ ð_z, gÞT :

ð2:97Þ

where

If the equilibrium of Eq. (2.94) has a nonzero solution, i.e., z 6¼ 0 ) ða0 , b , c Þ 6¼ 0,

ð2:98Þ

and in a neighborhood of u the linearized equation with Δu ¼ u  u is Δu_ ¼ Dfðu ÞΔu

ð2:99Þ

with 



Dfðu Þ ¼

0nð2Nþ1Þ nð2Nþ1Þ

Inð2Nþ1Þ nð2Nþ1Þ

Gnð2Nþ1Þ nð2Nþ1Þ

Hnð2Nþ1Þ nð2Nþ1Þ

 ð2:100Þ

and   ∂g  ∂g  ,H ¼ G¼ ∂z u ∂_z u

ð2:101Þ

then the eigenvalues of Dfðu Þ determined by j Dfðu Þ  λI2nð2Nþ1Þ 2nð2Nþ1Þ j¼ 0

ð2:102Þ

yield the stability of the steady-state motion xðtÞ and equilibrium x of ðn1 : n2 : n3 jn4 : n5 : n6 Þ accordingly for Σ6r¼1 nr ¼ 2nð2N þ 1Þ.

ð2:103Þ

Reference

85

The periodic motion xðtÞ generated by the Hopf bifurcation of the equilibrium x is determined as follows: (i) If Reλl < 0 (l ¼ 1, 2, . . ., 2n(2N + 1)), the periodic motion xðtÞ near the equilibrium x is stable. (ii) If Reλl > 0 (l 2 {1, 2, . . . ,2n(2N + 1)}), the periodic motion xðtÞ near the equilibrium x is unstable. (iii) The boundary between the stable and unstable periodic motions xðtÞ is determined by the higher singularity. The equilibrium x ¼ x þ a0 generated by the hyperbolic bifurcation of the equilibrium x is determined as follows: (iv) If Reλl < 0 (l ¼ 1, 2, . . ., 2n), the equilibrium x ¼ x þ a0 is stable. (v) If Reλl > 0 (l 2 {1, 2, . . . ,2n}), the equilibrium x ¼ x þ a0 is unstable. (vi) The boundary between the stable and unstable equilibrium x is determined by the higher singularity.

Reference Luo, A.C.J., 2019, On stability and bifurcation of equilibriums in nonlinear systems, Journal of Vibration Testing and System Dynamics, 3(2), pp. 147–232.

Chapter 3

Low-Dimensional Dynamical Systems

In this chapter, low-dimensional nonlinear dynamical systems are discussed. The stability and bifurcations of the 1-dimensional systems are presented. The higher order singularity and stability for 1-dimensional nonlinear systems are developed. The stability and bifurcations of 2-dimensional systems are discussed and the Hopf bifurcation of equilibriums is presented.

3.1

1-Dimensional Nonlinear Systems

In this section, the stability and bifurcation with singularity for equilibriums will be discussed in 1-dimensional nonlinear systems and a few examples will be presented as in Luo (2012, 2019).

3.1.1

Stability and Singularity

The stability of equilibrium in 1-dimensional systems will be discussed for a better understanding of the stability and singularity theory in nonlinear dynamical systems. Definition 3.1 Consider a 1-dimensional, autonomous, nonlinear dynamical system x_ ¼ f ðx, pÞ

ð3:1Þ

with an equilibrium x of f(x, p) ¼ 0 and f(x, p) is Cr-continuous (r  1) in a neighborhood of x (i.e., U(x) ¼ (x  δ, x + δ)  Ω for an arbitrary δ > 0). The corresponding solution is x(t) ¼ Φ(x0, t  t0, p). Let y ¼ x  x.

© Springer Nature Switzerland AG 2019 A. C. J. Luo, Bifurcation and Stability in Nonlinear Dynamical Systems, Nonlinear Systems and Complexity 28, https://doi.org/10.1007/978-3-030-22910-8_3

87

88

3 Low-Dimensional Dynamical Systems

(i) x(t) at the equilibrium x is stable if xðt þ εÞ  xðtÞ < 0 for y ¼ xðtÞ  x > 0; xðt þ εÞ  xðtÞ > 0 for y ¼ xðtÞ  x < 0,

ð3:2Þ

for all x 2 U(x)  Ω and all t 2 [t0, 1). The equilibrium x is called a sink (or stable node). (ii) x(t) at the equilibrium x is unstable if xðt þ εÞ  xðtÞ > 0 for y ¼ xðtÞ  x > 0; xðt þ εÞ  xðtÞ < 0 for y ¼ xðtÞ  x < 0,

ð3:3Þ

for all x 2 U(x)  Ω and all t 2 [t0, 1). The equilibrium x is called a source (or unstable node). (iii) x(t) at the equilibrium x is increasingly unstable if xðt þ εÞ  xðtÞ > 0 for y ¼ xðtÞ  x > 0; xðt þ εÞ  xðtÞ > 0 for y ¼ xðtÞ  x < 0,

ð3:4Þ

for all x 2 U(x)  Ω and all t 2 [t0, 1). The equilibrium x is called an increasing saddle (or upper saddle). (iv) x(t) at the equilibrium x is decreasingly unstable if xðt þ εÞ  xðtÞ < 0 for y ¼ xðtÞ  x > 0; xðt þ εÞ  xðtÞ < 0 for y ¼ xðtÞ  x < 0,

ð3:5Þ

for all x 2 U(x)  Ω and all t 2 [t0, 1). The equilibrium x is called a decreasing saddle (or lower saddle). (v) x(t) at the equilibrium x is invariant if xðt þ εÞ  xðtÞ ¼ 0 for y ¼ xðtÞ  x 6¼ 0,

ð3:6Þ

for all x 2 U(x)  Ω and all t 2 [t0, 1). x(t) in domain x 2 U(x)  Ω is said to be static. Theorem 3.1 Consider a 1-dimensional, autonomous, nonlinear dynamical system x_ ¼ f ðx, pÞ in Eq. (3.1) with an equilibrium x and f(x, p) is Cr-continuous (r  1) in a neighborhood of x (i.e., U(x)  Ω). The corresponding solution is x(t) ¼ Φ(x0, t  t0, p). In a neighborhood of x, y ¼ x  x.

3.1 1-Dimensional Nonlinear Systems

89

(i) x(t) at the equilibrium x is stable if and only if Gðx, pÞ ¼ x_ ¼ f ðx, pÞ < 0 for y ¼ xðtÞ  x > 0; Gðx, pÞ ¼ x_ ¼ f ðx, pÞ > 0 for y ¼ xðtÞ  x < 0

ð3:7Þ

for all x 2 U(x)  Ω and all t 2 [t0, 1). (ii) x(t) at the equilibrium x is unstable if and only if Gðx, pÞ ¼ x_ ¼ f ðx, pÞ > 0 for y ¼ xðtÞ  x > 0; Gðx, pÞ ¼ x_ ¼ f ðx, pÞ < 0 for y ¼ xðtÞ  x < 0

ð3:8Þ

for all x 2 U(x)  Ω and all t 2 [t0, 1). (iii) x(t) at the equilibrium x is increasingly unstable if and only if Gðx, pÞ ¼ x_ ¼ f ðx, pÞ > 0 for y ¼ xðtÞ  x > 0; Gðx, pÞ ¼ x_ ¼ f ðx, pÞ > 0 for y ¼ xðtÞ  x < 0

ð3:9Þ

for all x 2 U(x)  Ω and all t 2 [t0, 1). (iv) x(t) at the equilibrium x is decreasingly unstable if and only if Gðx, pÞ ¼ x_ ¼ f ðx, pÞ < 0 for y ¼ xðtÞ  x > 0; Gðx, pÞ ¼ x_ ¼ f ðx, pÞ < 0 for y ¼ xðtÞ  x < 0

ð3:10Þ

for all x 2 U(x)  Ω and all t 2 [t0, 1). (v) x(t) at the equilibrium x is invariant if Gðx, pÞ ¼ x_ ¼ f ðx, pÞ ¼ 0 for all x 2 U(x)  Ω and all t 2 [t0, 1). Proof Because _ xðt þ εÞ  xðtÞ ¼ xðtÞ þ xðtÞε þ oðεÞ  xðtÞ _ ¼ xðtÞε þ oðεÞ, and x_ ¼ f ðx, pÞ, we have xðt þ εÞ  xðtÞ ¼ f ðx, pÞε þ oðεÞ ¼ Gðx, pÞε þ oðεÞ:

ð3:11Þ

90

3 Low-Dimensional Dynamical Systems

(i) Due to any selection of ε > 0, for y ¼ x(t)  x < 0, xðt þ εÞ  xðtÞ > 0 if Gðx, pÞ ¼ f ðx, pÞ > 0, and vice versa; and for y ¼ x(t)  x > 0, xðt þ εÞ  xðtÞ < 0 if Gðx, pÞ ¼ f ðx, pÞ < 0, and vice versa. (ii) For y ¼ x(t)  x > 0, xðt þ εÞ  xðtÞ > 0 if Gðx, pÞ ¼ f ðx, pÞ > 0, and vice versa; and for y ¼ x(t)  x < 0, xðt þ εÞ  xðtÞ < 0 if Gðx, pÞ ¼ f ðx, pÞ < 0, and vice versa. (iii) For y ¼ x(t)  x > 0, xðt þ εÞ  xðtÞ > 0 if Gðx, pÞ ¼ f ðx, pÞ > 0, and vice versa; and for y ¼ x(t)  x < 0, xðt þ εÞ  xðtÞ > 0 if Gðx, pÞ ¼ f ðx, pÞ > 0, and vice versa. (iv) For y ¼ x(t)  x > 0, xðt þ εÞ  xðtÞ < 0 if Gðx, pÞ ¼ f ðx, pÞ < 0, and vice versa; and for y ¼ x(t)  x < 0, xðt þ εÞ  xðtÞ < 0 if Gðx, pÞ ¼ f ðx, pÞ < 0, and vice versa. (v) For y ¼ x(t)  x > 0, xðt þ εÞ  xðtÞ ¼ 0 if Gðx, pÞ ¼ f ðx, pÞ ¼ 0, and vice versa. Similarly, for y ¼ x(t)  x < 0, xðt þ εÞ  xðtÞ ¼ 0 if Gðx, pÞ ¼ f ðx, pÞ ¼ 0, and vice versa. The theorem is proved.

∎

3.1 1-Dimensional Nonlinear Systems

91

Theorem 3.2 Consider a 1-dimensional, autonomous, nonlinear dynamical system x_ ¼ f ðx, pÞ in Eq. (3.1) with an equilibrium x and f(x, p) is Cr-continuous (r  1) in a neighborhood of x (i.e., U(x)  Ω). The corresponding solution is x(t) ¼ Φ(x0, t  t0, p). In a neighborhood of x, y ¼ x  x. Under the condition in Eq. (1.24), the linearized equation of Eq. (3.1) is y_ ¼ Df ðx , pÞy:

ð3:12Þ

There is a real eigenvalue λ ¼ Df(x, p) ¼ G(1)(x, p) for a specific p. Suppose jG(2)(x, p) j < 1 where G(2)(x, p) ¼ D2f(x, p). (i) x(t) at the equilibrium x is stable if and only if Gð1Þ ðx , pÞ ¼ Df ðx , pÞ ¼ λ < 0

ð3:13Þ

for all x 2 U(x)  Ω and all t 2 [t0, 1). (ii) x(t) at the equilibrium x is unstable if and only if Gð1Þ ðx , pÞ ¼ Df ðx , pÞ ¼ λ > 0

ð3:14Þ

for all x 2 U(x)  Ω and all t 2 [t0, 1). (iii) x(t) at the equilibrium x is increasingly unstable if and only if Gð1Þ ðx , pÞ ¼ Df ðx , pÞ ¼ λ ¼ 0 and Gð2Þ ðx , pÞ ¼ D2 f ðx , pÞ > 0

ð3:15Þ

for all x 2 U(x)  Ω and all t 2 [t0, 1). (iv) x(t) at the equilibrium x is decreasingly unstable if and only if Gð1Þ ðx , pÞ ¼ Df ðx , pÞ ¼ λ ¼ 0 and Gð2Þ ðx , pÞ ¼ D2 f ðx , pÞ < 0

ð3:16Þ

for all x 2 U(x)  Ω and all t 2 [t0, 1). (v) x(t) at the equilibrium x is invariant if and only if GðmÞ ðx , pÞ ¼ Dm f ðx , pÞ ¼ 0 ðm ¼ 0, 1, 2, . . .Þ for all x 2 U(x)  Ω and all t 2 [t0, 1). Proof For x ¼ x, y ¼ 0. Using the Taylor series expansion gives y_ ¼ f ðx þ y, pÞ  f ðx , pÞ ¼ Df ðx , pÞy þ oðyÞ ¼ Gð1Þ ðx , pÞy þ oðyÞ,

ð3:17Þ

92

3 Low-Dimensional Dynamical Systems

and Gð1Þ ðx , pÞ ¼ Df ðx , pÞ ¼ λ: Thus, y_ ¼ Gð1Þ ðx , pÞy þ oðyÞ ¼ λy þ oðyÞ: (i) For y ¼ x  x > 0, Gðx, pÞ ¼ y_ ¼ λy < 0, and for y ¼ x  x < 0, Gðx, pÞ ¼ y_ ¼ λy > 0: Thus, G(1)(x, p) ¼ Df(x, p) ¼ λ < 0. (ii) For y ¼ x  x > 0, Gðx, pÞ ¼ y_ ¼ λy > 0, and for y ¼ x  x < 0, Gðx, pÞ ¼ y_ ¼ λy < 0, Thus, G(1)(x, p) ¼ Df(x, p) ¼ λ > 0. (iii) For y ¼ x  x > 0, Gðx, pÞ ¼ y_ ¼ λy > 0, and for y ¼ x  x < 0, Gðx, pÞ ¼ y_ ¼ λy > 0: Thus, G(1)(x, p) ¼ Df(x, p) ¼ λ ¼ 0 and the higher order derivatives should be considered. The higher order Taylor series expansion gives y_ ¼ f ðx, pÞ  f ðx , pÞ 1 ¼ Df ðx , pÞy þ D2 ðx , pÞy2 þ oðy2 Þ 2! 1 ¼ D2 ðx , pÞy2 þ oðy2 Þ 2! 1 ¼ Gð2Þ ðx , pÞy2 þ oðy2 Þ: 2!

3.1 1-Dimensional Nonlinear Systems

93

For y ¼ x  x < 0, Gðx, pÞ ¼ y_ ¼

1 ð2Þ  G ðx , pÞy2 > 0, 2!

and for y ¼ x  x > 0, Gðx, pÞ ¼ y_ ¼

1 ð2Þ  G ðx , pÞy2 > 0: 2!

So we have Gð2Þ ðx , pÞ ¼ D2 f ðx , pÞ > 0: ð1Þ

(iv) Similar to (iii), we have Gk ðx , pÞ ¼ λk ¼ 0. In addition, for y ¼ x  x < 0, Gðx, pÞ ¼ y_ ¼

1 ð2Þ  G ðx , pÞy2 < 0: 2!

and for y ¼ x  x > 0, Gðx, pÞ ¼ y_ ¼

1 ð2Þ  G ðx , pÞy2 < 0: 2!

So we have Gð2Þ ðx , pÞ ¼ D2 f ðx , pÞ < 0: (v) Using the Taylor series expansion yields y_ ¼ f ðx, pÞ  f ðx , pÞ XN 1 Dm f ðx , pÞym þ oðyN Þ ¼ m¼1 m! XN 1 ¼ GðmÞ ðx , pÞym þ oðyN Þ m¼1 m! ¼0 ðN ¼ 1, 2, . . .Þ: Thus only if GðmÞ ðx , pÞ ¼ Dm f ðx , pÞ ¼ 0 ðm ¼ 1, 2, . . .Þ, the above equation of y_ ¼ 0 holds, and vice versa. The theorem is proved.

∎

94

3 Low-Dimensional Dynamical Systems

Definition 3.2 Consider a 1-dimensional, autonomous, nonlinear dynamical system x_ ¼ f ðx, pÞ in (3.1) with an equilibrium x and f(x, p) is Cr-continuous (r  1) in a neighborhood of x (i.e., U(x)  Ω). The corresponding solution is x(t) ¼ Φ(x0, t  t0, p). In a neighborhood of x, y ¼ x  x. Under the condition in Eq. (1.24), the linearized equation of Eq. (3.1) is in Eq. (3.12). There is a real eigenvalue λ ¼ Df(x, p) ¼ G(1)(x, p) for a specific p. Suppose jG(m)(x, p) j < 1 where G(r)(x, p) ¼ Drf(x, p) (r ¼ 1, 2, . . .). (i) x(t) at the equilibrium x is stable of the (2m + 1)th order if GðrÞ ðx , pÞ ¼ Dr f ðx , pÞ ¼ 0, r ¼ 0, 1, 2, . . . , 2m; xðt þ εÞ  xðtÞ < 0 for y ¼ xðtÞ  x > 0; xðt þ εÞ  xðtÞ > 0 for y ¼ xðtÞ  x < 0

ð3:18Þ

for all x 2 U(x)  Ω and all t 2 [t0, 1). The equilibrium x is called the sink (or stable node) of the (2m + 1)th order. (ii) x(t) at the equilibrium x is unstable of the (2m + 1)th order if GðrÞ ðx , pÞ ¼ Dr f ðx , pÞ ¼ 0, r ¼ 0, 1, 2, . . . , 2m; xðt þ εÞ  xðtÞ > 0 for y ¼ xðtÞ  x > 0;

ð3:19Þ



xðt þ εÞ  xðtÞ < 0 for y ¼ xðtÞ  x < 0 for all x 2 U(x)  Ω and all t 2 [t0, 1). The equilibrium x is called the source (or unstable node) of the (2m + 1)th order. (iii) x(t) at the equilibrium x is increasingly unstable of the (2m)th order if GðrÞ ðx , pÞ ¼ Dr f ðx , pÞ ¼ 0, r ¼ 0, 1, 2, . . . , 2m  1; xðt þ εÞ  xðtÞ > 0 for y ¼ xðtÞ  x > 0; xðt þ εÞ  xðtÞ > 0 for y ¼ xðtÞ  x < 0

ð3:20Þ

for all x 2 U(x)  Ω and all t 2 [t0, 1). The equilibrium x is called the increasing saddle (or upper saddle) of the (2m)th order. (iv) x(t) at the equilibrium x is decreasingly unstable of the (2m)th order if GðrÞ ðx , pÞ ¼ Dr f ðx , pÞ ¼ 0, r ¼ 0, 1, 2, . . . , 2m  1; xðt þ εÞ  xðtÞ < 0 for y ¼ xðtÞ  x > 0;

ð3:21Þ



xðt þ εÞ  xðtÞ < 0 for y ¼ xðtÞ  x < 0 for all x 2 U(x)  Ω and all t 2 [t0, 1). The equilibrium x is called the decreasing saddle (or lower saddle) of the (2m)th order.

3.1 1-Dimensional Nonlinear Systems

95

Theorem 3.3 Consider a 1-dimensional, autonomous, nonlinear dynamical system x_ ¼ f ðx, pÞ in Eq. (3.1) with an equilibrium x and f(x, p) is Cr-continuous (r  1) in a neighborhood of x (i.e., U(x)  Ω). The corresponding solution is x(t) ¼ Φ(x0, t  t0, p). In a neighborhood of x, y ¼ x  x. Under the condition in Eq. (1.24), the linearized equation of Eq. (3.1) is in Eq. (3.12). There is a real eigenvalue λ ¼ Df(x, p) ¼ G(1)(x, p) for a specific p. Suppose jG(r)(x, p) j < 1 where G(r)(x, p) ¼ Drf(x, p) (r ¼ 1, 2, . . .). (i) x(t) at the equilibrium x is stable of the (2m + 1)th order if and only if GðrÞ ðx , pÞ ¼ Dr f ðx , pÞ ¼ 0, r ¼ 0, 1, 2, . . . , 2m; Gð2mþ1Þ ðx , pÞ ¼ D2mþ1 f ðx , pÞ < 0

ð3:22Þ

for all x 2 U(x)  Ω and all t 2 [t0, 1). (ii) x(t) at the equilibrium x is unstable of the (2mk + 1)th order if and only if GðrÞ ðx , pÞ ¼ Dr f ðx , pÞ ¼ 0, r ¼ 0, 1, 2, . . . , 2m; Gð2mþ1Þ ðx , pÞ ¼ D2mþ1 f ðx , pÞ > 0

ð3:23Þ

for all x 2 U(x)  Ω and all t 2 [t0, 1). (iii) x(t) at the equilibrium x is increasingly unstable of the (2m)th order if and only if GðrÞ ðx , pÞ ¼ Dr f ðx , pÞ ¼ 0, r ¼ 0, 1, 2, . . . , 2m  1; Gð2mÞ ðx , pÞ ¼ D2m f ðx , pÞ > 0

ð3:24Þ

for all x 2 U(x)  Ω and all t 2 [t0, 1). (iv) x(t) at the equilibrium x is decreasingly unstable of the (2m)th order if and only if GðrÞ ðx , pÞ ¼ Dr f ðx , pÞ ¼ 0, r ¼ 0, 1, 2, . . . , 2m  1; Gð2mÞ ðx , pÞ ¼ D2m f ðx , pÞ < 0

ð3:25Þ

for all x 2 U(x)  Ω and all t 2 [t0, 1). Proof For x ¼ x, y ¼ 0. Using the Taylor series expansion gives y_ ¼ f ðx, pÞ  f ðx , pÞ X2m 1 1 Dr f ðx , pÞyr þ D2mþ1 f ðx , pÞy2mþ1 þ oðy2mþ1 Þ ¼ r¼1 r! ð2m þ 1Þ! X2m 1 1 GðrÞ ðx , pÞyr þ Gð2mþ1Þ ðx , pÞy2mþ1 þ oðy2mþ1 Þ, ¼ r¼1 r! ð2m þ 1Þ!

96

3 Low-Dimensional Dynamical Systems

and GðrÞ ðx , pÞ ¼ Dr f ðx , pÞ ¼ 0 for r ¼ 0, 1, 2, . . . , 2m; 1 D2mþ1 f ðx , pÞy2mþ1 ð2m þ 1Þ! 1 ¼ Gð2mþ1Þ ðx , pÞy2mþ1 : ð2m þ 1Þ!

y_ ¼

(i) For y > 0, Gðx, pÞ ¼ y_ ¼

1 Gð2mþ1Þ ðx , pÞy2mþ1 < 0, ð2m þ 1Þ!

and for y < 0 Gðx, pÞ ¼ y_ ¼

1 Gð2mþ1Þ ðx , pÞy2mþ1 > 0: ð2m þ 1Þ!

Thus, G(2m + 1)(x, p) ¼ D2m + 1f(x, p) < 0. (ii) For y > 0, Gðx, pÞ ¼ y_ ¼

1 Gð2mþ1Þ ðx , pÞy2mþ1 > 0, ð2m þ 1Þ!

and for y < 0, Gðx, pÞ ¼ y_ ¼

1 Gð2mþ1Þ ðx , pÞy2mþ1 < 0: ð2m þ 1Þ!

Thus, G(2m + 1)(x, p) ¼ D2m + 1f(x, p) > 0. (iii) For x ¼ x, sk ¼ 0. Using the Taylor series expansion gives y_ ¼ f ðx, pÞ  f ðx , pÞ X2m1 1 1 ¼ Dr f ðx , pÞyr þ D2m f ðx , pÞy2m þ oðy2m Þ r¼1 r! ð2mÞ! X2m 1 1 GðrÞ ðx , pÞyr þ Gð2mÞ ðx , pÞy2m þ oðy2m Þ ¼ r¼1 r! ð2mÞ! and GðrÞ ðx , pÞ ¼ Dr f ðx , pÞ ¼ 0 for r ¼ 0, 1, . . . , 2m  1, y_ ¼

1 1 Gð2mÞ ðx , pÞy2m : D2m f ðx , pÞy2m ¼ ð2mÞ! ð2mÞ!

3.1 1-Dimensional Nonlinear Systems

97

For y > 0, Gðx, pÞ ¼ y_ ¼

1 Gð2mÞ ðx , pÞy2m > 0, ð2mÞ!

Gðx, pÞ ¼ y_ ¼

1 Gð2mÞ ðx , pÞy2m > 0: ð2mÞ!

and for y < 0,

Thus, G(2m)(x, p) ¼ D2mf(x, p) > 0. (iv) Similar to (iii), for y > 0, Gðx, pÞ ¼ y_ ¼

1 Gð2mÞ ðx , pÞy2m < 0, ð2mÞ!

Gðx, pÞ ¼ y_ ¼

1 Gð2mÞ ðx , pÞy2m < 0: ð2mÞ!

and for y < 0,

Thus, G(2m)(x, p) ¼ D2mf(x, p) < 0. The theorem is proved.

∎

To illustrate dynamical behaviors of 1-dimensional systems in the vicinity of equilibrium, the sink (unstable node), source (unstable node), increasing saddle (upper saddle), deceasing saddle (lower saddle), and invariant points are presented in Fig. 3.1a–e, respectively. Uδ(x) ¼ (xδ, x+δ) is a neighborhood of equilibrium x. The circular symbols are for initially selected points. The arrows are the moving directions of x(t). Saddle point is the half sink and half source.

3.1.2

Bifurcations

In this section, the local bifurcation theory for 1-dimensional system is presented. Definition 3.3 Consider a 1-dimensional, autonomous, nonlinear dynamical system x_ ¼ f ðx, pÞ in Eq. (3.1) with an equilibrium x and f(x, p) is Cr-continuous (r  1) in a neighborhood of x (i.e., U(x)  Ω). The corresponding solution is x(t) ¼ Φ(x0, tt0, p). In a neighborhood of x, y ¼ xx. Under the condition in Eq. (1.24), the linearized equation of Eq. (3.1) is in Eq. (3.12). There is a real eigenvalue λ ¼ Df(x, p) ¼ G(1)(x, p) for a specific p. Suppose jG(m)(x, p) j < 1 where G(r)(x, p) ¼ D(r)f(x, p) (r ¼ 1, 2, . . .). In the vicinity of point ðx0 , p0 Þ, f(x, p) can be expanded as

98

3 Low-Dimensional Dynamical Systems

a

b

c

d

e Fig. 3.1 Equilibrium x in a 1-dimensional nonlinear system: (a) stable node (sink), (b) unstable node (source), (c) increasing saddle (upper saddle), (d) decreasing saddle (lower saddle), (e) invariant point

f ðx , pÞ ¼ ay þ bT  ðp  p0 Þ þ

q m X X 1 r ðqr, rÞ qr Cq a y ðp  p0 Þr q! q¼2 r¼0

1 ½y∂x þ ðp  p0 Þ∂p mþ1 ðf ðx0 þ θy , p0 þ θΔpÞÞ, þ ðm þ 1Þ!

ð3:26Þ

where y ¼ x  x0 , a ¼ ∂x f ðx; pÞjðx ;p0 Þ ¼ Df ðx; pÞjðx ;p0 Þ ¼ λ0 , 0

0

bT ¼ ∂p f ðx; pÞjðx ;p0 Þ , 0

a

ðr;sÞ

¼

ðrÞ ðsÞ ∂x ∂p f ðx; pÞjðx ;p Þ , 0 0

γ 2 ð0; 1Þ, θ ¼ ðθ1 ; θ2 ; . . . ; θm ÞT , θr 2 ð0; 1Þ, ðr ¼ 1; 2; . . . ; mÞ:

ð3:27Þ

3.1 1-Dimensional Nonlinear Systems

99

If λ0 ¼ 0 and p ¼ p0, the stability of current equilibrium x changes from stable to unstable state (or from unstable to stable state). The bifurcation branch of equilibrium is determined by

)

f ðx0 þ y , pÞ ¼ 0, or bTk  ðp  p0 Þ þ

q s X X 1 r ðqr, rÞ qr Cq a y ðp  p0 Þr  0; q! q¼2 r¼0

equivalent to f ðx0 þ y , pÞ ¼ A0 ys þ A1 ys1 þ    þ As ¼ 0,

) ð3:28Þ

Ai ¼ Ai ðx0 , pÞ ði ¼ 0, 1, 2, . . . , sÞ, A0 ¼ aðs, 0Þ ¼ Ds f ðx0 , p0 Þ 6¼ 0;

)

λ0 ¼ Df ðx0 , p0 Þ ¼ 0, Dr f ðx0 , p0 Þ ¼ 0 ðr ¼ 2, 3, . . . , s  1Þ, Ds f ðx0 , p0 Þ 6¼ 0:

In the neighborhood of ðx0 , p0 Þ, Eq. (3.1) possesses l-equilibrium y (0 < l s) P with s ¼ li¼1 αi : Such l-branch solutions are called the bifurcation solutions of an equilibrium x in the neighborhood of ðx0 , p0 Þ. Such a bifurcation at point ðx0 , p0 Þ is called a hyperbolic bifurcation of the sth order. (i) Consider f ðx0 þ y , pÞ ¼ 0 at point ðx0 , p0 Þ approximated by f ðx0 þ y , pÞ ¼ 0 with λ0 ¼ Df ðx0 , p0 Þ ¼ 0, or bTk  ðp  p0 Þ þ

2mþ1 P q¼2

q P 1 r ðqr, rÞ qr Cq a y ðp  p0 Þr  0; q! r¼0

equivalent to f ðx0



þ y , pÞ ¼ A0 y Ai ¼

)

2mþ1

Ai ðx0 , pÞ

þ A1 y

2m

þ    þ A2mþ1 ¼ 0,

)

ð3:29Þ

ði ¼ 0, 1, 2,   , 2m þ 1Þ,

A0 ¼ að2mþ1, 0Þ ¼ D2mþ1 f ðx0 , p0 Þ 6¼ 0: (i1) If λ0 ¼ Df ðx0 , p0 Þ ¼ 0, Dr f ðx0 , p0 Þ ¼ 0 ðr ¼ 2, 3, . . . , 2mÞ, 2mþ1

D

f ðx0 , p0 Þ

< 0,

) ð3:30Þ

100

3 Low-Dimensional Dynamical Systems

the bifurcation of equilibrium at point ðx0 p0 Þ is called the pitchfork bifurcation of the (2m+1)th order. The special bifurcation point is the (2m+1)th order sink (stable node). (i2) If λ0 ¼ Df ðx0 , p0 Þ ¼ 0,

)

Dr f ðx0 , p0 Þ ¼ 0 ðr ¼ 2, 3, . . . , 2mÞ, 2mþ1

D

f ðx0 , p0 Þ

ð3:31Þ

> 0,

the bifurcation of equilibrium at point ðx0 , p0 Þ is called an unstable pitchfork bifurcation of the (2m+1)th order. The special bifurcation is a (2m+1)thorder source (unstable node). (ii) Consider f ðx0 þ y , pÞ ¼ 0 at point ðx0 , p0 Þ approximated by f ðx0 þ y , pÞ ¼ 0 with λ0 ¼ Df ðx0 , p0 Þ ¼ 0, or bTk  ðp  p0 Þ þ

q 2m P P 1 r ðqr, rÞ qr Cq a y ðp  p0 Þr  0; q! q¼2 r¼0

equivalent to f ðx0



 2m

þ y , pÞ ¼ A0 ðy Þ Ai ¼

)

Ai ðx0 , pÞ

 2m1

þ A1 ðy Þ

þ    þ A2m

) ¼ 0,

ð3:32Þ

ði ¼ 0, 1, 2, . . . , 2m  1Þ,

A0 ¼ að2m, 0Þ ¼ D2m f ðx0 , p0 Þ 6¼ 0: (ii1) If λ0 ¼ Df ðx0 , p0 Þ ¼ 0, Dr f ðx0 , p0 Þ ¼ 0 ðr ¼ 2, 3, . . . , 2m  1Þ, 2m

D

f ðx0 , p0 Þ

) ð3:33Þ

> 0,

the bifurcation of equilibrium at point ðx0 , p0 Þ is called an increasing saddlenode (or an upper-saddle-node) bifurcation of the (2m)th order. The special bifurcation point is a (2m)th-order increasing saddle (or upper saddle). (ii2) If λ0 ¼ Df ðx0 , p0 Þ ¼ 0, Dr f ðx0 , p0 Þ ¼ 0 ðr ¼ 2, 3, . . . , 2m  1Þ, 2m

D

f ðx0 , p0 Þ

)

ð3:34Þ

< 0,

the bifurcation at point ðx0 , p0 Þ is called a decreasing saddle-node (or a lower-saddle-node) bifurcation of the (2m)th order. The special bifurcation point is a (2m)th-order decreasing saddle (or lower saddle).

3.1 1-Dimensional Nonlinear Systems

101

Definition 3.4 Consider a 1-dimensional, autonomous, nonlinear dynamical system x_ ¼ f ðx, pÞ in Eq. (3.1) with an equilibrium x and f(x, p) is Cr-continuous (r  1) in a neighborhood of x (i.e., U(x)  Ω). The corresponding solution is x(t) ¼ Φ(x0, tt0, p). In a neighborhood of x, y ¼ xx. Under the condition in Eq. (1.24), the linearized equation of Eq. (3.1) is in Eq. (3.12). There is a real eigenvalue λ ¼ Df(x, p) ¼ G(1)(x, p) for a specific p. Suppose jG(m)(x, p) j < 1 where G(r)(x, p) ¼ D(r)f(x, p) (r ¼ 1, 2,   ). In the vicinity of point ðx0 , p0 Þ, f(x, p) can be expended in Eq. (3.26), and three special cases are defined as follows. (i) Consider λ0 ¼ Df ðx0 , p0 Þ ¼ 0, f ðx0 þ y , p0 þ ΔpÞ ¼ 0

)

or 1 ð0, 2Þ a  ðp  p0 Þ2 2! 1 það1, 1Þ  ðp  p0 Þy2 þ að2, 0Þ y2  0; 2! equivalent to y_ ¼ A0 y2 þ A1 y þ A2 ¼ 0 with bT  ðp  p0 Þ þ

)

1 ð2, 0Þ a , 2! A1 ðx0 , p0 Þ ¼ að1, 1Þ  ðp  p0 Þ þ λ0 ,

A0 ðx0 , p0 Þ ¼

A2 ðx0 , p0 Þ ¼ bT  ðp  p0 Þ þ

ð3:35Þ

1 ð0, 2Þ  ðp  p0 Þ2 , a 2!

with ð2Þ ð0Þ

að2;0Þ ¼ ∂x ∂p f ðx; pÞjðx ;p 0

0Þ

¼ D f ðx; pÞjðx ;p0 Þ ¼ Gð2Þ ðx0 ; p0 Þ 6¼ 0, 2

0

a

ð1;1Þ

¼

ð1Þ ð1Þ ∂x ∂p f ðx; pÞjðx ;p Þ , 0

ð3:36Þ

0

ð0Þ ð2Þ

ð2Þ

að0;2Þ ¼ ∂x ∂p f ðx; pÞjðx ;p Þ ¼ ∂p f ðx; pÞjðx ;p Þ , 0

0

0

0

bT ¼ ∂p f ðx; pÞjðx ;p0 Þ : 0

(i1) If the following conditions are satisfied in Eq. (3.35) A1 ¼ 0 ) að1, 1Þ ¼ 0 and λ0 ¼ Df ðx0 , p0 Þ ¼ 0; að2, 0Þ A < 0,

ð3:37Þ

2

such a bifurcation at point ðx0 , p0 Þ is called the saddle-node-appearing/ vanishing bifurcation.

102

3 Low-Dimensional Dynamical Systems

(i1a) For að2, 0Þ ¼ D2 f ðx, pÞjðx , p0 Þ ¼ Gð2Þ ðx0 , p0 Þ > 0, 0

ð3:38Þ

the saddle-node bifurcation is called an increasing saddle-node (or an upper-saddle-node)-appearing/vanishing bifurcation of the second order. The bifurcation point ðx0 , p0 Þ is an increasing saddle (or an upper saddle) of the second order. (i1b) For að2, 0Þ ¼ D2 f ðx, pÞjðx , p0 Þ ¼ Gð2Þ ðx0 , p0 Þ < 0, 0

ð3:39Þ

the saddle-node bifurcation is called a decreasing saddle-node (or a lower-saddle-node)-appearing/vanishing bifurcation. The bifurcation point ðx0 , p0 Þ is a decreasing saddle (or a lower saddle) of the second order. (i2) If the following conditions are satisfied in Eq. (3.35) A2 ¼ 0 ) að0, 1Þ ¼ 0 and að0, 2Þ ¼ 0, A1 6¼ 0 ) að1, 1Þ 6¼ 0,

ð3:40Þ

such a bifurcation at point ðx0 , p0 Þ is called a transcritical switching bifurcation, which is a kind of saddle-node-switching bifurcation. (i2a) For að2, 0Þ ¼ D2 f ðx, pÞjðx , p0 Þ ¼ Gð2Þ ðx0 , p0 Þ > 0, 0

ð3:41Þ

the transcritical saddle-node bifurcation is called an increasing saddlenode (or an upper-saddle-node)-switching bifurcation of the second order. The bifurcation point ðx0 , p0 Þ is an increasing saddle (or an upper saddle) of the second order. (i2b) For að2, 0Þ ¼ D2 f ðx, pÞjðx , p0 Þ ¼ Gð2Þ ðx0 , p0 Þ < 0, 0

ð3:42Þ

the transcritical saddle-node bifurcation is called a decreasing saddlenode (or a lower-saddle-node)-switching bifurcation of the second order. The bifurcation point ðx0 , p0 Þ is a decreasing saddle (or a lower saddle) of the second order.

3.1 1-Dimensional Nonlinear Systems

103

)

(ii) Consider λ0 ¼ Df ðx0 , p0 Þ ¼ 0, f ðx0 þ y , p0 þ ΔpÞ ¼ 0 or 1 ð0, 2Þ  ðp  p0 Þ2 þ að1, 1Þ  ðp  p0 Þy a 2! 1 1 1 þ að2, 0Þ y2 þ að0, 3Þ  ðp  p0 Þ3 þ að2, 1Þ  ðp  p0 Þy2 2! 3! 2! 1 1 þ að1, 2Þ  ðp  p0 Þ2 y þ að3, 0Þ y3  0; 2! 3! equivalent to bT  ðp  p0 Þ þ

ð3:43aÞ

)

y_ ¼ A0 y3 þ A1 y2 þ A2 y þ A3 ¼ 0 with 1 A0 ðx0 , p0 Þ ¼ að3, 0Þ , 3! 1 1 A1 ðx0 , p0 Þ ¼ að2, 1Þ  ðp  p0 Þ þ að2, 0Þ , 2! 2! 1 A2 ðx0 , p0 Þ ¼ λ0 þ að1, 1Þ  ðp  p0 Þ þ að1, 2Þ  ðp  p0 Þ2 , 2! 1 1 A3 ðx0 , p0 Þ ¼ bT  ðp  p0 Þ þ að0, 2Þ  ðp  p0 Þ2 þ að0, 3Þ  ðp  p0 Þ3 , 2! 3! ð3:43bÞ where ð3Þ ð0Þ

ð3Þ

að3;0Þ ¼ ∂x ∂p f ðx; pÞjðx ;p Þ ¼ ∂x f ðx; pÞjðx ;p0 Þ 0

0

0

¼ D3 f ðx; pÞjðx ;p0 Þ ¼ Gð3Þ ðx0 ; p0 Þ 6¼ 0,

ð3:44Þ

0

ðrÞ ðsÞ

aðr;sÞ ¼ ∂x ∂p f ðx; pÞjðx ;p Þ ðr; s ¼ 0; 1; 2; 3; r þ s ¼ 3Þ: 0

0

If the following conditions are satisfied, ð2, 0Þ

A1 ðx0 , p0 Þ ¼ 0 ) ak

ð2, 1Þ

¼ 0 and ak

A3 ðx0 , p0 Þ ¼ 0 ) b ¼ 0 and að0, 2Þ ¼ 0

¼ 0, ð3:45Þ

A1 y þ A0 y3  0 with A1 A0 < 0, such a bifurcation at point ðx0 , p0 Þ is called a pitchfork-switching and -appearing bifurcation of the third order.

104

3 Low-Dimensional Dynamical Systems

(ii1) For að3, 0Þ ¼ D3 f ðx, pÞjðx , p0 Þ ¼ Gð3Þ ðx0 , p0 Þ > 0, 0

ð3:46Þ

the pitchfork bifurcation is called a switching and appearing bifurcation of the third-order source. The corresponding bifurcation point is a third-order source (or unstable node). (iii2) For að3, 0Þ ¼ D3 f ðx, pÞjðx , p0 Þ ¼ Gð3Þ ðx0 , p0 Þ < 0, 0

ð3:47Þ

the pitchfork bifurcation is called a switching and appearing bifurcation of the third-order sink. The corresponding bifurcation point is a third-order sink (or stable node).

3.1.3

Sampled Systems

As in Luo (2012), the above three special cases can be discussed through 1-dimensional systems and intuitive illustrations are presented in Fig. 3.2. The bifurcation point is also represented by a solid circular symbol. The stable and unstable equilibrium branches are given by solid and dashed curves, respectively. The vector fields are represented by lines with arrows. If no equilibriums exist, such a region is shaded.

3.1.3a Saddle-Node-Appearing Bifurcation Consider a saddle-node bifurcation in 1-dimensional system: x_ ¼ f ðx, pÞ  p  x2 :

ð3:48Þ

pffiffiffi The equilibriums of the foregoing equation are x ¼ p ( p > 0) and no equilibriums exist for p < 0. From Eq. (3.48), the linearized equation in the vicinity of the equilibrium with y ¼ xx is y_ ¼ Gð1Þ ðx , pÞy ¼ Df ðx , pÞy ¼ 2x y:

ð3:49Þ

pffiffiffi pffiffiffi For the branch of x ¼ p ( p > 0), due to Df ðx , pÞ ¼ 2 p < 0, the equilibrium is a stable node because y_ > 0 for y < 0 and y_ < 0 for y > 0. However, for the branch of pffiffiffi pffiffiffi x ¼  p ( p > 0), due to Df ðx , pÞ ¼ 2 p > 0, such an equilibrium is an unstable node because y_ > 0 for y > 0 and y_ < 0 for y < 0.

3.1 1-Dimensional Nonlinear Systems

105

x

x

LS

US

p

a

p

b x

x

US

LS

p

c

p

d x

x

3rd SO

3rd SI

p

p

e

f

Fig. 3.2 Bifurcation diagrams: (a, b) saddle-node-appearing/vanishing bifurcation, (c, d) transcritical switching bifurcation, (e) pitchfork-switching/appearing bifurcation for stable symmetry, and (f) pitchfork bifurcation for unstable symmetry

106

3 Low-Dimensional Dynamical Systems

For p ¼ p0 ¼ 0, x ¼ x0 ¼ 0 and Df ðx0 , p0 Þ ¼ 0. D2 f ðx0 , p0 Þ ¼ 2 is obtained. Thus y_ ¼

1 2  D f ðx0 , p0 Þy2 ¼ y2 : 2!

ð3:50Þ

In the vicinity of ðx0 , p0 Þ ¼ ð0, 0Þ, the flow vector field in Eq. (3.50) is always less than zero except for y ¼ 0. The equilibrium point ðx0 , p0 Þ ¼ ð0, 0Þ is a bifurcation point, which is a decreasing saddle (or lower saddle) of the second order. For p < 0, the vector field of Eq. (3.48) is always less than zero without any equilibriums. The equilibrium varying with parameter p is sketched in Fig. 3.2a. On the left side of xaxis, no equilibrium exists, so only the vector field is presented. Consider a dynamical system as x_ ¼ f ðx, pÞ  p þ x2

ð3:51Þ

pffiffiffi and the corresponding equilibriums are x ¼ p with the variational equation pffiffiffi y_ ¼ Gð1Þ ðx , pÞy ¼ Df ðx , pÞy ¼ 2 py

ð3:52Þ

pffiffiffi Gð1Þ ðx , pÞ ¼ Df ðx , pÞ ¼ 2x ¼ 2 p:

ð3:53Þ

where

pffiffiffi For Gð1Þ ðx , pÞ ¼ 2 p > 0, y_ < 0 for y < 0 and y_ > 0 for y > 0. So such an pffiffiffi pffiffiffi equilibrium of x ¼ p is an unstable node. If Gð1Þ ðx , pÞ ¼ 2 p < 0, y_ > 0 for p ffiffi ffi y < 0 and y_ < 0 for y > 0. So such an equilibrium of x ¼  p is a stable node. pffiffiffi For Gð1Þ ¼ Df ðx , pÞ ¼ 2 p ¼ 0 at x ¼ x0 ¼ 0 and p ¼ p0 ¼ 0, we have y_ ¼

1 ð2Þ  1 G ðx0 , p0 Þy2 ¼ D2 f ðx0 , p0 Þy2 ¼ y2 2! 2!

ð3:54Þ

Gð2Þ ðx0 , p0 Þ ¼ D2 f ðx0 , p0 Þ ¼ 2 > 0:

ð3:55Þ

where

Thus, the bifurcation point at ðx0 , p0 Þ ¼ ð0, 0Þ has y_ > 0 except for y ¼ 0. The bifurcation point is the increasing saddle (or upper saddle) of the second order. For p < 0, we have x_ > 0 always without any equilibrium. So the flow increases always for a specific p < 0. The bifurcation and flow behaviors are presented in Fig. 3.2b.

3.1 1-Dimensional Nonlinear Systems

107

3.1.3b Saddle-Node-Switching Bifurcation Consider a saddle-node-switching bifurcation through the following system: x_ ¼ f ðx, pÞ  px  x2

ð3:56Þ

The equilibriums of the foregoing equation are x ¼ 0, p. From Eq. (3.56), the linearized equation in the vicinity of the equilibriums with y ¼ xx is y_ ¼ Gð1Þ ðx , pÞy ¼ Df ðx , pÞy ¼ ðp  2x Þy:

ð3:57Þ

For the branch of x ¼ 0 ( p > 0), due to Df(x, p) ¼ p > 0 the equilibrium is an unstable node because y_ < 0 for y < 0 and y_ > 0 for y > 0. For the branch of x ¼ p ( p > 0), such an equilibrium is a stable node due to Df(x, p) ¼ p < 0. However, for the branch of x ¼ 0 ( p < 0), the equilibrium is a stable node due to Df(x, p) ¼ p < 0. For the branch of x ¼ p ( p < 0), such an equilibrium is an unstable node due to Df(x, p) ¼ p > 0, which causes y_ < 0 for y < 0 and y_ > 0 for y > 0. For p ¼ p0 ¼ 0, x ¼ x0 ¼ 0 and Df ðx0 , p0 Þ ¼ p0 ¼ 0. D2 f ðx0 , p0 Þ ¼ 2 is obtained. Thus the higher order variational equation at the equilibrium is y_ ¼

1 ð2Þ  1 G ðx0 , p0 Þy2 ¼ D2 f ðx0 , p0 Þy2 ¼ y2 : 2! 2!

ð3:58Þ

At ðx0 , p0 Þ ¼ ð0, 0Þ, the flow vector field is always less than zero except for y ¼ 0. The equilibrium point ðx0 , p0 Þ ¼ ð0, 0Þ is a bifurcation point. This transcritical bifurcation is a decreasing saddle (lower saddle) of the second order. The equilibrium varying with parameter p is sketched in Fig. 3.2c. Consider another transcritical switching bifurcation of a dynamical system via x_ ¼ f ðx, pÞ  px þ x2 :

ð3:59Þ

The equilibriums of the foregoing equation are x ¼ 0, p with the variational equation as y_ ¼ Gð1Þ ðx , pÞy ¼ Df ðx , pÞy ¼ ðp þ 2x Þy,

ð3:60Þ

Gð1Þ ðx , pÞ ¼ Df ðx , pÞ ¼ p þ 2x :

ð3:61Þ

where

For G(1)(x, p) ¼ p < 0, y_ > 0 for y < 0 and y_ < 0 for y > 0. So such an equilibrium of x ¼ 0 is a stable node. For G(1)(x, p) ¼ p > 0, y_ < 0 for y < 0 and y_ > 0 for y > 0. So such an equilibrium of x ¼ 0 is an unstable node. If G(1)(x, p) ¼ p > 0, y_ < 0 for y < 0 and y_ > 0 for y > 0. So such an equilibrium of x ¼ p is an unstable

108

3 Low-Dimensional Dynamical Systems

node. If G(1)(x, p) ¼ p < 0, y_ > 0 for y < 0 and y_ < 0 for y > 0. So such an equilibrium of x ¼ p is a stable node. For G(1) ¼ Df(x, p) ¼ 0 at x ¼ x0 ¼ 0 and p ¼ p0 ¼ 0, the higher order variational equation is y_ ¼

1 ð2Þ  1 G ðx0 , p0 Þy2 ¼ D2 f ðx0 , p0 Þy2 ¼ y2 2! 2!

ð3:62Þ

Gð2Þ ðx0 , p0 Þ ¼ D2 f ðx0 , p0 Þ ¼ 2 > 0:

ð3:63Þ

where

Thus, the bifurcation point at ðx0 , p0 Þ ¼ ð0, 0Þ has y_ > 0 except for y ¼ 0. The transcritical bifurcation point is the increasing saddle (or upper saddle) of the second order. The bifurcation and flow behaviors are presented in Fig. 3.2d.

3.1.3c Pitchfork-Switching/Appearing Bifurcation Consider the pitchfork bifurcation with stable symmetry through x_ ¼ px  x3 :

ð3:64Þ

pffiffiffi Setting x_ ¼ 0 gives the equilibriums of x ¼ 0, p ( p > 0) and x ¼ 0 ( p 0).  From Eq. (3.64), in the vicinity of x with y ¼ xx, the variational equation is y_ ¼ Gð1Þ ðx , pÞy ¼ Df ðx , pÞy ¼ ½p  3ðx Þ2 y:

ð3:64Þ

For the branch of x ¼ 0 ( p > 0), because of Df(x, p) ¼ p > 0, the equilibrium is pffiffiffi unstable. For the branches of x ¼ p ( p > 0), such equilibriums are stable due to Df(x, p) ¼ 2p < 0. However, for the branch of x ¼ 0 ( p < 0), the equilibrium is stable due to Df(x, p) ¼ p < 0. For p ¼ p0 ¼ 0, x ¼ x0 ¼ 0 and Df ðx0 , p0 Þ ¼ 0 are obtained. Because of ð2Þ  G ðx0 , p0 Þ ¼ D2 f ðx0 , p0 Þ ¼ 6x0 ¼ 0, Gð3Þ ðx0 , p0 Þ ¼ D3 f ðx0 , p0 Þ ¼ 6 < 0 is needed. Further, the variational equation at ðx0 , p0 Þ ¼ ð0, 0Þis y_ ¼

1 ð3Þ  1 G ðx0 , p0 Þy3 ¼ D3 f ðx0 , p0 Þy3 ¼ y3 : 3! 3!

ð3:65Þ

In the vicinity of ðx0 , p0 Þ ¼ ð0, 0Þ, y_ < 0 for y > 0 and y_ > 0 for y < 0. The equilibrium point ðx0 , p0 Þ ¼ ð0, 0Þ is a bifurcation point, which is a sink of the third order. The equilibrium varying with parameter p is sketched in Fig. 3.2e.

3.2 2-Dimensional Nonlinear Systems

109

Consider the pitchfork bifurcation for unstable symmetry as x_ ¼ px þ x3 :

ð3:66Þ

pffiffiffiffiffiffiffi Setting x_ ¼ 0 gives equilibriums of x ¼ 0, p ( p < 0) and x ¼ 0 ( p  0). in the vicinity of x, the linearized equation with y ¼ xx is y_ ¼ Gð1Þ ðx , pÞy ¼ Df ðx , pÞy ¼ ½p þ 3ðx Þ2 y:

ð3:67Þ

For the branch of x ¼ 0 ( p < 0), due to Df(x, p) ¼ p < 0, the equilibrium is stable. In the vicinity of x, we have y_ < 0 for y > 0 and y_ > 0 for y < 0. For the branches of pffiffiffiffiffiffiffi x ¼ p ( p < 0), such two equilibriums are unstable due to Df(x, p) ¼ 2p > 0. However, for the branch of x ¼ 0 ( p > 0), the equilibrium is unstable due to Df(x, p) ¼ p > 0. For ðx , pÞ ¼ ðx0 , p0 Þ ¼ ð0, 0Þ, Df ðx0 , p0 Þ ¼ 0 and D2 f ðx0 , p0 Þ ¼ 6x0 ¼ 0 are obtained. Thus, D3 f ðx0 , p0 Þ ¼ 6 > 0 is needed, and the variational equation at ðx0 , p0 Þ ¼ ð0, 0Þ is y_ ¼

1 ð3Þ  1 G ðx0 , p0 Þy3 ¼ D3 f ðx0 , p0 Þy3 ¼ y3 : 3! 3!

ð3:68Þ

In the vicinity of ðx0 , p0 Þ ¼ ð0, 0Þ, y_ > 0 for y > 0 and y_ < 0 for y < 0. The equilibrium point ðx0 , p0 Þ ¼ ð0, 0Þ is a bifurcation point, which is a source of the third order. The equilibrium varying with parameter p is sketched in Fig. 3.2f. From the analysis, the bifurcation points possess the higher order singularity of the flow in dynamical systems. For the saddle-node bifurcation, the (2m)th-order singularity of the flow at the bifurcation point exists as a saddle of the (2m)th order. For the transcritical bifurcation, the (2m)th-order singularity of the flow at the bifurcation point exists as a saddle of the (2m)th order. However, for the stable pitchfork bifurcation, the (2m+1)th-order singularity of the flow at the bifurcation point exists as a sink of the (2m+1)th order. For the unstable pitchfork bifurcation, the (2m+1)th-order singularity of the flow at the bifurcation point exists as a source of the (2m+1)th order.

3.2

2-Dimensional Nonlinear Systems

In this section, the stability and bifurcation of equilibriums in a 2-dimensional dynamical system will be discussed for a better understanding of stability and bifurcation theory as in Luo (2019).

110

3.2.1

3 Low-Dimensional Dynamical Systems

Stability and Singularity

Consider a 2-dimensional nonlinear system as x_ ¼ fðx, pÞ 2 R2

ð3:70Þ

f ¼ ðf 1 , f 2 ÞT , x ¼ ðx1 , x2 ÞT , p ¼ ðp1 , p2 , . . . , pm ÞT :

ð3:71Þ

where

The equilibrium x is determined by fðx , pÞ ¼ 0 ) f 1 ðx1 , x2 , pÞ ¼ 0 and f 2 ðx1 , x2 , pÞ ¼ 0:

ð3:72Þ

In the vicinity of x, the linearized equation is y_ ¼ Dfðx , pÞy,

ð3:73Þ

where y ¼ xx with kyk δ for a small δ > 0: 2

∂f 1 6 ∂x1  Dfðx , pÞ ¼ 6 4 ∂f 2 ∂x1

3 ∂f 1  a ∂x2 7 7 ¼ 11 ∂f 2 5 a21 ∂x2 x

a12 a22

 :

ð3:74Þ

The corresponding eigenvalues are determined by    a11  λ a12  ¼ 0, j Dfðx , pÞ  λI2 2 j¼ 0 )  a21 a22  λ 

ð3:75Þ

λ2  TrðDfÞλ þ DetðDfÞ ¼ 0,

ð3:76Þ

and

where   a11 TrðDfÞ ¼ a11 þ a12 , DetðDfÞ ¼  a21

 a12  : a 

ð3:77Þ

22

So the eigenvalues are TrðDfÞ λ1, 2 ¼ 2

pffiffiffiffi Δ

,

ð3:78Þ

3.2 2-Dimensional Nonlinear Systems

111

where Δ ¼ ðTrðDfÞÞ2  4DetðDfÞ:

ð3:79Þ

If Δ > 0, λ1, 2 are real eigenvalues. (i) If Tr(Df) < 0 and Det(Df) > 0, then λ1, 2 < 0. Thus the equilibrium x is a stable node. (ii) If Tr(Df) < 0 and Det(Df) < 0, then λ1 < 0 and λ2 < 0. Thus the equilibrium x is a saddle. (iii) If Tr(Df) > 0 and Det(Df) > 0, then λ1, 2 > 0. Thus the equilibrium x is an unstable node. (iv) If Tr(Df) > 0 and Det(Df) < 0, then λ1 < 0 and λ2 > 0. Thus the equilibrium x is a saddle. (v) If Tr(Df) ¼ 0, then λ1 < 0 and λ2 > 0. Thus the equilibrium x is a saddle. If Δ ¼ 0, λ1, 2 are repeated real eigenvalues: ðTrðDfÞÞ2  4DetðDfÞ ¼ 0

ð3:80Þ

A parabolic curve between Tr(Df) and Det(Df) exists. (vi) If Tr(Df) > 0, then λ1, 2 ¼ λ > 0. Thus the equilibrium x is an unstable node. (vii) If Tr(Df) < 0, then λ1, 2 ¼ λ < 0. Thus the equilibrium x is a stable node. (viii) If Tr(Df) ¼ 0, then λ1, 2 ¼ λ ¼ 0. Thus, for Df ¼ 0, then the equilibrium x is a critical case. For Df 6¼ 0, then the equilibrium x is an unstable node, and flow moves in one eigenvector direction. If Δ < 0, λ1, 2 are complex eigenvalues, i.e., λ1, 2 ¼ α þ βi,

ð3:81Þ

pffiffiffiffiffiffiffi 1, α ¼ TrðDfÞ=2, pffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi β ¼ Δ ¼ 4DetðDfÞ  ðTrðDfÞÞ :

ð3:82Þ

where i¼

(ix) If Tr(Df) < 0, then Reλ1, 2 < 0. Thus the equilibrium x is a stable focus. (x) If Tr(Df) < 0, then Reλ1, 2 > 0. Thus the equilibrium x is an unstable focus. (xi) If Tr(Df) ¼ 0, then λ1, 2 ¼ βi. Thus the equilibrium x is a center. From the previous discussion, the equilibrium stability for stable node (sink), unstable node (source), saddle, unstable focus, stable focus, and centers is presented in Fig. 3.3.

112

3 Low-Dimensional Dynamical Systems

a

b

c

d

e

f

Fig. 3.3 Equilibrium stability: (a) stable node (sink), (b) unstable node (source), (c) saddle, (d) stable focus, (e) unstable focus, and (f) center. The eigenvectors may not always be orthogonal

3.2 2-Dimensional Nonlinear Systems

113

From the eigenvalues, the eigenvectors vj ( j ¼ 1, 2) are determined by ðDfðx , pÞ  λj I2 2 Þvj ¼ 0 ) Dfðx , pÞvj ¼ λj vj ðj ¼ 1, 2Þ

ð3:83Þ

or 

a11  λj

a12

 v 

a21

a22  λj

vj2

j1

¼

( ) 0

ðj ¼ 1, 2Þ:

ð3:84Þ

0

Thus a new transformation based on the covariant eigenvectors vj ( j ¼ 1, 2) is defined as x ¼ cj vj ¼ c1 v1 þ c2 v2 ðsummation of j ¼ 1, 2Þ,

ð3:85Þ

and the matrix of the covariant eigenvectors is P ¼ ðv1 , v2 Þ,

ð3:86Þ

and the contravariant eigenvectors v j ( j ¼ 1, 2) are defined as v1 ¼ ðb11 , b12 ÞT , v2 ¼ ðb21 , b22 ÞT and Q ¼ ðv1 , v2 Þ,

ð3:87Þ

QT ¼ P1 ¼ ðbij Þ2 2 :

ð3:88Þ

ck ¼ ðvk ÞT  x, ck ¼ ðvk ÞT  x ðk ¼ 1, 2Þ,

ð3:89Þ

c_k ¼ ðvk ÞT  x_ ¼ ðvk ÞT  fðx, pÞ ¼ Gk ðx, pÞ:

ð3:90Þ

zk ¼ ck  ck ¼ ðvk ÞT  ðx  x Þ, z_k ¼ c_k ¼ ðvk ÞT  x_ ¼ ðvk ÞT  fðx þ Δx, pÞ ðk ¼ 1, 2Þ:

ð3:91Þ

where

Therefore,

and

Let

114

3 Low-Dimensional Dynamical Systems

In the neighborhood of equilibrium x, the linearized equation on the covariant eigenvector vk (k ¼ 1, 2) is ð1Þ

z_k ¼ GkðjÞ ðx , pÞzj ¼ λj δkj zj ¼ λk zk ð summation on j but not kÞ,

ð3:92Þ

where GkðjÞ ðx , pÞ ¼ ðvk ÞT  Dfðx , pÞvj ¼ ðvk ÞT  λj vj ¼ λj δkj :

ð3:93Þ

zk ¼ zk0 eλk ðtt0 Þ :

ð3:94Þ

Thus

(i) If λk < 0, zk ! 0 as t ! 1. Thus, the equilibrium x on the covariant eigenvector vk (k 2 {1, 2}) is a stable node. (ii) If λk > 0, zk ! 1 as t ! 1. Thus, the equilibrium x on the covariant eigenvector vk (k 2 {1, 2}) is an unstable node. (iii) If λk ¼ 0, zk ¼ const. For linear systems, this is static. For a nonlinear system, in the vicinity of the equilibrium x, the higher order variation equations on the covariant eigenvector vk (k 2 {1, 2}) should be considered. In the neighborhood of equilibrium x, the higher order variational equation on the covariant eigenvector vk (k ¼ 1, 2) is ðr Þ 1 j2 ...jr k Þ

Gkðjk

z_k ¼

¼ 0, r k ¼ 1, 2, . . . , sk  1

1 ðsk Þ ðx , pÞzj1 zj1 . . . zjsk G mk ! kðj1 j2 ...jsk Þ

ð3:95Þ

ðsummation on j1 , j2 , . . . , jsk ¼ 1, 2Þ, where ðs Þ ðx , pÞ 1 j2 jsk Þ

Gkðjk

¼ ðvk ÞT  Dðsk Þ fðx , pÞvj1 vj2 . . . vjsk :

ð3:96Þ

3.2 2-Dimensional Nonlinear Systems

115

For sk ¼ 2, ð1Þ

ð1Þ

Gkð1Þ ¼ λ1 δk1 ¼ 0, Gkð2Þ ¼ λ2 δk2 ¼ 0, 1 ð2Þ G ðx , pÞzj1 zj2 2! kðj1 j2 Þ 1 ð2Þ ð2Þ ¼ ½Gkð11Þ ðx , pÞz1 z1 þ Gkð12Þ ðx , pÞz1 z2 2!

z_k ¼

ð2Þ

ð3:97Þ

ð2Þ

þ Gkð21Þ ðx , pÞz2 z1 þ Gkð22Þ ðx , pÞz2 z2  ðk ¼ 1, 2Þ, where ð2Þ

Gkðj

1 j2 Þ

ðx , pÞ ¼ ðvk ÞT  D2 fðx , pÞvj1 vj2

ðj1 , j2 ¼ 1, 2Þ:

ð3:98Þ

(i) If ð2Þ

Gkðj

1 j2 Þ

ðx , pÞzj1 zj2 > 0,

ð3:99Þ

then the equilibrium x on the covariant eigenvector vk (k 2 {1, 2}) is an increasing saddle (or upper saddle) of the second order. (ii) If ð2Þ

Gkðj

1 j2 Þ

ðx , pÞzj1 zj2 < 0,

ð3:100Þ

then the equilibrium x on the covariant eigenvector vk (k 2 {1, 2}) is a decreasing saddle (or lower saddle) of the second order. (iii) If ð2Þ

zk Gkðj

1 j2 Þ

ðx , pÞzj1 zj2 > 0,

ð3:101Þ

then the equilibrium x on the covariant eigenvector vk (k 2 {1, 2}) is a source of the second order. (iv) If ð2Þ

zk Gkðj

1 j2 Þ

ðx , pÞzj1 zj2 < 0,

ð3:102Þ

then the equilibrium x on the covariant eigenvector vk (k 2 {1, 2}) is a sink of the second order.

116

3 Low-Dimensional Dynamical Systems

(v) If ð2Þ

Gkðj

1 j2 Þ

ðx , pÞ ¼ 0,

ð3:103Þ

then the equilibrium x on the covariant eigenvector vk (k 2 {1, 2}) is a critical case of the second order. For this case, the higher order singularity should be considered for sk ¼ 3: ð1Þ

ð2Þ

Gkðj Þ ¼ λj1 δkj1 ¼ λk ¼ 0, Gkðj

1 j2 Þ

1

ðx , pÞ ¼ 0,

1 ð3Þ G ðx , pÞzj1 zj2 zj3 3! kðj1 j2 j3 Þ 1 ð3Þ ð3Þ ¼ ½Gkð111Þ ðx , pÞz1 z1 z1 þ Gkð112Þ ðx , pÞz1 z1 z2 3!

z_k ¼

ð3Þ

ð3Þ

ð3Þ

ð3Þ

ð3Þ

ð3Þ

ð3:104Þ

þ Gkð121Þ ðx , pÞz1 z2 z1 þ Gkð122Þ ðx , pÞz1 z2 z2 þ Gkð211Þ ðx , pÞz2 z1 z1 þ Gkð212Þ ðx , pÞz2 z1 z2 þ Gkð221Þ ðx , pÞz2 z2 z1 þ Gkð222Þ ðx , pÞz2 z2 z2 , where ð3Þ

Gkðj

1 j2 j3 Þ

ðx , pÞ ¼ ðvk ÞT  D3 fðx , pÞvj1 vj2 vj3

ðj1 , j2 , j3 ¼ 1, 2Þ:

ð3:105Þ

(i) If ð3Þ

Gkðj

1 j2 j3 Þ

ðx , pÞzj1 zj2 zj3 > 0,

ð3:106Þ

then the equilibrium x on the covariant eigenvector vk (k 2 {1, 2}) is an increasing saddle of the third order (or an upper-saddle of the third-order). (ii) If ð3Þ

Gkðj

1 j2 j3 Þ

ðx , pÞzj1 zj2 zj3 < 0,

ð3:107Þ

then the equilibrium x on the covariant eigenvector vk (k 2 {1, 2}) is a deceasing saddle of the third order (or an lower-saddle of the third-order). (iii) If ð3Þ

zk Gkðj

1 j2 j3 Þ

ðx , pÞzj1 zj2 zj3 > 0,

ð3:108Þ

3.2 2-Dimensional Nonlinear Systems

117

then the equilibrium x on the covariant eigenvector vk (k 2 {1, 2}) is an unstable node of the third order (or a source of the third-order). (iv) If ð3Þ

zk Gkðj

1 j2 j3 Þ

ðx , pÞzj1 zj2 zj3 < 0,

ð3:109Þ

then the equilibrium x on the covariant eigenvector vk (k 2 {1, 2}) is a stable node of the third order (or a sink of the third-order).

3.2.2

Hopf Bifurcation

Consider a dynamical system as x_ ¼ ½αd þ aðx2 þ y2 Þx þ ½β þ αc þ bðx2 þ y2 Þy, y_ ¼ ½β þ αc þ bðx2 þ y2 Þy þ ½αd þ aðx2 þ y2 Þy:

ð3:110Þ

r 2 ¼ x2 þ y2 with x ¼ rcosθ and y ¼ rsinθ,

ð3:111Þ

Setting

Eq. (3.110) becomes r_ ¼ f r ðr, αÞ ¼ ½αd þ ar 2 r, θ_ ¼ f θ ðr, αÞ ¼ β0 þ αc þ br 2 :

ð3:112Þ

The equilibrium of Eq. (3.112) is r 1 ¼ 0 for α 2 ð1, þ1Þ, r 2 ¼ ðαd=aÞ1=2 for ðαdÞ a < 0:

ð3:113Þ

If d 6¼ 0, we have Df r ðr  , αÞ ¼ αd þ 3ar 2 :

ð3:114Þ

(i) For r 1 ¼ 0, Dfr ¼ αd. For d > 0, such an equilibrium is stable as α < 0 or unstable as α > 0. The equilibrium is a critical point for α ¼ 0. However, for d < 0, the equilibrium is stable as α > 0 or unstable as α < 0. (ii) The equilibrium of r 2 ¼ ðαd=aÞ1=2 requires (αd) a < 0. For a d > 0, such an equilibrium solution exists for α < 0. For a d < 0, the equilibrium existence condition is α > 0. From Dfr ¼ 2αd, for a d > 0, the equilibrium is stable for

118

3 Low-Dimensional Dynamical Systems

(d > 0, a > 0) and unstable for (d < 0, a < 0) because of α < 0. For a d < 0, the equilibrium is stable for (d < 0, a > 0) and unstable for (d > 0, a < 0) because of α > 0. (iii) For α ¼ 0, we have r 2 ¼ r 1  r 0 ¼ 0 and Df r ðr 0 , αÞ ¼ αd ¼ 0 and Dα Df r ðr 0 , αÞ ¼ d:

ð3:115Þ

Therefore, for α ¼ 0, r 0 ¼ 0 is stable for d > 0 and r 0 ¼ 0 is unstable for d < 0. (iv) The bifurcation of equilibrium at point (r, α) ¼ (0, 0) is the Hopf bifurcation. On the other hand, D2fr(r0, α) ¼ 6ar0 ¼ 0 for r ¼ 0. D3fr(r0, α) ¼ 6a is obtained. Further, z_ ¼

1 3 D f r ðr 0 , αÞz3 ¼ az3 : 3!

ð3:116Þ

where z ¼ rr. (iv1) For a > 0, the vector field in Eq. (3.116) is greater than zero if y > 0 (r > r 0 ) and less than zero if y < 0 (r < r 0 ). If a > 0 and α ! 0+, r 2 ¼ ðαd=aÞ1=2 ! 0þ is unstable with d < 0. If a > 0 and α ! 0+, r 1 ! 0þ is stable. If a > 0 and α ! 0, r 1 ! 0þ is unstable. Thus, the bifurcation point possesses a source flow of the third order, and the bifurcation branch is unstable. From Eq. (3.115), we have αd < 0 for such an unstable bifurcation because of (αd) a < 0. (iv2) If a < 0, the vector field in Eq. (3.116) is less than zero if y > 0 (r > r 0 ) and greater than zero if y < 0 (r > r 0 ). If a < 0 and α ! 0+, r 2 ¼ ðαd=aÞ1=2 ! 0þ is stable with d > 0. If a > 0 and α ! 0+, r 1 ! 0þ is unstable. If a > 0 and α ! 0, r 1 ! 0þ is stable. Thus, for such a case, the bifurcation point possesses a sink flow of the third order. The bifurcation branch is stable. From Eq. (3.115), we have αd > 0 for such a stable bifurcation due to (αd) a < 0. The Hopf bifurcation with stable focus (d > 0) is supercritical. The Hopf bifurcation with unstable focus (d < 0) is subcritical. The supercritical and subcritical Hopf bifurcation is shown in Fig. 3.4a, b. The solid lines and curves represent stable equilibrium. The dashed lines and curves represent unstable equilibrium. Since θ_ ¼ β is constant and r 6¼ 0, one gets a periodic motion on the circle. Consider a dynamical system in a polar coordinate frame as r_ ¼ f r ðr, αÞ, θ_ ¼ f θ ðr, αÞ 6¼ 0

ð3:117Þ

3.2 2-Dimensional Nonlinear Systems

a

119

b

Fig. 3.4 Hopf bifurcations: (a) supercritical (d > 0, a > 0) and (b) subcritical (d < 0, a > 0)

and f r ðr  , αÞ ¼ 0 ) r l ¼ Δl ðl ¼ 1, 2, . . . , l1 Þ

ð3:118Þ

1 m D f r ðr l , αÞðΔr l Þm for Δr l ¼ r l  r l , m! Drl f r ðr l , αÞ ¼ 0 for r l ¼ 1, 2, . . . , m  1,

ð3:119Þ

with Δr_l ¼

which can be used to determine the stability of periodic motions near the equilibrium. At the bifurcation point r 0 , the corresponding variational equation is z_ ¼

1 m D f r ðr 0 , αÞzm , m!

ð3:120Þ

where z ¼ r   r 0 . (i) For m ¼ 2l1, the periodic flow is stable of the (2l1)th order if Gð2l1Þ ¼ D2l1 f r ðr 0 , αÞ < 0: r

ð3:121Þ

The periodic flow is called a periodic flow of the (2l1)th-order sink. (ii) For m ¼ 2l1, the periodic flow is unstable of the (2l1)th order if Gð2l1Þ ¼ D2l1 f r ðr 0 , αÞ > 0: r

ð3:122Þ

The periodic flow is called a periodic flow of the (2l1)th-order source.

120

3 Low-Dimensional Dynamical Systems

(iii) For m ¼ 2l, the periodic flow is unstable of the (2l)th-order lower saddle if Gð2lÞ ¼ D2l f r ðr 0 , αÞ < 0: r

ð3:123Þ

The periodic flow is called a periodic flow of the (2l )th-order lower saddle. (iv) For m ¼ 2l, the periodic flow is of the (2l)th-order upper saddle if Gð2l1Þ ¼ D2l1 f r ðr 0 , αÞ > 0: r

ð3:124Þ

The periodic flow is called a periodic flow of the (2l )th-order upper saddle. From the above similar analysis, the Hopf bifurcation points possess the higher order singularity of the flow in dynamical system in the corresponding radial direction, as shown in Fig. 3.5. (i) For the stable Hopf bifurcation, the (2l1)th-order singularity of the flow at the bifurcation point exists as a sink of the (2l1)th order in the radial direction. þ  Both upper and lower branches (r þ 1 and r 1 ¼ r 1 ) are stable. (ii) For the unstable Hopf bifurcation, the (2l1)th-order singularity of the flow at the bifurcation point exists as a source of the (2l1)th order in the radial þ  direction. Both upper and lower branches (r þ 1 and r 1 ¼ r 1 ) are unstable. (iii) For the lower saddle Hopf bifurcation, the (2l)th-order singularity of the flow at the bifurcation point exists as a lower saddle of the (2l)th order in the radial þ  direction. The upper branch r þ 1 is stable and the lower branch r 1 ¼ r 1 is unstable. (iv) For the upper saddle Hopf bifurcation, the (2l)th-order singularity of the flow at the bifurcation point exists as a source of the (2l )th order in the radial direction. þ  The upper branch r þ 1 is unstable and the lower branch r 1 ¼ r 1 is stable. The stability and bifurcation of equilibrium for 2D nonlinear dynamic system are summarized with detðDfÞ ¼ detðDfðx0 , p0 ÞÞ and trðDfÞ ¼ trðDfðx0 , p0 ÞÞ in Fig. 3.6. The thick dashed lines are bifurcation lines. The stability of equilibriums is given by the eigenvalues in complex plane. The stability of equilibriums for higher dimensional systems can be identified by using a naming of stability for linear dynamical systems. The saddle-node bifurcation possesses stable saddle-node bifurcation (critical) and unstable saddle-node bifurcation (degenerate).

3.2 2-Dimensional Nonlinear Systems

121

a

b

c

d

Fig. 3.5 Hopf bifurcations: (a) (2l  1)th-order sink Hopf bifurcation and (b) (2l  1)th-order source Hopf bifurcation, (c) (2l )th-order lower-saddle-appearing/vanishing Hopf bifurcation, (d) (2l )th-order upper-saddle-appearing/vanishing Hopf bifurcation. The black branch is for existence solution. The red branch is the asymmetric branch of the black branch

122

3 Low-Dimensional Dynamical Systems

Im

Im

Re

Re tr( Df ) Saddle

Re

Re Im

Im

Im

Im

Unstable node

Unstable focus

Re

Re

Im

Hopf bifurcation

det( Df )

Im

Im

Re Re

Re

Stable focus

Im Stable node

Saddle-node bifurcation

Im

Im

Re Node-focus separatrix

Re

Re

Fig. 3.6 Stability and bifurcation diagrams through the complex plane of eigenvalues for 2D dynamical systems

References Luo, A.C.J., 2012, Continuous Dynamical Systems, HEP/L&H Scientific: Beijing/Glen Carbon. Luo, A.C.J., 2019, On stability and bifurcation of equilibriums in nonlinear systems, Journal of Vibration Testing and System Dynamics, 3(2). pp. 147–232.

Chapter 4

Equilibrium Stability in 1-Dimensional Systems

In this chapter, a global analysis of equilibrium stability in 1-dimensional nonlinear dynamical systems is presented. The classification of dynamical systems is given first, and infinite-equilibrium systems are defined. The 1-dimensional dynamical systems with single equilibrium are discussed first. The 1-dimensional dynamical systems with two and three equilibriums are discussed. Simple equilibriums and higher order equilibriums in 1-dimensional dynamical systems are analyzed, and herein a higher order equilibrium is an equilibrium with higher order singularity. The separatrix flow of equilibriums in 1-dimensional systems in phase space is illustrated for a better understanding of the global stability of equilibriums in 1-dimensional dynamical systems.

4.1

System Classifications

Definition 4.1 Consider a dynamical system as x_ ¼ fðx, t, pÞ for t 2 I, x 2 Ω and p 2 Λ

ð4:1Þ

f ¼ ðf 1 , f 2 , . . . , f n ÞT :

ð4:2Þ

where

(i) The dynamical system is static if f(x, t, p)  0 for t 2 I  R and x 2 Ω. (ii) The dynamical system is of the constant velocity if f(x, t, p)  constant for t 2 I  R and x 2 Ω.

© Springer Nature Switzerland AG 2019 A. C. J. Luo, Bifurcation and Stability in Nonlinear Dynamical Systems, Nonlinear Systems and Complexity 28, https://doi.org/10.1007/978-3-030-22910-8_4

123

124

4 Equilibrium Stability in 1-Dimensional Systems

(iii) The dynamical system is independent of state space if f(x, t, p)  g(t, p) for t 2 I  R and x 2 Ω. (iv) The dynamical system is autonomous with equilibrium if f(x, t, p)  g(x, p) and gðx, pÞjx ¼ x  ¼ 0

ð4:3Þ

for t 2 I  R and x 2 Ω. (v) The dynamical system is autonomous without equilibrium if f(x, t, p)  g(x, p) and gðx, pÞjx¼x 6¼ 0

ð4:4Þ

for t 2 I  R and x ¼ x 2 Ω. (vi) The dynamical system is nonautonomous if f(x, t, p) is explicitly relative to time for t 2 I  R and x 2 Ω. Definition 4.2 Consider an autonomous dynamical system as x_ ¼ gðx, pÞ for t 2 I, x 2 Ω and p 2 Λ

ð4:5Þ

g ¼ ðg1 , g2 , . . . , gn ÞT :

ð4:6Þ

where

(i) Such a dynamical system is called an autonomous system without equilibrium in r-dimensional directions on domain Ω if for x 2 Ω, gk ðx, pÞjx ¼ x 6¼ 0 for k 2 fl1 , l2 , . . . , lr g  f1, 2, . . . , ng:

ð4:7Þ

(ii) Such a dynamical system is called an autonomous system without equilibrium in all directions if for x 2 Ω, gk ðx, pÞjx ¼ x 6¼ 0 for k ¼ 1, 2, . . . , n:

ð4:8Þ

(iii) Such a dynamical system is called an autonomous system with equilibrium in r-dimensional directions if for x 2 Ω, gk ðx, pÞjx ¼ x ¼ 0 for k 2 fl1 , l2 , . . . , lr g  f1, 2, . . . , ng:

ð4:9Þ

(iv) Such a dynamical system is called an autonomous system with equilibrium in all directions if for x 2 Ω, gk ðx, pÞjx ¼ x ¼ 0 for k ¼ 1, 2, . . . , n:

ð4:10Þ

4.1 System Classifications

125

Definition 4.3 Consider an autonomous dynamical system as x_ ¼ gðx; pÞ and gðx; pÞjx¼x ¼ 0 for t 2 I, x 2 Ω and p 2 Λ

ð4:11Þ

where g ¼ ðg1 , g2 , . . . , gn ÞT , A ¼


∂g
:
∂x x¼x

ð4:12Þ

(i) Such a dynamical system is called an autonomous system with finite equilibriums if gðx , pÞ ¼ 0, and detðAÞ 6¼ 0:

ð4:13Þ

(ii) Such a dynamical system is called an autonomous system with infinite equilibriums if gðx , pÞ ¼ 0, and detðAÞ ¼ 0:

ð4:14Þ

(iii) Such a dynamical system is called an autonomous system with at least r-sets of infinite equilibriums if gðx , pÞ ¼ 0, and detðAÞ ¼ 0 with rankðAÞ ¼ n  r:

ð4:15Þ

Definition 4.4 Consider an autonomous dynamical system as x_i ¼ gi1 ðx, pi1 Þgi2 ðx, pi2 Þ . . . giri ðx, piri Þ ði ¼ 1, 2, . . . , nÞ, r i  1

ð4:16Þ

where gi ðx, pÞ  gi1 ðx, pi1 Þgi2 ðx, pi2 Þ . . . giri ðx, piri Þ:

ð4:17Þ

(i) Such a dynamical system is called an autonomous system without equilibrium in the direction of the ith component if gi ðx, pÞ 6¼ 0 or gij ðx, pij Þ 6¼ 0 for i 2 f1, 2, . . . , ng, j ¼ 1, 2, . . . , r i :

ð4:18Þ

126

4 Equilibrium Stability in 1-Dimensional Systems

(ii) Such a dynamical system is called an autonomous system with equilibrium in the direction of the ith component if gi ðx, pÞ ¼ 0 or gij ðx, pij Þ ¼ 0 for i 2 f1, 2, . . . , ng, j 2 f1, 2, . . . , r i g:

ð4:19Þ

(iii) There is an infinite-equilibrium surface of the zeroth order in such a dynamical system if gi ðx, pÞ ¼ 0, i ¼ 1, 2, . . . , n; gi1 j1 ðx, pi1 j1 Þ  gi2 j2 ðx, pi2 j2 Þ ¼ 0 for i1 , i2 2 f1, 2, . . . , ng, i1 6¼ i2 ;

ð4:20Þ

j1 2 f1, 2, . . . , r i1 g, j2 2 f1, 2, . . . , r i2 g: (iv) There is an infinite-equilibrium surface of the first order in such a dynamical system as gi ðx, pÞ ¼ 0, i ¼ 1, 2, . . . , n; φðx, qÞ ¼ 0

ð4:21Þ

if lim gi1 j1 ðx, pi1 j1 Þ ¼ 0,

φðx, qÞ!0

lim Dφ gi1 j1 ðx, pi1 j1 Þ ¼

φðx, qÞ!0

lim Dφ gi2 j2 ðx, pi2 j2 Þ ¼

φðx, qÞ!0

lim gi2 j2 ðx, pi2 j2 Þ ¼ 0,

φðx, qÞ!0

lim

d

φðx, qÞ!0 dφ

lim

d

φðx, qÞ!0 dφ

gi1 j1 ðx, pi1 j1 Þ 6¼ 0, gi2 j2 ðx, pi2 j2 Þ 6¼ 0

ð4:22Þ

gi1 j1 ðx, pi1 j1 Þ Dφ gi1 j1 ðx, pi1 j1 Þ with lim ¼ lim ¼ Δ, φðx, qÞ!0 gi2 j2 ðx, pi2 j2 Þ φðx, qÞ!0 Dφ gi2 j2 ðx, pi2 j2 Þ 0 6¼j Δ j< 1 for i1 , i2 2 f1, 2, . . . , ng, i1 6¼ i2 ;j1 , j2 2 f1, 2, . . . , r i g: (v) There is an infinite-equilibrium surface of the rth order in such a dynamical system as gi ðx, pÞ ¼ 0, i ¼ 1, 2, . . . , n; φðx, qÞ ¼ 0 if

ð4:23Þ

4.2 Equilibrium Stability

127

lim gi1 j1 ðx, pi1 j1 Þ ¼ 0,

φðx, qÞ!0

lim gi2 j2 ðx, pi2 j2 Þ ¼ 0

φðx, qÞ!0

dk lim Dkφ gi1 j1 ðx, pi1 j1 Þ ¼ lim gi1 j1 ðx, pi1 j1 Þ ¼ 0, φðx, qÞ!0 φðx, qÞ!0 dφk ð k ¼ 1, 2, . . . , r  1Þ dk lim Dkφ gi2 j2 ðx, pi2 j2 Þ ¼ lim gi2 j2 ðx, pi2 j2 Þ ¼ 0, φðx, qÞ!0 φðx, qÞ!0 dφk ðk ¼ 1, 2, . . . , r  1Þ dr lim Drφ gi1 j1 ðx, pi1 j1 Þ ¼ lim r gi1 j1 ðx, pi1 j1 Þ 6¼ 0, φðx, qÞ!0 φðx, qÞ!0 dφ dr lim Drφ gi2 j2 ðx, pi2 j2 Þ ¼ lim r gi2 j2 ðx, pi2 j2 Þ 6¼ 0 φðx, qÞ!0 φðx, qÞ!0 dφ Drφ gi1 j1 ðx, pi1 j1 Þ gi1 j1 ðx, pi1 j1 Þ with lim ¼ lim ¼ Δ, r φðx, qÞ!0 gi2 j2 ðx, pi2 j2 Þ φðx, qÞ!0 Dφ gi2 j2 ðx, pi2 j2 Þ j Δ j< 1 but Δ 6¼ 0 for i1 , i2 2 f1, 2, . . . , ng, i1 6¼ i2 ; j1 , j2 2 f1, 2, . . . , r i g: ð4:24Þ

4.2

Equilibrium Stability

In this section, 1-dimensional nonlinear dynamical systems will be discussed first to help one understand the stability and bifurcation of nonlinear systems as in Luo (2019). Consider a dynamical system decomposed by x_ ¼ f ðx, pÞ ¼ g0 ðxÞg1 ðx  a1 Þg2 ðx  a2 Þ . . . gm1 ðx  am1 Þgm ðx  am Þ

ð4:25Þ

where g0 ðxÞ 6¼ 0 for x 2 R gk ðx  ak Þ ¼ 0 with xk ¼ ak ðk ¼ 1, 2, . . . , mÞ: Thus, such a system has equilibriums as

ð4:26Þ

128

4 Equilibrium Stability in 1-Dimensional Systems

xk ¼ ak for k ¼ 1, 2, . . . , m:

ð4:27Þ

Note that gk(x  ak) can be any type of functions but not necessary to be a polynomial function. Consider a polynomial dynamical system as x_ ¼ f ðx, pÞ ¼ g0 ðxÞðx  a1 Þr1 ðx  a2 Þr2 . . . ðx  am1 Þrm1 ðx  am Þrm

ð4:28Þ

where g0 ðxÞ 6¼ 0 for x 2 R ¼ ð1, 1Þ, ak < akþ1 with  1 < a1 and am < 1,

ð4:29Þ

ai 6¼ aj and r k 2 f1, 2, . . .g for i, j, k 2 f1, 2, . . . , mg: Thus, such a system has equilibriums as xk ¼ ak with the r k th order roots for k ¼ 1, 2, . . . , m:

ð4:30Þ

From local analysis, equilibriums of Eq. (4.28) are presented in Fig. 4.1. In Fig. 4.1a, b, g0(x) > 0 and g0(x) < 0. The acronyms LS, US, SI, and SO represent lower saddle, upper saddle, sink (stable node), and source (unstable node), respectively. The circular symbols are equilibrium points. For the case of g0(x) > 0, there are four basic equilibriums. (i) For an equilibrium point xk (k ¼ 1, 2, . . ., m), if m Σm j¼kþ1 r j ¼ 2lk1 and Σj¼k r j ¼ 2lk2 for lk1 , lk2 2 f1, 2, 3, . . .g,

ð4:31Þ

then such an equilibrium is an upper saddle of the rkth order because x_ > 0 always for x 2 ðxk1 , xk Þ and ðxk , xkþ1 Þ. (ii) For an equilibrium point xk (k ¼ 1, 2, . . ., m), if m Σm j¼kþ1 r j ¼ 2lk1  1 and Σj¼k r j ¼ 2lk2  1 for lk1 , lk2 2 f1, 2, 3, . . .g,

ð4:32Þ

then such an equilibrium is a lower saddle of the rkth order because x_ < 0 always for x 2 ðxk1 , xk Þ and ðxk , xkþ1 Þ. (iii) For an equilibrium point xk (k ¼ 1, 2, . . ., m), if m Σm j¼kþ1 r j ¼ 2lk1 and Σj¼k r j ¼ 2lk2  1 for lk1 , lk2 2 f1, 2, 3, . . .g,

ð4:33Þ

then such an equilibrium is a source of the rkth order because x_ > 0 for x 2 ðxk , xkþ1 Þ and x_ < 0 for x 2 ðxk1 , xk Þ. (iv) For an equilibrium point xk (k ¼ 1, 2, . . ., m), if

4.2 Equilibrium Stability

129

a

b Fig. 4.1 Possible distributions of equilibriums stability of the first-order dynamical system (a) g0 > 0 and (b) g0 < 0. LS: lower saddle, US: upper saddle, SI: sink (stable node), SO: source (unstable node) m Σm j¼kþ1 r j ¼ 2lk1  1 and Σj¼k r j ¼ 2lk2 for lk1 , lk2 2 f1, 2, 3,   g,

ð4:34Þ

then such an equilibrium is a sink of the rkth order because x_ < 0 for x 2 ðxk , xkþ1 Þ and x_ > 0 forx 2 ðxk1 , xk Þ. For the case of g0(x) < 0, similarly, there are four basic equilibriums. (i) For an equilibrium point xk (k ¼ 1, 2, . . ., m), if m Σm j¼kþ1 r j ¼ 2lk1  1 and Σj¼k r j ¼ 2lk2  1 for lk1 , lk2 2 f1, 2, 3, . . .g,

ð4:35Þ

then such an equilibrium is an upper saddle of the rkth order because x_ > 0 for x 2 ðxk , xkþ1 Þ and x_ > 0 for x 2 ðxk1 , xk Þ. (ii) For an equilibrium point xk (k ¼ 1, 2, . . ., m), if

130

4 Equilibrium Stability in 1-Dimensional Systems m Σm j¼kþ1 r j ¼ 2lk1 and Σj¼k r j ¼ 2lk2 for lk1 , lk2 2 f1, 2, 3, . . .g,

ð4:36Þ

then such an equilibrium is a lower saddle of the rkth order because x_ < 0 for x 2 ðxk , xkþ1 Þ and x_ < 0 for x 2 ðxk1 , xk Þ. (iii) For an equilibrium point xk (k ¼ 1, 2, . . . ,m), if m Σm j¼kþ1 r j ¼ 2lk1  1 and Σj¼k r j ¼ 2lk2 for lk1 , lk2 2 f1, 2, 3, . . .g,

ð4:37Þ

then such an equilibrium is a source of the rkth order because x_ > 0 for x 2 ðxk , xkþ1 Þ and x_ < 0 for x 2 ðxk1 , xk Þ. (iv) For an equilibrium point xk (k ¼ 1, 2, . . . ,m), if m Σm j¼kþ1 r j ¼ 2lk1 and Σj¼k r j ¼ 2lk2  1 for lk1 , lk2 2 f1, 2, 3, . . .g,

ð4:38Þ

then such an equilibrium is a sink of the rkth order because x_ < 0 for x 2 ðxk , xkþ1 Þ and x_ > 0 for x 2 ðxk1 , xk Þ. From the above discussions, the properties of distributed equilibriums can be determined. Thus, a few examples are presented as follows.

4.3

One-Equilibrium Systems

Consider a 1-dimensional system with one equilibrium point as x_ ¼ a0 ðx  a1 Þr with r ¼ 2m  1, 2m for m ¼ 1, 2, . . .

ð4:39Þ

where a0 6¼ 0. The rth-order equilibrium of x ¼ a1 is obtained easily and the corresponding stability is discussed. Thus, we have x€ ¼ ra0 ðx  a1 Þr1 x_ ¼ ra20 ðx  a1 Þ2r1 for x 6¼ x ¼ a1

ð4:40Þ

and f ðx, pÞ ¼ a0 ðx  a1 Þr ¼ 0, DðjÞ f ¼ a0 rðr  1Þ . . . ðr  j þ 1Þðx  a1 Þrj ¼ 0, j ¼ 1, 2, . . . , r,

ð4:41Þ

_ 2, . . . : D f ¼ a0 r! with r ¼ 2m  1, 2m for m ¼ 1, ðrÞ

From Eqs. (4.39) and (4.40), the signs of x_ and x€ can be determined for x < a1 and x > a1, as tabulated in Table 4.1. From such signs, the flow direction in phase space _ can be determined. For a0 > 0, x_ ¼ a0 ðx  a1 Þr > 0 if x > a1. For a0 > 0, x_ ¼ of ðx, xÞ a0 ðx  a1 Þr > 0 if x < a1 for r ¼ 2m, but x_ ¼ a0 ðx  a1 Þr < 0 if x1 < a1 for r ¼ 2m1.

4.4 Two-Equilibrium Systems

131

Table 4.1 Signs of x_ and x€ for the 1-dimensional system in Eq. (4.39)

r ¼ 2m  1

r ¼ 2m

x > a1 x < a1 x ¼ a1 x > a1 x < a1 x ¼ a1

a0 > 0 _ x x, x€, x_ +, ! +, " , , # (2m  1)th-order source (SO) +, ! +, " +, ! , # (2m)th-order upper saddle (US)

a0 < 0 _ x x, x€, x_ , +, " +, ! , # (2m  1)th-order sink (SI) , +, " , , # (2m)th-order lower saddle (LS)

Note: x_ ¼ a0 ðx  a1 Þr and x€ ¼ ra20 ðx  a1 Þ2r1 . LS: lower saddle, US: upper saddle, SI: sink (stable node), SO: source (unstable node)

For a0 < 0, x_ ¼ a0 ðx  a1 Þ < 0 if x > a1. For a0 < 0, x_ ¼ a0 ðx  a1 Þr > 0 if x < a1 for r ¼ 2m but x_ ¼ a0 ðx  a1 Þr < 0 if x < a1 for r ¼ 2m  1. For acceleration x€, for a0 > 0 and a0 < 0, x€ ¼ ra20 ðx  a1 Þ2r1 > 0 if x > a1, but x€ ¼ ra20 ðx  a1 Þ2r1 < 0 if x < a1. Because Drf ¼ a0r! with r ¼ 2m  1, 2m (m ¼ 1, 2, . . .), Drf ¼ a0r! > 0 if a0 > 0 and Drf ¼ a0r! < 0 if a0 < 0. Thus, the equilibrium should be four basic equilibriums, i.e., a source (unstable node) of the (2m  1)th order, a sink (stable node) of the (2m  1)th order, an upper saddle of the (2m)th order, and a lower saddle of the (2m  1)th order. _ the equilibrium and phase trajectory are presented in Fig. 4.2. In a plane of ðx, xÞ,

4.4

Two-Equilibrium Systems

Consider a 1-dimensional system with two equilibrium points as x_ ¼ a0 ðx  a1 Þr1 ðx  a2 Þr2 , a0 6¼ 0, ai 2 R with a1 < a2 ,

ð4:42Þ

for r i ¼ 2mi  1, 2mi , i 2 f1, 2g and mi ¼ 1, 2, . . . : Thus x€ ¼ a0 ðr 1 þ r 2 Þðx  a1 Þr1 1 ðx  a2 Þr2 1 ðx  a12 Þx_ ¼ a20 ðr 1 þ r 2 Þðx  a1 Þ2r1 1 ðx  a12 Þðx  a2 Þ2r2 1 ,

ð4:43Þ

where a12 ¼

r 1 a2 þ r 2 a1 with a1 < a12 < a2 : r1 þ r2

ð4:44Þ

Using x_ and x€, trajectories and equilibriums are presented in Figs. 4.3–4.6. If a1 ¼ a2 ¼ a12, a bifurcation point exists. The two-equilibrium system becomes one

132

4 Equilibrium Stability in 1-Dimensional Systems

a

b

c

d

Fig. 4.2 Single equilibrium: (a) a source of the (2m  1)th order (SO) for a0 > 0, (b) a sink of the (2m  1)th order (SI) for a0 < 0, (c) an upper saddle of the (2m)th order (US) for a0 > 0, (d) a lower saddle of the (2m)th order (LS) for a0 < 0. (m ¼ 1, 2, . . .). LS: lower saddle, US: upper saddle, SI: sink (stable node), SO: source (unstable node)

equilibrium system with the (r1 + r2)th-order singularity. This bifurcation generates two equilibriums with the r1th- and r2th-order singularity. The (1:1)-equilibrium points (i.e., r1 ¼ 1, r2 ¼ 1) are called simple equilibriums because the two equilibriums are without any singularity. For a0 > 0, using x_ ¼ a0 ðx  a1 Þðx  a2 Þ, x_ > 0 for x > a2, x_ < 0 for a1 < x < a2, and x_ > 0 for x < a1. Thus, the equilibrium of x ¼ a2 is a source (unstable node) and the equilibrium of x ¼ a1 is a sink (stable node), as shown in Fig. 4.3a. The (1:2)-equilibrium points (i.e., r1 ¼ 1, r2 ¼ 2) possess the first-order and second-order singularity for x ¼ a1 and x ¼ a2, respectively. For a0 > 0, with x_ ¼ a0 ðx  a1 Þðx  a2 Þ2 , x_ > 0 for x > a2, x_ > 0 for a1 < x < a2, and x_ < 0 for x < a1. Thus, the equilibrium of x ¼ a2 is an upper-saddle of the second-order and the equilibrium of x ¼ a1 is a source (unstable node), as shown in Fig. 4.3b. The (2:1)-equilibrium points (i.e., r1 ¼ 2, r2 ¼ 1) possess the second-order and firstorder singularity for x ¼ a1 and x ¼ a2, respectively. For a0 > 0, with x_ ¼ a0 ðx  a1 Þ2 ðx  a2 Þ, x_ > 0 for x > a2, x_ < 0 for a1 < x < a2, and x_ < 0 for x < a1.

4.4 Two-Equilibrium Systems

133

a

b

c

d

e

f

Fig. 4.3 Two equilibriums (a0 > 0): (a) (1 : 1)-(SI:SO), (b) (1 : 2)-(SO:US), (c) (2 : 1)-(LS:SO), (d) (2 : 2)-(US:US), (e) (2 : 3)-(LS:SO), (f) (3 : 2)-(SO:US). LS: lower saddle, US: upper saddle, SI: sink (stable node), SO: source (unstable node)

Therefore, the equilibrium of x ¼ a2 is a source (unstable node) and the equilibrium of x ¼ a1 is a lower saddle of the second order, as shown in Fig. 4.3c. The (2 : 2)-equilibrium points (i.e., r1 ¼ 2, r2 ¼ 2) possess the two second-order singularity for x ¼ a1 and x ¼ a2. For a0 > 0, with x_ ¼ a0 ðx  a1 Þ2 ðx  a2 Þ2 , obtained are (i) x_ > 0 for x > a2, (ii) x_ > 0 for a1 < x < a2, and (iii) x_ > 0 for x < a1. Thus, the equilibriums of x ¼ a1 and x ¼ a2 are two upper saddles of the second order (US), as shown in Fig. 4.3d.

134

4 Equilibrium Stability in 1-Dimensional Systems

a

b

c

d

e

f

Fig. 4.4 Two equilibriums (a0 < 0): (a) (1 : 1)-(SO:SI), (b) (1 : 2)-(SI:LS), (c) (2 : 1)-(US:SI), (d) (2 : 2)-(LS:LS), (e) (2 : 3)-(US:SI), (f) (3 : 2)-(SI:LS). SO: source (or unstable node), SI: sink (stable node), US: upper saddle, LS: lower saddle

The (2 : 3)-equilibrium points (i.e., r1 ¼ 2, r2 ¼ 3) possess the second-order and third-order singularity for x ¼ a1 and x ¼ a2, respectively. For a0 > 0, with x_ ¼ a0 ðx  a1 Þ2 ðx  a2 Þ3 , in three intervals, x_ > 0 for x > a2, x_ < 0 for a1 < x < a2, and x_ < 0 for x < a1 are determined. Thus, the equilibrium of x ¼ a2 is a source (unstable node) of the third order and the equilibrium of x ¼ a1 is a lower

4.4 Two-Equilibrium Systems

135

a

b

c

d

Fig. 4.5 Two equilibriums with general higher singularity: (a0 > 0) (a) (2m1  1 : 2m2  1)-(SI: SO), (b) (2m1  1 : 2m2)-(SO:US), (c) (2m1 : 2m2  1)-(LS:SO), (d) (2m1 : 2m2)-(US:US). LS: lower saddle, US: upper saddle, SI: sink (stable node), SO: source (unstable node)

saddle of the second order, as shown in Fig. 4.3e. The (2 : 3)-equilibrium points are similar to the (2 : 1)-equilibrium points without singularity. The (3 : 2)-equilibrium points (i.e., r1 ¼ 3, r2 ¼ 2) possess the third-order and second-order singularity for x ¼ a1 and x ¼ a2, respectively. For a0 > 0, with x_ ¼ a0 ðx  a1 Þ3 ðx  a2 Þ2 , in three intervals, x_ > 0 for x > a2, x_ > 0 for a1 < x < a2, and x_ < 0 for x < a1 are obtained. Thus, the equilibrium of x ¼ a2 is an upper saddle (unstable node) of the second order and the equilibrium of x ¼ a1 is a source of the third order, as shown in Fig. 4.3f. The (3 : 2)-equilibrium points are similar to the (1 : 2)-equilibrium points without singularity. The stability and singularity of two equilibriums of x_ ¼ a0 ðx  a1 Þr1 ðx  a2 Þr2 for a0 > 0 are summarized in Table 4.2. Such analysis can help one understand the stability and singularity of equilibrium, which will be used to discuss the bifurcations of equilibriums later. For a0 < 0, the (1 : 1)-simple equilibrium points (i.e., r1 ¼ 1, r2 ¼ 1) are different from those for a0 > 0. Using x_ ¼ a0 ðx  a1 Þðx  a2 Þ, for a0 < 0, obtained are x_ < 0 for x > a2, x_ > 0 for a1 < x < a2, and x_ < 0 for x < a1. Thus, the equilibrium of x ¼ a2

136

4 Equilibrium Stability in 1-Dimensional Systems

a

b

c

d

Fig. 4.6 Two equilibriums with general higher singularity: (a0 < 0) (a) (2m1  1 : 2m2  1)-(SO: SI), (b) (2m1  1 : 2m2)-(SI:LS), (c) (2m1 : 2m2  1)-(US:SI), (d) (2m1 : 2m2)-(LS:LS). LS: lower saddle, US: upper saddle, SI: sink (stable node), SO: source (unstable node) Table 4.2 Equilibrium stability and multiplicity for two-equilibrium systems (a0 > 0) (r1 : r2) (1:1) (1:2) (2:1) (2:2) (2:3) (3:2)

x  ¼ a1 Stability 1st-order sink (SI) 1st-order source (SO) 2nd-order lower saddle (LS) 2nd-order upper saddle (US) 2nd-order lower saddle (LS) 3rd-order source (SO)

Repeated r1 ¼ 1 r1 ¼ 1 r1 ¼ 2 r1 ¼ 2 r1 ¼ 2 r1 ¼ 3

x ¼ a2 Stability 1st-order source (SO) 2nd-order upper saddle (US) 1st-order source (SO) 2nd-order upper saddle (US) 3rd-order source (SO) 2nd-order upper saddle (US)

Repeated r2 ¼ 1 r2 ¼ 2 r2 ¼ 1 r2 ¼ 2 r2 ¼ 3 r2 ¼ 2

LS: lower saddle, US: upper saddle, SI: sink (stable node), SO: source (unstable node)

is a sink (stable node) and the equilibrium of x ¼ a1 is a source (stable node), as shown in Fig. 4.4a. For a0 < 0, the (1 : 2)-equilibrium points (i.e., r1 ¼ 1, r2 ¼ 2) possess the secondorder and first-order singularity for x ¼ a1 and x ¼ a2, respectively. With x_ ¼ a0 ðx  a1 Þðx  a2 Þ2 , x_ < 0 for x > a2, x_ < 0 for a1 < x < a2, and x_ > 0 for x < a1 are obtained. Thus, the equilibrium of x ¼ a2 is a lower saddle of the second

4.4 Two-Equilibrium Systems

137

Table 4.3 Equilibrium stability and singularity for two-equilibrium systems (a0 < 0) (r1 : r2) (1:1) (1:2) (2:1) (2:2) (2:3) (3:2)

x ¼ a1 Stability 1st-order source (SO) 1st-order sink (SI) 2nd-order upper saddle (US) 2nd-order lower saddle (LS) 2nd-order upper saddle (US) 3rd-order sink (SI)

Repeated r1 ¼ 1 r1 ¼ 1 r1 ¼ 2 r1 ¼ 2 r1 ¼ 2 r1 ¼ 3

x ¼ a2 Stability 1st-order sink (SI) 2nd-order lower saddle (LS) 1st-order sink (SI) 2nd-order lower saddle (LS) 3rd-order sink (SI) 2nd-order lower saddle (LS)

Repeated r2 ¼ 1 r2 ¼ 2 r2 ¼ 1 r2 ¼ 2 r2 ¼ 3 r2 ¼ 2

LS: lower saddle, US: upper saddle, SI: sink (stable node), SO: source (unstable node)

order and the equilibrium of x ¼ a1 is a sink of the first order (stable node) (SI), as shown in Fig. 4.4b. For a0 < 0, the (2 : 1)-equilibrium points (i.e., r1 ¼ 2, r2 ¼ 1) possess the secondorder and first-order singularity for x ¼ a1 and x ¼ a2, respectively. With x_ ¼ a0 ðx  a1 Þ2 ðx  a2 Þ, x_ < 0 for x > a2, x_ > 0 for a1 < x < a2, and x_ > 0 for x < a1. Therefore, the equilibrium of x ¼ a2 is a sink of the first order (stable node) (SI) and the equilibrium of x ¼ a1 is an upper saddle of the second order (US), as shown in Fig. 4.4c. For a0 < 0, the (2 : 2)-equilibrium points (i.e., r1 ¼ 2, r2 ¼ 2) also possess the two second-order singularity for x ¼ a1 and x ¼ a2. But the two second-order singularity is for the two lower saddles. With x_ ¼ a0 ðx  a1 Þ2 ðx  a2 Þ2 , obtained are x_ < 0 for x > a2, x_ < 0 for a1 < x < a2, and x_ < 0 for x < a1. Thus, the equilibriums of x ¼ a2 and x ¼ a1 are two lower saddles of the second order, as shown in Fig. 4.4d. For a0 < 0, the (2 : 3)-equilibrium points (i.e., r1 ¼ 2, r2 ¼ 3) possess the secondorder and third-order singularity. With x_ ¼ a0 ðx  a1 Þ2 ðx  a2 Þ3 , in three intervals, x_ < 0 for x > a2, x_ > 0 for a1 < x < a2, and x_ > 0 for x < a1. Thus, the equilibrium of x ¼ a2 is a sink (stable node) of the third order and the equilibrium of x ¼ a1 is an upper saddle of the second order, as shown in Fig. 4.4e. The two equilibriums are also similar to the (2 : 1)-equilibrium points. For a0 < 0, the (3 : 2)-equilibrium points (i.e., r1 ¼ 3, r2 ¼ 2) possess the thirdorder and second-order singularity. With x_ ¼ a0 ðx  a1 Þ3 ðx  a2 Þ2 , in three intervals, x_ < 0 for x > a2, x_ < 0 for a1 < x < a2, and x_ > 0 for x < a1 are determined. Thus, the equilibrium of x ¼ a2 is a lower saddle of the second order and the equilibrium of x ¼ a1 is a sink (stable node) of the third order, as shown in Fig. 4.4f. The two equilibriums are also similar to the (1 : 2)-equilibrium points. The stability and singularity of two equilibriums of x_ ¼ a0 ðx  a1 Þr1 ðx  a2 Þr2 for a0 < 0 are summarized in Table 4.3, which are different from the stability and singularity for a0 > 0. From the foregoing discussion, the general cases for stability and singularity for the two-equilibrium system of x_ ¼ a0 ðx  a1 Þr1 ðx  a2 Þr2 are discussed as follows. For a0 > 0, the (2m1  1 : 2m2  1)-equilibrium points (i.e., r1 ¼ 2m1  1, r2 ¼ 2m2  1) possess the (2m1  1)th-order and (2m2  1)th-order singularity for

138

4 Equilibrium Stability in 1-Dimensional Systems

Table 4.4 Equilibrium stability and singularity of a two-equilibrium system in Eq. (4.42) (a0 > 0) (r1 : r2) (2m1  1 : 2m2  1) (2m1  1 : 2m2) (2m1 : 2m2  1) (2m1 : 2m2)

x ¼ a1 (2m1  1)th-order sink (SI) (2m1  1)th-order source (SO) (2m1)th-order lower saddle (LS) (2m1)th-order upper saddle (US)

x ¼ a2 (2m2  1)th-order source (SO) (2m2)th-order upper saddle (US) (2m2  1)th-order source (SO) (2m2)th-order upper saddle (US)

LS: lower saddle, US: upper saddle, SI: sink (stable node), SO: source (unstable node)

x ¼ a1 and x ¼ a2, respectively. With x_ ¼ a0 ðx  a1 Þ2m1 1 ðx  a2 Þ2m2 1 , x_ > 0 for x > a2, x_ < 0 for a1 < x < a2, and x_ > 0 for x < a1. Thus, the equilibrium of x ¼ a2 is a source (unstable node) of the (2m1  1)th order and the equilibrium of x ¼ a1 is a sink (stable node) of (2m2  1)th order, as shown in Fig. 4.5a. For a0 > 0, the (2m1  1 : 2m2)-equilibrium points (i.e., r1 ¼ 2m1  1, r2 ¼ 2m2) possess the (2m1  1)th-order and (2m2)th-order singularity for x ¼ a1 and x ¼ a2, respectively. With x_ ¼ a0 ðx  a1 Þ2m1 1 ðx  a2 Þ2m2 , x_ > 0 for x > a2, x_ > 0 for a1 < x < a2, and x_ < 0 for x < a1 are obtained. Thus, the equilibrium of x ¼ a2 is an upper saddle of the (2m2)th order and the equilibrium of x ¼ a1 is a source (unstable node) of the (2m1  1)th order, as shown in Fig. 4.5b. For a0 > 0, the (2m1 : 2m2  1)-equilibrium points (i.e., r1 ¼ 2m1, r2 ¼ 2m2  1) possess the (2m1)th-order and (2m2  1)th-order singularity for x ¼ a1 and x ¼ a2, respectively. With x_ ¼ a0 ðx  a1 Þ2m1 ðx  a2 Þ2m2 1 , obtained are x_ > 0 for x > a2, x_ < 0 for a1 < x < a2, and x_ < 0 for x < a1. Thus, the equilibrium of x ¼ a2 is a source (unstable node) of the (2m2  1)th order and the equilibrium of x ¼ a1 is a lower saddle of the (2m1)th order (LS), as shown in Fig. 4.5c. For a0 > 0, the (2m1 : 2m2)-equilibrium points (i.e., r1 ¼ 2m1, r2 ¼ 2m2) possess the (2m1)th-order and (2m2)th-order singularity for x ¼ a1 and x ¼ a2, respectively. With x_ ¼ a0 ðx  a1 Þ2m1 ðx  a2 Þ2m2 , obtained are x_ > 0 for x > a2, x_ > 0 for a1 < x < a2, and x_ > 0 for x < a1. Thus, as shown in Fig. 4.5d, the equilibriums of x ¼ a1 and x ¼ a2 are two upper saddles of the (2m1)th order and (2m2)th order, respectively. The stability and higher order singularity of the two-equilibrium system for a0 > 0 are summarized in Table 4.4. Similarly, for a0 < 0, the stability and higher order singularity of the two-equilibrium system are discussed as follows. The (2m1  1:2m2  1)-equilibrium points (i.e., r1 ¼ 2m11, r2 ¼ 2m21) for a0 < 0 possess the (2m1  1)th-order and (2m2  1)th-order singularity for x ¼ a1 and x ¼ a2, respectively. With x_ ¼ a0 ðx  a1 Þ2m1 1 ðx  a2 Þ2m2 1 , obtained are x_ < 0 for x > a2, x_ > 0 for a1 < x < a2, and x_ < 0 for x < a1. Thus, the equilibrium of x ¼ a2 is a sink (stable node) of the (2m2  1)th order and the equilibrium of x ¼ a1 is a source (unstable node) of (2m1  1)th order, as shown in Fig. 4.6a. For a0 < 0, the (2m1 – 1:2m2)-equilibrium points (i.e., r1 ¼ 2m1 – 1, r2 ¼ 2m2) possess the (2m1  1)th-order and (2m2)th-order singularity for x ¼ a1 and x ¼ a2, respectively. With x_ ¼ a0 ðx  a1 Þ2m1 1 ðx  a2 Þ2m2 , x_ < 0 for x > a2, x_ < 0 for

4.5 Three-Equilibrium Systems

139

Table 4.5 Stability and singularity of a two-equilibrium system in Eq. (4.42) (a0 < 0) (r1 : r2) (2m1  1 : 2m2  1) (2m1  1 : 2m2) (2m1 : 2m2  1) (2m1 : 2m2)

x ¼ a1 (2m1  1)th-order source (SO) (2m1  1)th-order sink (SI) (2m1)th-order upper saddle (US) (2m1)th-order lower saddle (LS)

x ¼ a2 (2m2  1)th-order sink (SI) (2m2)th-order lower saddle (LS) (2m2  1)th-order sink (SI) (2m2)th-order lower saddle (LS)

LS: lower saddle, US: upper saddle, SI: sink (stable node), SO: source (unstable node)

a1 < x < a2, and x_ > 0 for x < a1 are obtained. Thus, the equilibrium of x ¼ a2 is a lower saddle of the (2m2)th order and the equilibrium of x ¼ a1 is a sink (stable node) of the (2m1  1)th order, as shown in Fig. 4.6b. For a0 < 0, the (2m1:2m2  1)-equilibrium points (i.e., r1 ¼ 2m1, r2 ¼ 2m2  1) possess the (2m1)th-order and (2m2  1)th-order singularity for x ¼ a1 and x ¼ a2, respectively. With x_ ¼ a0 ðx  a1 Þ2m1 ðx  a2 Þ2m2 1 , obtained are x_ < 0 for x > a2, x_ > 0 for a1 < x < a2, and x_ > 0 for x < a1. Thus, the equilibrium of x ¼ a2 is a sink (stable node) of the (2m2  1)th order and the equilibrium of x ¼ a1 is an upper saddle of the (2m1)th order, as shown in Fig. 4.6c. For a0 < 0, the (2m1:2m2)-equilibrium points (i.e., r1 ¼ 2m1, r2 ¼ 2m2) possess the (2m1)th-order and (2m2)th-order singularity for x ¼ a1 and x ¼ a2, respectively. With x_ ¼ a0 ðx  a1 Þ2m1 ðx  a2 Þ2m2 , obtained are x_ < 0 for x > a2, x_ < 0 for a1 < x < a2, and x_ < 0 for x < a1. Thus, as shown in Fig. 4.6d, the equilibriums of x ¼ a1 and x ¼ a2 are two lower saddles of the (2m1)th order and (2m2)th order, respectively. For a0 < 0, the stability and higher order singularity of two equilibriums of x_ ¼ a0 ðx  a1 Þr1 ðx  a2 Þr2 are summarized in Table 4.5. The odd-order singularity is relative to the stable and unstable nodes (i.e., sink and source). The even-order singularity is relative to the upper and lower saddles.

4.5

Three-Equilibrium Systems

Consider a 1-dimensional system with three equilibrium points as x_ ¼ a0 ðx  a1 Þr1 ðx  a2 Þr2 ðx  a3 Þr3 , a0 6¼ 0, ai 2 R with a1 < a2 < a3 , for r i ¼ 2mi  1, 2mi , i 2 f1, 2, 3g and mi ¼ 1, 2, . . . :

ð4:45Þ

140

4 Equilibrium Stability in 1-Dimensional Systems

Thus x€ ¼ a0 ðr 1 þ r 2 þ r 3 Þðx  a1 Þr1 1 ðx  a2 Þr2 1 ðx  a3 Þr3 1 ðx2 þ a12 x þ a123 Þx_ ¼ a20 ðr 1 þ r 2 þ r 3 Þðx  a1 Þ2r1 1 ðx  a2 Þ2r2 1 ðx  a3 Þ2r3 1 ðx2 þ a12 x þ a123 Þ ð4:46Þ where a12 ¼  a123 ¼

r 1 ða2 þ a3 Þ þ r 2 ða1 þ a3 Þ þ r 3 ða1 þ a2 Þ , r1 þ r2 þ r3

r 1 a2 a3 þ r 2 a1 a3 þ r 3 a1 a2 : r1 þ r2 þ r3

ð4:47Þ

Except for equilibrium points, the existence of the three-equilibrium system in Eq. (4.45) requires two points to make x€ ¼ 0, i.e., x2 þ a12 x þ a123 ¼ 0:

ð4:48Þ

Thus, the existence condition for such three-equilibrium systems is Δ ¼ a212  4a123 > 0,

ð4:49Þ

with x1, 2 ¼

a12 pffiffiffiffi Δ , a1 < x 1 < a2 < x 2 < a3 : 2

ð4:50Þ

If x1 ¼ a1 ¼ a2, the equilibrium is a bifurcation point with the (r1 + r2)th-order singularity. If x2 ¼ a2 ¼ a3, the equilibrium is a bifurcation point with the (r2 + r3)thorder singularity. If x1 ¼ x2 ¼ a2, the equilibrium of x ¼ a2 is a bifurcation point with the r2th-order singularity from the three-equilibrium systems becoming a two-equilibrium system. If x1 ¼ x2 ¼ a1 ¼ a2 ¼ a3, the equilibrium is a bifurcation point with the (r1 + r2 + r3)th-order singularity from the three-equilibrium systems becoming a one-equilibrium system. Consider a few lower order singularity of three equilibriums in the threeequilibrium system to demonstrate stability and singularity of equilibriums. First, consider a simple three-equilibrium system as x_ ¼ a0 ðx  a1 Þðx  a2 Þðx  a3 Þ

ð4:51Þ

with ai < ai+1 (i ¼ 1, 2, 3) and a0 > 0. For such a three-equilibrium system, there are three equilibriums x ¼ ai (i ¼ 1, 2, 3). The equilibrium of x ¼ a3 is a source point. The equilibrium of x ¼ a2 is a sink point. The equilibrium of x ¼ a1 is a source point. Two equilibriums are unstable and one equilibrium is stable. The three equilibriums without any singularity are named the (1 : 1 : 1)-equilibriums, which means that three _ for the (1 : 1 : 1)-equilibrium dynamical equilibriums are simple. The plane of ðx, xÞ

4.5 Three-Equilibrium Systems

141

a

b

c

d

e

f

Fig. 4.7 Three-equilibrium systems (a0 > 0): (a) (1 : 1 : 1)-(SO:SI:SO), (b) (2 : 1 : 1)-(US: SI:SO), (c) (1 : 2 : 1)-(SI: US:SO), (d) (1:1:2)-(SI:SO:US), (e) (2:2:1)-(LS:LS:SO), (f) (2:1:2)-(LS:SO:US), (g) (1:2:2)-(SO:US:US), (h) (3:1:1)-(SO:SI:SO), (i) (1:3:1)-(SO:SI:SO), (j) (1:1:3)-(SO:SI:SO)

system for a0 > 0 is presented in Fig. 4.7a from Eq. (4.51). Such a simple threeequilibrium has a third-degree polynomial. Consider a three-equilibrium system with a second-order singularity as x_ ¼ a0 ðx  a1 Þ2 ðx  a2 Þðx  a3 Þ

ð4:52Þ

with ai < ai+1 (i ¼ 1, 2, 3) and a0 > 0. For such a three-equilibrium system, the equilibrium of x ¼ a3 is a source point, the equilibrium of x ¼ a2 is a sink point,

142

4 Equilibrium Stability in 1-Dimensional Systems

g

h

i

j

Fig. 4.7 (continued)

and the equilibrium of x ¼ a1 is an upper saddle point. The three-equilibrium system possesses the (2:1:1)-equilibriums. Two equilibriums are unstable and one equilib_ for the (2:1;1)-equilibrium rium is stable. From Eq. (4.52), the plane of ðx, xÞ dynamical system for a0 > 0 is presented in Fig. 4.7b. If the double repeated equilibrium is considered as two independent equilibriums, such a system should be four equilibriums, and the corresponding equation is a fourth-degree polynomial. Consider a three-equilibrium system with the second-order singularity on the middle of equilibrium as x_ ¼ a0 ðx  a1 Þðx  a2 Þ2 ðx  a3 Þ

ð4:53Þ

with ai < ai+1 (i ¼ 1, 2, 3) and a0 > 0. In fact, such a three-equilibrium system in Eq. (4.53) is the same class as in Eq. (4.52). Just the double repeated equilibrium point is different from two simple equilibriums. For such a three-equilibrium system, the equilibrium of x ¼ a3 is a source point, the equilibrium of x ¼ a2 is a lower saddle point, and the equilibrium of x ¼ a1 is a sink point. The three-equilibrium system possesses the (1:2;1)-equilibriums. Two equilibriums are unstable and one equilib_ for the (1:2:1)-equilibrium rium is stable. From Eq. (4.53), the plane of ðx, xÞ

4.5 Three-Equilibrium Systems

143

dynamical system for a0 > 0 is presented in Fig. 4.7c. The three-equilibrium system also has four equilibriums with a fourth-degree polynomial. Consider a three-equilibrium system with the second-order singularity of x ¼ a3 as x_ ¼ a0 ðx  a1 Þðx  a2 Þðx  a3 Þ2

ð4:54Þ

with ai < ai+1 (i ¼ 1, 2, 3) and a0 > 0. The three-equilibrium system in Eq. (4.54) is a class of the dynamical systems in Eqs. (4.52 and 4.53). The three systems possess one double-equilibrium point with a fourth-degree polynomial. However, the threeequilibrium systems with the second-order singularity demonstrate the different dynamical behaviors. For such a three-equilibrium system, the equilibrium of x ¼ a3 is an upper saddle point, the equilibrium of x ¼ a2 is a source point, and the equilibrium of x ¼ a1 is a sink point. The three-equilibrium system possesses the (1 : 1 : 2)-equilibriums. Two equilibriums are unstable and one equilibrium is stable. _ for the (1 : 1 : 2)-equilibrium dynamical system From Eq. (4.54), the plane of ðx, xÞ for a0 > 0 is presented in Fig. 4.7d. From the aforementioned three plots, the three three-equilibrium dynamical systems have different dynamical characteristics. Consider a three-equilibrium system with two second-order singularities as x_ ¼ a0 ðx  a1 Þ2 ðx  a2 Þ2 ðx  a3 Þ

ð4:55Þ

with ai < ai+1 (i ¼ 1, 2, 3) and a0 > 0. For such a three-equilibrium system, the equilibrium of x ¼ a3 is a source point, the equilibrium of x ¼ a2 is a lower saddle point, and the equilibrium of x ¼ a1 is a lower saddle point. The three-equilibrium system possesses the (2 : 2 : 1)-equilibriums. The three equilibriums are unstable. _ for the (2 : 2 : 1)-equilibrium dynamical system is From Eq. (4.55), the plane of ðx, xÞ presented in Fig. 4.7e for a0 > 0. If a double repeated equilibrium is considered as two independent equilibriums, such a three-equilibrium system has five equilibriums and the corresponding equation should be a fifth-degree polynomial. Consider a three-equilibrium system with two second-order singularities as x_ ¼ a0 ðx  a1 Þ2 ðx  a2 Þðx  a3 Þ2

ð4:56Þ

with ai < ai+1 (i ¼ 1, 2, 3) and a0 > 0. For such a three-equilibrium system, the equilibrium of x ¼ a3 is an upper saddle point, the equilibrium of x ¼ a2 is a source point, and the equilibrium of x ¼ a1 is a lower saddle point. The three-equilibrium system possesses the (2:1:2)-equilibriums. Three equilibriums are unstable. From _ for the (2:1:2)-equilibrium dynamical system is Eq. (4.56), the plane of ðx, xÞ presented in Fig. 4.7f for a0 > 0. Such a three-equilibrium system also has five equilibriums with a fifth-degree polynomial. Consider a three-equilibrium system with two second-order singularities as x_ ¼ a0 ðx  a1 Þðx  a2 Þ2 ðx  a3 Þ2

ð4:57Þ

144

4 Equilibrium Stability in 1-Dimensional Systems

with ai < ai+1 (i ¼ 1, 2, 3) and a0 > 0. For such a three-equilibrium system, the equilibrium of x ¼ a3 is an upper saddle point, the equilibrium of x ¼ a2 is an upper point, and the equilibrium of x ¼ a1 is a source point. The three-equilibrium system possesses the (1 : 2 : 2)-equilibriums. Three equilibriums are also unstable. From _ for the (1 : 2 : 2)-equilibrium dynamical system is Eq. (4.57), the plane of ðx, xÞ presented in Fig. 4.7g for a0 > 0. Such a three-equilibrium system also has five equilibriums with a fifth-degree polynomial. For a three-equilibrium system with a fifth-degree polynomial, consider the following three cases. Case I A three-equilibrium dynamical system with the third-order singularity is x_ ¼ a0 ðx  a1 Þ3 ðx  a2 Þðx  a3 Þ

ð4:58Þ

with ai < ai+1 (i ¼ 1, 2, 3) and a0 > 0. For such a three-equilibrium system, the equilibrium of x ¼ a3 is a source point, and the equilibrium of x ¼ a2 is a sink point, and the equilibrium of x ¼ a1 is a source point of the third order. The threeequilibrium system possesses the (3 : 1 : 1)-equilibriums. Two equilibriums are _ for the unstable and one equilibrium is stable. From Eq. (4.58), the plane of ðx, xÞ (3 : 1 : 1)-equilibrium dynamical system is presented in Fig. 4.7h for a0 > 0. Such a three-equilibrium system has five equilibriums with a fifth-degree polynomial. Case II A three-equilibrium dynamical system with the third-order singularity is x_ ¼ a0 ðx  a1 Þðx  a2 Þ3 ðx  a3 Þ

ð4:59Þ

with ai < ai+1 (i ¼ 1, 2, 3) and a0 > 0. For such a three-equilibrium system, the equilibrium of x ¼ a3 is a source point, and the equilibrium of x ¼ a2 is a sink point of the third order, and the equilibrium of x ¼ a1 is a source point. The threeequilibrium system possesses the (1 : 3 : 1)-equilibriums. Two equilibriums are also _ for unstable and one equilibrium is also stable. From Eq. (4.59), the plane of ðx, xÞ the (1 : 3 : 1)-equilibrium dynamical system is presented in Fig. 4.7i for a0 > 0. Such a three-equilibrium system also has five equilibriums with a fifth-degree polynomial. Case III A three-equilibrium dynamical system with the third-order singularity is x_ ¼ a0 ðx  a1 Þðx  a2 Þðx  a3 Þ3

ð4:60Þ

with ai < ai+1 (i ¼ 1, 2, 3) and a0 > 0. For the three-equilibrium system, the equilibrium behaviors are changed. The equilibrium of x ¼ a3 is a source point of the third order, and the equilibrium of x ¼ a2 is a sink point, and the equilibrium of x ¼ a1 is a source point. The three-equilibrium system possesses the (1:1:3)equilibriums. Two equilibriums are also unstable and one equilibrium is also stable. _ for the (1:1:3)-equilibrium dynamical system is From Eq. (4.60), the plane of ðx, xÞ presented in Fig. 4.7j for a0 > 0. Such a three-equilibrium system also has five equilibriums with a fifth-degree polynomial.

4.5 Three-Equilibrium Systems

145

Table 4.6 Equilibrium stability and singularity of a three-equilibrium system in Eq. (4.45) (a0 > 0) (1:1:1) (2:1:1) (1:2:1) (1:1:2) (2:2:1) (2:1:2) (1:2:2) (3:1:1) (1:3:1) (1:1:3)

x  ¼ a1 Source (SO) Upper saddle (2nd) Sink (SI) Sink (SI) Lower saddle (2nd) Lower saddle (2nd) Source (SO) Source (3rd) Source (SO) Source (SO)

x  ¼ a2 Sink (SI) Sink (SI) Lower saddle (2nd) Source (SO) Lower saddle (2nd) Source (SO) Upper saddle (2nd) Sink (SI) Sink (3rd) Sink (SI)

x  ¼ a3 Source (SO) Source (SO) Source (SO) Upper saddle (2nd) Source (SO) Upper saddle (2nd) Upper saddle (2nd) Source (SO) Source (SO) Source (3rd)

LS: lower saddle, US: upper saddle, SI: sink (stable node), SO: source (unstable node)

The stability and singularity of equilibriums for the three-equilibrium systems are summarized in Table 4.6 for a0 > 0. Similarly, equilibrium and trajectories in the _ for the three-equilibrium systems for a0 < 0 are presented in Fig. 4.8. plane of ðx, xÞ Only the stability and singularity of three equilibriums for a0 < 0 are summarized in Table 4.7. With singularity, stability for the other three-equilibrium systems can be discussed as for two-equilibrium dynamical systems. The readers can do the similar discussion as an exercise, which can help readers better understand stability and singularity of equilibriums. The equilibrium stability and singularity for the general cases of the three-equilibrium systems are tabulated in Tables 4.8 and 4.9 for a0 > 0 and a0 < 0, respectively. The odd-number-order singularity is for sink (stable node) and source (unstable node). The even-number-order singularity is for lower saddles and upper saddles. For an n-equilibrium dynamical system with a (r1 : r2 : . . . : rn) type, there are 2n-potential singularities of 2mi  1 and 2mj number combinations for given mi and mj (i, j ¼ 1, 2, . . . ,n).

146

4 Equilibrium Stability in 1-Dimensional Systems

a

b

c

d

e

f

Fig. 4.8 Three-equilibrium systems (a0 < 0): (a) (1 : 1 : 1)-(SI:SO:SI), (b) (2 : 1 : 1)-(LS: SO:SI), (c) (1 : 2 : 1)-(SO:US:SI), (d) (1:1:2)-(SO:SI:LS), (e) (2:2:1)-(US:US:SI), (f) (2:1:2)-(US:SI:LS), (g) (1:2:2)-(SI:LS:LS), (h) (3:1:1)-(SI:SO:SI), (i) (1:3:1)-(SI:SO:SI), (j) (1:1:3)-(SI:SO:SI)

4.5 Three-Equilibrium Systems

147

g

h

i

j

Fig. 4.8 (continued)

Table 4.7 Equilibrium stability and singularity of a three-equilibrium system in Eq. (4.45) (a0 < 0) (r1 : r2 : r3) (1:1:1) (2:1:1) (1:2:1) (1:1:2) (2:2:1) (2:1:2) (1:2:2) (3:1:1) (1:3:1) (1:1:3)

x ¼ a1 Sink (SI) Lower saddle (2nd) Source (SO) Source (SO) Upper saddle (2nd) Upper saddle (2nd) Sink (SI) Sink (3rd) Sink (SI) Sink (SI)

x ¼ a2 Source (SO) Source (SO) Upper saddle (2nd) Sink (SI) Upper saddle (2nd) Sink (SI) Lower saddle (2nd) Source (SO) Source (3rd) Source (SO)

x ¼ a3 Sink (SI) Sink (SI) Sink (SI) Lower saddle (2nd) Sink (SI) Lower saddle (2nd) Lower saddle (2nd) Sink (SI) Sink (SI) Sink (3rd)

LS: lower saddle, US: upper saddle, SI: sink (stable node), SO: source (unstable node)

148

4 Equilibrium Stability in 1-Dimensional Systems

Table 4.8 Equilibrium stability and singularity of a three-equilibrium system in Eq. (4.45) (a0 > 0) (r1 : r2 : r3) (2m1  1 : 2m2  1 : 2m3  1) (2m1 : 2m2  1 : 2m3  1) (2m1  1 : 2m2 : 2m3  1) (2m1  1 : 2m2  1 : 2m3) (2m1 : 2m2 : 2m3  1) (2m1 : 2m2  1 : 2m3) (2m1  1 : 2m2 : 2m3) (2m1 : 2m2 : 2m3)

x ¼ a1 Source Upper saddle Sink Sink Lower saddle Lower saddle Source Upper saddle

x  ¼ a2 Sink Sink Lower saddle Source Lower saddle Source Upper saddle Upper saddle

x ¼ a3 Source Source Source Upper saddle Source Upper saddle Upper saddle Upper saddle

LS: lower saddle, US: upper saddle, SI: sink (stable node), SO: source (unstable node)

Table 4.9 Equilibrium stability and singularity of a three-equilibrium system in Eq. (4.45) (a0 < 0) (r1 : r2 : r3) (2m1  1 : 2m2  1 : 2m3  1) (2m1 : 2m2  1 : 2m3  1) (2m1  1 : 2m2 : 2m3  1) (2m1  1 : 2m2  1 : 2m3) (2m1 : 2m2 : 2m3  1) (2m1 : 2m2  1 : 2m3) (2m1  1 : 2m2 : 2m3) (2m1 : 2m2 : 2m3)

x ¼ a1 Sink Lower saddle Source Source Upper saddle Upper saddle Sink Lower saddle

x ¼ a2 Source Source Upper saddle Sink Upper saddle Sink Lower saddle Lower saddle

x ¼ a3 Sink Sink Sink Lower saddle Sink Lower saddle Lower saddle Lower saddle

LS: lower saddle, US: upper saddle, SI: sink (stable node), SO: source (unstable node)

Reference Luo, A.C.J., 2019, The global analysis of equilibrium stability in 1-dimensional systems, Journal of Vibration Testing and System Dynamics, 3(3), pp. 329–346.

Chapter 5

Low-Degree Polynomial Systems

In this chapter, the global stability and bifurcation of equilibriums in low-degree polynomial systems are discussed. Appearing and switching bifurcations of simple and higher order equilibriums are discussed, and such bifurcations of equilibriums are not only for simple equilibriums but also for higher order equilibriums. The thirdorder sink and source bifurcations for simple equilibriums are presented. The thirdorder sink and source switching bifurcations for saddle and nodes are discovered, and the fourth-order upper saddle and lower saddle switching and appearing bifurcations are obtained for two second-order upper saddles and two second-order lower saddles, respectively. Graphical illustrations of global stability and bifurcations of equilibriums are presented.

5.1

Linear Systems

In this section, the stability and stability switching of equilibriums in linear systems are discussed. The sink and source equilibriums are discussed. Definition 5.1 Consider a 1-dimensional linear dynamical system: x_ ¼ AðpÞx þ BðpÞ

ð5:1Þ

where two scalar constants A(p) and B(p) are determined by a vector parameter p ¼ ðp1 , p2 , . . . , pm ÞT :

ð5:2Þ

(i) If A(p) 6¼ 0, there is an equilibrium point of

© Springer Nature Switzerland AG 2019 A. C. J. Luo, Bifurcation and Stability in Nonlinear Dynamical Systems, Nonlinear Systems and Complexity 28, https://doi.org/10.1007/978-3-030-22910-8_5

149

150

5 Low-Degree Polynomial Systems

x ¼ a1 ðpÞ ¼ 

BðpÞ , with a0 ðpÞ ¼ AðpÞ AðpÞ

ð5:3Þ

and the corresponding dynamical system becomes x_ ¼ a0 ðx  a1 Þ:

ð5:4Þ

(ii) If A(p) ¼ 0, equation (5.1) becomes x_ ¼ BðpÞ:

ð5:5Þ

For B(p) 6¼ 0, the 1-dimensional system is called a constant velocity system. For B(p) ¼ 0, the 1-dimensional system is called a permanent static system with zero velocity. (iii) For kpk ! kp0k ¼ β, if the following relations hold AðpÞ ¼ a0 ¼ ε ! 0, BðpÞ ¼ εa1 ðpÞ ! 0,

ð5:6Þ

then there is an instant equilibrium to the vector parameter p: x ¼ a1 ðpÞ:

ð5:7Þ

Theorem 5.1 Under assumption (5.6), a standard form of the 1-dimensional dynamical system in Eq. (5.1) is x_ ¼ f ðxÞ ¼ a0 ðx  a1 Þ

ð5:8Þ

(i) If a0(p) < 0 (or df =dxjx¼a1 < 0), then equilibrium x ¼ a1(p) is stable. Such a stable equilibrium is called a sink or a stable node. (ii) If a0(p) > 0 (or df =dxjx¼a1 > 0), then equilibrium x ¼ a1(p) is unstable. Such an unstable equilibrium is called a source or an unstable node. (iii) If a0(p) ¼ 0, then the flow in the neighborhood of equilibrium x ¼ a1(p) is static (critical). Such a static equilibrium is called a critical case. _ Thus, equation (5.8) becomes Proof Let y ¼ xa1 and x_ ¼ y. y_ ¼ a0 y: The corresponding solution is y ¼ y0 ea0 ðtt0 Þ where y0 ¼ y(t0) ¼ x0a is an initial condition.

5.2 Quadratic Nonlinear Systems

151

a

b

Fig. 5.1 Stability of single equilibrium in the 1-dimensional linear dynamical system: (a) left stable equilibrium and (b) right stable equilibrium. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The stability switching is labeled by a circular symbol

(i) If a0(p) < 0, we have lim ðx  a1 Þ ¼ lim y ¼ lim y0 ea0 ðtt0 Þ ¼ 0 ) lim xðtÞ ¼ a1

t!1

t!1

t!1

t!1

So equilibrium x ¼ a1(p) is stable. (ii) If a0(p) > 0, we have lim ðx  a1 Þ ¼ lim y ¼ lim y0 ea0 ðtt0 Þ ¼ 1 ) lim xðtÞ ¼ 1:

t!1

t!1

t!1

t!1

So equilibrium x ¼ a1(p) is unstable. (iii) If a0(p) ¼ 0, we have lim ðx  a1 Þ ¼ lim y ¼ lim y0 ea0 ðtt0 Þ ¼ y0 ) lim xðtÞ ¼ x0 :

t!1

t!1

t!1

t!1

So the flow in the neighborhood of equilibrium x ¼ a1(p) is static. ∎

The theorem is proved. 

To illustrate the stability of equilibrium, one equilibrium point of x ¼ a1(p) changes with a vector parameter p. The stability of such an equilibrium is determined by the constant a0(p). The stability switching at the boundary p0 2 ∂Ω12 with a0 ¼ 0. The stable equilibrium on the left and right sides is presented in Fig. 5.1a, b, respectively. The stable and unstable portions of the equilibrium are presented by the solid and dash curves, respectively.

5.2

Quadratic Nonlinear Systems

In this section, the stability of equilibriums in 1-dimensional quadratic nonlinear systems is discussed. The upper-saddle-node and lower-saddle-node appearing and switching bifurcations are discussed as in Luo (2019).

152

5 Low-Degree Polynomial Systems

Definition 5.2 Consider a 1-dimensional quadratic nonlinear dynamical system: x_ ¼ AðpÞx2 þ BðpÞx þ CðpÞ

ð5:9Þ

where three scalar constants A(p) 6¼ 0, B(p), and C(p) are determined by a vector parameter p ¼ ðp1 , p2 , . . . , pm ÞT :

ð5:10Þ

Δ ¼ B2  4AC < 0 for p 2 Ω1  Rm ,

ð5:11Þ

(i) If

then the quadratic nonlinear system does not have any equilibriums. The flow without equilibrium is called a nonequilibrium flow. (i1) If a0(p) ¼ A(p) > 0, the nonequilibrium flow is called a positive flow. (i2) If a0(p) ¼ A(p) < 0, the nonequilibrium flow is called a negative flow. (ii) If Δ ¼ B2  4AC > 0 for p 2 Ω2  Rm ,

ð5:12Þ

then the quadratic nonlinear system has two different simple equilibriums as x ¼ a1 and x ¼ a2 ,

ð5:13Þ

and the corresponding standard form is x_ ¼ a0 ðx  a1 Þðx  a2 Þ,

ð5:14Þ

where pffiffiffiffi BðpÞ  Δ with a1 < a2 : 2AðpÞ

ð5:15Þ

Δ ¼ B2  4AC ¼ 0 for p ¼ p0 2 ∂Ω12  Rm1 ,

ð5:16Þ

a0 ¼ AðpÞ, a1, 2 ¼ (iii) If

then the 1-dimensional dynamical system has a double-repeated equilibrium, i.e., x ¼ a1 and x ¼ a1 ,

ð5:17Þ

5.2 Quadratic Nonlinear Systems

153

with the corresponding standard form of x_ ¼ a0 ðx  a1 Þ2 ,

ð5:18Þ

where a0 ¼ Aðp0 Þ, and a1 ¼ a2 ¼ 

Bðp0 Þ : 2Aðp0 Þ

ð5:19Þ

Such a flow with the equilibrium of x ¼ a1 is called a saddle flow of the second order. (iii1) If a0(p) > 0, then the equilibrium x ¼ a1(p) is an upper saddle. (iii2) If a0(p) < 0, then the equilibrium x ¼ a1(p) is a lower saddle. (iv) The equilibrium of x ¼ a1 for two equilibriums vanishing or appearance is called a saddle-node bifurcation point of equilibrium at a point p ¼ p0 2 ∂Ω12, and the bifurcation condition is Δ ¼ B2  4AC ¼ 0:

ð5:20Þ

(iv1) If a0(p) > 0, the bifurcation at x ¼ a1(p) for two equilibriums appearing or vanishing is called an upper-saddle-node bifurcation. (iv2) If a0(p) < 0, the bifurcation at x ¼ a1(p) for two equilibriums appearing or vanishing is called a lower-saddle-node bifurcation. Theorem 5.2

(i) Under a condition of Δ ¼ B2  4AC < 0,

ð5:21Þ

a standard form of the 1-dimensional dynamical system in Eq. (5.9) is   1B2 1 Þ þ 2 ðΔÞ x_ ¼ a0 ðx  2A 4A

ð5:22Þ

with a0 ¼ A(p), which has a nonequilibrium flow. (i1) If a0(p) > 0, the nonequilibrium flow is called a positive flow. (i2) If a0(p) > 0, the non-equilibrium flow is called a negative flow. (ii) Under a condition of Δ ¼ B2  4AC > 0,

ð5:23Þ

154

5 Low-Degree Polynomial Systems

a standard form of the 1-dimensional dynamical system in equation (5.9) is x_ ¼ f ðx, pÞ ¼ a0 ðx  a1 Þðx  a2 Þ:

ð5:24Þ

(ii1) If a0(p) > 0, then equilibrium x ¼ a1(p) is stable (sink) df =dxjx¼a1 < 0, and equilibrium x ¼ a2(p) is unstable (source) df =dxjx¼a2 > 0. (ii2) If a0(p) < 0, then equilibrium x ¼ a1(p) is unstable (source) df =dxjx¼a1 > 0, and equilibrium x ¼ a2(p) is stable (sink) df =dxjx¼a2 < 0.

with with with with

(iii) Under a condition of Δ ¼ B2  4AC ¼ 0,

ð5:25Þ

a standard form of the 1-dimensional dynamical system in Eq. (5.9) is x_ ¼ f ðx, pÞ ¼ a0 ðx  a1 Þ2 :

ð5:26Þ

(iii1) If a0(p) > 0, then the equilibrium x ¼ a1(p) is an upper saddle of the second order with d2 f =dx2 jx¼a1 > 0. The bifurcation at x ¼ a1(p) for two equilibriums appearing or vanishing is called an upper-saddle-node bifurcation. (iii2) If a0(p) < 0, then the equilibrium x ¼ a1(p) is a lower saddle of the second order with d2 f =dx2 jx¼a1 < 0. The bifurcation at x ¼ a1(p) for two equilibriums appearing or vanishing is called a lower-saddle-node bifurcation. Proof

(i) For Δ ¼ B2  4AC < 0,

(i1) If a0 > 0, we have x_ ¼ a0 ½ðx 

1B2 1 Þ þ 2 ðΔÞ > 0: 2A 4A

Thus such a nonequilibrium flow is called a positive flow. (i2) If a0 < 0, we have x_ ¼ a0 ½ðx 

1 1B2 Þ þ 2 ðΔÞ < 0: 2A 4A

5.2 Quadratic Nonlinear Systems

155

Thus such a nonequilibrium flow is called a negative flow. Let Δxi ¼ xai (i ¼ 1, 2) and x_ ¼ Δx_i . Thus, equation (5.24) becomes Δx_i ¼ a0 ðai  aj ÞΔxi þ a0 Δx2i ði, j 2 f1, 2g, j 6¼ iÞ: Because Δxi is arbitrarily small, we have Δx_i  λi Δxi for λi  a0 ðai  aj Þ: The corresponding solution is Δxi ¼ Δxi0 eλi ðtt0 Þ where Δxi0 ¼ x0ai is an initial condition. (ii) If λi < 0, we have lim ðx  ai Þ ¼ lim Δxi ¼ lim Δxi0 eλi ðtt0 Þ ¼ 0 ) lim xðtÞ ¼ ai

t!1

t!1

t!1

t!1

So equilibrium x ¼ ai(p) is stable. (ii1) For λ1 ¼ a0(a1a2) < 0, due to a1a2 < 0, such an equilibrium of x ¼ a1(p) is stable (sink) for a0 > 0. (ii2) For λ2 ¼ a0(a2a1) < 0, due to a2a1 > 0, such an equilibrium of x ¼ a2(p) is stable (sink) for a0 < 0. (iii) If λi > 0, we have lim ðx  ai Þ ¼ lim Δxi ¼ lim Δxi0 eλi ðtt0 Þ ¼ 1 ) lim xðtÞ ¼ 1

t!1

t!1

t!1

t!1

So equilibrium x ¼ ai(p) is unstable. (iii1) For λ1 ¼ a0(a1a2) > 0, due to a1a2 < 0, such an equilibrium of x ¼ a1(p) is unstable (source) for a0 < 0. (iii2) For λ2 ¼ a0(a2a1) > 0, due to a2a1 > 0, such an equilibrium of x ¼ a2(p) is unstable (source) for a0 > 0. (iv) If a1(p) ¼ a2(p), we have Δx_i ¼ a0 Δx2i ði, j 2 f1, 2g, j 6¼ iÞ For a0 > 0, Δx_i > 0 exists. So a flow of x reaches to x ¼ ai from the initial point of x0 < ai and it goes to the positive infinity from x0 > ai. Such an equilibrium is unstable of the second order, which is called an upper saddle. Similarly, for a0 < 0, Δx_i < 0 exists. So a flow of x reaches x ¼ ai from the initial point of

156

5 Low-Degree Polynomial Systems

x0 > ai and it goes to the negative infinity from x0 < ai. Such an equilibrium is unstable of the second order, which is called a lower saddle. ∎

The theorem is proved.

The stability and bifurcation of equilibriums for the 1-dimensional system in Eq. (5.9) are illustrated in Fig. 5.2. The stable and unstable equilibriums varying with the vector parameter are depicted by the solid and dashed curves, respectively. The bifurcation point of equilibriums occurs at the double-repeated equilibrium at the boundary of p0 2 ∂Ω12. In Fig. 5.2a, for a0 > 0, the equilibriums of x ¼ a2 and x ¼ a1 for Δ > 0 are unstable and stable, respectively. The bifurcation of equilibriums also occurs at Δ ¼ 0. The flow of x(t) is a forward upper flow for a0 > 0, and Bðp0 Þ the equilibrium point x ¼  2Aðp at Δ ¼ 0 is termed an upper saddle. Such a 0Þ bifurcation is termed an upper-saddle-node bifurcation. For Δ < 0, no any equilibrium exists. Such a 1-dimensional system is termed the nonequilibrium system. For Δ < 0 and a0 > 0, the flow of x(t) is always toward the positive direction because of B 2 x_ ¼ a0 ½ðx þ 2A Þ þ ð 4AΔ2 Þ > 0. The corresponding-phase portrait is presented in Fig. 5.2b. In Fig. 5.2c, the equilibriums of x ¼ a2 and x ¼ a1 for a0 < 0 are stable and unstable, respectively. The bifurcation of equilibriums also occurs at Δ ¼ 0. The

a

b

c

d

Fig. 5.2 Stability and bifurcation of two equilibriums in the quadratic dynamical system: (a) an upper-saddle-node bifurcation and (b) phase portrait (a0 > 0), (c) a lower-saddle-node bifurcation and (d) phase portrait (a0 < 0). Stable and unstable equilibriums are represented by solid and dashed curves, respectively

5.2 Quadratic Nonlinear Systems

157

Fig. 5.3 Stability and bifurcation of a double equilibrium of the second order in the quadratic dynamical system. Unstable equilibriums are represented by a dashed curve. The stability switching from the lower saddle to upper saddle is labeled by a circular symbol

flow of x(t) is a forward lower flow for a0 < 0, and the equilibrium point of x ¼ Bðp0 Þ at Δ ¼ 0 is termed a lower saddle. Such a bifurcation of equilibrium is  2Aðp 0Þ termed a lower-saddle-node bifurcation. For Δ < 0 and a0 < 0, the flow of x(t) is B 2 always toward the negative direction because of x_ ¼ a0 ½ðx þ 2A Þ þ ð 4AΔ2 Þ < 0. The corresponding-phase portrait is presented in Fig. 5.2d. To illustrate the stability and bifurcation of equilibrium with singularity in a 1-dimensional, quadratic nonlinear system, the equilibrium of x_ ¼ a0 ðx  a1 Þ2 is presented in Fig. 5.3. The upper saddle and lower saddle of equilibrium of x ¼ a1 with the second-order multiplicity are unstable, which are depicted by dashed curves. At a0 ¼ 0, the upper saddle and lower saddle equilibriums will be switched, which is marked by a circular symbol.

Definition 5.3 Consider a 1-dimensional dynamical system in Eq. (5.9) as x_ ¼ AðpÞx2 þ BðpÞx þ CðpÞ ¼ a0 ðpÞðx  aðpÞÞðx  bðpÞÞ:

ð5:27Þ

(i) For a < b, the corresponding standard form is x_ ¼ a0 ðx  aÞðx  bÞ

ð5:28Þ

with two equilibriums x ¼ a1 ¼ a and x ¼ a2 ¼ b with Δ ¼ a20 ða  bÞ2 > 0:

ð5:29Þ

158

5 Low-Degree Polynomial Systems

(ii) For a > b, the corresponding standard form is x_ ¼ a0 ðx  bÞðx  aÞ

ð5:30Þ

with two equilibriums of x ¼ a1 ¼ b and x ¼ a2 ¼ a with Δ ¼ a20 ða  bÞ2 > 0:

ð5:31Þ

(iii) For a ¼ b, the corresponding standard form is x_ ¼ a0 ðx  aÞ2

ð5:32Þ

with a double-repeated equilibrium of x ¼ a. Such an equilibrium point is called a saddle of the second order. (iii1) If a0 > 0, the equilibrium is an upper saddle of the second order. (iii2) If a0 < 0, the equilibrium is a lower saddle of the second order. (iv) The equilibrium of x ¼ a for two equilibriums switching is called a saddlenode bifurcation point of equilibrium at a point p ¼ p0 2 ∂Ω12, and the bifurcation condition is Δ ¼ a20 ða  bÞ2 ¼ 0 or a ¼ b: Theorem 5.3

ð5:33Þ

(i) Under a condition of a < b and Δ ¼ a20 ða  bÞ2 > 0

ð5:34Þ

a standard form of the 1-dimensional dynamical system in Eq. (5.27) is x_ ¼ f ðx, pÞ ¼ a0 ðx  aÞðx  bÞ:

ð5:35Þ

(i1) If a0(p) > 0, then equilibrium x ¼ a is stable with df =dxjx ¼a < 0 and equilibrium x ¼ b is unstable with df =dxjx ¼b > 0. (i2) If a0(p) < 0, then equilibrium x ¼ a is unstable with df =dxjx ¼a > 0 and equilibrium x ¼ b is stable with df =dxjx ¼b < 0. (ii) Under a condition of a > b and Δ ¼ a20 ða  bÞ2 > 0

ð5:36Þ

5.2 Quadratic Nonlinear Systems

159

a standard form of the 1-dimensional dynamical system in Eq. (5.9) is x_ ¼ a0 ðx  bÞðx  aÞ:

ð5:37Þ

(ii1) If a0(p) > 0, then equilibrium x ¼ b is unstable with df =dxjx ¼a > 0 and equilibrium x ¼ b < 0 is stable df =dxjx ¼b < 0. (ii2) If a0(p) < 0, then equilibrium x ¼ a is stable with df =dxjx ¼a < 0 and equilibrium x ¼ b is unstable df =dxjx ¼b > 0. (iii) For a ¼ b, the corresponding standard form with Δ ¼ 0 is x_ ¼ f ðx, pÞ ¼ a0 ðx  aÞ2

ð5:38Þ

(iii1) If a0(p) > 0, then the equilibrium x ¼ a is an upper saddle of the second order with d2 f =dx2 jx ¼a > 0. The equilibrium x ¼ a for two equilibriums switching is an upper-saddle-node bifurcation of the second order. (iii2) If a0(p) < 0, then the equilibrium x ¼ a is a lower saddle of the second order with d 2 f =dx2 jx ¼a < 0. The equilibrium x ¼ a for two equilibriums switching is a lower-saddle-node bifurcation of the second order. Proof The theorem can be proved as for Theorem 5.2.

∎

Definition 5.4 If C(p) ¼ 0 in Eq. (5.9), a 1-dimensional quadratic system is x_ ¼ AðpÞx2 þ BðpÞx:

ð5:39Þ

(i) For A(p) B(p) < 0, the corresponding standard form is x_ ¼ a0 xðx  aÞ

ð5:40Þ

x ¼ a1 ¼ 0 and x ¼ a2 ¼ a > 0 BðpÞ with a0 ¼ AðpÞ and a ¼  : AðpÞ

ð5:41Þ

with two equilibriums

(ii) For A(p) B(p) > 0, the corresponding standard form is x_ ¼ a0 ðx  aÞx

ð5:42Þ

x ¼ a1 ¼ a < 0 and x ¼ a2 ¼ 0:

ð5:43Þ

with two equilibriums of

160

5 Low-Degree Polynomial Systems

(iii) For B(p) ¼ 0, the corresponding standard form is x_ ¼ a0 x2

ð5:44Þ

with a double-repeated equilibrium of x ¼ 0. Such an equilibrium is called a saddle of the second order. If a0 > 0, the equilibrium is an upper saddle of the second order. If a0 < 0, the equilibrium is a lower saddle of the second order. (iv) The bifurcation of x ¼ 0 for two equilibriums switching is called a saddle-node bifurcation at a point p ¼ p0 2 ∂Ω12, and the bifurcation condition is Bðp0 Þ ¼ 0: Theorem 5.4

ð5:45Þ

(i) Under a condition of AðpÞ BðpÞ < 0,

ð5:46Þ

a standard form of the 1-dimensional dynamical system in Eq. (5.39) is x_ ¼ f ðx, pÞ ¼ a0 xðx  aÞ:

ð5:47Þ

(i1) If a0(p) > 0, then equilibrium x ¼ 0 is stable with df =dxjx ¼0 < 0 and equilibrium x ¼ a > 0 is unstable with df =dxjx ¼a > 0. (i2) If a0(p) < 0, then equilibrium x ¼ 0 is unstable with df =dxjx ¼0 > 0 and equilibrium x ¼ a > 0 is stable with df =dxjx ¼a < 0. (ii) Under a condition of AðpÞ BðpÞ > 0,

ð5:48Þ

a standard form of the 1-dimensional quadratic system in Eq. (5.9) is x_ ¼ a0 ðx  aÞx:

ð5:49Þ

(ii1) If a0(p) > 0, then equilibrium x ¼ 0 is unstable with df =dxjx ¼0 > 0 and equilibrium x ¼ a < 0 is stable with df =dxjx ¼a < 0. (ii2) If a0(p) < 0, then equilibrium x ¼ 0 is stable with df =dxjx ¼a < 0 and equilibrium x ¼ a < 0 is unstable with df =dxjx ¼a > 0. (iii) For B(p) ¼ 0, the corresponding standard form with Δ ¼ 0 is x_ ¼ f ðx, pÞ ¼ a0 x2 :

ð5:50Þ

5.2 Quadratic Nonlinear Systems

161

(iii1) If a0(p) > 0, then the equilibrium x ¼ 0 is an upper saddle of the second order with d2 f =dx2 jx ¼0 > 0. The equilibrium x ¼ 0 for two equilibriums switching is an upper-saddle-node bifurcation of the second order. (iii2) If a0(p) < 0, then the equilibrium x ¼ 0 is a lower saddle of the second order with d 2 f =dx2 jx ¼0 < 0. The equilibrium x ¼ 0 for two equilibriums switching is a lower-saddle-node bifurcation of the second order. Proof The theorem can be proved as for Theorem 5.2.

∎

The stability and bifurcation of two equilibriums for the 1-dimensional system in Eq. (5.27) with Δ ¼ B2  4AC ¼ a20 ða  bÞ2 0 are presented in Fig. 5.4. The stable and unstable equilibriums varying with the vector parameter are depicted by solid and dashed curves, respectively. The bifurcation point of equilibriums occurs at the double-repeated equilibrium at the boundary of p0 2 ∂Ω12. With varying parameters, the two equilibriums of x ¼ a, b equal each other (i.e., x ¼ a ¼ b). Such an equilibrium is a bifurcation point at x ¼ a ¼ b for Δ ¼ 0. The equilibriums of x ¼ a, b with Δ 0 are presented in Fig. 5.4a, b for a0 > 0 and a0 < 0, respectively. The dynamical system in Eq. (5.39) is a special case of the dynamical system in Eq. (5.9) with C(p) ¼ 0. Thus Δ ¼ B24AC ¼ B2 0. The equilibriums exist in the entire domain. In Fig. 5.4c, for a0 > 0 and B < 0, the equilibriums of x ¼ 0 and x ¼ a are unstable and stable, respectively. However, for a0 > 0 and B > 0, the equilibriums of x ¼ 0 and x ¼ a are stable and unstable, respectively. The

a

b

c

d

Fig. 5.4 Stability and bifurcation of two equilibriums in the quadratic dynamical system: (a) an upper-saddle-node bifurcation (a0 > 0), (b) a lower-saddle-node bifurcation (a0 < 0), (c) an uppersaddle-node bifurcation (a0 > 0), (d) a lower-saddle-node bifurcation (a0 < 0). Stable and unstable equilibriums are represented by solid and dashed curves, respectively

162

5 Low-Degree Polynomial Systems

bifurcation of equilibriums occurs at B ¼ 0. The flow of x(t) is a forward upper flow for a0 > 0, and the equilibrium point x ¼ 0 at B ¼ 0 is termed an upper saddle. Such a bifurcation is termed an upper-saddle-node bifurcation. In Fig. 5.4d, for a0 < 0 and B < 0, the equilibriums of x ¼ 0 and x ¼ a are stable and unstable, respectively. However, for a0 < 0 and B > 0, the equilibriums of x ¼ 0 and x ¼ a are unstable and stable, respectively. The bifurcation of equilibriums also occurs at B ¼ 0. The flow of x(t) is a forward lower flow for a0 < 0, and the equilibrium point of x ¼ 0 at B ¼ 0 is termed a lower saddle. Such a bifurcation is termed a lower-saddle-node bifurcation. Definition 5.5 If B(p) ¼ 0 in Eq. (5.9), a 1-dimensional quadratic system is x_ ¼ AðpÞx2 þ CðpÞ:

ð5:51Þ

(i) For A(p) C(p) > 0, the system does not have any equilibriums. The nonequilibrium flow of the system is called a positive flow if A(p) > 0. The nonequilibrium flow is called a negative flow if A(p) < 0. (ii) For A(p) C(p) < 0, the corresponding standard form is x_ ¼ a0 ðx þ aÞðx  aÞ

ð5:52Þ

with two symmetric equilibriums x ¼ a and x ¼ a,

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Cðp0 Þ with a0 ¼ Aðp0 Þ and a ¼  : Aðp0 Þ

ð5:53Þ

(iii) For C(p0) ¼ 0, the corresponding standard form with Δ ¼ 0 is x_ ¼ a0 x2

ð5:54Þ

x ¼ a ¼ 0 and x ¼ a ¼ þ0:

ð5:55Þ

with two equilibriums of

Such an equilibrium point of x ¼ 0 is called a saddle of the second order. If a0 > 0, the equilibrium is an upper saddle of the second order. If a0 < 0, the equilibrium is a lower saddle of the second order. (iv) The equilibrium of x ¼ 0 for two equilibriums appearaing or vanishing is called a saddle-node bifurcation point of equilibrium at a point p ¼ p0 2 ∂Ω12, and the bifurcation condition is Cðp0 Þ ¼ 0:

ð5:56Þ

5.2 Quadratic Nonlinear Systems

163

Theorem 5.5 (i) Under a condition of AðpÞ CðpÞ < 0,

ð5:57Þ

a standard form of the 1-dimensional dynamical system in Eq. (5.51) is x_ ¼ f ðx, pÞ ¼ a0 ðx þ aÞðx  aÞ:

ð5:58Þ

(i1) If a0(p) > 0, then equilibrium x ¼ a is stable with df =dxjx ¼a < 0 and equilibrium x ¼ a is unstable with df =dxjx ¼a > 0. (i2) If a0(p) < 0, then equilibrium x ¼ a is unstable with df =dxjx ¼a > 0 and equilibrium x ¼ a is stable with df =dxjx ¼a < 0. (ii) Under a condition of CðpÞ ¼ 0,

ð5:59Þ

a standard form of the 1-dimensional dynamical system in Eq. (5.38) is x_ ¼ f ðx, pÞ ¼ a0 x2 :

ð5:60Þ

(ii1) If a0(p) > 0, then the equilibrium of x ¼ 0 is an upper saddle of the second order with d2 f =dx2 jx ¼0 > 0. Such a bifurcation for two equilibriums appearing or vanishing is an upper-saddle-node bifurcation of the second order. (ii2) If a0(p) < 0, then the equilibrium x ¼ 0 is a lower saddle of the second order with d 2 f =dx2 jx ¼0 < 0. Such a bifurcation for two equilibriums appearing or vanishing is a lower-saddle-node bifurcation of the second order. Proof The proof is similar to Theorem 5.2. The theorem is proved.

∎

The stability and bifurcation of equilibriums for the quadratic nonlinear system in Eq. (5.51) are illustrated in Fig. 5.5 as a special case of the dynamical system in Eq. (5.9) with B(p) ¼ 0. The stable and unstable equilibriums varying with the vector parameter are depicted by solid and dashed curves, respectively. The bifurcation point of equilibrium occurs at the double equilibrium at the boundary of p0 2 ∂Ω12. In Fig. 5.5a, for Δ ¼ 4AC > 0 and a0 ¼ A > 0, the equilibriums of x ¼ a < 0 and x ¼ a > 0 for C < 0 are stable and unstable, respectively. The bifurcation of equilibrium also occurs at C ¼ 0. The flow of x(t) is a forward upper flow for a0 > 0, and the equilibrium point of x ¼ 0 at C ¼ 0 is termed the upper saddle. Such a bifurcation is termed the upper-saddle-node bifurcation. For Δ ¼ 4AC < 0 and a0 ¼ A > 0, we have C > 0. Thus, no equilibrium exists because of x_ ¼ Ax2 þ C > 0: Such a 1-dimensional system is termed a nonequilibrium system. For a0 ¼ A > 0 and

164

5 Low-Degree Polynomial Systems

a

b

Fig. 5.5 Stability and bifurcation of two equilibriums in the quadratic dynamical system: (a) an upper-saddle-node bifurcation (a0 > 0), (b) a lower-saddle-node bifurcation (a0 < 0). Stable and unstable equilibriums are represented by solid and dashed curves, respectively. SO: source, SI: sink

C > 0, the flow of x(t) is always toward the positive direction. In Fig. 5.5b, for Δ ¼ 4AC > 0 and a0 ¼ A < 0, the equilibriums of x ¼ a and x ¼ a for C > 0 are unstable and stable, respectively. The bifurcation of equilibrium also occurs at C ¼ 0. The flow of x(t) for the bifurcation point is a forward lower flow for a0 ¼ A < 0, and the equilibrium bifurcation point of x ¼ 0 at C ¼ 0 is termed a lower saddle. Such a bifurcation is termed a lower-saddle-node bifurcation. For Δ ¼ 4AC < 0 and a0 ¼ A < 0, we have C < 0. For a0 ¼ A < 0 and C < 0, the flow of x(t) is always toward the negative direction without any equilibrium because of x_ ¼ Ax2 þ C < 0.

5.3

Cubic Nonlinear Systems

In this section, the stability and stability switching of equilibriums in cubic polynomial systems are discussed. As in Luo (2019), the upper-saddle-node and lowersaddle-node appearing and switching bifurcations are discussed and the third-order sink and source switching bifurcations are discussed as well. Definition 5.6 Consider a cubic nonlinear dynamical system x_ ¼ AðpÞx3 þ BðpÞx2 þ CðpÞx þ DðpÞ  a0 ðpÞðx  aðpÞÞ½x2 þ B1 ðpÞx þ C1 ðpÞ

ð5:61Þ

where four scalar constants A(p) 6¼ 0, B(p), C(p), and D(p) are determined by A ¼ a0 , B ¼ ða þ B1 Þa0 , C ¼ ðaB1 þ C 1 Þa0 , D ¼ aa0 C 1 , p ¼ ðp1 , p2 , . . . , pm ÞT :

ð5:62Þ

5.3 Cubic Nonlinear Systems

165

(i) If Δ1 ¼ B21  4C 1 < 0 for p 2 Ω1  Rm

ð5:63Þ

then the cubic nonlinear system has a simple equilibrium only as x ¼ a for p 2 Ω1  Rm

ð5:64Þ

and the standard form of such a 1-dimensional system is 1 1 2 x_ ¼ a0 ðx  aÞ½ðx  B1 Þ þ ðΔ1 Þ: 2 4

ð5:65Þ

Δ1 ¼ B21  4C 1 > 0 for p 2 Ω2  Rm

ð5:66Þ

(ii) If

then there are three equilibriums with pffiffiffiffiffiffi 1 a0 ¼ AðpÞ, b1, 2 ¼  ðB1 ðpÞ  Δ1 Þ with b1 > b2 ; 2 a1 ¼ min fa, b1 , b2 g, a3 ¼ max fa, b1 , b2 g, a2 2 fa, b1 , b2 g 6¼ fa1 , a3 g, Δij ¼ ðai  aj Þ2 > 0 for i, j 2 f1, 2, 3g but i 6¼ j:

ð5:67Þ

(ii1) If ai 6¼ aj with Δij ¼ ðai  aj Þ2 > 0 for i, j 2 f1, 2, 3g but i 6¼ j:

ð5:68Þ

the cubic nonlinear system has three different, simple equilibriums as x ¼ a1 , x ¼ a2 , and x ¼ a3

ð5:69Þ

and the corresponding standard form is x_ ¼ a0 ðx  a1 Þðx  a2 Þðx  a3 Þ:

ð5:70Þ

(ii2) If at p ¼ p1 a1 ¼ b2 , a2 ¼ a, a3 ¼ b1 ; Δ12 ¼ ða1  a2 Þ ¼ ða  b2 Þ2 ¼ 0,

ð5:71Þ

166

5 Low-Degree Polynomial Systems

the cubic nonlinear system has a double-repeated equilibrium and a simple equilibrium as x ¼ a1 , x ¼ a1 and x ¼ a2

ð5:72Þ

and the corresponding standard form is x_ ¼ a0 ðx  a1 Þ2 ðx  a2 Þ:

ð5:73Þ

Such a flow at the equilibrium of x ¼ a1 is called a saddle flow of the second order.The equilibrium of x ¼ a1 for two different equilibriums switching is called a bifurcation point of equilibrium at a point p ¼ p1 with the second-order multiplicity, and the bifurcation condition is pffiffiffiffiffiffi pffiffiffiffiffiffi 1 1 a ¼ b1 ¼ min f ðB1 ðpÞ þ Δ1 Þ,  ðB1 ðpÞ  Δ1 Þg: 2 2

ð5:74Þ

If at p ¼ p2, a2 ¼ b1 , a3 ¼ a, a1 ¼ b2 , Δ23 ¼ ða2  a3 Þ ¼ ða  b1 Þ2 ¼ 0,

ð5:75Þ

the cubic nonlinear system has three equilibriums as x ¼ a1 , x ¼ a2 , and x ¼ a2

ð5:76Þ

and the corresponding standard form is x_ ¼ a0 ðx  a1 Þðx  a2 Þ2 :

ð5:77Þ

Such a flow at the equilibrium of x ¼ a2 is called a saddle flow of the second order.The equilibrium of x ¼ a2 for two different equilibriums switching is called a bifurcation point of equilibrium at a point p ¼ p1 with the second-order multiplicity, and the bifurcation condition is pffiffiffiffiffiffi pffiffiffiffiffiffi 1 1 a ¼ b2 ¼ max f ðB1 ðpÞ þ Δ1 Þ,  ðB1 ðpÞ  Δ1 Þg: 2 2

ð5:78Þ

If at p ¼ p3, a1 ¼ b2 , a2 ¼ a, a3 ¼ b1 , Δ12 ¼ ða1  a2 Þ2 ¼ ða  b2 Þ2 ¼ 0, Δ23 ¼ ða2  a3 Þ2 ¼ ða  b1 Þ2 ¼ 0, Δ13 ¼ ða1  a3 Þ2 ¼ ðb2  b1 Þ2 ¼ 0,

ð5:79Þ

5.3 Cubic Nonlinear Systems

167

the cubic nonlinear system has three repeated equilibriums as x ¼ a1 ¼ a, x ¼ a2 ¼ a and x ¼ a3 ¼ a

ð5:80Þ

and the corresponding standard form is x_ ¼ a0 ðx  aÞ3 :

ð5:81Þ

Such a flow at the equilibrium of x ¼ a is called a sink or source flow of the third order. The equilibrium of x ¼ a at a point p ¼ p3 for three different equilibriums switching is called a bifurcation point of equilibrium with the thirdorder multiplicity, and the bifurcation condition is 1 a ¼ b ¼  B1 ðpÞ: 2

ð5:82Þ

Δ1 ¼ B21  4A1 C 1 ¼ 0 for p ¼ p0 2 ∂Ω12  Rm1 ,

ð5:83Þ

(iii) If

then there exist 1 a0 ¼ Aðp0 Þ, and b1 ¼ b2 ¼ b ¼  B1 ðp0 Þ: 2

ð5:84Þ

(iii1) For a < b, the cubic nonlinear system has a double-repeated equilibrium plus a lower simple equilibrium: x ¼ a1 ¼ a, x ¼ a2 ¼ b and x ¼ a2 ¼ b

ð5:85Þ

with the corresponding standard form of x_ ¼ a0 ðx  a1 Þðx  a2 Þ2 :

ð5:86Þ

Such a flow at the equilibrium of x ¼ a2 is called a saddle flow of the second order. The equilibrium of x ¼ a2 for two different equilibrium switching is called a bifurcation point of equilibrium at a point p ¼ p0 2 ∂Ω12 with the second-order multiplicity, and the bifurcation condition is Δ1 ¼ B21  4C1 ¼ 0 with a < b:

ð5:87Þ

(iii2) For a > b, the cubic nonlinear system has a lower double-repeated equilibrium plus an upper simple equilibrium:

168

5 Low-Degree Polynomial Systems

x ¼ a1 ¼ b and x ¼ a1 ¼ b, x ¼ a2 ¼ a

ð5:88Þ

with the corresponding standard form of x_ ¼ a0 ðx  a1 Þ2 ðx  a2 Þ:

ð5:89Þ

Such a flow at the equilibrium of x ¼ a1 is called a saddle flow of the second order. The equilibrium of x ¼ a1 ¼ b for two different equilibriums switching is called a bifurcation point of equilibrium at a point p ¼ p0 2 ∂Ω12 with the second-order multiplicity, and the bifurcation condition is also Δ1 ¼ B21  4C1 ¼ 0 with a > b:

ð5:90Þ

(iii3) For a ¼ b, the cubic nonlinear system has a triple-repeated equilibrium as x ¼ a1 ¼ a and x ¼ a1 ¼ a, x ¼ a2 ¼ a

ð5:91Þ

with the corresponding standard form of x_ ¼ a0 ðx  a1 Þ3 :

ð5:92Þ

Such a flow at the equilibrium of x ¼ a1 is called a source or sink flow of the third order. The equilibrium of x ¼ a1 ¼ a for three equilibriums switching or two equilibriums switching is called a bifurcation point of equilibrium at a point p ¼ p0 2 ∂Ω12 with the third-order multiplicity, and the bifurcation condition is Δ1 ¼ B21  4C1 ¼ 0 with a ¼ b:

ð5:93Þ

Theorem 5.6 (i) Under a condition of Δ1 ¼ B21  4C 1 < 0

ð5:94Þ

a standard form of the 1-dimensional dynamical system in Eq. (5.61) is 1 1 x_ ¼ f ðx, pÞ ¼ a0 ðx  a1 Þ½ðx þ B1 Þ2 þ ðΔ1 Þ: 2 4

ð5:95Þ

(i1) If a0(p) > 0, then equilibrium x ¼ a1 is unstable (df =dxjx ¼a1 > 0). (i2) If a0(p) < 0, then equilibrium x ¼ a1 is stable (df =dxjx ¼a1 < 0).

5.3 Cubic Nonlinear Systems

169

(ii) Under the conditions of Δ1 ¼ B21  4C 1 > 0, a1 , a2 , a3 ¼ sortfb2 , a, b1 g, ai 6¼ aj , ai < aiþ1 ;

ð5:96Þ

2

Δij ¼ ðai  aj Þ 6¼ 0 for i, j 2 f1, 2, 3g, a standard form of the 1-dimensional dynamical system in Eq. (5.61) is x_ ¼ f ðx, pÞ ¼ a0 ðx  a1 Þðx  a2 Þðx  a3 Þ:

ð5:97Þ

(ii1) If a0(p) > 0, then the equilibriums of x ¼ a1, x ¼ a2, and x ¼ a3 are unstable (df =dxjx ¼a1 > 0), stable (df =dxjx ¼a2 < 0), and unstable (df =dxjx ¼a3 > 0), respectively. (ii2) If a0(p) < 0, then the equilibriums of x ¼ a1, x ¼ a2, and x ¼ a3 are stable (df =dxjx ¼a1 < 0), unstable (df =dxjx ¼a2 > 0), and stable (df =dxjx ¼a3 < 0), respectively. (iii) Under a condition of Δ1 ¼ B21  4C 1 > 0, a1 , a2 , a3 ¼ sortfb2 , a, b1 g, ai 6¼ aj , ai aiþ1

ð5:98Þ

Δ12 ¼ ða1  a2 Þ2 ¼ 0, for i, j 2 f1, 2, 3g a standard form of the 1-dimensional dynamical system in Eq. (5.61) is x_ ¼ f ðx, pÞ ¼ a0 ðx  a1 Þ2 ðx  a3 Þ:

ð5:99Þ

(iii1) If a0(p) > 0, then the equilibriums of x ¼ a1 and x ¼ a3 are unstable (lower saddle, d 2 f =dx2 jx ¼a1 < 0) and unstable (source, df =dxjx ¼a3 > 0), respectively. The bifurcation of equilibrium at x ¼ a1 for the two different equilibriums switching is a lower-saddle-node bifurcation of the second order at a point p ¼ p1. (iii2) If a0(p) < 0, then the equilibriums of x ¼ a1 and x ¼ a3 are unstable (upper saddle, d2 f =dx2 jx ¼a1 > 0) and stable (sink, df =dxjx ¼a3 < 0), respectively. The bifurcation of equilibrium at x ¼ a1 for the two different equilibriums switching is an upper saddle of the second order at a point p ¼ p1.

170

5 Low-Degree Polynomial Systems

(iv) For Δ1 ¼ B21  4C 1 > 0, a1 , a2 , a3 ¼ sortfb2 , a, b1 g, ai 6¼ aj , ai aiþ1

ð5:100Þ

2

Δ23 ¼ ða2  a3 Þ ¼ 0, for i, j 2 f1, 2, 3g a standard form of the 1-dimensional dynamical system in Eq. (5.61) is x_ ¼ f ðx, pÞ ¼ a0 ðx  a1 Þðx  a2 Þ2 :

ð5:101Þ

(iv1) If a0(p) > 0, then the equilibriums of x ¼ a1 and x ¼ a2 are unstable (source, df =dxjx ¼a1 > 0) and unstable (upper saddle, d2 f =dx2 jx ¼a2 > 0), respectively. The bifurcation of equilibrium at x ¼ a2 for two different equilibriums switching is an upper-saddlenode bifurcation of the second order at a point p ¼ p1. (iv2) If a0(p) < 0, then the equilibriums of x ¼ a1 and x ¼ a2 are stable (sink, df =dxjx ¼a1 < 0) and unstable (lower saddle, d2 f =dx2 jx ¼a2 < 0), respectively. The bifurcation of equilibrium at x ¼ a2 for two equilibrium switching is a lower-saddle-node bifurcation of the second order at a point p ¼ p1. (v) For Δ1 ¼ B21  4C1 0, b1 ¼ b2 a1 , a2 , a3 ¼ sortfb2 , a, b1 g, ai aiþ1 ,

ð5:102Þ

2

Δij ¼ ðai  aj Þ ¼ 0 for i, j ¼ 1, 2, 3 but i 6¼ j, a standard form of the 1-dimensional dynamical system in Eq. (5.61) is x_ ¼ f ðx, pÞ ¼ a0 ðx  a1 Þ3 :

ð5:103Þ

(v1) If a0(p) > 0, then the equilibrium of x ¼ a1 is unstable (third-order source, d3 f =dx3 jx ¼a1 > 0). The bifurcation of equilibrium at x ¼ a1 for three different equilibriums switching is a source switching bifurcation of the third order at a point p ¼ p1. (v2) If a0(p) < 0, then the equilibrium of x ¼ a1 is stable (third-order sink, d3 f =dx3 jx ¼a1 < 0). The bifurcation of equilibrium at x ¼ a1 for three different equilibrium switching is a sink switching bifurcation of the third order at a point p ¼ p1.

5.3 Cubic Nonlinear Systems

171

(vi) For Δ1 ¼ B21  4A1 C1 ¼ 0, a < b a1 ¼ a, a2 ¼ b, Δ12 ¼ ða1  a2 Þ2 6¼ 0

ð5:104Þ

at p ¼ p0 2 ∂Ω12  Rm1, a standard form of the 1-dimensional dynamical system is x_ ¼ f ðx, pÞ ¼ a0 ðx  a1 Þðx  a2 Þ2 :

ð5:105Þ

(vi1) If a0(p) > 0, then the equilibriums of x ¼ a1 and x ¼ a2 are unstable (source, df =dxjx ¼a1 > 0) and unstable (upper saddle, 2 2 d f =dx jx ¼a2 > 0), respectively. The bifurcation of equilibrium at x ¼ a2 for two different equilibriums vanishing or appearance is an uppersaddle-node bifurcation of the second order at a point p ¼ p0 2 ∂Ω12. (vi2) If a0(p) < 0, then the equilibriums of x ¼ a1 and x ¼ a2 are stable (sink, df =dxjx ¼a1 < 0) and unstable (lower saddle, d2 f =dx2 jx ¼a2 < 0), respectively. The bifurcation of equilibrium at x ¼ a2 for two different equilibriums vanishing or appearance is a lower saddle of the second order at a point p ¼ p0 2 ∂Ω12. (vii) For Δ1 ¼ B21  4A1 C1 ¼ 0, a > b a1 ¼ b, a2 ¼ a, Δ12 ¼ ða1  a2 Þ2 6¼ 0

ð5:106Þ

at p ¼ p0 2 ∂Ω12  Rm1, a standard form of the 1-dimensional dynamical system is x_ ¼ f ðx, pÞ ¼ a0 ðx  a1 Þ2 ðx  a2 Þ:

ð5:107Þ

(vii1) If a0(p) > 0, then the equilibriums of x ¼ a1 and x ¼ a2 are unstable (lower saddle, d2 f =dx2 jx ¼a1 < 0) and unstable (source, df =dxjx ¼a2 > 0), respectively. The bifurcation of equilibrium at x ¼ a1 for two different equilibriums switching is a lower-saddle-node bifurcation of the second order at a point p ¼ p0 2 ∂Ω12. (vii2) If a0(p) < 0, then the equilibriums of x ¼ a1 and x ¼ a2 are unstable (upper saddle, d2 f =dx2 jx ¼a1 > 0) and stable (sink, df =dxjx ¼a2 < 0), respectively. The bifurcation of equilibrium at x ¼ a1 for two different equilibriums switching is an upper-saddle-node bifurcation of the second order at a point p ¼ p0 2 ∂Ω12.

172

5 Low-Degree Polynomial Systems

(viii) For Δ1 ¼ B21  4A1 C 1 ¼ 0, a ¼ b a2 ¼ a, a2 ¼ a3 ¼ b,

ð5:108Þ

2

Δ12 ¼ ða1  a2 Þ ¼ 0 at p ¼ p0 2 ∂Ω12  Rm1, a standard form of the 1-dimensional dynamical system is x_ ¼ f ðx, pÞ ¼ a0 ðx  a1 Þ3 :

ð5:109Þ

(viii1) If a0(p) > 0, then the equilibrium of x ¼ a1 is unstable (third-order source, d3 f =dx3 jx ¼a1 > 0). The bifurcation of equilibrium at x ¼ a1 for one equilibrium to three different equilibriums switching is a source bifurcation of the third order at a point p ¼ p0 2 ∂Ω12. (viii2) If a0(p) < 0, then the equilibrium of x ¼ a1 is stable (third-order sink, d 3 f =dx3 jx ¼a1 < 0). The bifurcation of equilibrium at x ¼ a1 for one simple equilibrium to three different simple equilibriums switching is a sink bifurcation of the third order at a point p ¼ p0 2 ∂Ω12. Proof The proof is similar to Theorem 5.2.

∎

The 1-dimensional cubic nonlinear system can be expressed by a factor of (xa) and a quadratic form of a0(x2 + B1x + C1) as in Eq. (5.61). Three equilibriums do not have any intersections. Thus, only one bifurcation occurs at Δ1 ¼ B21  4C1 ¼ 0. The bifurcation of equilibrium occurs at the double-repeated equilibrium at the boundary of p0 2 ∂Ω12. For Δ1 ¼ B21  4C 1 > 0, x2 + B1x + C1 ¼ 0 gives two equilibriums of x ¼ b1, b2. For a0 > 0, if a > max {b1, b2}, then the equilibrium of x ¼ a3 ¼ a is unstable, and two equilibriums of x ¼ a2 ¼ max {b1, b2} and x  ¼ a1 ¼ min {b1, b2} are stable and unstable, respectively. For Δ1 ¼ B21  4C 1 < 0, x2 + B1x + C1 ¼ 0 does not have any real solutions. For Δ1 ¼ B21  4C 1 ¼ 0, x2 + B1x + C1 ¼ 0 has a double-repeated equilibrium of x ¼ b ¼  12 B1 . The condition of Δ1 ¼ B21  4C 1 ¼ 0 gives B1 2 ¼ 4C 1 :

ð5:110Þ

From Eq. (5.62), one obtains B1 ¼ a þ

B C B and C 1 ¼ þ aða þ Þ: A A A

ð5:111Þ

5.3 Cubic Nonlinear Systems

173

Thus, Eq. (5.110) gives a¼

ffi B 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  B2  3AC : 3A 3A

ð5:112Þ

Further, the double-repeated equilibrium of x ¼ b ¼  12 B1 is given by x ¼ b ¼ 

ffi B 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  B2  3AC : 3A 3A

ð5:113Þ

If B2 > 3AC, such a double-repeated equilibrium exists. If B2 < 3AC, such a double-repeated equilibrium does not exist. From (5.111), another equilibrium is x ¼ a, which is different from x ¼ b. If B2 ¼ 3AC, such a double-repeated B equilibrium with equilibrium of x ¼ a has an intersected point at x ¼  3A . The bifurcation diagram for a>max{b1, b2} and a0 > 0 is presented in Fig. 5.6a. The stable and unstable equilibriums varying with the vector parameter are presented by solid and dashed curves, respectively. Such an equilibrium of x ¼ b is a lower-saddle-node (LSN) bifurcation. The equilibrium of x ¼ a is a source, which is unstable. The equilibrium of x ¼ max {b1, b2} is a sink, which is stable. The equilibrium of x ¼ min {b1, b2} is a source, which is unstable. However, the bifurcation diagram for a >max {b1, b2} and a0 < 0 is presented in Fig. 5.6b. The equilibrium of x ¼ b is an upper-saddle-node (USN) bifurcation. The equilibrium of x ¼ a is a sink. The equilibrium of x ¼ max {b1, b2} is a source. The equilibrium of x ¼ min {b1, b2} is a sink. The bifurcation diagram for a < min {b1, b2} and a0 > 0 is presented in Fig. 5.6c. The equilibrium of x ¼ b is an upper-saddle-node (USN) bifurcation. The equilibrium of x ¼ max {b1, b2} is a source. The equilibrium of x ¼ min {b1, b2} is a sink. The equilibrium of x ¼ a is a source. The bifurcation diagram for a < min {b1, b2} and a0 < 0 is presented in Fig. 5.6d. The equilibrium of x ¼ b is a lower-saddle-node (LSN) bifurcation. The equilibrium of x ¼ max {b1, b2} is a sink. The equilibrium of x ¼ min {b1, b2} is a source. The equilibrium of x ¼ a is a sink. The stability and bifurcations of equilibriums of the 1-dimensional cubic nonlinear system are summarized in Table 5.1. For Δ1 ¼ B21  4C1 0, the 1-dimensional cubic nonlinear system in Eq. (5.61) has three equilibriums. Three equilibriums are x ¼ a, b1, b2. Assume ai ai + 1 for i ¼ 1, 2 with a1,2,3 ¼ sort(a, b1, b2). With varying parameters, two of three equilibriums (i.e., ai ¼ aj for i, j 2 {1, 2, 3} but i 6¼ j) will be intersected each other with the corresponding discriminant of Δij ¼ (aiaj)2 ¼ 0, and in the vicinity of the intersection point, Δij ¼ (aiaj)2 > 0. The two intersected points of a ¼ b1,2 give a¼

ffi B1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B1 2  4C1 ,  2 2

ð5:114Þ

174

5 Low-Degree Polynomial Systems

a

b

c

d

Fig. 5.6 Stability and bifurcation of three independent equilibriums in the 1-dimensional, cubic nonlinear dynamical system: For a > {b1, b2}: (a) a LSN bifurcation (a0 > 0), (b) an USN bifurcation (a0 < 0). For a < {b1, b2}: (c) an USN bifurcation (a0 > 0), (d) a LSN bifurcation (a0 < 0). LSN: lower saddle node, USN: upper saddle node. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

Table 5.1 Stability and bifurcation of a 1-dimensional cubic nonlinear system in Eq. (5.61) Fig. 5.6 (a) a0 > 0 (b) a0 < 0 (c) a0 < 0 (d) a0 < 0

a1 max {b1, b2} Unstable max {b1, b2} Stable min {b1, b2} Unstable min {b1, b2} Stable pffiffiffiffiffiffi Notice that b1, 2 ¼  12 ðB1  Δ1 Þ, Δ1 ¼ B21  4C 1 . saddle node, USN: upper saddle node a> a> a< a
3AC, such an intersected point of x ¼ a and x ¼ b1 or x ¼ b2 exists. If B2 < 3AC, such an intersected point does not exist. Such an intersection point is for the two equilibriums switching, which is called the saddle-node bifurcation. The stability and bifurcation diagrams for a0 > 0 and a0 < 0 are presented in Fig. 5.7a, b, respectively. Three equilibriums are intersected at a point with B Δij ¼ (aiaj)2 ¼ 0 and a1 ¼ a2 ¼ a3 ¼  3A , and in the vicinity of the intersection 2 point, Δij ¼ (aiaj) > 0 for i, j ¼ 1, 2, 3 but i 6¼ j. The intersection points for a0 > 0 and a0 < 0 are called the source and sink bifurcations of the third order, respectively. The corresponding stability and bifurcation diagrams for three equilibriums switching are presented in Fig. 5.7c, d.

a

b

c

d

Fig. 5.7 Stability and bifurcation of equilibriums switching in the 1-dimensional, cubic nonlinear dynamical system. For two equilibriums switching: (a) a0 > 0, (b) a0 < 0. For three equilibriums switching: (c) third-order source bifurcation (a0 > 0), (d) third-order sink bifurcation (a0 < 0). LSN: lower saddle node, USN: upper saddle node. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

176

5 Low-Degree Polynomial Systems

In the 1-dimensional cubic nonlinear system of Eq. (5.61), x2 + B1x + C1 ¼ 0 gives two equilibriums of x ¼ b1, b2 for Δ1 ¼ B21  4C1 > 0. One of the two equilibriums has one intersection with x ¼ a and there are three different equilibriums for a ¼ a2 2 (min{b1, b2}, max{b1, b2}). For this case, the intersection point occurs at a ¼ min {b1, b2} for p1 2 ∂Ω23 or a ¼ max {b1, b2} for p2 2 ∂Ω23. The bifurcation point of equilibrium occurs at the double-repeated equilibrium at Δ1 ¼ B21  4C1 ¼ 0 for p0 2 ∂Ω12. Such a bifurcation is a lower- or upper-saddle-node bifurcation. For a ¼  12 B1 with Δ1 ¼ B21  4C 1 ¼ 0, three equilibriums are repeated with three multiplicities. The intersected point of a ¼  12 B1 with Eq. (5.111) gives 1 B a ¼  ða þ Þ: 2 A

ð5:117Þ

Thus a¼

B : 3A

ð5:118Þ

Such a bifurcation at the intersection point is also a third-order source or sink bifurcation. The bifurcation diagrams for six cases of three equilibriums with one intersection are presented in Fig. 5.8a–f and the stability and bifurcations are listed in Table 5.2. The 1-dimensional cubic nonlinear system is expressed by a factor of (xa) and a quadratic form of a0(x2 + B1x + C1) as in Eq. (5.61). For Δ1 ¼ B21  4C 1 > 0, x2 + B1x + C1 ¼ 0 gives two equilibriums of x ¼ b1, b2. The two equilibriums do not have any intersections with x ¼ a. For Δ1 ¼ B21  4C 1 ¼ 0, there are two parameters of p1 2 ∂Ω12 and p2 2 ∂Ω12, and the two double-repeated equilibriums are at x ðpi Þ ¼  12 B1 ðpi Þ (i ¼ 1, 2). With the two repeated equilibriums, the two equilibriums of x ¼ b1, b2 formed a closed path in the bifurcation diagram. The bifurcation points of equilibrium occur at the two double-repeated equilibriums of Δ1 ¼ B21  4C 1 ¼ 0 for pi 2 ∂Ω12 (x ¼ b1, b2). Such a bifurcation at the intersection point is also a lower- or upper-saddle-node bifurcation. The stable and unstable equilibriums varying with the vector parameter are also represented by solid and dashed curves, respectively. The bifurcation diagrams for four cases of three equilibriums are presented in Fig. 5.9a–d, and the stability and bifurcations are summarized in Table 5.3. If the two repeated equilibriums have two intersections with x ¼ a(pi) (i ¼ 1, 2), i.e., aðpi Þ ¼  12 B1 ðpi Þ then there are two triple-repeated equilibriums at x ¼ a(pi) (i ¼ 1, 2), which are the third-order sink or source bifurcations. The stability and bifurcation diagrams of equilibriums are formed by the equilibrium of x ¼ a(p) and the closed loop of equilibriums of x ¼ b1, b2, as shown in Fig. 5.9e, f for a0 > 0 and a0 < 0, respectively. The stability and bifurcations are also summarized in Table 5.3. In the 1-dimensional cubic nonlinear system in Eq. (5.61), x2 + B1x + C1 ¼ 0 for Δ1 ¼ B21  4C 1 > 0 gives two equilibriums of x ¼ b1, b2, which have an

5.3 Cubic Nonlinear Systems

177

a

b

c

d

e

f

Fig. 5.8 Stability and bifurcation of equilibriums in the 1-dimensional, cubic nonlinear dynamical system: (a) the LSN (Δ1 ¼ 0) and USN (a ¼ max {b1, b2}) bifurcations (a0 > 0), (b) the USN (Δ1 ¼ 0) and LSN (a ¼ max {b1, b2}) bifurcations (a0 < 0), (c) the USN (Δ1 ¼ 0) and LSN (a ¼ min {b1, b2}) bifurcations (a0 > 0), (d) the LSN (Δ1 ¼ 0) and USN (a ¼ min {b1, b2}) bifurcations (a0 < 0), (e) the third-order SO bifurcation (Δ1 ¼ 0 and a ¼ b) (a0 > 0), (f) the thirdorder SI bifurcation (Δ1 ¼ 0 and a ¼ b) (a0 < 0). LSN: lower saddle node, USN: upper saddle node, SI: sink, SO: source. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

intersection with x ¼ a. The intersected point is at a ¼ b1 or a ¼ b2 with Eq. (5.114). The double-repeated equilibrium requires Δ1 ¼ B21  4C1 ¼ 0 and the two equilibriums of x ¼ a, b1 under Δ1 ¼ B21  4C1 > 0 and x ¼ b2 for Δ1 ¼ B21  4C1 < 0. Similarly, the two equilibriums of x ¼ a, b2 under Δ1 ¼ B21  4C1 > 0 and x ¼ b2

178

5 Low-Degree Polynomial Systems

Table 5.2 Stability and bifurcation of a 1-dimensional cubic (x_ ¼ a0 ðx  aÞ½x2 þ B1 ðpÞx þ C 1 ðpÞ,a 2 (min{b1, b2}, max{b1, b2})) Fig. 5.8 (a) a0 > 0 (b) a0 < 0 (c) a0 > 0 (d) a0 < 0 (e) a0 > 0

a1 Unstable Stable Unstable Stable Unstable

a2 Stable Unstable Stable Unstable Stable

a3 Unstable Stable Unstable Stable Unstable

B-I Second LSN Second USN Second USN Second LSN Δ1 ¼ 0

B-II Second USN Second LSN Second USN Second USN a ¼  12 B1

nonlinear

system

B-III a ¼ max {b1, b2} a ¼ max {b1, b2} a ¼ min {b1, b2} a ¼ min {b1, b2} Third order SO

Unstable Stable Δ1 ¼ 0 Third order SI a ¼  12 B1 pffiffiffiffiffiffi 2  Δ1 Þ, Δ1 ¼ B1  4C 1 . Bifurcation-I (B-I): Δ1 ¼ 0. Bifurcation-II Notice that b1, 2 ¼ B1 (B-II): a ¼ max {b1, b2} or a ¼ min {b1, b2}. Bifurcation-III (B-III): Δ1 ¼ 0and a ¼  2A . LSN: 1 lower saddle node, USN: upper saddle node. SO: source, SI: sink (f) a0 < 0

Stable

 12 ðB1

for Δ1 ¼ B21  4C1 < 0 are required. Such a bifurcation for two equilibriums appearing and vanishing is called a lower- or upper-saddle-node bifurcation. The stable and unstable equilibriums varying with the vector parameter are also represented by solid and dashed curves, respectively. The bifurcation diagrams for four cases of three equilibriums are presented in Fig. 5.10a–f. If the double-repeated B equilibrium has an intersection with x ¼ aðp0 Þ ¼  12 B1 ¼  3A , then there two  triple-repeated equilibriums at x ¼ a(p0), which is the third-order sink and source bifurcations for a0 > 0 and a0 < 0, respectively. The stability and bifurcation diagrams of equilibriums are shown in Fig. 5.10e, f. Consider a 1-dimensional, cubic nonlinear dynamical system with a doublerepeated equilibrium and one simple equilibrium. (i) For b < a, the 1-dimensional, cubic nonlinear dynamical system is x_ ¼ a0 ðpÞðx  bðpÞÞ2 ðx  aðpÞÞ:

ð5:119Þ

For such a system, if a0 > 0, the double-repeated equilibrium of x ¼ b is a lower saddle, which is unstable, and the simple equilibrium of x ¼ b is a source, which is unstable. If a0 < 0, the double-repeated equilibrium of x ¼ b is an upper saddle, which is unstable, and the simple equilibrium of x ¼ a is a sink, which is stable. (ii) For b > a, the 1-dimensional cubic nonlinear dynamical system is x_ ¼ a0 ðpÞðx  aðpÞÞðx  bðpÞÞ2 :

ð5:120Þ

For such a system, if a0 > 0, the double-repeated equilibrium of x ¼ b is an upper saddle, which is unstable, and the simple equilibrium of x ¼ a is a source, which is unstable. If a0 < 0, the double-repeated equilibrium of x ¼ b is a lower saddle, which is unstable, and the simple equilibrium of x ¼ a is a sink, which is stable.

5.3 Cubic Nonlinear Systems

179

a

b

c

d

e

f

Fig. 5.9 Stability and bifurcation of three equilibriums in the 1-dimensional, cubic nonlinear dynamical system: For a < {b1, b2}: (a) two USN bifurcations (a0 > 0), (b) two LSN bifurcations (a0 < 0). For a > {b1, b2}: (c) two LSN bifurcations (a0 > 0), (d) two USN bifurcations (a0 < 0). (e) Two third-order SO bifurcations (a0 > 0), (f) two third-order SI bifurcations (a0 < 0). LSN: lower saddle node, USN: upper saddle node. SO: source, SI: sink. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

180

5 Low-Degree Polynomial Systems

Table 5.3 Stability and bifurcation of a 1-dimensional cubic (x_ ¼ a0 ðx  aÞ½x2 þ B1 ðpÞx þ C 1 ðpÞ,a 2 (min{b1, b2}, max{b1, b2})) Fig. 5.9 (a) a0 > 0 (b) a0 < 0 (c) a0 > 0 (d) a0 < 0 (e) a0 < 0 (f) a0 < 0

nonlinear

system

a Unstable Stable Unstable Stable Unstable Stable

b1 b2 B-I B-I B-III Stable Unstable USN USN – Unstable Stable LSN LSN – Stable Unstable LSN LSN – Unstable Stable USN USN – Stable Unstable – – Third-order SO Unstable Stable – – Third-order SI pffiffiffiffiffiffi Notice that b1, 2 ¼  12 ðB1  Δ1 Þ, Δ1 ¼ B21  4C 1 . Bifurcation-I (B-I): Δ1 ¼ 0. Bifurcation-II (B-II): a ¼ max {b1, b2}. Bifurcation-III (B-III): a ¼ min {b1, b2}. LSN: lower-saddle node, USN: upper-saddle-node

(iii) For b ¼ a, the dynamical system on the boundary is x_ ¼ a0 ðpÞðx  bðpÞÞ3 :

ð5:121Þ

For such a system, if a0 > 0, the triple-repeated equilibrium of x ¼ b with the third-order multiplicity is a source bifurcation of the third-order for the (US: SO) to (SO:LS) equilibrium. If a0 < 0, the triple-repeated equilibrium of x ¼ b with the third-order multiplicity is a sink bifurcation of the third-order for the (LS:SI) to (SO:US) equilibrium. With parameter changes, the bifurcation diagram for the cubic nonlinear system is presented in Fig. 5.11. The acronyms LS, US, SI, and SO are for lower-saddle, upper-saddle, sink, and source, respectively. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation point is marked by a circular symbol. The third-order source bifurcation for the upper-saddle and source equilibriums to the source and lower-saddle equilibriums is presented in Fig. 5.11a. The third-order sink bifurcation for the lower-saddle and sink equilibriums to the sink and upper-saddle equilibriums is presented in Fig. 5.11b. To illustrate the stability and bifurcation of equilibrium with singularity in a 1-dimensional, cubic nonlinear system, the equilibrium of x_ ¼ a0 ðx  a1 Þ3 is presented in Fig. 5.12. The third-order sink and source of equilibrium of x ¼ a1 with the third-order multiplicity are stable and unstable, respectively. The stable and unstable equilibriums are depicted by solid and dashed curves, respectively. At a0 ¼ 0, the third-order sink and source equilibriums are switched, which is marked by a circular symbol. For the 1-dimensional, cubic nonlinear systems, the equilibrium stability and bifurcation can be described through an alternative way as follows. Definition 5.7 Consider a 1-dimensional, cubic nonlinear dynamical system x_ ¼ AðpÞx3 þ BðpÞx2 þ CðpÞx þ DðpÞ i h B B  a0 ðpÞ ðx þ Þ3 þ pðpÞðx þ Þ þ qðpÞ 3A 3A

ð5:122Þ

5.3 Cubic Nonlinear Systems

181

a

b

c

d

e

f

Fig. 5.10 Stability and bifurcation of equilibriums in the 1-dimensional, cubic nonlinear dynamical system: (a) the LSN (Δ1 ¼ 0) and USN (a ¼ max {b1, b2}) bifurcations (a0 > 0), (b) the USN (Δ1 ¼ 0) and LSN (a ¼ max {b1, b2}) bifurcations (a0 < 0), (c) the USN (Δ1 ¼ 0) and LSN (a ¼ min {b1, b2}) bifurcations (a0 > 0), (d) the LSN (Δ1 ¼ 0) and USN (a ¼ min {b1, b2}) bifurcations (a0 < 0), (e) the third-order SO bifurcation (Δ1 ¼ 0 and a ¼ b) (a0 > 0), (f) the thirdorder SI bifurcation (Δ1 ¼ 0 and a ¼ b) (a0 < 0). LSN: lower saddle node, USN: upper saddle node, SI: sink, SO: source. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

182

5 Low-Degree Polynomial Systems

a

b

Fig. 5.11 Stability and bifurcation of a triple-repeated equilibrium with a simple equilibrium in a 1-dimensional, cubic nonlinear dynamical system: (a) a third-order source bifurcation for (US:SO) to (SO:LS) switching (a0 > 0), (b) a third-order sink bifurcation (a0 < 0) for (LS:SI) to (SI:US) switching. Stable and unstable equilibriums are represented by solid and dashed curves, respectively

Fig. 5.12 Stability of a triple equilibrium in the 1-dimensional, cubic nonlinear dynamical system: Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The stability switching is labeled by a circular symbol

where four scalar constants A(p) 6¼ 0, B(p), C(p), and D(p) satisfy C B2 D BC 2B3  2,q ¼  2 þ A 3A A 3A 27A3 T p ¼ ðp1 , p2 , . . . , pm Þ : A ¼ a0 , p ¼

ð5:123Þ

(i) If Δ¼

q2 p3 þ >0 4 27

ð5:124Þ

5.3 Cubic Nonlinear Systems

183

the cubic nonlinear system has one equilibrium as q pffiffiffiffi 1=3 q pffiffiffiffi 1=3 B x ¼ a  ð þ ΔÞ þ ð  ΔÞ  2 2 3A

ð5:125Þ

and the corresponding standard form is 2 1 1 x_ ¼ a0 ðx  aÞ½ðx þ B1 Þ þ ðΔ1 Þ 2 4 ¼ a0 ðx  aÞ½x2 þ B1 x þ C 1 Þ:

ð5:126Þ

where A ¼ a0 , B ¼ ða þ B1 Þa0 , C ¼ ðaB1 þ C 1 Þa0 , D ¼ aa0 C1 :

ð5:127Þ

(ii) If Δ¼

q2 p3 þ b is x_ ¼ a0 ðx  bÞ2 ðx  aÞ:

ð5:136Þ

Such a flow at the equilibrium of x ¼ b is called a saddle flow of the second order. The bifurcation of equilibrium at x ¼ b for two equilibriums appearing or vanishing is called a saddle-node bifurcation of the second order at a point p ¼ p1 2 ∂Ω12, and the bifurcation condition is 1 1 1 1 Δ ¼ q2 þ p3 ¼ 0, q2 ¼  p3 6¼ 0, a > b: 4 27 4 27

ð5:137Þ

1 1 1 1 Δ ¼ q2 þ p3 ¼ 0, q2 ¼  p3 ¼ 0, 4 27 4 27

ð5:138Þ

(iv) If

the 1-dimensional dynamical system has a triple-repeated equilibrium as x ¼ a ¼ 

B  B B , x ¼ b1 ¼  , x  ¼ b2 ¼  : 3A 3A 3A

ð5:139Þ

5.3 Cubic Nonlinear Systems

185

The corresponding standard form is x_ ¼ a0 ðx  aÞ3 :

ð5:140Þ

Such a flow at the equilibrium of x ¼ a is called a source or sink flow of the B third order. The bifurcation of equilibrium at x ¼  3A for one equilibrium to three equilibriums is called a source or think bifurcation of the third order at p ¼ p1 2 ∂Ω12, and the bifurcation condition is 1 1 1 1 Δ ¼ q2 þ p3 ¼ 0, q2 ¼  p3 ¼ 0: 4 27 4 27

ð5:141Þ

From the afore-described stability and bifurcation of the 1-dimensional, cubic nonlinear dynamical systems, the equilibrium stability and bifurcations of equilibrium in Eq. (5.122) are similar to Theorem 5.6. The 1-dimensional cubic nonlinear system has the following four cases: (i) One real solution of equilibrium of x ¼ a requires Δ ¼ ðq2Þ2 þ ðp3Þ3 > 0 for Eq. (5.122), equivalent to Δ1 ¼ B21  4C 1 < 0 for Eq. (5.61): Aa3 þ Ba2 þ Ca þ D ¼ 0:

ð5:142Þ

(ii) Three different solutions of equilibriums of x ¼ a, b1, b2 require Δ < 0 for Eq. (5.122), equivalent to Δ1 ¼ B21  4C 1 > 0 for Eq. (5.61): 1 Aa3 þ Ba2 þ Ca þ D ¼ 0 and b1, 2 ¼ ðB1  2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B21  4C1 Þ:

ð5:143Þ

(iii) The double-repeated equilibriums require Δ ¼ 0 and ðq2Þ2 ¼ ðp3Þ3 6¼ 0 for Eq. (5.122), equivalent to Δ1 ¼ B21  4C 1 ¼ 0 or a ¼ b1,2 for Eq. (5.61): a ¼ b1 , 2 ¼ 

ffi B 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  B2  3AC with B2 > 3AC: 3A 3A

ð5:144Þ

(iv) The triple-repeated equilibriums require Δ ¼ 0 and ðq2Þ2 ¼ ðp3Þ3 ¼ 0 for Eq. (5.122), equivalent to Δ1 ¼ B21  4C 1 ¼ 0 and a ¼ b1,2 for Eq. (5.61): a ¼ b1 , 2 ¼ 

B with B2 ¼ 3AC: 3A

ð5:145Þ

186

5 Low-Degree Polynomial Systems

5.4

Quartic Nonlinear Systems

In this section, the stability and bifurcation of the quartic nonlinear systems will be presented as in Luo (2019). The fourth-order upper-saddle and lower-saddle appearing bifurcations of two second-order upper saddles and lower saddles will be presented. The third-order switching sink and source bifurcations of lower saddle with sink and upper saddle with source will be discussed. Definition 5.8 Consider a 1-dimensional, quartic nonlinear dynamical system x_ ¼ AðpÞx4 þ BðpÞx3 þ CðpÞx2 þ DðpÞx þ EðpÞ ¼ a0 ðpÞ½x2 þ B1 ðpÞx þ C 1 ðpÞ½x2 þ B2 ðpÞx þ C2 ðpÞ

ð5:146Þ

where A(p) 6¼ 0, and p ¼ ðp1 , p2 , . . . , pm ÞT :

ð5:147Þ

Δi ¼ B2i  4Ci < 0 for i ¼ 1, 2

ð5:148Þ

(i) If

the quartic nonlinear dynamical system does not have any equilibrium, and the corresponding standard form is 2 2 1 1 1 1 x_ ¼ a0 ½ðx þ B1 Þ þ ðΔ1 Þ½ðx þ B2 Þ þ ðΔ2 Þ: 2 4 2 4

ð5:149Þ

The flow of such a system without equilibriums is called a nonequilibrium flow. (i1) If a0 > 0, the nonequilibrium flow is called the positive flow. (i2) If a0 < 0, the nonequilibrium flow is called the negative flow. (ii) If Δi ¼ B2i  4C i > 0 and Δj ¼ B2j  4C j < 0 for i, j 2 f1, 2g, i 6¼ j

ð5:150Þ

the quartic polynomial system has two simple equilibriums, i.e., pffiffiffiffiffi pffiffiffiffiffi 1 1 ðiÞ ðiÞ x ¼ b1 ¼  ðBi þ Δi Þ, x ¼ b2 ¼  ðBi  Δi Þ: 2 2

ð5:151Þ

5.4 Quartic Nonlinear Systems

187

The corresponding standard form is 1 2 1 x_ ¼ a0 ðx  a1 Þðx  a2 Þ½ðx þ Bj Þ þ ðΔj Þ 2 4

ð5:152Þ

where ðiÞ

ðiÞ

ðiÞ

ðiÞ

a1 ¼ min fb1 , b2 g and a2 ¼ max fb1 , b2 g:

ð5:153Þ

Such a flow of equilibriums is called a flow of two simple equilibriums. (iii) If Δi ¼ B2i  4C i ¼ 0 and Δj ¼ B2j  4C j < 0 for i, j 2 f1, 2g, i 6¼ j

ð5:154Þ

the quartic polynomial system has a double-repeated equilibrium, i.e., 1 1 ðiÞ ðiÞ x ¼ b1 ¼  Bi , x ¼ b2 ¼  Bi : 2 2

ð5:155Þ

The corresponding standard form is 1 2 1 x_ ¼ a0 ðx  a1 Þ2 ½ðx þ Bj Þ þ ðΔj Þ 2 4

ð5:156Þ

ðiÞ

ð5:157Þ

where ðiÞ

a1 ¼ b1 ¼ b2 :

Such a flow of the equilibrium of x ¼ a1 is called a saddle flow of the second order. The equilibrium of x ¼ a1 for two equilibriums switching or appearing or vanishing is called a bifurcation point of equilibrium at p ¼ p1 2 ∂Ω12, and the bifurcation condition is 1 Δi ¼ B2i  4Ci ¼ 0 ði 2 f1, 2gÞ and a1 ¼  Bi : 2

ð5:158Þ

Δi ¼ B2i  4Ci 0 for i ¼ 1, 2

ð5:159Þ

(iv) If

the quartic nonlinear dynamical system has four equilibriums, i.e., pffiffiffiffiffi pffiffiffiffiffi 1 1 ðiÞ ðiÞ x ¼ b1 ¼  ðBi þ Δi Þ, x ¼ b2 ¼  ðBi  Δi Þ for i ¼ 1, 2: 2 2

ð5:160Þ

188

5 Low-Degree Polynomial Systems

(iv1) A standard form is x_ ¼ a0 ðx  a1 Þðx  a2 Þðx  a3 Þðx  a4 Þ

ð5:161Þ

where Δi ¼ B2i  4Ci > 0, i ¼ 1, 2; ð1Þ

ð2Þ

bk 6¼ bl

for k, l 2 f1, 2g;

a1 , 2 , 3 , 4 2

ð1Þ ð1Þ ð2Þ ð2Þ fb1 , b2 , b1 , b2 g

ð5:162Þ with am < amþ1 :

Such a flow of equilibriums is called a flow of four simple equilibriums. (iv2) The corresponding standard form is x_ ¼ a0 ðx  ai1 Þ2 ðx  ai2 Þðx  ai3 Þ

ð5:163Þ

where Δi ¼ B2i  4C i > 0, Δj ¼ B2j  4C j > 0 for i, j ¼ 1, 2; ðiÞ

ðjÞ

ai1 ¼ bk ¼ bl , ði, kÞ 6¼ ðj, lÞ;i, j, k, l 2 f1, 2g

ð5:164Þ

ai12 = fai2 , ai3 g for iα 2 f1, 2, 3, 4g and α 2 f1, 2, 3, 4g: Such a flow of equilibrium x ¼ ai1 is called a saddle flow of the second order. The equilibrium of x ¼ ai1 for two equilibriums switching or appearing or vanishing is called a saddle bifurcation of equilibrium at a point p ¼ p1 2 ∂Ω12, and the bifurcation condition is Δi ¼ B2i  4C i > 0 ði 2 f1, 2gÞ and Δj ¼ B2j  4C j > 0 ðj 2 f1, 2gÞ ðiÞ bk

¼

ðjÞ bl , ði, kÞ

ð5:165Þ

6¼ ðj, lÞ, ði, j, k, l 2 f1, 2gÞ:

(iv3) The corresponding standard form is x_ ¼ a0 ðx  ai1 Þ3 ðx  ai2 Þ

ð5:166Þ

where Δi ¼ B2i  4C i > 0, Δj ¼ B2j  4C j ¼ 0 for i, j ¼ 1, 2; 1 ðiÞ ðiÞ ai1 ¼  Bj ¼ bl , ai2 ¼ bk , k 6¼ l;k, l 2 f1, 2g 2 ai1 ¼ ai3 for iα 2 f1, 2, 3, 4g and α 2 f1, 2, 3, 4g:

ð5:167Þ

5.4 Quartic Nonlinear Systems

189

Such a flow of the equilibrium of x ¼ ai1 is called a source and sink flow of the third order. The equilibrium of x ¼ ai1 for one equilibrium to three equilibrium is called a third-order source or sink bifurcation of equilibrium at a point p ¼ p1 2 ∂Ω12, and the bifurcation condition is Δi ¼ B2i  4Ci > 0 ði 2 f1, 2gÞ and Δj ¼ B2j  4Cj ¼ 0 ðj 2 f1, 2gÞ 1 ðiÞ ðiÞ ðiÞ bk ¼  Bj , bk 6¼ bl , ðk 6¼ l, k, l 2 f1, 2gÞ: 2

ð5:168Þ

(iv4) The corresponding standard form is x_ ¼ a0 ðx  a1 Þ2 ðx  a2 Þ2

ð5:169Þ

where Δi ¼ B2i  4C i ¼ 0, i ¼ 1, 2 1 1 ð1Þ ð1Þ ð2Þ ð2Þ b1 ¼ b2 ¼  B1 , b1 ¼ b2 ¼  B2 , B1 6¼ B2 ; 2 2 1 1 1 1 a1 ¼ min f B1 ,  B2 g, a2 ¼ max f B1 ,  B2 g: 2 2 2 2

ð5:170Þ

Such a flow with the two equilibriums of x ¼ a1 and x ¼ a2 is called a (US:US) or (LS:LS) flow. The equilibriums of x ¼ a1 and x ¼ a2 for two equilibriums switching are called two bifurcations of equilibrium at a point p ¼ p1 2 ∂Ω12, and the bifurcation condition is Δi ¼ B2i  4Ci ¼ 0, i ¼ 1, 2 1 1 ð1Þ ð1Þ ð2Þ ð2Þ b1 ¼ b2 ¼  B1 , b1 ¼ b2 ¼  B2 : 2 2

ð5:171Þ

(iv5) The corresponding standard form is x_ ¼ a0 ðx  a1 Þ4

ð5:172Þ

Δi ¼ B2i  4Ci ¼ 0, i ¼ 1, 2 1 ðiÞ ðiÞ b1 ¼ b 2 ¼  B i , B 1 ¼ B 2 : 2

ð5:173Þ

where

Such a flow at the equilibrium of x ¼ a1 is called a saddle flow of the fourth order. The equilibrium of x ¼ a1 for two second-order equilibriums

190

5 Low-Degree Polynomial Systems

switching or four simple equilibriums appearing or from two equilibriums to four simple equilibriums switching is called a fourth-order saddle bifurcation of equilibrium at a point p ¼ p1 2 ∂Ω12, and the bifurcation condition is ðiÞ

ðiÞ

Δi ¼ B2i  4C i ¼ 0, a1 ¼ b1 ¼ b2 , i ¼ 1, 2: Theorem 5.7

ð5:174Þ

(i) Under conditions of Δi ¼ B2i  4Ci < 0 for i ¼ 1, 2

ð5:175Þ

a standard form of Eq. (5.146) is ih i h 1 1 1 1 x_ ¼ f ðx, pÞ ¼ a0 ðx þ B1 Þ2 þ ðΔ1 Þ ðx þ B2 Þ2 þ ðΔ2 Þ 2 4 2 4

ð5:176Þ

with a0 ¼ A(p), which has a nonequilibrium flow. (i1) If a0(p) > 0, the nonequilibrium flow is called a positive flow. (i2) If a0(p) > 0, the nonequilibrium flow is called a negative flow. (ii) Under a condition of Δi ¼ B2i  4C i > 0 and Δj ¼ B2j  4C j < 0 for i, j 2 f1, 2g, i 6¼ j

ð5:177Þ

a standard form of Eq. (5.146) is h i 1 1 x_ ¼ f ðx, pÞ ¼ a0 ðx  a1 Þðx  a2 Þ ðx þ Bj Þ2 þ ðΔj Þ 2 4

ð5:178Þ

where ðiÞ

ðiÞ

ðiÞ

ðiÞ

a1 ¼ min ðb1 , b2 Þ and a2 ¼ max ðb1 , b2 Þ, pffiffiffiffiffi ðiÞ pffiffiffiffiffi 1 1 ðiÞ b1 ¼  ðBi þ Δi Þ, b2 ¼  ðBi  Δi Þ: 2 2

ð5:179Þ

(ii1) For a0(p) > 0, the equilibriums of x ¼ a1 and x ¼ a2 are stable (sink, df =dxjx ¼a1 < 0) and unstable (source, df =dxjx ¼a2 > 0), respectively. (ii2) For a0(p) < 0, the equilibriums of x ¼ a1 and x ¼ a2 are unstable (source, df =dxjx ¼a1 > 0) and stable (sink, df =dxjx ¼a2 < 0), respectively. (iii) Under conditions of

5.4 Quartic Nonlinear Systems

191

Δi ¼ B2i  4C i ¼ 0 and Δj ¼ B2j  4C j < 0 for i, j 2 f1, 2g, i 6¼ j

ð5:180Þ

a standard form of Eq. (5.146) is i h 1 1 x_ ¼ f ðx, pÞ ¼ a0 ðx  a1 Þ2 ðx þ Bj Þ2 þ ðΔj Þ 2 4

ð5:181Þ

1 ðiÞ ðiÞ a1 ¼ b1 ¼ b2 ¼  B i : 2

ð5:182Þ

where

(iii1) For a0(p) > 0, the equilibrium of x ¼ a1 is unstable (an upper saddle, d2 f =dx2 jx ¼a1 > 0). Such a flow at the equilibrium of x ¼ a1 is called an upper-saddle flow of the second order. The bifurcation of equilibrium of at x ¼ a1 for two equilibriums appearing or vanishing is called an upper-saddle-node bifurcation of the second order at a point p ¼ p1 2 ∂Ω12. (iii2) For a0(p) < 0, the equilibrium of x ¼ a1 is unstable (a lower saddle, d2 f =dx2 jx ¼a1 < 0). Such a flow at the equilibrium of x ¼ a1 is called a lower-saddle flow of the second order. The bifurcation of equilibrium of at x ¼ a1 for two equilibriums appearing or vanishing is called a lowersaddle-node bifurcation of the second order at a point p ¼ p1 2 ∂Ω12. (iv) Under conditions of Δi ¼ B2i  4Ci > 0, i ¼ 1, 2 ð1Þ

ð2Þ

bk 6¼ bl ðiÞ b1

for k, l 2 f1, 2g; pffiffiffiffiffi ðiÞ pffiffiffiffiffi 1 1 ¼  ðBi þ Δi Þ, b2 ¼  ðBi  Δi Þ for i ¼ 1, 2 2 2

ð5:183Þ

a standard form is x_ ¼ f ðx, pÞ ¼ a0 ðx  a1 Þðx  a2 Þðx  a3 Þðx  a4 Þ

ð5:184Þ

where ðiÞ

ðiÞ

a1, 2, 3, 4 2 [2i¼1 fb1 , b2 g with am < amþ1 :

ð5:185Þ

(iv1) For a0(p) > 0, the equilibriums of x ¼ a1, a2, a3, a4 are stable, unstable, stable, and unstable, respectively. The flow is called a (SI:SO:SI:SO) flow.

192

5 Low-Degree Polynomial Systems

(iv2) For a0(p) < 0, the equilibriums of x ¼ a1, a2, a3, a4 are unstable, stable, unstable, and stable, respectively. The flow is called a (SO:SI:SO:SI) flow. The equilibrium of x ¼ ai (i ¼ 1, 2, 3, 4) is unstable (source, df =dxjx ¼ai > 0) and stable (sink, df =dxjx ¼ai < 0). (v) Under conditions of Δi ¼ B2i  4C i > 0 ði 2 f1, 2gÞ and Δj ¼ B2j  4C j > 0 ðj 2 f1, 2gÞ, pffiffiffiffiffiffi ðαÞ pffiffiffiffiffiffi 1 1 ðαÞ b1 ¼  ðBα þ Δα Þ, b2 ¼  ðBα  Δα Þ for α ¼ i, j 2 2 ðiÞ ðjÞ bk ¼ bl , ði, kÞ 6¼ ðj, lÞ, ði, j, k, l 2 f1, 2gÞ,

ð5:186Þ

a standard form of Eq. (5.146) is x_ ¼ f ðx, pÞ ¼ a0 ðx  ai1 Þ2 ðx  ai2 Þðx  ai3 Þ

ð5:187Þ

where ðiÞ

ðjÞ

ðiÞ

ðiÞ

ai1 ¼ bk ¼ bl 2 [2i¼1 fb1 , b1 g, ði, kÞ 6¼ ðj, lÞ;i, j, k, l 2 f1, 2g ðiÞ

ðiÞ

ai12 = fai2 , ai3 g  [2i¼1 fb1 , b1 g for iα 2 f1, 2, 3g and α 2 f1, 2, 3g:

ð5:188Þ

The equilibriums of x ¼ ai2 , ai3 are unstable (source, df =dxjx ¼ai , ai > 0) and stable (sink, df =dxjx ¼ai , ai < 0). 2 3 2 3 (v2) The equilibrium of x ¼ ai1 is unstable (an upper saddle, d 2 f =dx2 jx ¼ai > 0) and unstable (a lower saddle, d 2 f =dx2 jx ¼ai < 0). 1 1 The bifurcation of equilibrium at x ¼ ai1 for two equilibriums switching or vanishing is called the upper-saddle-node or lower-saddle-node bifurcation of the second order at a point p ¼ p1 2 ∂Ω12.

(v1)

(vi) Under conditions of Δi ¼ B2i  4C i > 0 ði 2 f1, 2gÞ and Δj ¼ B2j  4C j ¼ 0 ðj 2 f1, 2gÞ pffiffiffiffiffi ðiÞ pffiffiffiffiffi 1 1 ðiÞ b1 ¼  ðBi þ Δi Þ, b2 ¼  ðBi  Δi Þ 2 2 1 ðjÞ ðiÞ ðiÞ b1, 2 ¼  Bj , bk ¼ bl , ðk 6¼ l, k, l 2 f1, 2gÞ 2 a standard form of Eq. (5.146) is

ð5:189Þ

5.4 Quartic Nonlinear Systems

193

x_ ¼ f ðx, pÞ ¼ a0 ðx  ai1 Þ3 ðx  ai2 Þ

ð5:190Þ

where ðjÞ

ðiÞ

ðiÞ

ai1 ¼ b1, 2 ¼ bl , ai2 ¼ bk , a1 < a2 ; for i, j, l 2 f1, 2g, iα 2 f1, 2g and α 2 f1, 2g:

ð5:191Þ

(vi1) The equilibrium of x ¼ ai2 is unstable (source, df =dxjx ¼ai > 0) or 2 stable (sink, df =dxjx ¼ai < 0). 2 (vi2) The equilibrium of x ¼ ai1 is unstable (the third-order source, d3 f =dx3 jx ¼ai > 0) and stable (the third-order sink, d 3 f =dx3 jx ¼ai 1 1 < 0). The bifurcation of equilibrium at x ¼ ai1 for one equilibrium to three equilibriums is called the source or sink bifurcation of the third order at a point p ¼ p1 2 ∂Ω12. (vii) Under conditions of Δi ¼ B2i  4Ci ¼ 0 ði 2 f1, 2gÞ and Δj ¼ B2j  4Cj ¼ 0 ðj 2 f1, 2gÞ 1 ðαÞ ðαÞ b1 ¼ b2 ¼  Bα for α ¼ i, j 2 B1 6¼ B2 ,

ð5:192Þ

a standard form of Eq. (5.146) is x_ ¼ a0 ðx  a1 Þ2 ðx  a2 Þ2

ð5:193Þ

where 1 1 1 1 a1 ¼ min f B1 ,  B2 g, a2 ¼ max f B1 ,  B2 g: 2 2 2 2

ð5:194Þ

(vii1) For a0(p) > 0, the equilibriums of x ¼ ai (i ¼ 1, 2) are unstable (upper saddle, d2 f =dx2 jx ¼ai > 0). The equilibrium of x ¼ ai for two equilibriums vanishing and appearing are called an upper-saddle-node bifurcation of the second order at a point p ¼ p1 2 ∂Ω12. (vii2) For a0(p) < 0, the equilibriums of x ¼ ai (i ¼ 1, 2) are unstable (lower saddle, d2 f =dx2 jx ¼ai < 0). The equilibrium of x ¼ a1 for two equilibriums vanishing and appearing are called a lower-saddle-node bifurcation of the second order at a point p ¼ p1 2 ∂Ω12.

194

5 Low-Degree Polynomial Systems

(viii) Under conditions of Δi ¼ B2i  4Ci ¼ 0 ði 2 f1, 2gÞ and Δj ¼ B2j  4Cj ¼ 0 ðj 2 f1, 2gÞ 1 ðαÞ ðαÞ b1 ¼ b2 ¼  Bα for α ¼ i, j 2 B1 6¼ B2 ,

ð5:195Þ

the corresponding standard form is x_ ¼ a0 ðx  a1 Þ4

ð5:196Þ

1 1 a1 ¼  B1 ¼  B2 : 2 2

ð5:197Þ

where

(viii1) For a0(p) > 0, the equilibriums of x ¼ a1 are unstable (upper saddle, d 4 f =dx4 jx ¼a1 > 0). The equilibrium of x ¼ a1 for four equilibriums vanishing and appearing are called an upper-saddle-node bifurcation of the fourth order at a point p ¼ p1 2 ∂Ω12. (viii2) For a0(p) < 0, the equilibriums of x ¼ a1 are unstable (lower saddle, d4 f =dx4 jx ¼a1 < 0). The equilibrium of x ¼ a1 for four equilibriums vanishing and appearing are called a lower-saddle-node bifurcation of the fourth order at a point p ¼ p1 2 ∂Ω12. As discussed before, a quartic nonlinear system is expressed by the product of two quadratic polynomials, i.e., x_ ¼ a0 ðpÞ½x2 þ B1 ðpÞx þ C1 ðpÞ½x2 þ B2 ðpÞx þ C2 ðpÞ:

ð5:198Þ

Thus, for x_ ¼ 0, the equilibriums are determined by the roots of two quadratic polynomial equations, i.e., x2 þ B1 ðpÞx þ C 1 ðpÞ ¼ 0 and=or x2 þ B2 ðpÞx þ C2 ðpÞ ¼ 0:

ð5:199Þ

The roots of such quadratic polynomial equations are determined by the corresponding discriminant of the quadratic equations, i.e., Δi ¼ B2i  4C i for i ¼ 1, 2:

ð5:200Þ

If Δi < 0, the quadratic equation of x2 + Bi(p)x + Ci(p) ¼ 0 does not have any roots. If Δi > 0, the quadratic equation of x2 + Bi(p)x + Ci(p) ¼ 0 possesses two roots.

5.4 Quartic Nonlinear Systems

195

If Δi ¼ 0, the quadratic equation of x2 + Bi(p)x + Ci(p) ¼ 0 has a repeated root. With parameter variation, suppose one of the two quadratic polynomial equations has one root intersected with the roots of the other quadratic polynomial equation. There are ðiÞ ðjÞ ðjÞ ðiÞ ðiÞ ðiÞ ðjÞ six cases for a0 > 0: (a) b2 ¼ b1 , (b) b1 ¼ b1 ¼ b2 ¼  12 Bi , (c) b1 ¼ b1 , (d) ðiÞ

ðjÞ

ðjÞ

ðiÞ

ðiÞ

ðiÞ

ðjÞ

b2 ¼ b2 , (e) b2 ¼ b1 ¼ b2 ¼  12 Bi , and (f) b1 ¼ b2 , as presented in Fig. 5.13. The intersected point for non-repeated roots is a saddle-node bifurcation for the subcritical case. The lower-saddle-node and upper-saddle-node bifurcations are shown in Fig. 5.13a, c and d, f, respectively. The bifurcation dynamics for the 1-dimensional quartic system is determined by x_ ¼ a0 ðx  ai1 Þ2 ðx  ai2 Þðx  ai3 Þ with iα, α 2 {1, 2, 3} for four equilibriums or x_ ¼ a0 ðx  ai Þ2 ½ðx þ 12 Bj Þ2  14 Δj  with i, j 2 {1, 2} for two equilibriums. If the intersected point occurs at the repeated root, the third-order source and sink bifurcations are presented in Fig. 5.13b, e, respectively. The corresponding bifurcation dynamics for the 1-dimensional quartic system is determined by x_ ¼ a0 ðx  ai1 Þ2 ðx  ai2 Þ with iα, α 2 {1, 2}. The stable and unstable equilibriums are presented by the solid and dashed curves, respectively. The intersected points are marked by circular symbols, which are the bifurcation points. Without losing generality, suppose the two roots of the quadratic polynomial equaðiÞ ðiÞ tion have a relation of b1 > b2 for i ¼ 1, 2. The repeated roots of the two quadratic polynomial equations are also the upper- or lower-saddle-node bifurcations for two equilibriums appearing and vanishing. Similarly, the six cases of stability and bifurcation diagrams varying with parameter for a0 < 0 are presented in Fig. 5.14. The stability and bifurcation conditions for a0 < 0 are opposite to a0 > 0. If the roots of two quadratic equations do not have any intersections, the open loops for stability and bifurcation diagrams of equilibriums for a0 > 0 and a0 < 0 are presented in Fig. 5.15. There are four cases of open loops for a0 > 0: (a) Bi < Bj, ðjÞ ðjÞ (b) Bi > Bj, (c) b2 <  12 Bi < b1 , and (d) Δi ¼ Δj, Bi 6¼ Bj and four cases of open ðjÞ

ðjÞ

loops for a0 < 0: (e) Bi < Bj, (f) Bi > Bj, (g) b2 <  12 Bi < b1 , and (h) Δi ¼ Δj ¼ 0, Bi 6¼ Bj. The two bifurcations occur at the same time because the quadratic equations have Δi ¼ Δj ¼ 0, Bi 6¼ Bj. The bifurcation points are only for two equilibriums appearing or vanishing from the discriminants of the quadratic equations. The bifurcation dynamics for the 1-dimensional quartic system is determined by x_ ¼ a0 ðx  ai1 Þ2 ðx  ai2 Þðx  ai3 Þ with iα, α 2 {1, 2, 3}. With varying vector parameter, the open loops of stability and bifurcation diagrams will become closed loops. Thus, the closed loops of stability and bifurcation diagrams of equilibriums for a0 > 0 and a0 < 0 are presented in Fig. 5.16. There are ðjÞ ðjÞ six cases of closed loops: (a) Bi < Bj, (b) Bi > Bj, and (c) b2 <  12 Bi < b1 for a0 > 0 ðjÞ

ðjÞ

and (d) Bi < Bj, (e) Bi > Bj, and (f) b2 <  12 Bi < b1 for a0 < 0 For such a closed loop, the bifurcation points are the upper- or lower-saddle bifurcations of the second order at both ends. The bifurcation points are determined from the discriminants of the quadratic equations. If Δi ¼ Δj ¼ 0 occurs at the same parameter, the bifurcation dynamics for the quartic dynamical system is determined by x_ ¼ a0 ðx  a1 Þ4 , as shown in Fig. 5.17.

196

5 Low-Degree Polynomial Systems

a

d

b

e

c

f

Fig. 5.13 Open loops for stability and bifurcations of equilibriums in the 1-dimensional, quartic ðiÞ ðjÞ ðjÞ ðiÞ ðiÞ ðiÞ ðjÞ nonlinear dynamical system (a0 > 0): (a) b2 ¼ b1 , (b) b1 ¼ b1 ¼ b2 ¼  12 Bi , (c) b1 ¼ b1 , ðiÞ

ðjÞ

ðjÞ

ðiÞ

ðiÞ

ðiÞ

ðjÞ

(d) b2 ¼ b2 , (e) b2 ¼ b1 ¼ b2 ¼  12 Bi , (f) b1 ¼ b2 . LSN: lower saddle node, USN: upper saddle node, SI: sink, SO: source. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

There are six cases: two closed loops (Δi > Δj, Bi ¼ Bj): (a) a0 > 0 and (b) a0 > 0; two open loops (Δi > Δj, Bi ¼ Bj): (c) a0 > 0 and (d) a0 > 0; and two open loops (Bi Bj): (e) a0 > 0 and (f) a0 > 0. The bifurcation points are the upper- and lower-saddle-node bifurcations of the fourth order. In Fig. 5.17a, b, the fourth-order upper- and lower-

5.4 Quartic Nonlinear Systems

197

a

d

b

e

c

f

Fig. 5.14 Open loops for stability and bifurcation of equilibriums in the 1-dimensional, quartic ðiÞ ðjÞ ðjÞ ðiÞ ðiÞ ðiÞ ðjÞ nonlinear dynamical system (a0 < 0): (a) b2 ¼ b1 , (b) b1 ¼ b1 ¼ b2 ¼  12 Bi , (c) b1 ¼ b1 , ðiÞ

ðjÞ

ðjÞ

ðiÞ

ðiÞ

ðiÞ

ðjÞ

(d) b2 ¼ b2 , (e) b2 ¼ b1 ¼ b2 ¼  12 Bi , (f) b1 ¼ b2 . LSN: lower saddle node, USN: upper saddle node, SI: sink, SO: source. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

saddle-node bifurcations in the closed loop of bifurcation diagrams for a0 > 0 and a0 > 0 are presented, respectively. However, for the open loop of bifurcation diagrams, the fourth-order upper- and lower-saddle-node bifurcations for a0 > 0 and a0 > 0 are presented in Fig. 5.17c, d, respectively. The fourth-order saddle-node

198

5 Low-Degree Polynomial Systems

Fig. 5.15 Open loops of stability and bifurcation of equilibriums in the 1-dimensional, quartic ðjÞ ðjÞ nonlinear dynamical system. (a0 > 0): (a) Bi < Bj, (b) Bi > Bj, (c) b2 <  12 Bi < b1 , (d) Δi ¼ Δj ¼ 0, ðjÞ

ðjÞ

Bi 6¼ Bj. (a0 < 0): (e) Bi < Bj, (f) Bi > Bj, (g) b2 <  12 Bi < b1 , (h) Δi ¼ Δj ¼ 0, Bi 6¼ Bj. LSN: lower saddle node, USN: upper saddle node, SI: sink, SO: source. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

5.4 Quartic Nonlinear Systems

199

a

d

b

e

c

f

Fig. 5.16 Closed loops for stability and bifurcation of equilibriums in the 1-dimensional, quartic ðjÞ ðjÞ nonlinear dynamical system: (a0 > 0): (a) Bi < Bj, (b) Bi > Bj, (c) b2 <  12 Bi < b1 . (a0 < 0): ðjÞ

ðjÞ

(d) Bi < Bj, (e) Bi > Bj, (f) b2 <  12 Bi < b1 . LSN: lower saddle node, USN: upper saddle node, SI: sink, SO: source. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

bifurcation possesses four branches rather than two branches for the second-order saddle-node bifurcation. For Bi Bj, the fourth-order upper- and lower-saddle-node bifurcations are presented in Fig. 5.17e, f. The quadratic equation gives the two equilibriums, which are of the same stability. That is, both equilibriums for the same quadratic equation are stable or unstable.

200

5 Low-Degree Polynomial Systems

a

b

c

d

e

f

Fig. 5.17 Stability and bifurcation of equilibriums in the 1-dimensional, quartic nonlinear dynamical system. Closed loop (Δi > Δj, Bi ¼ Bj) for (a) a0 > 0 and (b) a0 > 0; open loop (Δi > Δj, Bi ¼ Bj) for (c) a0 > 0 and (d) a0 > 0; open loop (Bi < Bj) for (e) a0 > 0 and (f) a0 > 0. LSN: lower saddle node, USN: upper saddle node, SI: sink, SO: source. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

If Δi ¼ 0 (i 2 {1, 2}) occurs for new equilibrium appearing or vanishing only, the open loop of the bifurcation diagrams for a0 > 0 is presented in Fig. 5.18. There are ðiÞ ðjÞ ðiÞ ðjÞ ðiÞ ðjÞ ðiÞ ðjÞ four cases for a0 > 0: (a) b2 ¼ b1 , (b) b1 ¼ b1 , (c) b2 ¼ b2 , and (d) b1 ¼ b2 . The bifurcation points are the upper- and lower-saddle-node bifurcations of the

5.4 Quartic Nonlinear Systems

201

a

c

b

d

Fig. 5.18 Open loops of stability and bifurcation of equilibriums in the 1-dimensional, quartic ðiÞ ðjÞ ðiÞ ðjÞ ðiÞ ðjÞ ðiÞ ðjÞ nonlinear dynamical system (a0 > 0): (a) b2 ¼ b1 , (b) b1 ¼ b1 , (c) b2 ¼ b2 , (d) b1 ¼ b2 . LSN: lower saddle node, USN: upper saddle node, SI: sink, SO: source. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

second order. In Fig. 5.19a–d, the upper- and lower-saddle-node bifurcations of the second order in the open loop of bifurcation diagrams for a0 < 0 are presented. Such a diagram of stability and bifurcation possesses three saddle-node bifurcations. If Δi ¼ 0 and Δj > 0 (i, j 2 {1, 2}, i 6¼ j) exist for equilibrium appearing or vanishing with the source and sink bifurcations of the third order, one closed loop of the bifurcation diagrams for a0 > 0 is presented in Fig. 5.20. Four cases for a0 > 0 ðiÞ ðjÞ exist with the four saddle-node bifurcations in one closed loop: (a) b2 ¼ b1 and ðiÞ

ðjÞ

ðiÞ

ðjÞ

ðiÞ

ðjÞ

ðiÞ

ðjÞ

ðiÞ

ðjÞ

ðiÞ

ðjÞ

b1 ¼ b2 , (b) b1 ¼ b1 and b2 ¼ b2 , (c) b2 ¼ b2 and b1 ¼ b1 , (d) b1 ¼ b2 ðiÞ

ðjÞ

and b2 ¼ b1 . Two cases for a0 > 0 exist with the three saddle-node bifurcations ðjÞ

plus the source and sink bifurcations of the third order in two closed loops: (e) b1 ¼ ðiÞ

ðiÞ

ðiÞ

ðjÞ

ðiÞ

ðjÞ

ðjÞ

ðiÞ

ðiÞ

b1 ¼ b2 ¼  12 Bi with b1 ¼ b1 and b2 ¼ b2 , and (f) b2 ¼ b1 ¼ b2 ¼  12 Bi ðiÞ

ðjÞ

ðiÞ

ðjÞ

with b1 ¼ b1 and b2 ¼ b2 . However, in Fig. 5.21a–f, the upper- and lower-

202

5 Low-Degree Polynomial Systems

a

c

b

d

Fig. 5.19 Open loops for stability and bifurcation of equilibriums in the 1-dimensional, quartic ðiÞ ðjÞ ðiÞ ðjÞ ðiÞ ðjÞ ðiÞ ðjÞ nonlinear dynamical system (a0 < 0): (a) b2 ¼ b1 , (b) b1 ¼ b1 , (c) b2 ¼ b2 , (d) b1 ¼ b2 . LSN: lower saddle-node, USN: upper-saddle-node, SI: sink, SO: source. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

saddle-node bifurcations of the second order plus the source and sink bifurcations of the third order in the closed loop of bifurcation diagrams for a0 < 0 are presented. Definition 5.9 Consider a 1-dimensional, quartic nonlinear dynamical system: x_ ¼ AðpÞx4 þ BðpÞx3 þ CðpÞx2 þ DðpÞx þ EðpÞ ¼ a0 ðpÞðx  b1 Þ2 ½x2 þ B2 ðpÞx þ C 2 ðpÞ

ð5:201Þ

where A(p) 6¼ 0, and p ¼ ðp1 , p2 , . . . , pm ÞT :

ð5:202Þ

5.4 Quartic Nonlinear Systems

203

a

d

b

e

c

f

Fig. 5.20 Closed loops of stability and bifurcation of equilibriums in the 1-dimensional, quartic ðiÞ ðjÞ ðiÞ ðjÞ ðjÞ ðiÞ ðiÞ nonlinear dynamical system (a0 > 0): (a) b2 ¼ b1 and b1 ¼ b2 , (b) b1 ¼ b1 ¼ b2 ¼  12 Bi ðiÞ

ðjÞ

ðiÞ

ðjÞ

ðiÞ

ðjÞ

ðiÞ

ðjÞ

ðiÞ

ðjÞ

ðiÞ

ðjÞ

ðjÞ

with b1 ¼ b1 and b2 ¼ b2 , (c) b1 ¼ b1 and b2 ¼ b2 , (d) b2 ¼ b2 and b1 ¼ b1 , (e) b2 ¼ ðiÞ b1

ðiÞ b2

ðiÞ b1

ðjÞ b1

ðiÞ b2

ðjÞ b2 ,

ðiÞ b1

ðjÞ b2

ðiÞ b2

ðjÞ b1 .

¼ ¼  12 Bi with ¼ and ¼ (f) ¼ and ¼ LSN: lower saddle node, USN: upper saddle node, SI: sink, SO: source. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

204

5 Low-Degree Polynomial Systems a0 < 0

a0 < 0

b1(i )

SI

SI

LSN

b1( j ) LSN

b2(i )

SO

SO LSN

USN

LSN

b2( j )

x*

Dj < 0 Dj = 0 Dj > 0

LSN

Dj > 0

Dj < 0 Dj = 0

|| p ||

a

b2(i )

SO

x*

b2( j )

SO

|| p ||

SI

b1( j )

SI LSN

b1(i )

USN

d

a0 < 0

a0 < 0

b1(i )

LSN

SI

b1( j )

SI

b1(i )

b1( j )

3rd SI

SO

LSN

b2(i )

SI

LSN

SO SI

3rd SO

b2( j )

LSN SO

x*

b2( j )

SO

x*

Dj < 0 Dj = 0 Dj > 0

|| p ||

b2(i )

LSN

Dj < 0 Dj = 0 Dj > 0

|| p ||

b

e

a0 < 0 SI

b1( j )

a0 < 0

b1( j )

SI

LSN

b1(i ) LSN

SO LSN

SI

b

SO

b2( j )

SO

b2( j )

SI

b1(i )

USN

USN (i ) 2

LSN

x*

x*

LSN

Dj < 0 Dj = 0 Dj > 0

|| p ||

LSN

LSN

b2(i ) SO

Dj < 0 Dj = 0 Dj > 0

|| p ||

c

f

Fig. 5.21 A closed loop for stability and bifurcations of equilibriums in the 1-dimensional, quartic ðiÞ ðjÞ ðiÞ ðjÞ ðjÞ ðiÞ ðiÞ nonlinear dynamical system (a0 < 0): (a0 > 0): (a) b2 ¼ b1 and b1 ¼ b2 , (b) b1 ¼ b1 ¼ b2 ¼ ðiÞ

ðjÞ

ðiÞ

ðjÞ

ðiÞ

ðjÞ

ðiÞ

ðjÞ

ðiÞ

ðjÞ

ðiÞ

ðjÞ

 12 Bi with b1 ¼ b1 and b2 ¼ b2 , (c) b1 ¼ b1 and b2 ¼ b2 , (d) b2 ¼ b2 and b1 ¼ b1 , ðjÞ b2

ðiÞ b1

ðiÞ b2

 12 Bi

ðiÞ b1

ðjÞ b1

ðiÞ b2

ðjÞ b2 ,

ðiÞ b1

ðjÞ b2

ðiÞ b2

ðjÞ b1 .

¼ ¼ ¼ with ¼ and ¼ (f) ¼ and ¼ LSN: (e) lower saddle node, USN: upper saddle node, SI: sink, SO: source. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

(i) If Δ2 ¼ B22  4C 2 < 0 the corresponding standard form is

ð5:203Þ

5.4 Quartic Nonlinear Systems

205

1 1 x_ ¼ a0 ðx  b1 Þ2 ½ðx þ B2 Þ2 þ ðΔ2 Þ 2 4

ð5:204Þ

with a double equilibrium: x  ¼ a1 ¼ b1 :

ð5:205Þ

(a) For a0 > 0, the flow is called an upper-saddle (US) flow. (b) For a0 < 0, the flow is called a lower-saddle (LS) flow. (ii) If Δ2 ¼ B22  4C 2 > 0

ð5:206Þ

the quartic nonlinear dynamical system has two more equilibriums as pffiffiffiffiffiffi pffiffiffiffiffiffi 1 1 ð2Þ ð2Þ x ¼ b1 ¼  ðB2 þ Δ2 Þ, x ¼ b2 ¼  ðB2  Δ2 Þ: 2 2

ð5:207Þ

(ii1) The corresponding standard form is x_ ¼ a0 ðx  a1 Þ2 ðx  a2 Þðx  a3 Þ

ð5:208Þ

where ð2Þ

ð2Þ

ð2Þ

ð2Þ

a1 ¼ b1 < min fb1 , b2 g, a2 ¼ min fb1 , b2 g, ð2Þ

ð2Þ

a3 ¼ max fb1 , b2 g:

ð5:209Þ

(a) For a0 > 0, the flow is called a (US:SI:SO) flow. (b) For a0 < 0, the flow is called an (LS:SO:SI) flow. (ii2) The corresponding standard form is x_ ¼ a0 ðx  a1 Þðx  a2 Þ2 ðx  a3 Þ

ð5:210Þ

where ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

a1 ¼ min fb1 , b2 g, a2 ¼ b1 > min fb1 , b2 g, a3 ¼ max fb1 , b2 g > b1 : (a) For a0 > 0, the flow is called an (SI:LS:SO) flow. (b) For a0 < 0, the flow is called an (SO:US:SI) flow. (ii3) The corresponding standard form is

ð5:211Þ

206

5 Low-Degree Polynomial Systems

x_ ¼ a0 ðx  a1 Þðx  a2 Þðx  a3 Þ2

ð5:212Þ

where ð2Þ

ð2Þ

ð2Þ

ð2Þ

a1 ¼ min fb1 , b2 g, a2 ¼ max fb1 , b2 g, ð2Þ

ð2Þ

a3 ¼ b1 > max fb1 , b2 g:

ð5:213Þ

(a) For a0 > 0, the flow is called an (SI:SO:US) flow. (b) For a0 < 0, the flow is called an (SO:SI:LS) flow. (ii4) The corresponding standard form is x_ ¼ a0 ðx  a1 Þ3 ðx  a2 Þ

ð5:214Þ

where ð2Þ

ð2Þ

ð2Þ

ð2Þ

a1 ¼ b1 ¼ min fb1 , b2 g, a2 ¼ max fb1 , b2 g:

ð5:215Þ

(a) For a0 > 0, the flow is called a (3rd order-SI:SO) flow. The bifurcation of equilibrium for (US:SI:SO)-equilibriums to (SI:LS:SO)-equilibrium is called the sink bifurcation of the third order. (b) For a0 < 0, the flow is called a (3rd order-SO:SI) flow. The bifurcation of equilibrium for (LS:SO:SI)-equilibrium to (SO:US:SI)-equilibrium is called the source bifurcation of the third order. (ii5) The corresponding standard form is x_ ¼ a0 ðx  a1 Þðx  a2 Þ3

ð5:216Þ

where ð2Þ

ð2Þ

ð2Þ

ð2Þ

a1 ¼ min fb1 , b2 g, a2 ¼ b1 ¼ max fb1 , b2 g:

ð5:217Þ

(a) For a0 > 0, the flow is called a (SI:3rd order-SO) flow. The bifurcation of equilibrium for (SI:SO:US)-equilibriums to (SI:US:SO)-equilibrium is called the source bifurcation of the third order. (b) For a0 < 0, the flow is called a (SO:3rd order-SI) flow. The bifurcation of equilibrium for (SO:SI:LS)-equilibriums to (SO:US:SI)-equilibriums is called the sink bifurcation of the third order. (iii) If Δ2 ¼ B22  4C 2 ¼ 0 the quartic nonlinear dynamical system has two equilibriums as

ð5:218Þ

5.4 Quartic Nonlinear Systems

207

1 ð2Þ ð2Þ x ¼ b1 ¼ b2 ¼ b2 ¼  B2 : 2

ð5:219Þ

(iii1) The corresponding standard form for b1 < b2 is x_ ¼ a0 ðx  b1 Þ2 ðx  b2 Þ2 :

ð5:220Þ

(a) For a0 > 0, the flow is called a (US:US) flow. The bifurcation of equilibrium for the (US:SI:SO)-equilibrium appearance is called the upper-saddle bifurcation of the second order. (b) For a0 < 0, the flow is called an (LS:LS) flow. The bifurcation of equilibrium for (LS:SO:SI)-equilibrium appearance is called the lowersaddle bifurcation of the second order. (iii2) The corresponding standard form for b1 > b2 is x_ ¼ a0 ðx  b2 Þ2 ðx  b1 Þ2 :

ð5:221Þ

(a) For a0 > 0, the flow is called a (US:US) flow. The bifurcation of equilibrium for the (SI:SO:US)-equilibrium appearance is called the upper-saddle bifurcation of the second order. (b) For a0 < 0, the flow is called an (LS:LS) flow. The bifurcation of equilibrium for (SO:SI:LS)-equilibrium appearance is called the lowersaddle bifurcation of the second order. (iii3) The corresponding standard form with b1 ¼ b2 ¼ a1 is x_ ¼ a0 ðx  a1 Þ4 :

ð5:222Þ

(a) For a0 > 0, the flow is called a fourth-order upper-saddle (US) flow. The bifurcation of equilibrium for the upper-saddle (US) equilibrium to (SO:LS:SI)-equilibriums is called the upper-saddle-node bifurcation of the fourth order. (b) For a0 < 0, the flow is called a fourth-order lower-saddle (LS) flow. The bifurcation of equilibrium for the lower-saddle (LS) equilibrium to (SO: US:SI)-equilibrium appearance is called the lower-saddle bifurcation of the fourth order. Definition 5.10 Consider a 1-dimensional, quartic nonlinear dynamical system x_ ¼ AðpÞx4 þ BðpÞx3 þ CðpÞx2 þ DðpÞx þ EðpÞ ¼ a0 ðpÞ½x2 þ B1 ðpÞx þ C1 ðpÞ2 where A(p) 6¼ 0, and

ð5:223Þ

208

5 Low-Degree Polynomial Systems

p ¼ ðp1 , p2 , . . . , pm ÞT :

ð5:224Þ

Δ1 ¼ B21  4C1 < 0,

ð5:225Þ

(i) If

the quartic nonlinear dynamical system does not have any equilibrium. (a) For a0 > 0, the nonequilibrium flow is called a positive flow. (b) For a0 < 0, the nonequilibrium flow is called a negative flow. (ii) If Δ1 ¼ B21  4C 1 > 0

ð5:226Þ

the 1-dimensional quartic nonlinear dynamical system has two double-repeated equilibriums as pffiffiffiffiffiffi pffiffiffiffiffiffi 1 1 ð1Þ ð1Þ x ¼ b1 ¼  ðB1 þ Δ1 Þ, x ¼ b1 ¼  ðB1  Δ1 Þ: 2 2

ð5:227Þ

The corresponding standard form is x_ ¼ a0 ðx  a1 Þ2 ðx  a2 Þ2

ð5:228Þ

where ð1Þ

ð1Þ

ð1Þ

ð1Þ

a1 ¼ min fb1 , b2 g, a2 ¼ max fb1 , b2 g:

ð5:229Þ

(a) For a0 > 0, the flow is called a (US:US) flow. (b) For a0 < 0, the flow is called an (LS:LS) flow. (iii) If Δ1 ¼ B21  4C 1 ¼ 0

ð5:230Þ

the 1-dimensional quartic nonlinear dynamical system has two equilibriums as 1 ð2Þ ð2Þ x  ¼ b1 ¼ b2 ¼  B 2 ¼ a1 : 2

ð5:231Þ

The corresponding standard form is x_ ¼ a0 ðx  a1 Þ4 :

ð5:232Þ

5.4 Quartic Nonlinear Systems

209

(a) For a0 > 0, the flow is called a fourth-order US flow. The bifurcation of equilibrium for (US:US)-equilibrium appearance is called the upper-saddle bifurcation of the fourth order. (b) For a0 < 0, the flow is called a fourth-order LS flow. The bifurcation of equilibrium for the LS equilibrium to (LS:LS)-equilibrium appearance is called the lower-saddle bifurcation of the fourth order. From a 1-dimensional, quartic nonlinear system with singularity, the saddle equilibrium with and without intersection with simple equilibriums is presented in Figs. 5.22 and 5.23. In Fig. 5.22a, d, the upper-saddle equilibrium for a0 > 0 does not intersect with any branch of the simple equilibriums. In Fig. 5.23b, c, the uppersaddle equilibrium for a0 > 0 intersects with one branch of the simple equilibriums, and the upper-saddle equilibrium switches to the lower-saddle equilibrium with source and sink equilibriums, which are called the source and sink bifurcations of the third order, accordingly. In Fig. 5.22e, the upper-saddle equilibrium for a0 > 0 intersects with a double-repeated equilibrium with upper saddle. The intersected point is an unstable equilibrium, which is called a fourth-order upper-saddle-node bifurcation. In Fig. 5.22f, the two second-order upper-saddle equilibriums are presented. The two upper-saddle equilibriums appear at the bifurcation of the fourth-order upper-saddle bifurcation. Similarly, the lower-saddle equilibrium for a0 < 0 does not intersect with any branch of the simple equilibriums, as shown in Fig. 5.23a, d. In Fig. 5.23b, c, the lower-saddle equilibrium for a0 < 0 intersects with one branch of the simple equilibriums, and the lower-saddle equilibrium switches to the upper-saddle equilibrium with source and sink equilibriums, which are called the source and sink bifurcations of the third order, accordingly. In Fig. 5.23e, the lower-saddle equilibrium for a0 < 0 intersects with a repeated equilibrium with a lower saddle. The intersection point is an unstable equilibrium, which is called a fourth-order lowersaddle-node bifurcation. In Fig. 5.23f, the two second-order lower-saddle equilibriums are presented for a0 < 0. The two lower-saddle equilibriums appear at the fourthorder lower-saddle bifurcation. Consider a 1-dimensional, quartic nonlinear dynamical system with two double equilibriums. (i) For b 6¼ a, the dynamical system is x_ ¼ a0 ðpÞðx  bðpÞÞ2 ðx  aðpÞÞ2 :

ð5:233Þ

For such a system, if a0 > 0, two double-repeated equilibriums of x ¼ a, b are two upper saddles, which are unstable. If a0 < 0, two double-repeated equilibriums of x ¼ a, b are two lower saddles, which are unstable. (ii) For b ¼ a, the dynamical system on the boundary is x_ ¼ a0 ðpÞðx  bðpÞÞ4 :

ð5:234Þ

210

5 Low-Degree Polynomial Systems

a

d

b

e

c

f

Fig. 5.22 Stability and bifurcation of three equilibriums with and without intersection in the ð2Þ 1-dimensional, quartic nonlinear dynamical system (a0 > 0): (a) without intersection b1 > b1 , ð2Þ

ð2Þ

ð2Þ

(b) an intersection at b1 ¼ b1 , (c) an intersection at b1 ¼ b2 , (d) without intersection b1 < b1 , (e) an intersection at b1 ¼  12 Bi , (f) Δ1 ¼ 0. LSN: lower saddle node, USN: upper saddle node, SI: sink, SO: source. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

With parameter changes, the bifurcation diagram for the quartic nonlinear system is presented in Fig. 5.24. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation point is marked by a circular symbol. In Fig. 5.24a, if a0 > 0, two double-repeated equilibriums of x ¼ a, b are

5.4 Quartic Nonlinear Systems

211

a

d

b

e

c

f

Fig. 5.23 Stability and bifurcation of four equilibriums with intersection in the 1-dimensional, ð2Þ quartic nonlinear dynamical system (a0 < 0): (a) without intersection b1 > b1 , (b) an intersection ð2Þ ð2Þ ð2Þ at b1 ¼ b1 , (c) an intersection at b1 ¼ b1 , (d) without intersection b1 < b1 , (e) an intersection at b1 ¼  12 Bi , (f) Δ1 ¼ 0. LSN: lower saddle node, USN: upper saddle node, SI: sink, SO: source. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

212

a

5 Low-Degree Polynomial Systems

b

Fig. 5.24 Stability and bifurcation of two US or LS equilibriums with intersection in the 1-dimensional, quartic nonlinear dynamical system: (a) (US:US)-flow (a0 > 0), (b) (LS:LS)-flow (a0 < 0). Fourth LS: fourth-order lower-saddle bifurcation, fourth US: fourth-order upper-saddle bifurcation. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

Fig. 5.25 Stability of a repeated equilibrium with the fourth multiplicity in the 1-dimensional, quartic nonlinear dynamical system: Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The stability switching is labeled by a circular symbol

the upper saddles of the second order. The two upper saddles intersect at a point of x ¼ a ¼ b with the fourth-order multiplicity, which is an upper-saddle bifurcation of the fourth order for the (US:US) to (US:US) equilibriums. If a0 < 0, two double equilibriums of x ¼ a, b are the lower saddle of the second order, which are intersected at a point of x ¼ a ¼ b, as shown in Fig. 5.24b. Such a quartic equilibrium with the fourth-order multiplicity is called a lower-saddle bifurcation of the fourth order for the (LS:LS) to (LS:LS) equilibrium. To illustrate the stability and bifurcation of equilibrium with singularity in a 1-dimensional, quadratic nonlinear system, the equilibrium of x_ ¼ a0 ðx  a1 Þ4 is presented in Fig. 5.25. The fourth-order upper and lower saddles of equilibrium of

5.4 Quartic Nonlinear Systems

213

x ¼ a1 with the fourth-order multiplicity are unstable, and the fourth-order upperand lower-saddle equilibriums of the fourth order are invariant. At a0 ¼ 0, the lowersaddle equilibrium switches to the upper-saddle equilibrium, which is a switching point marked by a circular symbol. For further discussion on the switching bifurcations in the quartic nonlinear system, the following definitions are presented. Definition 5.11 Consider a 1-dimensional, quartic nonlinear dynamical system x_ ¼ AðpÞx4 þ BðpÞx3 þ CðpÞx2 þ DðpÞx þ EðpÞ ¼ a0 ðpÞðx  aÞðx  bÞ½x2 þ B2 ðpÞx þ C 2 ðpÞ

ð5:235Þ

where A(p) 6¼ 0 and p ¼ ðp1 , p2 , . . . , pm ÞT :

ð5:236Þ

Δ2 ¼ B22  4C2 < 0, fa1 , a2 g ¼ sortfa, bg, a1 a2 ,

ð5:237Þ

(i) If

the quartic nonlinear dynamical system has two equilibriums. The standard form is 1 2 1 x_ ¼ a0 ðpÞðx  a1 Þðx  a2 Þ½ðx2 þ B2 Þ þ ðΔ2 Þ: 2 4

ð5:238Þ

(i1) For a0 > 0, the equilibrium flow is an (SI:SO) flow. The equilibrium of x = a1 is stable (sink, df =dxjx ¼a1 < 0) and the equilibrium of x = a2 is unstable (source, df =dxjx ¼a2 > 0). (i2) For a0 < 0, the equilibrium flow is an (SO:SI) flow. The equilibrium of x = a1 is unstable (source, df =dxjx ¼a1 > 0) and the equilibrium of x = a2 is stable (sink, df =dxjx ¼a2 < 0). (i3) Under Δ12 ¼ ða1  a2 Þ2 ¼ 0 with a1 ¼ a2 ,

ð5:239Þ

the quartic nonlinear dynamical system has a standard form as 1 2 1 x_ ¼ f ðx, pÞ ¼ a0 ðx  a1 Þ2 ½ðx2 þ B2 Þ þ ðΔ2 Þ: 2 4

ð5:240Þ

214

5 Low-Degree Polynomial Systems

(a) For a0 (p) > 0, the equilibrium of x = a1 is unstable (an upper saddle of second order, d2 f =dx2 jx ¼a1 > 0). Such a flow is called an upper-saddle flow. The bifurcation of equilibrium at x = a1 for two equilibriums switching of x = a1, a2 is called an upper-saddle-node switching bifurcation of the second order at a point p = p1. (b) For a0 (p) < 0, the equilibrium of x = a1 is unstable (a lower saddle of the second order, d 2 f =dx2 jx ¼a1 < 0). Such a flow is called a lowersaddle flow. The bifurcation of equilibrium at x = a1 for two equilibriums switching of x = a1, a2 is called a lower-saddle-node switching bifurcation of the second order at a point p = p1. (ii) If Δ2 ¼ B22  4C2 > 0,

ð5:241Þ

the 1-dimensional quartic nonlinear dynamical system has four equilibriums as pffiffiffiffiffiffi pffiffiffiffiffiffi 1 1 ð2Þ ð2Þ xk ¼ b1 ¼  ðB2 þ Δ2 Þ, xk ¼ b2 ¼  ðB2  Δ2 Þ 2 2 ð2Þ ð2Þ fa1 , a2 , a3 :a4 g ¼ sortfa, b, b1 , b2 g, ai < aiþ1

ð5:242Þ

The corresponding standard form is x_ ¼ a0 ðx  a1 Þðx  a2 Þðx  a3 Þðx  a4 Þ:

ð5:243Þ

(ii1) For a0 > 0, the flow is called an (SI:SO:SI:SO) flow. (ii2) For a0 < 0, the flow is called an (SO:SI:SO:SO) flow. (ii3) Under Δi1 i2 ¼ ðai1  ai2 Þ2 ¼ 0, ai1 ¼ ai2 , i1 , i2 2 f1, 2, 3, 4g, i1 6¼ i2 ,

ð5:244Þ

the standard form is x_ ¼ f ðx, pÞ ¼ a0 ðx  ai1 Þ2 ðx  ai3 Þðx  ai4 Þ iα 2 f1, 2, 3, 4g, α ¼ 1, 3, 4

ð5:245Þ

(a) The equilibrium of x ¼ ai1 is unstable (an upper saddle of the second order, d2 f =dx2 jx ¼ai > 0). Such a flow is called an upper-saddle flow at 1 x ¼ ai1. The bifurcation of equilibrium at x ¼ ai1 for two equilibriums switching of x ¼ ai1 , ai2 is called an upper-saddle-node switching bifurcation of the second order at a point p = p1.

5.4 Quartic Nonlinear Systems

215

(b) The equilibrium of x ¼ ai1 is unstable (a lower saddle of the second order, d2 f =dx2 jx ¼ai < 0). Such a flow is called a lower-saddle flow at 1 x ¼ ai1. The bifurcation of equilibrium at x ¼ ai1 for two equilibriums switching of x ¼ ai1 , ai2 is called a lower-saddle-node switching bifurcation of the second order at a point p = p1. (ii4) Under Δi1 i2 ¼ ðai1  ai2 Þ2 ¼ 0, Δi2 i3 ¼ ðai2  ai3 Þ2 ¼ 0 ai1 ¼ ai2 ¼ a3 , i1 , i2 , i3 2 f1, 2, 3, 4g, i1 6¼ i2 6¼ i3 ,

ð5:246Þ

the standard form is x_ ¼ f ðx, pÞ ¼ a0 ðx  ai1 Þ3 ðx  ai4 Þ iα 2 f1, 2, 3, 4g, α ¼ 1, 4:

ð5:247Þ

(a) The equilibrium of x ¼ ai1 is unstable (a source of the third order, d 3 f =dx3 jx ¼ai > 0). Such a flow is called a third-order source flow at 1 x ¼ ai1. The bifurcation of equilibrium at x ¼ ai1 for three simple equilibrium bundle-switching of x ¼ ai1 , ai2 , ai3 is called a third-order source bundle-switching bifurcation at a point p = p1. (b) The equilibrium of x ¼ ai1 is stable (a sink of the third order, d 3 f =dx3 jx ¼ai < 0). Such a flow is called a third-order sink flow at 1 x ¼ ai1. The bifurcation of equilibrium at x ¼ ai1 for three simple equilibrium bundle-switching of x ¼ ai1 , ai2 , ai3 is called a third-sink bundle-switching bifurcation at a point p = p1. (ii5) Under Δi1 i2 ¼ ðai1  ai2 Þ2 ¼ 0, Δi2 i3 ¼ ðai2  ai3 Þ2 ¼ 0, Δi3 i4 ¼ ðai4  ai4 Þ2 ¼ 0, ai1 ¼ ai2 ¼ ai3 ¼ ai4 ,

ð5:248Þ

i1 , i2 , i3 , i4 2 f1, 2, 3, 4g, i1 6¼ i2 6¼ i3 6¼ i4 , the standard form is x_ ¼ f ðx, pÞ ¼ a0 ðx  ai1 Þ4 :

ð5:249Þ

(a) The equilibrium of x ¼ ai1 is unstable (an upper saddle of the fourth order, d4 f =dx4 jx ¼ai > 0). Such a flow is called a fourth-order upper1 saddle flow at x ¼ ai1. The bifurcation of equilibrium at x ¼ ai1 for four simple equilibriums bundle-switching of x = a1,2,3,4 is called a fourth-order upper-saddle-node bundle-switching bifurcation at a point p = p1.

216

5 Low-Degree Polynomial Systems

(b) The equilibrium of x ¼ ai1 is unstable (a lower-saddle of the fourth order, d 4 f =dx4 jx ¼ai < 0). Such a flow is called a fourth-order lower1 saddle flow at x ¼ ai1. The bifurcation of equilibrium at x ¼ ai1 for four simple equilibriums bundle-switching of x = a1,2,3,4 is called a fourth-order lower-saddle bundle-switching bifurcation at a point p = p1. (iii) If Δ2 ¼ B22  4C2 ¼ 0,

ð5:250Þ

the 1-dimensional quartic nonlinear dynamical system has three equilibrium as 1 ð2Þ ð2Þ xk ¼ b1 ¼ b2 ¼  B2 ; 2 ð2Þ ð2Þ fa1 , a2 , a3 g ¼ sortfa, b, b1 ¼ b2 g, ai < aiþ1 ; ð2Þ b1

ð2Þ b2 , ai3

ð5:251Þ

ai1 , i2 ¼ ¼ ¼ a, ai4 ¼ b; iα 2 f1, 2, 3g, α 2 f1, 2, 3, 4g, and the corresponding standard form is x_ ¼ a0 ðx  ai1 Þ2 ðx  ai2 Þðx  ai3 Þ:

ð5:252Þ

(iii1) The equilibrium of x ¼ ai1 is unstable (an upper saddle, d2 f =dx2 jx ¼a1 > 0), the flow is an upper-saddle flow at x ¼ ai1 . The bifurcation of equilibrium at x ¼ ai1 for the appearing or vanishing of two simple equilibriums is called the upper-saddle-node bifurcation of the second order. (iii2) The equilibrium of x ¼ ai1 is unstable (a lower saddle, d2 f =dx2 jx ¼a1 < 0), the flow is a lower-saddle flow at x ¼ ai1 . The bifurcation of equilibrium at x ¼ ai1 for the appearing or vanishing of two simple equilibriums is called the lower-saddle-node bifurcation of the second order. (iii3) Under Δi3 i4 ¼ ðai3  ai4 Þ2 ¼ 0, ai3 ¼ ai4 , ai1 6¼ ai3 , iα 2 f1, 2, 3g, α 2 f1, 2, 3, 4g;

ð5:253Þ

the corresponding standard form is x_ ¼ f ðx, pÞ ¼ a0 ðx  ai1 Þ2 ðx  ai3 Þ2 iα 2 f1, 2g, α ¼ 1, 3

ð5:254Þ

5.4 Quartic Nonlinear Systems

217

The equilibriums of x ¼ ai1 , ai3 are unstable (an upper saddle of the second order, d2 f =dx2 jx ¼ai , ai > 0) and unstable (a lower saddle of the second 1 3 order, d 2 f =dx2 jx ¼ai , ai < 0). Such a flow is called an (US:US) or a (LS:LS) 1 3 flow. The bifurcation of equilibrium at x ¼ ai1 for two simple equilibrium onset of x ¼ ai1, ai2 and at x ¼ ai3 for two equilibrium switching of x ¼ ai3 , ai4 is called an (US:US) or a (LS:LS) bifurcation at a point p = p1. (iii4) Under Δi1 i3 ¼ ðai1  ai3 Þ2 ¼ 0, ai1 ¼ ai2 , ai1 ¼ ai3 ai1 6¼ ai4 , iα 2 f1, 2, 3g, α 2 f1, 2, 3, 4g;

ð5:255Þ

the standard form is x_ ¼ f ðx, pÞ ¼ a0 ðx  ai1 Þ3 ðx  ai4 Þ iα 2 f1, 2g, α ¼ 1, 4:

ð5:256Þ

(a) The equilibrium of x ¼ ai1 is unstable (a third-order source, d 3 f =dx3 jx ¼ai > 0). Such a flow is called a third-order source flow. 1 The bifurcation of equilibrium at x ¼ ai1 for one simple equilibrium of x ¼ ai3 switching to three simple equilibriums of x ¼ ai1 , i2 , i3 is called a third-order source pitchfork bifurcation at a point p = p1. (b) The equilibrium of x ¼ ai1 is stable (a third-order sink, d 3 f =dx3 jx ¼ai < 0). Such a flow is called a third-order sink flow. 1 The bifurcation of equilibrium at x ¼ ai1 for one simple equilibrium of x ¼ ai1 switching to three simple equilibriums of x ¼ ai1 , i2 , i3 is called a third-order sink pitchfork-switching bifurcation at a point p = p1. (iii5) Under Δi1 i3 ¼ ðai1  ai3 Þ2 ¼ 0, Δi3 i4 ¼ ðai3  ai4 Þ2 ¼ 0; ai1 ¼ ai2 , ai1 ¼ ai3 ai1 ¼ ai4 , iα 2 f1, 2, 3g, α 2 f1, 2, 3, 4g;

ð5:257Þ

the standard form is x_ ¼ f ðx, pÞ ¼ a0 ðx  ai1 Þ4 iα 2 f1, 2g, α ¼ 1, 3

ð5:258Þ

218

5 Low-Degree Polynomial Systems

(a) For a0 > 0, the equilibrium of x ¼ ai1 is unstable (a fourth-order upper saddle, d4 f =dx4 jx ¼ai > 0). Such a flow is called a fourth-order upper1 saddle flow. The bifurcation of equilibrium at x ¼ ai1 for two simple equilibriums switching to four simple equilibriums is called a fourthorder upper-saddle-node flower-bundle-switching bifurcation at a point p = p1. (b) For a0 > 0, the equilibrium of x ¼ ai1 is unstable (a fourth-order lower saddle, d 4 f =dx4 jx ¼ai < 0). Such a flow is called a fourth-order lower1 saddle flow. The bifurcation of equilibrium at x ¼ ai1 for two simple equilibriums switching to four simple equilibriums is called a fourthorder lower-saddle-node flower-bundle-switching bifurcation at a point p = p1. Based on the previous definition, the stability and bifurcations of equilibriums in the 1-dimensional, quartic nonlinear dynamical system (a0 > 0) are presented in Fig. 5.26. In Fig. 5.26a–c, upper-saddle-node (USN) and lower-saddle-node (LSN) switching bifurcations are at two locations for two simple equilibriums, and one upper-saddle-node (USN) appearing bifurcation is for two simple equilibriums. In Fig. 5.26d, a third-order sink (3rd SI) pitchfork-switching bifurcation for a switching of one sink equilibrium to three simple equilibriums is presented, and one uppersaddle-node (USN) switching bifurcation for two simple equilibriums switching is also presented. In Fig. 5.26e, a third-order source (3rd SO) bundle-switching bifurcation for three equilibrium bundle-switching is presented, and an upper-saddlenode (USN) appearing bifurcation for two equilibrium onset is also presented. In Fig. 5.26f, a fourth-order upper-saddle (4th US) flower-bundle-switching bifurcation for four simple equilibriums is presented. Similarly, the stability and bifurcations of equilibriums in the 1-dimensional, quartic nonlinear dynamical system (a0 < 0) are presented in Fig. 5.27. In Fig. 5.27 (a–c), lower-saddle-node (LSN) and upper-saddle-node (USN) switching bifurcations are at two locations for two simple equilibriums, and one lower-saddle-node (LSN) appearing bifurcation is for two simple equilibriums appearing. In Fig. 5.27 (d), a third-order source (3rd SO) pitchfork-switching bifurcation for a switching of one source equilibrium to three simple equilibriums is presented, and one lowersaddle-node (LSN) switching bifurcation for two simple equilibriums switching is also presented. In Fig. 5.27(e), a third-order sink (3rd SI) bundle-switching bifurcation for three equilibrium bundle-switching is presented, and a lower-saddle-node (LSN) appearing bifurcation for two equilibrium onset is also presented. In Fig. 5.27 (f), a fourth-order lower-saddle (4th LS) flower-bundle-switching bifurcation for four simple equilibriums is presented. For further discussion on the switching bifurcation, the following definition is given for the 1-dimensional, quartic nonlinear dynamical system.

5.4 Quartic Nonlinear Systems

219 b

a0 > 0

a0 > 0

SO USN SI

LSN

b

SO USN

b1(1)

SI

a

3rd SI

SO

b1(1)

SO

a

USN SI

x* || p ||

Δ2 < 0 Δ2 = 0

SI

b2(1) x*

Δ2 > 0

Δ2 < 0

|| p ||

a

b

(1) 2

Δ2 = 0 Δ2 > 0

d a0 > 0

a0 > 0

b USN

SO LSN SI

SO

b1(1)

SO

a

SI

3rd SO

SI

SO

b1(1) b a

SO USN

USN

b2(1)

SI

x*

x*

|| p ||

Δ2 < 0 Δ2 = 0

Δ2 > 0

SI

Δ2 < 0 Δ2 = 0

|| p ||

b

b2(1)

Δ2 > 0

e b

a0 > 0 USN

SO LSN SI

a0 > 0

SO

b1(1)

SI SO

a

b1(1) b

4thUS SO

SO

a

USN SI

x* || p ||

c

SI

Δ1 < 0 Δ1 = 0

b2(1)

SI

x*

Δ1 > 0

|| p ||

Δ2 < 0

Δ2 = 0

b2(1)

Δ2 > 0

f

Fig. 5.26 Stability and bifurcations of equilibriums in the 1-dimensional, quartic nonlinear dynamical system (a0 > 0): (a–c) Two (USN and LSN) switching and one (USN) appearing bifurcations, (d) 3rd SI pitchfork-switching bifurcation, (e) 3rd SO bundle-switching bifurcation, (f) 4th US flower-bundle-switching bifurcation. LSN: lower-saddle-node, USN: upper-saddle-node, SI: sink, SO: source. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

220

5 Low-Degree Polynomial Systems b

a0 < 0

a0 < 0

SI LSN SO

USN

b

SI LSN

b1(1)

SO

a

3rd SO

SI

b1(1)

SI

a

LSN

b2(1)

SO

x* || p ||

Δ2 < 0 Δ2 = 0

SO

x*

Δ2 > 0

Δ2 < 0

|| p ||

a

b

(1) 2

Δ2 = 0 Δ2 > 0

d a0 < 0

a0 < 0

b LSN

SI USN SO

SI

b1(1)

SI

a

SO

3rd SI

b1(1)

SO

b

SI

a

SI LSN

x* || p ||

Δ2 < 0 Δ2 = 0

LSN

b2(1)

SO

x*

Δ2 > 0

SO

Δ2 < 0 Δ2 = 0

|| p ||

b

b2(1)

Δ2 > 0

e a0 < 0

a0 < 0

b LSN

SI USN SO

SI

b1(1)

SO SI

a

b1(1) b

4th LS SI

SI LSN SO

x* || p ||

c

Δ2 < 0 Δ2 = 0

a SO

b

(1) 2

SO

x*

Δ2 > 0

|| p ||

Δ2 < 0

Δ2 = 0

b2(1)

Δ2 > 0

f

Fig. 5.27 Stability and bifurcations of equilibriums in the 1-dimensional, quartic nonlinear dynamical system (a0 < 0): (a–c) Two (LSN and USN) switching and one LSN appearing bifurcations, (d) 3rd SO pitchfork-switching bifurcation, (e) 3rd SI bundle-switching bifurcation, (f) 4th LSN flower-bundle-switching bifurcation. LSN: lower-saddle-node, USN: upper-saddlenode, SI: sink, SO: source. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

5.4 Quartic Nonlinear Systems

221

Definition 5.12 Consider a 1-dimensional, quartic nonlinear dynamical system x_ ¼ AðpÞx4 þ BðpÞx3 þ CðpÞx2 þ DðpÞx þ EðpÞ ¼ a0 ðpÞðx  aÞðx  bÞðx  cÞðx  dÞ

ð5:259Þ

where A(p) 6¼ 0, and p ¼ ðp1 , p2 , . . . , pm ÞT :

ð5:260Þ

fa1 , a2 , a3 , a4 g ¼ sortfa, b, c, dg, ai aiþ1 ,

ð5:261Þ

(i) If

the quartic nonlinear dynamical system has four simple equilibriums, and the standard form is x_ ¼ a0 ðpÞðx  a1 Þðx  a2 Þðx  a3 Þðx  a4 Þ:

ð5:262Þ

(i1) For a0 > 0, the equilibrium flow is an (SI:SO:SI:SO) flow. The equilibrium of x = a1,3 is stable (sink, df =dxjx ¼a1, 3 < 0) and the equilibrium of x = a2,4 is unstable (source, df =dxjx ¼a2, 4 > 0). (i2) For a0 < 0, the equilibrium flow is an (SO:SI:SO:SI) flow. The equilibrium of x = a1 is unstable (source, df =dxjx ¼a1 > 0) and the equilibrium of x = a2 is stable (sink, df =dxjx ¼a2 < 0). (ii) If Δi1 i2 ¼ ðai1  ai2 Þ2 ¼ 0 with ai1 ¼ ai2 ;i1 , i2 2 f1, 2, 3, 4g,

ð5:263Þ

the quartic nonlinear dynamical system has a standard form as x_ ¼ f ðx, pÞ ¼ a0 ðx  ai1 Þ2 ðx  ai3 Þðx  ai4 Þ:

ð5:264Þ

(ii1) The equilibrium of x ¼ ai1 is unstable (an upper saddle of the second order, d2 f =dx2 jx ¼ai > 0). Such a flow is called an upper-saddle flow at 1 x ¼ ai1 . The bifurcation of equilibrium at x ¼ ai1 for two equilibriums switching of x ¼ ai1 , ai2 is called an upper-saddle-node bifurcation of the second order at a point p = p1. (ii2) The equilibrium of x ¼ ai1 is unstable (a lower saddle of the second order, d 2 f =dx2 jx ¼ai < 0). Such a flow is called a lower-saddle flow at x ¼ ai1 . 1 The bifurcation of equilibrium at x ¼ ai1 for two equilibriums switching

222

5 Low-Degree Polynomial Systems

of x ¼ ai1 , ai2 is called a lower-saddle-node bifurcation of the second order at a point p = p1. (iii) If Δi1 i2 ¼ ðai1  ai2 Þ2 ¼ 0, Δi2 i3 ¼ ðai2  ai3 Þ2 ¼ 0 ai1 ¼ ai2 ¼ ai3 ; i1 , i2 , i3 2 f1, 2, 3, 4g, i1 6¼ i2 6¼ i3 ,

ð5:265Þ

the corresponding standard form is x_ ¼ f ðx, pÞ ¼ a0 ðx  ai1 Þ3 ðx  ai4 Þ iα 2 f1, 2, 3, 4g, α ¼ 1, 4:

ð5:266Þ

(iii1) The equilibrium of x ¼ ai1 is unstable with d3 f =dx3 jx ¼ai > 0 (a third1 order source). Such a flow is called a third-order source flow at x ¼ ai1 . The bifurcation of equilibrium at x ¼ ai1 for a bundle switching of three simple equilibriums of x ¼ ai1 , ai2 , ai3 is called a third-order source bundle-switching bifurcation at a point p = p1. (iii2) The equilibrium of x ¼ ai1 is stable with d3 f =dx3 jx ¼ai < 0 (a third-order 1 sink). Such a flow is called a third-order sink flow at x ¼ ai1 . The bifurcation of equilibrium at x ¼ ai1 for a bundle switching of three simple equilibriums of x ¼ ai1 , ai2 , ai3 is called a third-order sink bundle-switching bifurcation at a point p = p1. (iv) If Δi1 i2 ¼ ðai1  ai2 Þ2 ¼ 0, Δi2 i3 ¼ ðai2  ai3 Þ2 ¼ 0, Δi3 i4 ¼ ðai4  ai4 Þ2 ¼ 0, ai1 ¼ ai2 ¼ ai3 ¼ ai4 ,

ð5:267Þ

i1 , i2 , i3 , i4 2 f1, 2, 3, 4g, i1 6¼ i2 6¼ i3 6¼ i4 the corresponding standard form is x_ ¼ f ðx, pÞ ¼ a0 ðx  ai1 Þ4

ð5:268Þ

(iv1) The equilibrium of x ¼ ai1 is unstable (a fourth-order upper saddle, d4 f =dx4 jx ¼ai > 0). Such a flow is called a fourth-order upper-saddle 1 flow at x ¼ ai1 . The bifurcation of equilibrium at x ¼ ai1 for a bundle switching of four simple equilibriums of x = a1,2,3,4 is called a fourthorder upper-saddle-node bundle-switching bifurcation at a point p = p1. (iv2) The equilibrium of x ¼ ai1 is unstable (a fourth-order lower saddle, d4 f =dx4 jx ¼ai < 0). Such a flow is called a fourth-order lower-saddle 1 flow at x ¼ ai1 . The bifurcation of equilibrium at x ¼ ai1 for a bundle

5.4 Quartic Nonlinear Systems

223

switching of four simple equilibriums of x = a1,2,3,4 is called a fourthorder lower-saddle bundle-switching bifurcation at a point p = p1. From the previous definition, stability and bifurcations of equilibriums in the 1-dimensional, quartic nonlinear dynamical system are presented in Fig. 5.28. For a0 > 0, the bifurcations and stability of equilibriums are presented in Fig. 5.28a–c. In Fig. 5.28a, four upper-saddle-node (USN) and two lower-saddle-

Fig. 5.28 Stability and bifurcations of equilibriums in the 1-dimensional, quartic nonlinear dynamical system (a0 > 0): (a) Four USN and two LSN switching bifurcation network, (b) thirdorder SI bundle-switching bifurcation, (c) fourth-order US bundle-switching bifurcation (a0 < 0): (d) Four LSN and two USN switching bifurcation network, (e) third-order SO bundle-switching bifurcation, (f) fourth-order LS bundle-switching bifurcation. LSN: lower-saddle-node, USN: upper-saddle-node, SI: sink, SO: source. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

224

5 Low-Degree Polynomial Systems

node (LSN) switching bifurcation network are presented for all possible switching bifurcation between two simple equilibriums. In Fig. 5.28b, a thirdorder sink (3rd SI) bundle-switching bifurcation for three simple equilibrium is presented, and there are three possible upper-saddle-node (USN) and lowersaddle-node (LSN) switching bifurcations for two simple equilibriums. In Fig. 5.28c a fourth-order upper-saddle (4th US) bundle-switching bifurcation for four simple equilibriums is presented. Similarly, For a0 < 0, the bifurcations and stability of equilibriums are presented in Fig. 5.28d–f. In Fig. 5.28d, four lower-saddle-node (LSN) and two upper-saddle-node (USN) switching bifurcation network are presented for all possible switching bifurcation between two simple equilibriums. In Fig. 5.28e, a third-order source (3rd SO) bundleswitching bifurcation for three simple equilibrium is presented, and there are three possible lower-saddle-node (LSN) and upper-saddle-node (USN) switching bifurcations for two simple equilibriums. In Fig. 5.28f a fourthorder lower-saddle (4th LS) bundle-switching bifurcation for four simple equilibriums is presented. For the switching bifurcation between the second-order and simple equilibriums, the following definition is given for the 1-dimensional, quartic nonlinear dynamical system. Definition 5.13 Consider a 1-dimensional, quartic nonlinear dynamical system x_ ¼ AðpÞx4 þ BðpÞx3 þ CðpÞx2 þ DðpÞx þ EðpÞ ¼ a0 ðpÞðx  aÞ2 ðx  bÞðx  cÞ

ð5:269Þ

where A(p) 6¼ 0, and p ¼ ðp1 , p2 , . . . , pm ÞT :

ð5:270Þ

fa1 , a2 , a3 g ¼ sortfa, b, cg, ai < aiþ1 i1 , i2 , i3 2 f1, 2, 3g,

ð5:271Þ

(i) If

the quartic nonlinear dynamical system has a standard form as x_ ¼ f ðx, pÞ ¼ a0 ðx  ai1 Þ2 ðx  ai2 Þðx  ai3 Þ:

ð5:272Þ

(i1) The equilibrium of x ¼ ai1 is unstable (an upper saddle of the second order, d 2 f =dx2 jx ¼ai > 0). Such a flow is called an upper-saddle flow at 1 x  ¼ ai 1 . (i2) The equilibrium of x ¼ ai1 is unstable (a lower saddle of the second order, d2 f =dx2 jx ¼ai < 0). Such a flow is called a lower-saddle flow at x ¼ ai1 . 1

5.4 Quartic Nonlinear Systems

225

(ii) If a ¼ ai1 , b ¼ ai2 , c ¼ ai3 ; Δi2 i3 ¼ ðai2  ai3 Þ2 ¼ 0, ai2 ¼ ai3 ; i1 , i2 , i3 2 f1, 2g, i1 6¼ i2 6¼ i3 ;

ð5:273Þ

the corresponding standard form is x_ ¼ f ðx, pÞ ¼ a0 ðx  ai1 Þ2 ðx  ai2 Þ2 iα 2 f1, 2g, α ¼ 1, 2:

ð5:274Þ

The equilibriums of x ¼ ai1 , ai2 are unstable (an upper saddle of the second order, d 2 f =dx2 jx ¼ai , ai > 0 or a lower saddle of the second order, 1

2

d2 f =dx2 jx ¼ai , ai < 0). Such a flow is called an (US:US) or a (LS:LS) flow. 2 1 The bifurcation of equilibrium at x ¼ ai2 for two simple equilibriums switching of x = b, c is called an upper-saddle or lower-saddle switching bifurcation at a point p = p1. (iii) If a ¼ ai1 ;ai3 , ai2 2 fb, cg; Δi1 i3 ¼ ðai1  ai3 Þ2 ¼ 0, ai1 ¼ ai3 , ai3 6¼ ai2 ;

ð5:275Þ

i1 , i2 , i3 2 f1, 2, 3g, i1 6¼ i2 6¼ i3 , the corresponding standard form is x_ ¼ f ðx, pÞ ¼ a0 ðx  ai1 Þ3 ðx  ai2 Þ iα 2 f1, 2g, α ¼ 1, 2:

ð5:276Þ

(iii1) The equilibrium of x ¼ ai1 is unstable (a third-order source, d3 f =dx3 jx ¼ai > 0). Such a flow is called a third-order source flow at 1 x ¼ ai1 . The bifurcation of equilibrium at x ¼ ai1 for a switching of one second-order and one simple equilibriums of x ¼ ai1 , ai2 is called a third-order source switching bifurcation at a point p = p1. (iii2) The equilibrium of x ¼ ai1 is stable (a third-order sink, d 3 f =dx3 jx ¼ai < 0). Such a flow is called a third-order sink flow at 1 x ¼ ai1 . The bifurcation of equilibrium at x ¼ ai1 for a switching of one second-order and one simple equilibriums of x ¼ ai1 , ai2 is called a third-order sink switching bifurcation at a point p = p1.

226

5 Low-Degree Polynomial Systems

(iv) If Δi1 i2 ¼ ðai1  ai2 Þ2 ¼ 0, Δi2 i3 ¼ ðai2  ai3 Þ2 ¼ 0, i1 , i2 , i3 2 f1, 2, 3g, i1 6¼ i2 6¼ i3 ,

ð5:277Þ

the corresponding standard form is x_ ¼ f ðx, pÞ ¼ a0 ðx  ai1 Þ4 :

ð5:278Þ

(iv1) The equilibrium of x ¼ ai1 is unstable (a fourth-order upper saddle, d4 f =dx4 jx ¼ai > 0). Such a flow is called a fourth-order upper-saddle 1 flow at x ¼ ai1 . The bifurcation of equilibrium at x ¼ ai1 for a bundle switching of one second-order and two simple equilibriums of x = a1,2,3 is called a fourth-order upper-saddle bundle-switching bifurcation at a point p = p1. (iv2) The equilibrium of x ¼ ai1 is unstable (a fourth-order lower saddle, d4 f =dx4 jx ¼ai < 0). Such a flow is called a fourth-order lower-saddle 1 flow at x ¼ ai1 . The bifurcation of equilibrium at x ¼ ai1 for a bundle switching of one second-order and two simple equilibriums of x = a1,2,3 is called a fourth-order lower-saddle bundle-switching bifurcation at a point p = p1. From Definition 5.13, stability and bifurcations of equilibriums in the 1dimensional, quartic nonlinear dynamical system are presented in Fig. 5.29. For a0 > 0, the bifurcations and stability of equilibriums are presented in Fig. 5.29a–c. In Fig. 5.29a, there is a switching bifurcation network with two third-order source switching bifurcations and one upper-saddle-node bifurcation. The two third-order source (3rd SO) switching bifurcations are for (LS:SO) switching to (US:SO) equilibriums and for (SO:US) switching to (SO:LS)equilibriums. The upper-saddle-node (USN) switching bifurcation is for two simple equilibriums. In Fig. 5.29b, a fourth-order upper-saddle (4th US) bundleswitching bifurcation for (SI:LS:SO) switching to (SO:LS:SI) equilibriums. In Fig. 5.29c, a fourth-order upper-saddle (4th US) bundle-switching bifurcation for (SI:SO:US) switching to (SO:SI:US) equilibriums. Similarly, for a0 < 0, the bifurcations and stability of equilibriums are presented in Fig. 5.29d–f. In Fig. 5.29d, the switching bifurcation network consists of two third-order sink switching bifurcations and one lower-saddle-node bifurcation. The two thirdorder sink (3rd SI) switching bifurcations are for the (LS:SO) switching to (US:SO)-equilibriums and for the (SO:US) switching to (SO:LS)-equilibriums. The upper-saddle-node (USN) switching bifurcation is for two simple equilibriums. In Fig. 5.29e, a fourth-order lower-saddle (4th LS) bundle-switching bifurcation for (SO:US:SI) switching to (SI:US:SO) equilibriums. In Fig. 5.29f, a fourth-order lower-saddle (4th LS) bundle-switching bifurcation for (SO:SI: LS) switching to (SI:SO:LS) equilibriums.

5.4 Quartic Nonlinear Systems

227

Fig. 5.29 Stability and bifurcations of equilibriums in the 1-dimensional, quartic nonlinear dynamical system (a0 > 0): (a) two third-order SO and USN switching bifurcation network, (b) fourth-order bundle-switching bifurcation, (c) fourth-order US bundle-switching bifurcation; (a0 < 0): (d) two third-order SI and LSN switching bifurcation network, (e) fourth-order LS bundle-switching bifurcation, (f) fourth-order LS bundle switching. LSN: lower-saddle-node, USN: upper-saddle-node, SI: sink, SO: source. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

228

5 Low-Degree Polynomial Systems

For the switching bifurcation between the third-order and simple equilibriums, the following definition is given for the 1-dimensional, quartic nonlinear dynamical system. Definition 5.14 Consider a 1-dimensional, quartic nonlinear dynamical system x_ ¼ AðpÞx4 þ BðpÞx3 þ CðpÞx2 þ DðpÞx þ EðpÞ ¼ a0 ðpÞðx  aÞ3 ðx  bÞ

ð5:279Þ

where A(p) 6¼ 0, and p ¼ ðp1 , p2 , . . . , pm ÞT :

ð5:280Þ

fa1 , a2 g ¼ sortfa, bg, ai < aiþ1 i1 , i2 2 f1, 2g,

ð5:281Þ

(i) If

the quartic nonlinear dynamical system has a standard form as x_ ¼ f ðx, pÞ ¼ a0 ðx  ai1 Þ3 ðx  ai2 Þ:

ð5:282Þ

(i1) The equilibrium of x ¼ ai1 is unstable (a third-order source, d3 f =dx3 jx ¼ai > 0). The equilibrium of x ¼ ai2 is stable (a sink, 1 df =dxjx ¼ai < 0). Such a flow is called a (3rd SO:SI) or an (SI:3rd SO) flow. 2 (i2) The equilibrium of x ¼ ai1 is stable (a third-order sink, d3 f =dx3 jx ¼ai < 0). The equilibrium of x ¼ ai2 is unstable (a source, 1 df =dxjx ¼ai > 0). Such a flow is called a (3rd SI:SO) or an (SO:3rd SI) flow. 2

(ii) If a ¼ ai1 , b ¼ ai2 ; Δi1 i2 ¼ ðai1  ai2 Þ2 ¼ 0, ai1 ¼ ai2 ;

ð5:283Þ

i1 , i2 2 f1, 2g, i1 6¼ i2 ; the corresponding standard form is x_ ¼ f ðx, pÞ ¼ a0 ðx  ai1 Þ4

ð5:284Þ

(ii1) The equilibrium of x ¼ ai1 is unstable (a fourth-order upper saddle, d 4 f =dx4 jx ¼ai > 0). Such a flow is called a fourth-order upper-saddle 1 flow. The bifurcation of equilibrium at x ¼ ai1 for a switching of one

Reference

229 a0 < 0

a0 > 0

SI

SO 3rd SO

3rd SI

4th US

3rd SO SI

3rd SI

b

SO

a

4th LS

a x*

x* || p ||

b

D12 > 0 D12 = 0 D 21 > 0

a

|| p ||

D12 > 0 D12 = 0 D 21 > 0

b

Fig. 5.30 Stability and bifurcations of equilibriums in the 1-dimensional, quartic nonlinear dynamical system (a0 > 0): (a) fourth-order US switching bifurcation of (3rd SI:SO) to (SI:3rd SO), (a0 < 0): (b) fourth-order LS switching bifurcation of (3rd SO:SI) to (SO:3rd SI). LS: lower-saddle, US: upper-saddle, SI: sink, SO: source. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

third-order and one simple equilibriums of x = a1,2 is called a fourth-order upper-saddle switching bifurcation at a point p = p1. (ii2) The equilibrium of x ¼ ai1 is unstable (a fourth-order lower saddle, d 4 f =dx4 jx ¼ai < 0). Such a flow is called a fourth-order lower-saddle 1 flow. The bifurcation of equilibrium at x ¼ ai1 for a switching of one third-order and one simple equilibriums of x = a1,2 is called a fourth-order lower-saddle switching bifurcation at a point p = p1. From Definition 5.14, the stability and bifurcations of equilibriums in the 1dimensional, quartic nonlinear dynamical system are presented in Fig. 5.30. In Fig. 5.30(a), the fourth-order upper-saddle (4th US) switching bifurcation for a0 > 0 is presented for the third-order sink (3rd SI) with simple source (SO) equilibriums (i.e., (3rd SI:SO)) to the third-order source (3rd SO) with a simple sink (SI) equilibriums (i.e., (SI:3rd SO)). Similarly, in Fig. 5.30(b), the fourthorder lower-saddle (4th LS) switching bifurcation for a0 < 0 is presented for the third-order source (3rd SO) with simple sink (SI) equilibriums (i.e., (3rd SO:SI)) to the 3rd order sink (3rd SI) with a simple source (SO) equilibriums (i.e., (SO:3rd SI)). For the switching bifurcation between the two second-order equilibriums, the following definition was presented in Definition 5.10, and the corresponding illustrations are presented in Fig. 5.24.

Reference Luo, A.C.J., 2019, The stability and bifurcation of equilibriums in low-degree polynomial systems, Journal of Vibration Testing and System Dynamics, 3(4), pp. 403–451.

Chapter 6

(2m)th-Degree Polynomial Systems

In this chapter, the global stability and bifurcations of equilibriums in the (2m)thdegree polynomial system are discussed for a better understanding of the complexity of bifurcations and stability of equilibriums. The appearing and switching bifurcations for simple equilibriums are presented, and the appearing and switching bifurcations for higher order equilibriums are discussed as well. The parallel appearing bifurcations, spraying-appearing bifurcations, and sprinkler-spraying-appearing bifurcations for simple and higher order equilibriums are presented. The antennaswitching bifurcations for simple and higher order equilibriums are discussed and the parallel straw-bundle-switching bifurcations and flower-bundle-switching bifurcations for simple and higher order equilibriums are presented as well.

6.1

Global Stability and Bifurcations

In a similar fashion of low-degree polynomial systems in Chap. 5, the global stability and bifurcation of equilibriums in the (2m)th-degree polynomial nonlinear systems are discussed as in Luo (2020). The stability and bifurcation of each individual equilibrium are analyzed from the local analysis in Chaps. 1–3. Definition 6.1 Consider a (2m)th-degree polynomial nonlinear system: x_ ¼ A0 ðpÞx2m þ A1 ðpÞx2m1 þ    þ A2m2 ðpÞx2 þ A2m1 x þ A2m ðpÞ ¼ a0 ðpÞ½x2 þ B1 ðpÞx þ C1 ðpÞ  ½x2 þ Bm ðpÞx þ Cm ðpÞ

ð6:1Þ

where A0(p) 6¼ 0, and

© Springer Nature Switzerland AG 2019 A. C. J. Luo, Bifurcation and Stability in Nonlinear Dynamical Systems, Nonlinear Systems and Complexity 28, https://doi.org/10.1007/978-3-030-22910-8_6

231

6 (2m)th-Degree Polynomial Systems

232


T p ¼ p1 , p2 , . . . , pm 1 :

ð6:2Þ

Δi ¼ B2i  4C i < 0 for i ¼ 1, 2, . . . , m,

ð6:3Þ

(i) If

the 1-dimensional nonlinear dynamical system with a (2m)th-degree polynomial does not have any equilibrium, and the corresponding standard form is 1 2 1 1 2 1 x_ ¼ a0 ½ðx þ B1 Þ þ ðΔ1 Þ  ½ðx þ Bm Þ þ ðΔm Þ: 2 4 2 4

ð6:4Þ

The flow of such a system without equilibriums is called a nonequilibrium flow. (a) If a0 > 0, the nonequilibrium flow is called a positive flow. (b) If a0 < 0, the nonequilibrium flow is called a negative flow. (ii) If Δi ¼ B2i  4C i > 0, i ¼ i1 , i2 , . . . , il 2 f1, 2, . . . , mg, Δj ¼ B2j  4C j < 0, j ¼ ilþ1 , ilþ2 , . . . , im 2 f1, 2, . . . , mg

ð6:5Þ

with l 2 f0, 1, . . . , mg, the 1-dimensional, (2m)th-degree polynomial system has 2l-equilibriums as pffiffiffiffiffi pffiffiffiffiffi 1 1 ðiÞ ðiÞ x ¼ b1 ¼  ðBi þ Δi Þ, x ¼ b2 ¼  ðBi  Δi Þ 2 2 i 2 fi1 , i2 , . . . , il g  f1, 2, . . . , mg:

ð6:6Þ

(ii1) If ðjÞ bðiÞ r 6¼ bs for r,s 2 f1; 2g;i,j ¼ 1,2, . . . , l ð1Þ

ð1Þ

ðlÞ

ðlÞ

fa1 ; a2 ; . . . ; a2l g ¼ sortfb1 ; b2 ; . . . ; b1 ; b2 g, as < asþ1 ,

ð6:7Þ

then the corresponding standard form is x_ ¼ a0

l Y i¼1

ðx  a2i1 Þðx  a2i Þ

m Y

1 2 1 ½ðx þ Bik Þ þ ðΔik Þ: 2 4 k¼lþ1

ð6:8Þ

(a) If a0 > 0, the simple equilibrium separatrix flow is called a (SI : SO :   : SI : SO :   SI : SO) -flow.

6.1 Global Stability and Bifurcations

233

(b) If a0 < 0, the simple equilibrium separatrix flow is called a (SO : SI :   : SO : SI :   SO : SI) -flow. (ii2) If ð1Þ

ð1Þ

ðlÞ

ðlÞ

fa1 ; a2 ; . . . ; a2l g ¼ sortfb1 ; b2 ; . . . ; b1 ; b2 g, ai 1  a1 ¼    ¼ al 1 , ai2  al1 þ1 ¼    ¼ al1 þl2 ,

ð6:9Þ

⋮ air  aΣr1 ¼    ¼ aΣr1 ¼ a2l i¼1 li þ1 i¼1 li þlr r with Σs¼1 ls ¼ 2l,

then the corresponding standard form is x_ ¼ a0

r Y

ðx  ais Þls

s¼1

m Y

1 1 2 ½ðx þ Bik Þ þ ðΔik Þ: 2 4 k¼lþ1

ð6:10Þ

The equilibrium separatrix flow is called an (l1th XX : l2th XX :    : lrth XX)-flow. (a) For a0 > 0 and p ¼ 1, 2, . . . , r,

lp th XX ¼

f





2r p  1

th th

order source, for αp ¼ 2M p  1, lp ¼ 2r p  1;

2r p  1 order sink, for αp ¼ 2M p , lp ¼ 2r p  1; th 2r p order lower‐saddle, for αp ¼ 2M p  1, lp ¼ 2r p ;
th 2r p order upper‐saddle, for αp ¼ 2M p , lp ¼ 2r p ;


,

ð6:11Þ where αp ¼

Xr

l: s¼p s

ð6:12Þ

(b) For a0 < 0 and p ¼ 1, 2, . . . , r,

lp th XX ¼

f





2r p  1

th th

order sink, for αp ¼ 2M p  1, lp ¼ 2r p  1;

2r p  1 order source, for αp ¼ 2M p , lp ¼ 2r p  1; th 2r p order upper‐saddle, for αp ¼ 2M p  1, lp ¼ 2r p ;
th 2r p order lower‐saddle, for αp ¼ 2M p , lp ¼ 2r p :


ð6:13Þ

6 (2m)th-Degree Polynomial Systems

234

(c) The equilibrium of x ¼ aip for (lp
> 1)-repeated equilibriums switching is called an lpth XX bifurcation of lp1 th XX : lp2 th XX :    : lpβ th XXÞ equilibrium switching at a point p ¼ p1 2 ∂Ω12, and the bifurcation condition is aip  aΣp1 li þ1 ¼    ¼ aΣp1 li þlp , i¼1 i¼1 Xβ   l : aΣp1 l þ1 6¼    6¼ aΣp1 l þl ; lp ¼ i¼1 pi i¼1 i

i¼1 i

ð6:14Þ

p

(iii) If Δi ¼ B2i  4Ci ¼ 0, i 2 fi11 , i12 , . . . , i1s g  fi1 , i2 , . . . , il g  f1, 2, . . . , mg, Δk ¼ B2k  4C k > 0, k 2 fi21 , i22 , . . . , i2r g  fi1 , i2 , . . . , il g  f1, 2, . . . , mg, Δj ¼ B2j  4Cj < 0, j 2 filþ1 , ilþ2 , . . . , im g  f1, 2, . . . , mg, ð6:15Þ the 1-dimensional, (2m)th-degree polynomial system has 2l-equilibriums as 1 1 ðiÞ ðiÞ x ¼ b1 ¼  Bi , x ¼ b2 ¼  Bi for i 2 fi11 , i12 , . . . , i1s g, 2 2 pffiffiffiffiffiffi  pffiffiffiffiffiffi 1 1 ðk Þ ðk Þ  x ¼ b1 ¼  ðBk þ Δk Þ, x ¼ b2 ¼  ðBk  Δk Þ 2 2 for k 2 fi21 , i22 , . . . , i2r g:

ð6:16Þ

If ð1Þ

ð1Þ

ðlÞ

ðlÞ

fa1 ; a2 ; . . . ; a2l g ¼ sortfb1 ; b2 ; . . . ; b1 ; b2 g, ai 1  a1 ¼    ¼ al 1 , ai2  al1 þ1 ¼    ¼ al1 þl2 , ⋮

ð6:17Þ

air  aΣr1 ¼    ¼ aΣr1 ¼ a2l i¼1 li þ1 i¼1 li þlr r with Σs¼1 ls ¼ 2l,

then the corresponding standard form is x_ ¼ a0

r Y s¼1

ðx  ais Þls

m Y

1 2 1 ½ðx þ Bik Þ þ ðΔik Þ: 2 4 k¼lþ1

ð6:18Þ

6.1 Global Stability and Bifurcations

235

The equilibrium separatrix flow is called an (l1thXX : l2thXX :    : lrth XX)flow. (a) The equilibrium of x ¼ aip for (lp > 1)-repeated equilibriums appearing (or vanishing) is called an lpthXX bifurcation of equilibrium at a point p ¼ p1 2 ∂Ω12, and the bifurcation condition is 1 aip  aΣp1 li þ1 ¼    ¼ aΣp1 li þlp ¼  Biq , i¼1 i¼1 2
 2 with Δiq ¼ Biq  4C iq ¼ 0 iq 2 fi1 , i2 ,   , il g ,

ð6:19Þ

6¼    6¼ aþ or a 6¼    6¼ a : aþ Σp1 l þ1 Σp1 l þl Σp1 l þ1 Σp1 l þl i¼1 i

i¼1 i

i¼1 i

p

i¼1 i

p

(b) The equilibrium of x ¼ aiq for (lq > 1)-repeated equilibriums switching is called an lqth XX bifurcation of ðlq1 th XX : lq2 th XX :    : lqβ th XXÞ equilibrium switching at a point p ¼ p1 2 ∂Ω12, and the bifurcation condition is aiq  aΣq1 li þ1 ¼    ¼ aΣq1 li þlp , i¼1

a l þ1 Σq1 i¼1 i

i¼1

6¼    6¼

a ; l þlq Σq1 i¼1 i

lq ¼

Xβ

ð6:20Þ

l : i¼1 qi


  (c) The
equilibrium  of x ¼ aip for lp1 1 -repeated
equilibriums appearing and lp2 2 -repeated equilibriums switching of lp21 th XX : lp22 th XX :    : lp2β th XXÞ is called an lpth XX bifurcation of equilibrium at a point p ¼ p1 2 ∂Ω12, and the bifurcation condition is aip  aΣp1 li þ1 ¼    ¼ aΣp1 li þlp i¼1

with Δiq ¼

i¼1

B2iq

 4Ciq ¼ 0 ðiq 2 fi1 ; i2 ; . . . ; il gÞ

aþ 6¼    6¼ aþ or ap1 1 Σp1 l þj Σp1 l þj i¼1 i

1

i¼1 i

p1

Σi¼1 li þj1

for fj1 ; j2 ; . . . ; jp1 g  f1; 2; . . . ; lp g, a 6¼    6¼ a Σp1 l þk Σp1 l þk i¼1 i

1

i¼1 i

p2

for fk1 ; k2 ; . . . ; k p2 g  f1; 2; . . . ; lp g, Pβ with lp1 þ lp2 ¼ lp ; lp2 ¼ i¼1 lp2i :

6¼    6¼ ap1 1

Σi¼1 li þjp1

, ð6:21Þ

6 (2m)th-Degree Polynomial Systems

236

(iv) If Δi ¼ B2i  4Ci > 0 for i ¼ 1, 2, . . . , m

ð6:22Þ

the 1-dimensional, (2m)th-degree polynomial system has (2m) equilibriums as pffiffiffiffiffi pffiffiffiffiffi 1 1 ðiÞ ðiÞ x ¼ b1 ¼  ðBi þ Δi Þ, x ¼ b2 ¼  ðBi  Δi Þ 2 2 for i ¼ 1, 2, . . . , m:

ð6:23Þ

(iv1) If ðjÞ bðiÞ r 6¼ bs for r,s 2 f1; 2g;i,j ¼ 1,2, . . . , m ð1Þ

ð1Þ

ðmÞ

ðmÞ

fa1 ; a2 ; . . . ; a2m g ¼ sortfb1 ; b2 ; . . . ; b1 ; b2 g, as < asþ1 ,

ð6:24Þ

the corresponding standard form is x_ ¼ a0 ðx  a1 Þðx  a2 Þðx  a3 Þðx  a4 Þ  ðx  a2m1 Þðx  a2m Þ:

ð6:25Þ

Such a flow is formed with all the simple equilibriums. (a) If a0 > 0, the simple equilibrium separatrix flow is called a (SO : SI :   : SO : SI :   SO : SI) -flow. (b) If a0 < 0, the simple equilibrium separatrix flow is called a (SI : SO :   : SI : SO :   SI : SO) -flow. (iv2) If ð1Þ

ð1Þ

ðmÞ

ðmÞ

fa1 ; a2 ; . . . ; a2m g ¼ sortfb1 ; b2 ; . . . ; b1 ; b2 g, ai 1  a1 ¼    ¼ al 1 , ai2  al1 þ1 ¼    ¼ al1 þl2 , ⋮

ð6:26Þ

air  aΣr1 ¼    ¼ aΣr1 ¼ a2m i¼1 li þ1 i¼1 li þlr r with Σs¼1 ls ¼ 2m,

then the corresponding standard form is x_ ¼ a0

r Y

ðx  ais Þls :

ð6:27Þ

s¼1

The equilibrium separatrix flow is called an (l1th XX : l2th XX :  : lrth XX)flow. The equilibrium of x ¼ aip for lp-repeated equilibriums switching is

6.1 Global Stability and Bifurcations

237


called an lpth XX bifurcation of lp1 th XX : lp2 th XX :    : lpβ th XXÞ equilibrium switching at a point p ¼ p1 2 ∂Ω12, and the bifurcation condition is aip  aΣp1 li þ1 ¼    ¼ aΣp1 li þlp , i¼1

a l þ1 Σp1 i¼1 i

i¼1

6¼    6¼

a ; l þlp Σp1 i¼1 i

lp ¼

Xβ

l : i¼1 pi

ð6:28Þ

Definition 6.2 Consider a 1-dimensional, (2m)th-degree polynomial nonlinear dynamical system as x_ ¼ A0 ðpÞx2m þ A1 ðpÞx2m1 þ    þ A2m2 ðpÞx2 þ A2m1 x þ A2m ðpÞ n Y q ¼ a0 ðpÞ ½x2 þ Bi ðpÞx þ Ci ðpÞ i

ð6:29Þ

i¼1

where A0(p) 6¼ 0, and Xn
T p ¼ p1 , p2 , . . . , pm1 , m ¼ q: i¼1 i

ð6:30Þ

Δi ¼ B2i  4C i < 0 for i ¼ 1, 2, . . . , n

ð6:31Þ

(i) If

the 1-dimensional nonlinear dynamical system with a (2m)th-degree polynomial does not have any equilibriums, and the corresponding standard form is x_ ¼ a0

n Y

1 2 1 q ½ðx þ Bi Þ þ ðΔi Þ i : 2 4 i¼1

ð6:32Þ

The flow of such a system without equilibriums is called a nonequilibrium flow. (a) If a0 > 0, the nonequilibrium flow is called the positive flow. (b) If a0 < 0, the nonequilibrium flow is called the negative flow. (ii) If Δi ¼ B2i  4C i > 0, i 2 fi1 , i2 , . . . , il g  f1, 2, . . . , ng, Δj ¼ B2j  4C j < 0, j 2 filþ1 , ilþ2 , . . . , in g  f1, 2, . . . , ng

ð6:33Þ

6 (2m)th-Degree Polynomial Systems

238

the 1-dimensional, (2m)th-degree polynomial system has 2l-equilibriums as pffiffiffiffiffi pffiffiffiffiffi 1 1 ðiÞ ðiÞ x ¼ b1 ¼  ðBi þ Δi Þ, x ¼ b2 ¼  ðBi  Δi Þ 2 2 i 2 fi1 , i2 , . . . , il g  f1, 2, . . . , ng:

ð6:34Þ

(ii1) If ðjÞ bðiÞ r 6¼ bs for r,s 2 f1; 2g;i,j ¼ 1,2, . . . , l ð1Þ

ð1Þ

ðlÞ

ðlÞ

fa1 ; a2 ; . . . ; a2l g ¼ sortfb1 ; b2 ; . . . ; b1 ; b2 g, as < asþ1 ,

ð6:35Þ

then the corresponding standard form is x_ ¼ a0

2l Y

ðx  as Þls

s¼1

n Y

1 2 1 q ½ðx þ Bik Þ þ ðΔik Þ ik 2 4 k¼lþ1

ð6:36Þ

with ls 2 fqi1 ; qi2 ; . . . ; qil g: The equilibrium separatrix flow is called an (l1th XX : l2th XX :    : l2lth XX) -flow. (a) For a0 > 0 and p ¼ 1, 2, . . . , 2l,

lp th XX ¼

f





2r p  1

th th

order source, for αp ¼ 2M p  1, lp ¼ 2r p  1;

2r p  1 order sink, for αp ¼ 2M p , lp ¼ 2r p  1; th 2r p order lower‐saddle, for αp ¼ 2M p  1, lp ¼ 2r p ;
th 2r p order upper‐saddle, for αp ¼ 2M p , lp ¼ 2r p ;


,

ð6:37Þ where αp ¼

X2l

l: s¼p s

ð6:38Þ

(b) For a0 < 0 and p ¼ 1, 2, . . . , 2l,

lp th XX ¼

f





2r p  1

th th

order sink, for αp ¼ 2M p  1, lp ¼ 2r p  1;

2r p  1 order source, for αp ¼ 2M p , lp ¼ 2r p  1; th 2r p order upper‐saddle, for αp ¼ 2M p  1, lp ¼ 2r p ;
th 2r p order lower‐saddle, for αp ¼ 2M p , lp ¼ 2r p :


ð6:39Þ

6.1 Global Stability and Bifurcations

239

(ii2) If ð1Þ

ð1Þ

ðlÞ

ðlÞ

fa1 ; a2 ; . . . ; a2l g ¼ sortfb1 ; b2 ; . . . ; b1 ; b2 g, ai 1  a1 ¼    ¼ al 1 , ai2  al1 þ1 ¼    ¼ al1 þl2 ,

ð6:40Þ

⋮ ¼    ¼ aΣr1 ¼ a2l air  aΣr1 i¼1 li þ1 i¼1 li þlr r with Σs¼1 ls ¼ 2l,

then the corresponding standard form is x_ ¼ a0

r Y

ðx  ais Þls

s¼1

n Y

qi 1 2 1 ½ðx þ Bik Þ þ ðΔik Þ : 2 4 k¼lþ1

ð6:41Þ

The equilibrium separatrix flow is called an (l1th XX : l2th XX :    : lrth XX)-flow. (a) For a0 > 0 and s ¼ 1, 2, . . . , r,

lp th XX ¼

f





2r p  1

th th

order source, for αp ¼ 2M p  1, lp ¼ 2r p  1;

2r p  1 order sink, for αp ¼ 2M p , lp ¼ 2r p  1; th 2r p order lower‐saddle, for αp ¼ 2M p  1, lp ¼ 2r p ;
th 2r p order upper‐saddle, for αp ¼ 2M p , lp ¼ 2r p ;


ð6:42Þ

where αp ¼

Xr

l: s¼p s

ð6:43Þ

(b) For a0 < 0 and p ¼ 1, 2, . . . , r,

lp th XX ¼

f





2r p  1

th th

order sink, for αp ¼ 2M p  1, lp ¼ 2r p  1;

2r p  1 order source, for αp ¼ 2M p , lp ¼ 2r p  1; th 2r p order upper‐saddle, for αp ¼ 2M p  1, lp ¼ 2r p ;
th 2r p order lower‐saddle, for αp ¼ 2M p , lp ¼ 2r p :


ð6:44Þ

(c) The equilibrium of x ¼ aip for (lp > 1)-repeated equilibriums switching is called an lpth XX bifurcation of ðlp1 th XX : lp2 th XX :    : lpβ th XXÞ equilibrium switching at a point p ¼ p1 2 ∂Ω12, and the bifurcation condition is

6 (2m)th-Degree Polynomial Systems

240

aip  aΣp1 li þ1 ¼    ¼ aΣp1 li þlp , i¼1

a Σp1 l þ1 i¼1 i

i¼1

6¼    6¼

a ; Σp1 l þlp i¼1 i

lp ¼

Xβ

l : i¼1 pi

ð6:45Þ

(iii) If Δi ¼ B2i  4C i ¼ 0, i 2 fi11 , i12 , . . . , i1s g  fi1 , i2 , . . . , il g  f1, 2, . . . , ng, Δk ¼ B2k  4C k > 0, k 2 fi21 , i22 , . . . , i2r g  fi1 , i2 , . . . , il g  f1, 2, . . . , ng, Δj ¼ B2j  4C j < 0, j 2 filþ1 , ilþ2 , . . . , in g  f1, 2, . . . , ng with i 6¼ j 6¼ k, ð6:46Þ the 1-dimensional, (2m)th-degree polynomial system has 2l-equilibriums as 1 1 ðiÞ ðiÞ x ¼ b1 ¼  Bi , x ¼ b2 ¼  Bi for i 2 fi11 , i12 , . . . , i1s g, 2 2 pffiffiffiffiffiffi  pffiffiffiffiffiffi 1 1 ðk Þ ðk Þ  x ¼ b1 ¼  ðBk þ Δk Þ, x ¼ b2 ¼  ðBk  Δk Þ 2 2 for k 2 fi21 , i22 , . . . , i2r g:

ð6:47Þ

If ð1Þ

ð1Þ

ðlÞ

ðlÞ

fa1 ; a2 ; . . . ; a2l g ¼ sortfb1 ; b2 ; . . . ; b1 ; b2 g, ai 1  a1 ¼    ¼ al 1 , ai2  al1 þ1 ¼    ¼ al1 þl2 , ⋮

ð6:48Þ

¼    ¼ aΣr1 ¼ a2l air  aΣr1 i¼1 li þ1 i¼1 li þlr r with Σs¼1 ls ¼ 2l,

then the corresponding standard form is x_ ¼ a0

r Y s¼1

ðx  ais Þls

n Y

qi 1 1 2 ½ðx þ Bik Þ þ ðΔik Þ : 2 4 k¼lþ1

ð6:49Þ

The equilibrium separatrix flow is called an (l1th XX : l2th XX :    : lrth XX) -flow. (a) The equilibrium of x ¼ aip for (lp > 1)-repeated equilibriums appearing (or vanishing) is called an lpth XX bifurcation of equilibrium at a point p ¼ p1 2 ∂Ω12, and the bifurcation condition is

6.1 Global Stability and Bifurcations

241

1 aip  aΣp1 li þ1 ¼    ¼ aΣp1 li þlp ¼  Biq i¼1 i¼1 2 ð6:50Þ

with Δiq ¼ B2iq  4C iq ¼ 0 ðiq 2 fi1 ; i2 ;   ; il gÞ, aþ 6¼    6¼ aþ or a 6¼    6¼ a : Σp1 l þ1 Σp1 l þl Σp1 l þ1 Σp1 l þl i¼1 i

i¼1 i

i¼1 i

p

i¼1 i

p

(b) The equilibrium of x ¼ aiq for (lq > 1)-repeated equilibriums switching is called an lqth XX bifurcation of ðlq1 th XX : lq2 th XX :    : lqβ th XXÞequilibrium switching at a point p ¼ p1 2 ∂Ω12, and the bifurcation condition is aiq  aΣq1 li þ1 ¼    ¼ aΣq1 li þlq , i¼1

i¼1

a 6¼    6¼ a ; lq ¼ Σq1 l þ1 Σq1 l þl i¼1 i

i¼1 i

Xβ

l : i¼1 qi

q

ð6:51Þ

(iv) If Δi ¼ B2i  4C i > 0 for i ¼ 1, 2, . . . , n

ð6:52Þ

the 1-dimensional, (2m)th-degree polynomial system has (2n)-equilibriums as pffiffiffiffiffi pffiffiffiffiffi 1 1 ðiÞ ðiÞ x ¼ b1 ¼  ðBi þ Δi Þ, x ¼ b2 ¼  ðBi  Δi Þ 2 2 for i ¼ 1, 2, . . . , n:

ð6:53Þ

(iv1) If ðjÞ bðiÞ r 6¼ bs for r,s 2 f1; 2g;i,j ¼ 1,2, . . . , n ð1Þ

ð1Þ

ðnÞ

ðnÞ

fa1 ; a2 ; . . . ; a2n g ¼ sortfb1 ; b2 ; . . . ; b1 ; b2 g, as < asþ1 ,

ð6:54Þ

the corresponding standard form is x_ ¼ a0

2n Y

ðx  as Þls with ls 2 fqi1 ; qi2 ; . . . ; qin g:

ð6:55Þ

s¼1

The equilibrium separatrix flow is called an (l1th XX : l2th XX :    : l2nth XX)-flow.

6 (2m)th-Degree Polynomial Systems

242

(a) For a0 > 0 and p ¼ 1, 2, . . . , 2n, 8 ð2r p  1Þth order source, for αp ¼ 2M p  1,lp ¼ 2r p  1; > > > > < ð2r  1Þth order sink, for α ¼ 2M ,l ¼ 2r  1; p p p p p lp th XX ¼ > ð2r p Þth order lower‐saddle, for αp ¼ 2M p  1,lp ¼ 2r p ; > > > : ð2r p Þth order upper‐saddle, for αp ¼ 2M p ,lp ¼ 2r p ;

ð6:56Þ

where αp ¼

X2n

l: s¼p s

ð6:57Þ

(b) For a0 < 0 and p ¼ 1, 2, . . . , 2n, 8 ð2r p  1Þth order sink, for αp ¼ 2M p  1,lp ¼ 2r p  1; > > > > < ð2r  1Þth order source, for α ¼ 2M ,l ¼ 2r  1; p p p p p lp th XX ¼ th > ð2r p Þ order upper‐saddle, for αp ¼ 2M p  1,lp ¼ 2r p ; > > > : ð2r p Þth order lower‐saddle, for αp ¼ 2M p ,lp ¼ 2r p :

ð6:58Þ

(iv2) If ð1Þ

ð1Þ

ðnÞ

ðnÞ

fa1 ; a2 ; . . . ; a2n g ¼ sortfb1 ; b2 ; . . . ; b1 ; b2 g, ai1  a1 ¼    ¼ al1 , ai2  al1 þ1 ¼    ¼ al1 þl2 , ⋮

ð6:59Þ

¼    ¼ aΣr1 ¼ a2n , air  aΣr1 i¼1 li þ1 i¼1 li þlr r ls ¼ 2n, with Σs¼1

then the corresponding standard form is x_ ¼ a0

r Y

ðx  ais Þls :

ð6:60Þ

s¼1

The equilibrium separatrix flow is called an (l1th XX : l2th XX :    : lrth XX)-flow. The equilibrium of x ¼ aip for lp-repeated equilibriums switching is called a lpth XX bifurcation of ðlp1 th XX : lp2 th XX :    : lpβ th XXÞ equilibrium switching at a point p ¼ p1 2 ∂Ω12, and the bifurcation condition is

6.1 Global Stability and Bifurcations

243

aip  aΣp1 li þ1 ¼    ¼ aΣp1 li þlp , i¼1

a Σp1 l þ1 i¼1 i

i¼1

6¼    6¼

a ; Σp1 l þlp i¼1 i

lp ¼

Xβ

l : i¼1 pi

ð6:61Þ

Definition 6.3 Consider a 1-dimensional, (2m)th-degree polynomial nonlinear dynamical system x_ ¼ A0 ðpÞx2m þ A1 ðpÞx2m1 þ    þ A2m2 ðpÞx2 þ A2m1 x þ A2m ðpÞ r n  Y Y q ðx  cis ðpÞÞls x2 þ Bi ðpÞx þ Ci ðpÞ i ¼ a0 ðpÞ s¼1

ð6:62Þ

i¼rþ1

where A0(p) 6¼ 0, and Xr

l s¼1 s

¼ 2l,

Xn

q i¼rþ1 i


T ¼ ðm  lÞ, p ¼ p1 , p2 , . . . , pm1 :

ð6:63Þ

(i) If Δi ¼ B2i  4C i < 0 for i ¼ r þ 1, r þ 2, . . . , n, fa1 , a2 , . . . , ar g ¼ sortfc1 , c2 , . . . , cr g with ai < aiþ1

ð6:64Þ

the 1-dimensional nonlinear dynamical system with a (2m)th-degree polynomial has equilibriums of x ¼ ais ðpÞ (s ¼ 1, 2, . . . , r), and the corresponding standard form is x_ ¼ a0 ðpÞ

r Y s¼1

ðx  ais Þls

n Y

1 2 1 l ½ðx þ Bi Þ þ ðΔi Þ i : 2 4 i¼rþ1

ð6:65Þ

The equilibrium separatrix flow is called an (l1th XX : l2th XX :    : lrth XX)flow. (a) For a0 > 0 and s ¼ 1, 2, . . . , r, 8 > ð2r p  1Þth order source, for αp ¼ 2M p  1,lp ¼ 2r p  1; > > > > < ð2r  1Þth order sink, for α ¼ 2M ,l ¼ 2r  1; p p p p p lp th XX ¼ th > > ð2r p Þ order lower‐saddle, for αp ¼ 2M p  1,lp ¼ 2r p ; > > > : ð2r p Þth order upper‐saddle, for αp ¼ 2M p ,lp ¼ 2r p ; where

ð6:66Þ

6 (2m)th-Degree Polynomial Systems

244

Xr

αp ¼

l: s¼p s

ð6:67Þ

(b) For a0 < 0 and p ¼ 1, 2, . . . , r, 8 > ð2r p  1Þth order sink, for αp ¼ 2M p  1,lp ¼ 2r p  1; > > > > < ð2r  1Þth order source, for α ¼ 2M ,l ¼ 2r  1; p p p p p lp th XX ¼ th > > ð2r p Þ order upper‐saddle, for αp ¼ 2M p  1,lp ¼ 2r p ; > > > : ð2r p Þth order lower‐saddle, for αp ¼ 2M p ,lp ¼ 2r p :

ð6:68Þ

(ii) If Δi ¼ B2i  4C i > 0, i ¼ j1 , j2 , . . . , js 2 fl þ 1, l þ 2, . . . , ng, Δj ¼ B2j  4C j < 0, j ¼ jsþ1 , jsþ2 , . . . , jn 2 fl þ 1, l þ 2, . . . , ng

ð6:69Þ

with s 2 f1, . . . , n  lg, the 1-dimensional, (2m)th-degree polynomial system has 2n2-equilibriums as pffiffiffiffiffi pffiffiffiffiffi 1 1 ðiÞ ðiÞ x ¼ b1 ¼  ðBi þ Δi Þ, x ¼ b2 ¼  ðBi  Δi Þ 2 2 i 2 fj1 ; j2 ; . . . ; jn1 g  fl þ 1; l þ 2; . . . ; ng:

ð6:70Þ

If ðrþ1Þ

ðrþ1Þ

ðn Þ

ðn Þ

fa1 , a2 , . . . , a2n2 g ¼ sortfc1 , c2 , . . . , c2l , b1 , b2 , . . . , b1 1 , b2 1 g, |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} |fflfflfflfflfflffl{zfflfflfflfflfflffl} qrþ1 sets

ai1  a1 ¼    ¼ al1 ,

qn1 sets

ai2  al1 þ1 ¼    ¼ al1 þl2 , ⋮

ð6:71Þ

ain1  aΣn1 1 li þ1 ¼    ¼ aΣn1 1 li þln ¼ a2n2 i¼1

i¼1

1

1 with Σns¼1 ls ¼ 2n2 ,

then the corresponding standard form is x_ ¼ a0

n1 Y s¼1

ðx  ais Þls

n Y

qi 1 2 1 ½ðx þ Bi Þ þ ðΔi Þ : 2 4 i¼n þ1

ð6:72Þ

2

The equilibrium separatrix flow is called an ðl1 th XX : l2 thXX :    : ln1 th XXÞ-flow.

6.1 Global Stability and Bifurcations

245

(a) For a0 > 0 and p ¼ 1, 2, . . . , r, r + 1, . . ., n1, 8 > ð2r  1Þth order source, for αp ¼ 2M p  1,lp ¼ 2r p  1; > > p > > < ð2r  1Þth order sink, for α ¼ 2M ,l ¼ 2r  1; p p p p p lp th XX ¼ th > > ð2r p Þ order lower‐saddle, for αp ¼ 2M p  1,lp ¼ 2r p ; > > > : ð2r p Þth order upper‐saddle, for αp ¼ 2M p ,lp ¼ 2r p ;

ð6:73Þ

where αp ¼

Xn1

l: s¼p s

ð6:74Þ

(b) For a0 < 0 and p ¼ 1, 2, . . . , r, r + 1, . . ., n1, 8 > ð2r  1Þth order sink, for αp ¼ 2M p  1,lp ¼ 2r p  1; > > p > > < ð2r  1Þth order source, for α ¼ 2M ,l ¼ 2r  1; p p p p p lp th XX ¼ th > > ð2r p Þ order upper‐saddle, for αp ¼ 2M p  1,lp ¼ 2r p ; > > > : ð2r p Þth order lower‐saddle, for αp ¼ 2M p ,lp ¼ 2r p :

ð6:75Þ

(c) The equilibrium of x ¼ aip for (lp > 1)-repeated equilibriums switching is called an lpth XX switching bifurcation of ðlp1 th XX : lp2 th XX :    : lpβ th XXÞ equilibrium at a point p ¼ p1 2 ∂Ω12, and the bifurcation condition is aip  aΣp1 li þ1 ¼    ¼ aΣp1 li þlp , a 6¼    6¼ a : Σp1 l þ1 Σp1 l þl i¼1

i¼1

i¼1 i

i¼1 i

p

ð6:76Þ

(iii) If Δi ¼ B2i  4C i ¼ 0, for i 2 fi11 , i12 , . . . , i1s g  filþ1 , ilþ2 , . . . , in2 g  fl þ 1, l þ 2, . . . , ng, Δk ¼ B2k  4Ck > 0, for k 2 fi21 , i22 , . . . , i2r g  filþ1 , ilþ2 , . . . , in2 g  fl þ 1, l þ 2, . . . , ng, Δj ¼ B2j  4C j < 0, for j 2 fin2 þ1 , in2 þ2 , . . . , in g  fl þ 1, l þ 2, . . . , ng,

ð6:77Þ

6 (2m)th-Degree Polynomial Systems

246

the 1-dimensional, (2m)th-degree polynomial system has 2n2-equilibriums as 1 1 ðiÞ ðiÞ x ¼ b1 ¼  Bi , x ¼ b2 ¼  Bi for i 2 fi11 , i12 , . . . , i1s g, 2 2 pffiffiffiffiffiffi  pffiffiffiffiffiffi 1 1 ðk Þ ðk Þ  x ¼ b1 ¼  ðBk þ Δk Þ, x ¼ b2 ¼  ðBk  Δk Þ 2 2 for i 2 fi21 , i22 , . . . , i2r g:

ð6:78Þ

If ðrþ1Þ

ðrþ1Þ

ðn Þ

ðn Þ

fa1 , a2 , . . . , a2n2 g ¼ sortfc1 , c2 , . . . , c2l , b1 , b2 , . . . , b1 1 , b2 1 g, |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} |fflfflfflfflfflffl{zfflfflfflfflfflffl} qrþ1 sets

qn1 sets

ai1  a1 ¼    ¼ al1 , ai2  al1 þ1 ¼    ¼ al1 þl2 ,

ð6:79Þ

⋮ ain1  aΣn1 1 li þ1 ¼    ¼ aΣn1 1 li þln ¼ a2n2 i¼1

i¼1

1

1 with Σns¼1 ls ¼ 2n2 ,

then the corresponding standard form is x_ ¼ a0

n1 Y

ðx  ais Þls

n Y i¼n2

s¼1

qi 1 2 1 ½ðx þ Bi Þ þ ðΔi Þ : 2 4 þ1

ð6:80Þ

The equilibrium separatrix flow is called an l1 th XX : l2 th XX :    : ln1 th XXÞflow. (a) The equilibrium of x ¼ aip for (qp > 1)-repeated equilibriums appearing or vanishing is called an lpth XX bifurcation of equilibrium at a point p ¼ p1 2 ∂Ω12, and the bifurcation condition is 1 aip  aΣp1 li þ1 ¼    ¼ aΣp1 li þlp ¼  Biq i¼1 i¼1 2
 2 with Δiq ¼ Biq  4Ciq ¼ 0 iq 2 fi1 , i2 , . . . , il g

ð6:81Þ

6¼    6¼ aþ or a 6¼    6¼ a : aþ Σp1 q þ1 Σp1 q þq Σp1 q þ1 Σp1 q þq i¼1

i

i¼1

i

p

i¼1

i

i¼1

i

p

(b) The equilibrium of x ¼ aiq for (lp > 1)-repeated equilibriums switching is called an lpth XX bifurcation of ðlp1 th XX : lp2 th XX :    : lpβ th XXÞ equilibrium switching at a point p ¼ p1 2 ∂Ω12, and the bifurcation condition is

6.1 Global Stability and Bifurcations

247

aiq  aΣq1 li þ1 ¼    ¼ aΣq1 li þlp , a 6¼    6¼ a ,l ¼ Σq1 l þ1 Σq1 l þl p i¼1

i¼1 i

i¼1

i¼1 i

Xβ

l : i¼1 pi

q

ð6:82Þ

(c) The equilibrium of x ¼ aip for ðlp1 1Þ-repeated equilibriums appearing (or vanishing) and ðlp2 2Þ-repeated equilibriums switching of ðlp21 th XX : lp22 th XX :    : lp2β th XXÞ is called an qpth XX bifurcation of equilibrium at a point p ¼ p1 2 ∂Ω12, and the bifurcation condition is aip  aΣp1 q þ1 ¼    ¼ aΣp1 q þq i

i¼1

with Δiq ¼

B2iq

i¼1

i

p

 4C iq ¼ 0 ðiq 2 fi1 ; i2 ;   ; il gÞ

aþ 6¼    6¼ aþ or ap1 1 Σp1 q þj Σp1 q þj i¼1

i

1

i¼1

p1

i

Σi¼1 qi þj1

6¼    6¼ ap1 1

Σi¼1 qi þjp1

, ð6:83Þ

for fj1 , j2 , . . . , jp1g  f1; 2; . . . ; qp g, a 6¼    6¼ a Σp1 q þk Σp1 q þk i¼1

i

1

i¼1

i

p2

for fk 1 ; k 2 ; . . . ; kp2 g  f1; 2; . . . ; qp g, with lp1 þ lp2 ¼ lp : (iv) If Δi ¼ B2i  4Ci > 0 for i ¼ l þ 1, l þ 2, . . . , n

ð6:84Þ

the 1-dimensional, (2m)th-degree polynomial system has (2m)-equilibriums as pffiffiffiffiffi pffiffiffiffiffi 1 1 ðiÞ ðiÞ x ¼ b1 ¼  ðBi þ Δi Þ, x ¼ b2 ¼  ðBi  Δi Þ 2 2 for i ¼ l þ 1, l þ 2, . . . , n:

ð6:85Þ

If ðrþ1Þ

ðrþ1Þ

ðnÞ

ðnÞ

fa1 , a2 , . . . , a2m g ¼ sortfc1 , c2 , . . . , c2l , b1 , b2 , . . . , b1 , b2 g, |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} |fflfflfflffl{zfflfflfflffl} ai 1  a1 ¼    ¼ a l 1 , ai2  al1 þ1 ¼    ¼ al1 þl2 , ⋮ ain  aΣn1 ¼    ¼ aΣn1 ¼ a2m i¼1 li þ1 i¼1 li þlr with Σns¼1 ls ¼ 2m,

qrþ1 sets

qn sets

ð6:86Þ

6 (2m)th-Degree Polynomial Systems

248

then the corresponding standard form is n Y

x_ ¼ a0

ðx  ais Þls :

ð6:87Þ

s¼1

The equilibrium separatrix flow is called an (l1th XX : l2th XX :    : lnth XX)-flow. The equilibrium of x ¼ aip for lp- repeated equilibriums switching is called an lpth XX switching bifurcation of ðlp1 th XX : lp2 thXX :    : lpβ th XXÞ equilibrium at a point p ¼ p1 2 ∂Ω12, and the bifurcation condition is aip  aΣp1 li þ1 ¼    ¼ aΣp1 li þlp , i¼1

a Σp1 l þ1 i¼1 i

6.2

i¼1

6¼    6¼

a ; Σp1 l þlp i¼1 i

lp ¼

Xβ

l : i¼1 pi

ð6:88Þ

Simple Equilibrium Bifurcations

From the global analysis in Chap. 4, in this section, the bifurcations of simple equilibriums in the (2m)th-degree polynomial systems are discussed, which include appearing/vanishing bifurcations, switching bifurcations, and switching and appearing bifurcations as in Luo (2020).

6.2.1

Appearing Bifurcation

Consider a (2m)th-degree polynomial system in a form of x_ ¼ a0 QðxÞ

n
Y  x2 þ Bi x þ C i :

ð6:89Þ

i¼1

Without loss of generality, a function of Q(x) > 0 is either a polynomial function or a non-polynomial function. The roots of x2 + Bix + Ci ¼ 0 are 1 1 pffiffiffiffiffi ðiÞ Δi , Δi ¼ B2i  4Ci 0 ði ¼ 1, 2,   , nÞ; b1,2 ¼  Bi  2 2 ð1Þ

ð1Þ

ð2Þ

ð2Þ

ðnÞ

ðnÞ

fa1 , a2 ,   , a2l g sortfb1 , b2 , b1 , b2 ,   , b1 , b2 g, as  asþ1 ; Bi 6¼ Bj ði, j ¼ 1, 2,   , n; i 6¼ jÞ  at bifurcation: Δi ¼ 0 ði ¼ 1, 2,   , nÞ

ð6:90Þ

6.2 Simple Equilibrium Bifurcations

249

The second-order singularity bifurcation is for the birth of a pair of simple sink and source. There are two appearing bifurcations for i 2 {1, 2,   , n}:

ith quadratic factor

2 order US ! nd

appearing bifurcation



ith quadratic factor

2 order LS ! nd

appearing bifurcation

SO, for x ¼ a2i , SI, for x ¼ a2i1 : SI, for x ¼ a2i , SO, for x ¼ a2i1 :

ð6:91Þ ð6:92Þ

If Q(x) ¼ 1 and n ¼ m, a set of paralleled different simple upper-saddle appearing bifurcations in the (2m)th-degree polynomial nonlinear system is called the m-uppersaddle-node (m-USN) parallel appearing bifurcation. Such a bifurcation is also called the m-upper-saddle-node (m-USN) teeth comb appearing bifurcation. At the appearing bifurcation point, Δi ¼ 0 (i ¼ 1, 2,   , m), and the m-USN teeth comb appearing bifurcation structure is

m‐USN

f

mth bifurcation



US !

SI, for x ¼ a2m1 ;

appearing

⋮ ith bifurcation



US !

1st bifurcation

SO, for x ¼ a2i , SI, for x ¼ a2i1 ;

appearing

⋮

SO, for x ¼ a2m ,



US !

ð6:93Þ

SO, for x ¼ a2 , SI, for x ¼ a1 :

appearing

Similarly, a set of paralleled different simple lower-saddle appearing bifurcations is called the m-lower-saddle-node (m-LSN) parallel appearing bifurcation for the (2m)th-degree polynomial nonlinear system. The lower-saddle-node bifurcation is called the m-lower-saddle-node (m-LSN) teeth comb appearing bifurcation. At the bifurcation point, Δi ¼ 0 (i ¼ 1, 2, . . . , m), and the m-LSN appearing bifurcation structure is

m‐LSN

f

mth bifurcation



LS !

SO, for x ¼ a2m1 ;

appearing

⋮ ith bifurcation



LS !

1st bifurcation

LS ! appearing

SI, for x ¼ a2i , SO, for x ¼ a2i1 ;

appearing

⋮

SI, for x ¼ a2m ,



SI, for x ¼ a2 , SO, for x ¼ a1 :

ð6:94Þ

6 (2m)th-Degree Polynomial Systems

250

Consider an appearing bifurcation for a cluster of sink and source equilibriums with the following conditions: Bi ¼ Bj ði, j 2 f1, 2, . . . , ng; i 6¼ jÞ Δi ¼ 0 ði ¼ 1, 2, . . . , nÞ

 at bifurcation:

ð6:95Þ

Thus, the (2l )th-order appearing bifurcation is for a cluster of simple sinks and sources. Two (2l)th-order appearing bifurcations for l 2 {1, 2, . . ., s} are

th

cluster of l‐quadratics

f f

ð2lÞ order US ! appearing bifurcation

cluster of l‐quadratics

ð2lÞth order LS ! appearing bifurcation

SO, for x ¼ a2sl , SI, for x ¼ a2sl 1 , ⋮

ð6:96Þ

SO, for x ¼ a2s1 , SI, for x ¼ a2s1 1 :

SI, for x ¼ a2sl , SO, for x ¼ a2sl 1 , ⋮

ð6:97Þ 

SI, for x ¼ a2s1 , SO, for x ¼ a2s1 1 :

A set of paralleled, different, higher order upper-saddle-node bifurcations with multiplicity is the ((2l1)thUS : (2l2)thUS :    : (2ls)thUS) parallel appearing bifurcation in the (2m)th-degree polynomial system. (2li)thUS for (i ¼ 1, 2, . . . , s) with sources and sinks is the (2li)th-order upper saddle with li-pairs of simple source and sink equilibriums. With different orders of li-pairs of simple sources and sinks, the (2li)thUSN Pbifurcation possesses different spraying-appearing clusters of sinks and sources. si¼1 li ¼ n  m where s, li 2 {0, 1, 2,   , m}. If li ¼ 1 for i ¼ 1, 2,   , m with n ¼ m, the simple upper-saddle-node parallel bifurcation or the upper-saddlenode teeth comb appearing bifurcation is recovered. Introduce ðð2l1 Þth US : ð2l2 Þth US :    : ð2ls Þth USÞ  ð2l1 : 2l2 :    : 2ls Þth US:

ð6:98Þ

At the sprinkler-spraying-appearing bifurcation, Δi ¼ 0 (i ¼ 1, 2,   , s) and Bi ¼ Bj (i, j 2 {1, 2,   , s}; i 6¼ j). The sprinkler-spraying USN appearing bifurcation is

ð2l1 : 2l2 :    : 2ls Þth US ¼

f

ð2ls Þth order US, ⋮ ð2l2 Þth order US, ð2l1 Þth order US:

ð6:99Þ

6.2 Simple Equilibrium Bifurcations

251

Thus, the (2l1 : 2l2 :    : 2ls)th US appearing (or vanishing) bifurcation is called the (2l1 : 2l2 :    : 2ls)th USN sprinkler-spraying-appearing (or vanishing) bifurcation. Similarly, a set of paralleled different lower-saddle appearing bifurcations with multiplicity is the ((2l1)thLS : (2l2)thLS :    : (2ls)thLS) appearing bifurcation in the (2m)th-degree polynomial system. Thus, the (2l1 : 2l2 :    : 2ls)th LS appearing (or vanishing) bifurcation is also called the (2l1 : 2l2 :    : 2ls)th LS sprinklerspraying-appearing (or vanishing) bifurcation. Again, at the LS sprinkler-spraying bifurcation, Δi ¼ 0 (i ¼ 1, 2, . . . , n) and Bi ¼ Bj (i, j 2 {1, 2, . . ., n}; i 6¼ j). Thus, the sprinkler-spraying LSN appearing bifurcation is

ð2l1 : 2l2 :    : 2ls Þth LS ¼

f

ð2ls Þth order LS, ⋮ ð2l2 Þth order LS,

ð6:100Þ

ð2l1 Þth order LS:

Two m-USN and m-LSN teeth comb appearing bifurcations are presented in Fig. 6.1a, b for a0 > 0 and a0 < 0, respectively. The set of paralleled (4thUS :    : (2r)thUS :    : 4thUS : 6thUS) appearing bifurcations for simple sinks and sources is presented P in Fig. 6.1c for a0 > 0, where l1 ¼ 2, . . ., li ¼ r, . . ., and ls  1 ¼ 2, ls ¼ 3 with si¼1 li ¼ m. The (4 :    : 2r :    : 4 : 6)th-USN appearing bifurcation is a USN sprinkler-spraying-appearing bifurcation. However, for a0 < 0, the (4thLS :    : (2r)thLS :    : 4thLS : 6thLS) appearing bifurcations for simple sources and sinks are presented in Fig. 6.1d. The (4 :    : 2r :    : 4 : 6)th-LSN appearing bifurcation is a LSN sprinkler-spraying-appearing bifurcation. For a cluster of m-quadratics, Bi ¼ Bj (i, j 2 {1, 2, . . ., m}; i 6¼ j) and Δi ¼ 0 (i ¼ 1, 2, . . . , m). The (2m)th-order upper-saddle-node appearing bifurcation for m-pairs of sink and source equilibriums is

cluster of m‐quadratics

ð2mÞth order US ! appearing bifurcation

f

SO, for x ¼ a2m , SI, for x ¼ a2m1 , ⋮ SO, for x ¼ a2 , SI, for x ¼ a1 :

ð6:101Þ

6 (2m)th-Degree Polynomial Systems

252

a

c

b

d

Fig. 6.1 (a) m-USN parallel bifurcation (a0 > 0), (b) m-LSN parallel bifurcation (a0 < 0), (c) ((2l1)thUS : (2l2)thUS :    : (2ls)thUS) parallel bifurcation (a0 > 0), (d) ((2l1)thLS : (2l2)thLS :    : (2ls)thLS) parallel bifurcation (a0 < 0) in a (2m)th-degree polynomial system. LS: lower saddle, US: upper saddle, SI: sink, SO: source. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

6.2 Simple Equilibrium Bifurcations

253

The (2m)th-order lower-saddle-node appearing bifurcation for m-pairs of sink and source equilibriums is

th

cluster of m‐quadratics

ð2mÞ order LS ! appearing bifurcation

f

SI, for x ¼ a2m , SO, for x ¼ a2m1 , ⋮

ð6:102Þ

SI, for x ¼ a2 , SO, for x ¼ a1 :

The (2m)th-order upper-saddle-node appearing bifurcation with m-pairs of simple sources and sinks is a sprinkler-spraying cluster of the m-pairs of simple sources and sinks. The (2m)th-order lower-saddle-node appearing bifurcation with m-pairs of equilibriums is also a sprinkler-spraying cluster of the m-pairs of simple sources and sinks. Thus, the (2m)th-order USN appearing bifurcation (a0 > 0) and (2m)thorder LSN bifurcation (a0 < 0) are presented in Fig. 6.2a, b, respectively. The (2m)thorder upper-saddle-node appearing bifurcation is named the (2m)th-order USN sprinkler-spaying appearing bifurcation, and the (2m)th-order lower-saddle-node appearing bifurcation is named the (2m)th-order LSN sprinkler-spraying-appearing bifurcation. A series of the saddle-node bifurcations are aligned up with varying parameters, which is formed a special pattern. For m-quadratics in the (2m)th-order polynomial system, the following conditions should be satisfied: Bi Bj i, j 2 f1, 2, . . . , ng; i 6¼ j, Δi > Δiþ1 ði ¼ 1, 2, . . . , n; n  mÞ,

ð6:103Þ

Δi ¼ 0 with kpi k < kpiþ1 k:

a

b

Fig. 6.2 (a) (2m)th-order USN bifurcation (a0 > 0), (b) (2m)th-order LSN bifurcation (a0 < 0) in the (2m)th polynomial system. LS: lower saddle, US: upper saddle, SI: sink, SO: source. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

254

6 (2m)th-Degree Polynomial Systems

a

b Fig. 6.3 (a) m ‐ (US ‐ LS ‐ US ‐   ) series bifurcation (a0 > 0), (b) m ‐ (US ‐ LS ‐ US ‐   ) series bifurcation (a0 < 0) in the (2m)th-degree polynomial system. LS: lower saddle, US: upper saddle, SI: sink, SO: source. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

Thus, a series of m ‐ (USN ‐ LSN ‐ USN ‐   ) appearing bifurcations (a0 > 0) and a series of m ‐ (LSN ‐ USN ‐ LSN ‐   ) appearing bifurcations (a0 < 0) are presented in Fig. 6.3a, b. The bifurcation scenario is formed by the swapping pattern of USN and LSN appearing bifurcations. Such a bifurcation scenario is like the fish scale. Thus, such a bifurcation-swapping pattern of the USN and LSN is called the fishscale appearing bifurcation in the (2m)th-degree polynomial nonlinear system. There are two swapping bifurcations: (i) the USN-LSN fish-scale appearing bifurcation and (ii) the LSN-USN fish-scale appearing bifurcation.

6.2 Simple Equilibrium Bifurcations

6.2.2

255

Switching Bifurcations

Consider the roots of x2 + Bix + Ci ¼ 0 as ðiÞ

ðiÞ

ðiÞ

ðiÞ

Bi ¼ ðb1 þ b2 Þ,Δi ¼ ðb1  b2 Þ2 0, ðiÞ

ðiÞ

ðiÞ

ðiÞ



x1,2 ¼ b1,2 ,Δi > 0 if b1 6¼ b2 ði ¼ 1; 2; . . . ; nÞ; Bi 6¼ B j ði; j ¼ 1; 2; . . . ; n; i 6¼ jÞ ðiÞ

ðiÞ

Δi ¼ 0 at b1 ¼ b2 ði ¼ 1; 2; . . . ; nÞ

ð6:104Þ

at bifurcation:

The second-order singularity bifurcation is for the switching of a pair of simple sink and source. There are two switching bifurcations for i 2 {1, 2, . . ., n}: ith quadratic factor



2nd order US ! switching bifurcation



ith quadratic factor

2 order LS ! nd

switching bifurcation

ðiÞ

ðiÞ

SO, for a2i ¼ b2 ! b1 , ðiÞ

ðiÞ

SI, for a2i1 ¼ b1 ! b2 : ðiÞ

ð6:105Þ

ðiÞ

SI, for a2i ¼ b2 ! b1 , ðiÞ

ðiÞ

SO, for a2i1 ¼ b1 ! b2 :

ð6:106Þ

A set of m-paralleled pairs of different simple upper-saddle-node switching bifurcations in the (2m)th-degree polynomial nonlinear system is called the mupper-saddle-node (m-USN) parallel switching bifurcation. Such a bifurcation is also called the m-upper-saddle-node (m-USN) antenna-switching bifurcation. For ðiÞ ðiÞ non-switching point, Δi > 0 at b1 6¼ b2 (i ¼ 1, 2, . . . , n). At the bifurcation point, ðiÞ ðiÞ Δi ¼ 0 at b1 ¼ b2 (i ¼ 1, 2, . . . , n). The m-USN parallel switching bifurcation is

m‐USN

f

( mth bifurcation

US ! switching

⋮

(

US !

switching

¼ a2m1 " a2m ;

ðiÞ

SO # SI, for b2 ¼ a2i # a2i1 , SI " SO, for b1 ¼ a2i1 " a2i ;

(

US !

ðmÞ

ðiÞ

switching

1st bifurcation

¼ a2m # a2m1 ,

SI " SO, for b1

ith bifurcation

⋮

ðmÞ

SO # SI, for b2

ð6:107Þ

ð 1Þ

SO # SI, for b2 ¼ a2 # a1 , ð 1Þ

SI " SO, for b1 ¼ a1 " a2 :

Similarly, a set of paralleled different simple lower-saddle bifurcations is called the m-lower-saddle-node (m-LSN) parallel switching bifurcation for the (2m)thdegree polynomial nonlinear system. The lower-saddle-node switching bifurcation

6 (2m)th-Degree Polynomial Systems

256

is also called the m-lower-saddle-node (m-LSN) antenna switching bifurcation. For ðiÞ ðiÞ non-switching point, Δi > 0 at b1 6¼ b2 (i ¼ 1, 2, . . . , n). At the bifurcation point, ðiÞ ðiÞ Δi ¼ 0 at b1 ¼ b2 (i ¼ 1, 2, . . . , n). The m-LSN antenna switching bifurcation is

m‐LSN

f

( mth bifurcation

LS ! switching

⋮

(

LS !

switching

¼ a2m1 " a2m ;

ðiÞ

SI # SO, for b2 ¼ a2i # a2i1 , SO " SI, for b1 ¼ a2i1 " a2i ;

(

LS !

ðmÞ

ðiÞ

switching

1st bifurcation

¼ a2m # a2m1 ,

SO " SI, for b1

ith bifurcation

⋮

ðmÞ

SI # SO, for b2

ð6:108Þ

ð1Þ

SI # SO, for b2 ¼ a2 # a1 , ð1Þ

SO " SI, for b1 ¼ a1 " a2 :

Consider a switching bifurcation for a bundle of sink and source equilibriums with the following conditions: ðiÞ

ðiÞ

ðiÞ

ðiÞ

Bi ¼ ðb1 þ b2 Þ,Δi ¼ ðb1  b2 Þ2 0, ðiÞ

ðiÞ

ðiÞ

ðiÞ



x1,2 ¼ b1,2 ,Δi > 0 if b1 6¼ b2 ði ¼ 1; 2; . . . ; nÞ; Bi ¼ B j ði; j 2 f1; 2; . . . ; ng; i 6¼ jÞ ðiÞ

ðiÞ

Δi ¼ 0 at b1 ¼ b2 ði ¼ 1; 2; . . . ; nÞ

ð6:109Þ

at bifurcation:

Thus, the (2l)th-order switching bifurcation can be for a bundle of simple sinks and sources. Two (2l )th-order switching bifurcations for l 2 {1, 2, . . ., s} are

a bundle of ð2lÞ‐equilibriums

f f

ð2lÞth order US ! switching bifurcation

a bundle of ð2lÞ‐equilibriums

ð2lÞth order LS ! switching bifurcation

SO, for a2sl ! b2sl , SI, for a2sl 1 ! b2sl 1 , ⋮

ð6:110Þ

SO, for a2s1 ! b2s1 , SI, for a2s1 1 ! b2s1 1 :

SI, for a2sl ! b2sl , SO, for a2sl 1 ! b2sl 1 , ⋮ SI, for a2s1 ! b2s1 ,

SO, for a2s1 1 ! b2s1 1 :

ð6:111Þ

6.2 Simple Equilibrium Bifurcations

257

where Δij ¼ (ai  aj)2 ¼ (bi  bj)2 ¼ 0 with Bi ¼ Bj (i, j ¼ 2s1  1, 2s1, . . ., 2sl  1, 2sl) and fa2s1 1 ; a2s1 ; . . . ; a2sl 1 ; a2sl g fb2s1 1 ; b2s1 ; . . . ; b2sl 1 ; b2sl g

ð1Þ



before bifurcation

ð1Þ

ð1Þ



after bifurcation

ðnÞ

ðnÞ

sortfb1 ; b2 ; . . . ; b1 ; b2 g, ð1Þ

ðnÞ

ðnÞ

sortfb1 ; b2 ; . . . ; b1 ; b2 g:

ð6:112Þ

The (2l  1)th-order switching bifurcation can be for a bundle of simple sinks and sources. Two (2l  1)th-order switching bifurcations for l 2 {1, 2,   , s} are

a bundle of ð2l1Þ‐equilibriums

ð2l  1Þth order SO ! switching bifurcation

f f

SO, for a2sl 1 ! b2sl 1 , ⋮ SI, for a2s1 ! b2s1 ,

ð6:113Þ

SO, for a2s1 1 ! b2s1 1 :

a bundle of ð2l1Þ‐equilibriums

ð2l  1Þth order SI ! switching bifurcation

SI, for a2sl 1 ! b2sl 1 , ⋮ SO, for a2s1 ! b2s1 ,

ð6:114Þ

SI, for a2s1 1 ! b2s1 1 :

where Δij ¼ (ai  aj)2 ¼ (bi  bj)2 ¼ 0 with Bi ¼ Bj (i, j ¼ 2s1  1, 2s1, . . ., 2sl  1) and fa2s1 1 ; a2s1 ; . . . ; a2sl 1 g fb2s1 1 ; b2s1 ; . . . ; b2sl 1 g



ð1Þ

before bifurcation



after bifurcation

ð1Þ

ðnÞ

ðnÞ

sortfb1 ; b2 ; . . . ; b1 ; b2 g, ð1Þ

ð1Þ

ðnÞ

ðnÞ

sortfb1 ; b2 ; . . . ; b1 ; b2 g:

ð6:115Þ

A set of paralleled, different, higher order upper-saddle-node switching bifurcations with multiplicity is the ((α1)thXX : (α2)thXX :    : (αs)thXX) parallel switching bifurcation in the (2m)th-degree polynomial system. At the straw-bundle switching bifurcation, Δi ¼ 0 (i ¼ 1, 2, . . . , n) and Bi ¼ Bj ( i, j 2 {1, 2, . . ., n}; i 6¼ j). Thus, the parallel straw-bundle switching bifurcation is ððα1 Þth XX : ðα2 Þth XX :    : ðαs Þth XXÞ‐switching

¼

f

ðαs Þth order XX switching, ⋮

ð6:116Þ th

ðα2 Þ order XX switching, ðα1 Þth order XX switching;

6 (2m)th-Degree Polynomial Systems

258

where αi 2 f2li , 2li  1g with Σsi¼1 αi ¼ 2m, and XX 2 fUS, LS, SO, SIg:

ð6:117Þ

The (2li)thUS for (i ¼ 1, 2,   , s) with sinks and sources is the (2li)th-order upper saddle for a switching of li-pairs of simple sinks and sources. With different orders of li-pairs of simple sinks and sources, the (2li)thUSN switching bifurcation possesses different straw-bundle switching for a bundle of stable and unstable equilibriums. The (2l1 : 2l2 :    : 2ls)th USN bifurcation is called the (2l1 : 2l2 :    : 2ls)th USN straw-bundle switching bifurcation:

ð2l1 : 2l2 :    : 2ls Þth USN switching ¼

f

ð2ls Þth order USN switching, ⋮ ð2l2 Þth order USN switching,

ð6:118Þ

ð2l1 Þth order USN switching:

If li ¼ 1 for i ¼ 1, 2, . . . , m with n ¼ m, the simple upper-saddle-node parallel switching bifurcation or the upper-saddle-node antenna switching bifurcation is recovered. Similarly, a set of paralleled different lower-saddle switching bifurcations with multiplicity is the ((2l1)thLS : (2l2)thLS :    : (2ls)thLS) parallel switching bifurcation in the (2m)th-degree polynomial system. Thus, the (2l1 : 2l2 :    : 2ls)th LSN switching bifurcation is also called the (2l1 : 2l2 :    : 2ls)th LSN straw-bundle switching bifurcation. Again, at the LSN straw-bundle switching bifurcation, Δi ¼ 0 (i ¼ 1, 2, . . . , n) and Bi ¼ Bj (i, j 2 {1, 2, . . ., n}; i 6¼ j). Thus, the LSN straw-bundle switching bifurcation is

ð2l1 : 2l2 :    : 2ls Þth LSN switching ¼

f

ð2ls Þth order LSN switching, ⋮ ð2l2 Þth order LSN switching,

ð6:119Þ

ð2l1 Þth order LSN switching:

The set of m-upper-saddle-node (m-USN) parallel switching bifurcation is equivalent to the set of (2 : 2 :    : 2)nd-USN bifurcations. The set of m-lower-saddle-node (m-LSN) parallel switching bifurcation is equivalent to the set of (2 : 2 :    : 2)ndLSN bifurcations. Such two sets of parallel switching bifurcations are presented in Fig. 6.4a, b for a0 > 0 and a0 < 0, respectively. A set of paralleled (3rdSO : 2ndLS :    : 4thLS :    : 3rdSI) switching bifurcations for SI and SO equilibriums is presented in Fig. 6.4c for a0 > 0. However, for a0 < 0, the set of (3rdSI : 2ndUS :    : 4thUS :    : 3rdSI) switching bifurcations for sources and sinks is presented in Fig. 6.4d.

6.2 Simple Equilibrium Bifurcations

259

a

b

c

d

Fig. 6.4 Stability and bifurcations of equilibriums in a 1-dimensional (a0 < 0), (2m)th-degree polynomial system: (a) m-USN parallel switching bifurcation (a0 > 0), (b) m-LSN parallel switching bifurcation (a0 < 0), (c) (3rdSO : 2ndLS :    : 3rdSI) parallel switching bifurcation (a0 > 0), (d) (3rdSI : 2ndUS :    : 3rdSO) parallel switching bifurcation (a0 < 0). LS: lower saddle, US: upper saddle, SI: sink, SO: source. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

6 (2m)th-Degree Polynomial Systems

260

6.2.3

Switching and Appearing Bifurcations

Consider a (2m)th-degree 1-dimensional polynomial system in a form of x_ ¼ a0 QðxÞ

2n1 Y

ðx  ci Þ

i¼1

n2 Y

ðx2 þ B j x þ C j Þ:

ð6:120Þ

j¼1

Without loss of generality, a function of Q(x) > 0 is either a polynomial function or a non-polynomial function. The roots of x2 + Bjx + Cj ¼ 0 are 1 1 pffiffiffiffiffi ð jÞ Δj , Δj ¼ B2j  4Cj 0 ð j ¼ 1, 2, . . . , n2 Þ; b1,2 ¼  Bj  2 2

ð6:121Þ

either     fa 1 , a2 , . . . , a2n1 g ¼ sortfc1 , c2 , . . . , c2n1 g, as  asþ1 before bifurcation ð1Þ

ð1Þ

ðn Þ

ðn Þ

2 2 þ þ faþ 1 , a2 , . . . , a2n3 g ¼ sortfc1 , . . . , c2n1 ; b1 , b2 , . . . , b1 , b2 g,

þ aþ s  asþ1 , n3 ¼ n1 þ n2 after bifurcation;

ð6:122Þ or ð1Þ

ð1Þ

ðn Þ

ðn Þ

  2 2 fa 1 , a2 , . . . , a2n3 g ¼ sortfc1 , c2 , . . . , c2n1 ; b1 , b2 , . . . , b1 , b2 g,  a s  asþ1 , n3 ¼ n1 þ n2 before bifurcation;

ð6:123Þ

þ þ þ þ faþ 1 , a2 , . . . , a2n1 g ¼ sortfc1 , . . . , c2n1 g, as  asþ1 after bifurcation;

and
 Bj1 ¼ Bj2 ¼    ¼ Bjs jk1 2 f1, 2, . . . , ng; jk1 6¼ jk2 ðk 1 , k 2 2 f1, 2, . . . , sg; k 1 6¼ k 2 Þ Δj ¼ 0 ðj 2 U f1, 2, . . . , n2 gÞ 1 ci 6¼  Bj ði ¼ 1, 2, . . . , 2n1 , j ¼ 1, 2, . . . , n2 Þ 2

g

at bifurcation: ð6:124Þ

th     th  th Consider a just before bifurcation of ððα 1 Þ XX1 : ðα2 Þ XX2 : . . . : ðαs1 Þ XXs1 Þ    for simple sources and sinks. For αi ¼ 2li  1, XXi 2 fSO, SIg and for

6.2 Simple Equilibrium Bifurcations

261

  α i ¼ 2li , XXi 2 fUS, LSg (i ¼ 1, 2, . . . , s1). The detailed structures are as follows:

SI SO ⋮ SO SI SO SI ⋮ SO SI

g g

g g

SO !




2l i

1

th

SI

SI, and ⋮


th ! 2l i  1 SO;

SI

SO SI !




SO

th 2l US, i

and ⋮ SI SO

ð6:125Þ


th ! 2l LS: i

th     th  th The bifurcation set of ððα 1 Þ XX1 : ðα2 Þ XX2 : . . . : ðαs1 Þ XXs1 Þ at the same parameter point is called a left-parallel-bundle switching bifurcation. th þ þ th þ þ þ th Consider a just after bifurcation of ððαþ 1 Þ XX1 : ðα2 Þ XX2 : . . . : ðαs2 Þ XXs2 Þ þ þ þ for simple sources and sinks. For αi ¼ 2li  1, XXi 2 fSO, SIg and for þ  αþ i ¼ 2li , XXi 2 fUS, LSg. The four detailed structures are as follows:




2lþ i

1

th

SI !


þ th 2li US !

f f

SI SO
th ⋮ , and 2lþ i  1 SO ! SO SI

SO SI ⋮ ,


th and 2lþ LS ! i

SO SI

f

f

SO SI ⋮ ;

ð6:126aÞ

SI SO

SI SO ⋮ :

ð6:126bÞ

SI SO

þ þ þ þ þ The bifurcation set of ððαþ 1 Þ XX1 : ðα2 Þ XX2 : . . . : ðαs2 Þ XXs2 Þ at the same parameter point is called a right-parallel-bundle switching bifurcation. th

th

th

6 (2m)th-Degree Polynomial Systems

262

(i) For the just before and after bifurcation structure, if there exists a relation of th þ th þ  þ th   ðα i Þ XXi ¼ ðα j Þ XX j ¼ α XX, for x ¼ ai ¼ a j

ði 2 f1; 2; . . . ; s1 g; j 2 f1; 2; . . . ; s2 gÞ, XX 2 fUS; LS; SO; SIg

ð6:127Þ

then the bifurcation is a αthXX switching bifurcation for simple equilibriums. (ii) Just for the just before bifurcation structure, if there exists a relation of th th    ð2l i Þ XXi ¼ ð2lÞ XX, for x ¼ ai ¼ ai
i 2 f1; 2; . . . ; s1 g, XX 2 fUS; LSg

ð6:128Þ

then the bifurcation is a (2l)thXX left-appearing (or right-vanishing) bifurcation for simple equilibriums. (iii) Just for the just after bifurcation structure, if there exists a relation of th þ þ  ð2lþ i Þ XXi ¼ ð2lÞ XX, for x ¼ ai ¼ ai th

ði 2 f1; 2; . . . ; s1 gÞ, XX 2 fUS; LSg

ð6:129Þ

then the bifurcation is a (2l)thXX right-appearing (or left-vanishing) bifurcation for simple equilibriums. (iv) For the just before and after bifurcation structure, if there exists a relation of th þ th þ  þ   ðα i Þ XXi 6¼ ðα j Þ XX j for x ¼ ai ¼ a j þ XX i , XX j 2 fUS; LS; SO; SIg

ð6:130Þ

ði 2 f1; 2; . . . ; s1 g; j 2 f1; 2; . . . ; s2 gÞ, then there are two flower-bundle switching bifurcations of simple equilibriums: (iv1) For αj ¼ αi + 2l, the bifurcation is called a αth j XX right flower-bundle switching bifurcation for αi to αj-simple equilibriums with the appearance (birth) of 2l-simple equilibriums. (iv2) For αj ¼ αi  2l, the bifurcation is called a αth i XX left flower-bundle switching bifurcation for αi to αj-simple equilibriums with the vanishing (death) of 2l-simple equilibriums.

6.2 Simple Equilibrium Bifurcations

263

A general parallel switching bifurcation is switching

th   th   th  ððα 1 Þ XX1 : ðα2 Þ XX2 : . . . : ðαs1 Þ XXs1 Þ ! bifucation

th þ ððαþ 1 Þ XX1

:

th þ ðαþ 2 Þ XX2

: ... :

th þ ðαþ s1 Þ XXs2 Þ:

ð6:131Þ

Such a general, parallel switching bifurcation consists of the left and right parallel-bundle switching bifurcations. If the left and right parallel-bundle switching bifurcations are same in a parallel flower-bundle switching bifurcation, i.e., th  þ þ th ðα i Þ XXi ¼ ðαi Þ XXi ¼ α XX, th

þ for x ¼ a i ¼ ai ði ¼ 1, 2, . . . , sg

ð6:132Þ

then the parallel flower-bundle switching bifurcation becomes a parallel strawbundle switching bifurcation of ((α1)thXX : (α2)thXX :    : (αs)thXX). If the left and right parallel-bundle switching bifurcations are different in a parallel flower-bundle switching bifurcation, i.e., th  th þ th þ th  þ ðα i Þ XXi ¼ ð2li Þ XX, ðα j Þ XX j ¼ ð2l j Þ YY, þ for x ¼ a i 6¼ ai ði ¼ 1; 2; . . . ; sg

ð6:133Þ

XX 2 fUS; LSg, YY 2 fUS; LSg then the parallel flower-bundle switching bifurcation becomes a combination of two independent left and right parallel appearing bifurcations: th  th  th    parallel sprinkler(i) A ðð2l 1 Þ XX1 : ð2l2 Þ XX2 :    : ð2ls1 Þ XXs1 Þ-left spraying-appearing (or right vanishing) bifurcation th þ th þ th þ þ þ (ii) A ðð2lþ 1 Þ XX1 : ð2l2 Þ XX2 :    : ð2ls2 Þ XXs2 Þ-right parallel sprinklerspraying-appearing (or left vanishing) bifurcation.

The (6thUS : 4thLS :    : 4thUS : SI) appearing bifurcation for a0 > 0 is presented in Fig. 6.5a. Compared to the case of a0 > 0, the bifurcation and stability conditions of equilibriums for a0 < 0 will be swapped. The (6thLS : 4thUS :    : 4thLS : SO) parallel appearing bifurcation is shown in Fig. 6.5b. Such a kind of bifurcation is like a waterfall appearing bifurcation. The switching and appearing bifurcations of equilibriums exist at the same parameter. A set of paralleled, different switching and appearing bifurcations of higher order equilibrith th ums is also named the ðlth 1 XX : l2 XX :    : ls XXÞ parallel switching and appearing th bifurcation in the (2m) -degree polynomial system. The lith XX switching and

6 (2m)th-Degree Polynomial Systems

264

a0 > 0

SO

a0 < 0

b1(i2 ) b2(i1 )

6th US SO

b1(i1 )

SI

SO

SI

SI

b2(i2 )

b2(i1 )

SI

4th US •

SI

SO SO

4th LS

 

b1(i1 ) b1(i2 )

6th LS

b2(i2 )

SI



SI

• •

SO

SI

• • •

  

SO

SO

SI

SO (2r)th LS

(2r)th US •

 

• •



SO

SO



SI

SI

SO

SO

b2(im )

Δ iq > 0

Δ iq < 0 Δ iq = 0

|| p ||

x*

SI

SO

b2(im )

Δ iq > 0

b

a SO

a0 > 0

SO

b1(i1 )

a0 < 0

( i2 ) 1

b

b2(i1 )

6th US SI

SO SI

SI

SO

• • •

  

 

SI

|| p ||

• •

SI

  

•

SI

b1(im ) SO

SI

Δ iq < 0 Δ iq = 0

SI

SI

(2r)th LS

5th SI

x*

• •



(2r)th US



b2(i2 )

SO

•

 

SO

SO

6th US

SI

SO

b1(i1 ) b1(i2 ) b2(i1 )

SI

6th LS

SI

6th LS

b2(i2 )

SI

SO

c

b1(im )

US

Δ iq < 0 Δ iq = 0

|| p ||

SO

•

SO 4th

SI

SI

SI

b1(im )

4th LS

x*

• •

 

SI

• • •

SO SI

b1(im )

5th SO

x* SO

b2(im )

Δ iq > 0

|| p ||

Δ iq < 0 Δ iq = 0

b2(im )

Δ iq > 0

d

Fig. 6.5 Stability and bifurcation of equilibriums in a (2m)th-degree polynomial system: (a) (6thUS : SO : 4thLS :    : SI) appearing bifurcation (a0 > 0). (b) (6thLS : SI : 4thUS :    : SO) appearing bifurcation (a0 < 0). (c) (6thUS : SO : 6thLS :    : 5thSI) switching/appearing bifurcation (a0 > 0). (d) (6thLS : SI : 6thUS :    : 5thSO) switching/appearing bifurcation (a0 < 0). LS: lower saddle, US: upper saddle, SI: sink, SO: source. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

6.3 Higher Order Equilibrium Bifurcations

265

appearing bifurcation possesses different clusters of stable and unstable equilibriums before and after the bifurcation. The set of (5thSI :    : SO : 6thUS) flower-bundle switching bifurcation for SI and SO equilibriums is presented in Fig. 6.5c for a0 > 0. Such a flower-bundle switching bifurcation is from (SI : SO : SI : SO) to (5thSI :    : SO : 6thUS) with a waterfall appearing. The set of (5thSO :    : SI : 6thLS) flower-bundle switching bifurcation for SI and SO equilibriums is presented in Fig. 6.5d for a0 < 0. Such a flower-bundle switching bifurcation is from (SO : SI : SO : SI) to (5thSI :    : SO : 6thUS) with a waterfall appearing. After the bifurcation, the waterfall equilibrium birth can be observed. The equilibriums before such a bifurcation are much less than after the bifurcation.

6.3

Higher Order Equilibrium Bifurcations

The afore-discussed appearing and switching bifurcations in the (2m)th-degree polynomial system are relative to simple sources and sinks. As in Luo (92019), the higher order singularity bifurcations in the (2m)th-degree polynomial system can be for higher order equilibriums (i.e., sink, source, upper saddle, lower saddle).

6.3.1

Appearing Bifurcations

Consider a (2m)th-degree polynomial system as x_ ¼ a0 QðxÞ

s
Y

x2 þ Bi x þ C i

αi

,

ð6:134Þ

i¼1

where αi 2 {2l  1, 2l}. Without loss of generality, a function of Q(x) > 0 is either a polynomial function or a non-polynomial function. The roots of x2 + Bix + Ci ¼ 0 are 1 1pffiffiffiffiffi ðiÞ Δi ,Δi ¼ B2i  4C i 0; b1,2 ¼  Bi  2 2 ð1Þ ð1Þ ðsÞ ðsÞ fa1 ; a2 ; . . . ; a2s1 ; a2s g ¼ sortfb1 ; b2 ; . . . ; b1 ; b2 g, a j  a jþ1 :

ð6:135Þ

266

6 (2m)th-Degree Polynomial Systems

There are four higher order bifurcations as follows: ð2li 1Þth order quadratics

ð2ð2li  1ÞÞth order US ! appearing bifurcation ( th  ð2li  1Þ order SO, x ¼ a2i ,

ð6:136Þ

ð2li  1Þth order SI, x ¼ a2i1 ; ð2li 1Þth order quadratics

ð2ð2li  1ÞÞth order LS ! appearing bifurcation ( th  ð2li  1Þ order SI, x ¼ a2i ,

ð6:137Þ

ð2li  1Þth order SO, x ¼ a2i1 ; ð2li Þth ‐order quadratics

ð2ð2li ÞÞth order US ! appearing bifurcation ( th ð2li Þ order US, x ¼ a2i ,

ð6:138Þ

ð2li Þth order US, x ¼ a2i1 ; ð2li Þth ‐order quadratics

ð2ð2li ÞÞth order LS ! appearing bifurcation ( th  ð2li Þ order LS, x ¼ a2i ,

ð6:139Þ

ð2li Þth order LS, x ¼ a2i1 : (i) For αi ¼ 2li  1, the (2(2li  1))th-order upper-saddle (US) appearing bifurcation is for the onset of the (2li  1)th-order source (SO) (x ¼ a2i) and the (2l  1)th-order sink (SI) (x ¼ a2i  1) with a2i > a2i  1 for a0 > 0. (ii) For αi ¼ 2li  1, the (2(2li  1))th-order lower-saddle (LS) appearing bifurcation is for the onset of the (2li  1)th-order sink (SI) (x ¼ a2i) and the (2li  1)th-order source (SO) (x ¼ a2i  1) with a2i > a2i  1 for a0 < 0. (iii) For αi ¼ 2li, the (2(2li))th-order upper-saddle (US) appearing bifurcation is for the onset of two (2li)th-order upper saddles (US) (x ¼ a2i  1, a2i) with a2i 6¼ a2i  1 for a0 > 0. (iv) For αi ¼ 2li, the (2(2li))th-order lower-saddle (LS) appearing bifurcation is for the onset of two (2li)th-order lower saddles (LS) (x ¼ a2i  1, a2i) with a2i 6¼ a2i  1 for a0 < 0. From the higher order singular bifurcation conditions, in a (2m)th-degree polynomial system, the higher order saddle-node bifurcations for appearing and switching of the higher order equilibriums are discussed herein. A set of paralleled different higher order upper-saddle appearing bifurcations in the (2m)th-degree polynomial nonlinear system is called the ((2α1)thUS : (2α2)thUS :    : (2αs)thUS) parallel appearing bifurcation for a0 > 0.

6.3 Higher Order Equilibrium Bifurcations

267

Define ðð2α1 Þth US : ð2α2 Þth US :    : ð2αs Þth USÞ ¼ ð2α1 : 2α2 :    : 2αs Þth US ð6:140Þ where αi 2 {2li  1, 2li} for i ¼ 1, 2, . . . , s. Such an appearing bifurcation is called the (2α1 : 2α2 :    : 2αs)thUS teeth comb appearing bifurcation. Similarly, a set of paralleled different higher order lower-saddle appearing bifurcations in the (2m)th-degree polynomial nonlinear system is called the ((2α1)thLS : (2α2)thLS :    : (2αs)thLS) parallel appearing bifurcation for a0 < 0. Define ðð2α1 Þth LS : ð2α2 Þth LS :    : ð2αs Þth LSÞ ¼ ð2α1 : 2α2 :    : 2αs Þth LS

ð6:141Þ

where αi 2 {2li  1, 2li} for i ¼ 1, 2,   , s. Such an appearing bifurcation is called the (2α1 : 2α2 :    : 2αs)thLS teeth comb appearing bifurcation. Consider a 1-dimensional polynomial system as x_ ¼ a0 QðxÞ

n
Y

x2 þ Bi x þ C i

αi

:

ð6:142Þ

i¼1

where αi 2 {2ri  1, 2ri} (i ¼ 1, 2,   , n). Without loss of generality, a function of Q (x) > 0 is either a polynomial function or a non-polynomial function. The roots of x2 + Bix + Ci ¼ 0 are 1 1pffiffiffiffiffi ðiÞ Δi ,Δi ¼ B2i  4C i 0; x1,2 ¼  Bi  2 2 Bi ¼ B j ði; j ¼ 1; 2;   ; n; i 6¼ jÞ ð1Þ

ð1Þ

ð2Þ

ð2Þ

ð6:143Þ ðrÞ

ðrÞ

fa1 ; a2 ;   ; a2l g ¼ sortfx1 ; x2 ; x1 ; x2 ;   ; x1 ; x2 g, ai  aiþ1 : The higher order singularity bifurcation can be for a cluster of higher order sinks, sources, upper saddles, and lower saddles. There are four higher order bifurcations as follows: For the higher order upper-saddle appearing bifurcation, the cluster of higher order sinks, sources, upper saddles, and lower saddles is given by the following two cases: (i) The (2(2l  1))thorder US spraying-appearing bifurcation for a cluster of higher order sinks, sources, upper saddles, and lower saddles is

6 (2m)th-Degree Polynomial Systems

268

8 ðα2n Þth order XX for x ¼ a2n , > > > > > th < ðα  a cluster of 2n‐XX 2n1 Þ order XX for x ¼ a2n1 , th ð2ð2l  1ÞÞ order US ! appearing bifurcation > >⋮ > > > : ðα1 Þth order XX for x ¼ a1 ; ð6:144Þ where 2ð2l  1Þ ¼ satisfy

Pn

i¼1 αi

( th

ðα2n Þ order XX ¼ ( th

ðα1 Þ order XX ¼

and the minimum and maximum equilibriums

ð2r 2n Þth order US, for α2n ¼ 2r n , ð2r 2n  1Þth order SO, for α2n ¼ 2r n  1;

ð2r 1 Þth order US, for α1 ¼ 2r 1 ,

ð6:145Þ

ð2r 1  1Þth order SO, for α1 ¼ 2r 1  1:

(ii) The (2(2l))thorder US spraying-appearing bifurcation for a cluster of higher order sinks, sources, upper saddles, and lower saddles is 8 > ðα2n Þth order XX for x ¼ a2n , > > > < a cluster of 2n‐XX ðα2n1 Þth order XX for x ¼ a2n1 , ð2ð2lÞÞth order US ! appearing bifurcation > >⋮ > > : ðα1 Þth order XX for x ¼ a1 ; where 2ð2lÞ ¼

Pn

i¼1 αi

( th

ðα2n Þ order XX ¼ ( ðα1 Þth order XX ¼

ð6:146Þ

and the minimum and maximum equilibriums satisfy

ð2r 2n Þth order US, for α2n ¼ 2r n , ð2r 2n  1Þth order SO, for α2n ¼ 2r n  1;

ð2r 1 Þth order US, for α1 ¼ 2r 1 ,

ð6:147Þ

ð2r 1  1Þth order SI, for α1 ¼ 2r 1  1:

For the higher order lower-saddle bifurcation, the cluster of the higher order equilibriums is given by the following two cases. (iii) The (2(2l  1))thorder LS spraying-appearing bifurcation for a cluster of higher order sinks, sources, upper saddles, and lower saddles is

6.3 Higher Order Equilibrium Bifurcations

269

8 ðα2n Þth order XX, for x ¼ a2n , > > > > > th < ðα  a cluster of 2n‐XX 2n1 Þ order XX, for x ¼ a2n1 , th ð2ð2l  1ÞÞ order LS  ! appearing bifurcation > >⋮ > > > : ðα1 Þth order XX, for x ¼ a1 ; ð6:148Þ where 2ð2l  1Þ ¼ satisfy

Pn

i¼1 αi

( th

ðα2n Þ order XX ¼ ( th

ðα1 Þ order XX ¼

and the minimum and maximum equilibriums

ð2r 2n Þth order LS, for α2n ¼ 2r n , ð2r 2n  1Þth order SI, for α2n ¼ 2r n  1;

ð2r 1 Þth order LS, for α1 ¼ 2r 1 ,

ð6:149Þ

ð2r 1  1Þth order SI, for α1 ¼ 2r 1  1:

(iv) The (2(2l ))thorder LS spraying-appearing bifurcation for a cluster of higher order sinks, sources, upper saddles, and lower saddles is 8 > ðα2n Þth order XX, for x ¼ a2n , > > > < a cluster of 2n‐XX ðα2n1 Þth order XX, for x ¼ a2n1 , ð6:150Þ ð2ð2lÞÞth order LS ! appearing bifurcation > >⋮ > > : ðα1 Þth order XX, for x ¼ a1 ; where 2ð2lÞ ¼

Pn

i¼1 αi

( th

ðα2n Þ order XX ¼ ( ðα1 Þth order XX ¼

and the minimum and maximum equilibriums satisfy

ð2r 2n Þth order LS, for α2n ¼ 2r n , ð2r 2n  1Þth order SI, for α2n ¼ 2r n  1;

ð2r 1 Þth order LS, for α1 ¼ 2r 1 ,

ð6:151Þ

ð2r 1  1Þth order SO, for α1 ¼ 2r 1  1:

A set of paralleled, different, higher order upper-saddle-node appearing bifurcations with multiplicity in the (2m)th-degree polynomial system is the ((2β1)thUS : (2β2)thUS :    : (2βs)thUS) parallel appearing bifurcation for clusters of higher order sinks, sources, upper saddles, and lower saddles. For the (2βi)thUS ( th

ð2βi Þ US ¼

ð2ð2li  1ÞÞth order US, for βi ¼ 2li  1, ð2ð2li ÞÞth order US, for βi ¼ 2li ;

:

ð6:152Þ

6 (2m)th-Degree Polynomial Systems

270

Similarly, the following notation is introduced as


 ð2β1 Þth US : ð2β2 Þth US :    : ð2βs Þth US ¼ ð2β1 : 2β2 :    : 2βs Þth US: ð6:153Þ

Thus, the paralleled (2β1 : 2β2 :    : 2βs)th US spraying-appearing bifurcation is called the (2β1 : 2β2 :    : 2βs)th US sprinkler-spraying-appearing bifurcation for the higher order equilibriums. Similarly, a set of paralleled different lower-saddle appearing bifurcations for higher order singularity of equilibriums is called the ((2β1)thLS : (2β2)thLS :    : (2βs)thLS) parallel appearing bifurcation in the (2m)thdegree polynomial system. Thus, the paralleled (2β1 : 2β2 :    : 2βs)th LS bifurcation is also called the (2β1 : 2β2 :    : 2βs)th LS sprinkler-spraying-appearing bifurcation for higher order equilibriums. The (2α1 : 2α2 :    : 2αn)thUS and (2α1 : 2α2 :    : 2αn)thLS teeth comb appearing bifurcations for the higher order singularity of equilibriums are presented in Figs. 6.6a, b for a0 > 0 and a0 < 0, respectively. The components of the teeth comb appearing bifurcation are th

αj ¼2rj

(

ð2αj Þ US !

ð2r j Þth US

ð j ¼ i, n  1, . . .Þ, ð2r j Þth US ( ð2r k  1Þth SO αk ¼2r k 1 th ð2αk Þ US  ðk ¼ 1, n, . . .Þ; ! appearing ð2r k  1Þth SI appearing

ð6:154Þ

and th

αj ¼2rj

(

ð2αj Þ LS  ! appearing

( th

ð2r j Þth LS ð2r j Þth LS

ð j ¼ i, n  1, . . .Þ,

αk ¼2rk 1

ð2r k  1Þth SI

appearing

ð2r k  1Þth SO

ð2αk Þ LS  !

ð6:155Þ ðk ¼ 1, n, . . .Þ:

The (2β1 : 2β2 :    : 2βn)thUS and (2β1 : 2β2 :    : 2βs)thLS sprinkler-sprayingappearing bifurcations for the higher order singularity of equilibriums are presented in Fig. 6.6c, d for a0 > 0 and a0 < 0, respectively. The components of the sprinklerspraying-appearing bifurcation are ð2β1 : 2β2 :    : 2βn Þth US ¼ ðð2ð2l1  1Þ :    : 2ð2li Þ :    : 2ð2ln1 Þ : 2ð2ln ÞÞth US

ð6:156Þ

6.3 Higher Order Equilibrium Bifurcations

271

a

b

c

d

Fig. 6.6 The teeth comb appearing bifurcations of (2(2r1  1) :    : 2(2rn  1) : 2(2rn  1))thXX: (a) XX ¼ US (a0 > 0) and (b) XX ¼ LS (a0 < 0). The sprinkler-spraying-appearing bifurcations of (2(2l1  1) :    : (2(2ln  1) : 2(2ln))thXX: (c) XX ¼ US (a0 > 0) and (d) XX ¼ LS (a0 < 0). LS: lower saddle, US: upper saddle, SI: sink, SO: source. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

6 (2m)th-Degree Polynomial Systems

272

and ð2β1 : 2β2 :    : 2βn Þth LS ¼ ðð2ð2l1  1Þ :    : 2ð2li Þ :    : 2ð2ln1 Þ : 2ð2ln ÞÞth LS

ð6:157Þ

For a cluster of m-quadratics, Bi ¼ Bj (i, j 2 {1, 2,   , n}; i 6¼ j) and Δi ¼ 0 (i ¼ 1, 2,   , n). The (2m)th-order upper-saddle appearing bifurcation for n-pairs of the higher order singularity of equilibriums is 8 > ðα2n Þth order XX for x ¼ a2n , > > > < a cluster of 2n‐XX ðα2n1 Þth order XX for x ¼ a2n1 , ð2mÞth order US ! appearing bifurcation > >⋮ > > : ðα1 Þth order XX for x ¼ a1 ;

ð6:158Þ

where 2m ¼ 2ð2lÞ ¼

2n X i¼1

αi , 2m ¼ 2ð2l  1Þ ¼

2n X

αi :

ð6:159Þ

i¼1

The (2m)th-order lower-saddle-node appearing bifurcation for higher order equilibriums is 8 > ðα2n Þth order XX for x ¼ a2n , > > > < a cluster of 2n‐XX ðα2n1 Þth order XX for x ¼ a2n1 , ð2mÞth order LS ! appearing bifurcation > >⋮ > > : ðα1 Þth order XX for x ¼ a1 :

ð6:160Þ

The (2m)th-order upper-saddle appearing bifurcation with n-pairs of higher order singularity of equilibriums is a sprinkler-spraying cluster of the n-pairs of higher order singularity of equilibriums. The (2m)th-order lower-saddle appearing bifurcation with n-pairs of higher-order equilibriums is also a sprinkler-spraying cluster of the n-pairs of higher order singularity of equilibriums. Thus, the (2m)th-order US bifurcation (a0 > 0) and (2m)th-order LS bifurcation (a0 < 0) are presented in Fig. 6.7a–d, respectively. The (2m)th-order upper-saddle appearing bifurcation for higher order singularity of equilibriums is called the (2m)th-order US sprinkler-spraying-appearing bifurcation, and the (2m)th-order lower-saddle-node appearing bifurcation for higher order singularity of equilibriums is also called the (2m)th-order LS sprinkler-spraying-appearing bifurcation. A series of the saddle-node bifurcations for higher order singularity of equilibriums are aligned up with varying parameters, which is formed in a special pattern. For n-quadratics in the (2m)th-order polynomial systems, the following conditions should be satisfied:

6.3 Higher Order Equilibrium Bifurcations

273

Fig. 6.7 Spraying appearing bifurcations for higher-order equilibriums in the (2m)th polynomial system: (a) (2(2l  1))th US spraying-appearing bifurcation (a0 > 0), (b) (2(2l  1))th LS spraying appearing bifurcation (a0 < 0), (c) (2(2l ))th US spraying appearing bifurcation (a0 > 0), (d) (2(2l ))th US spraying appearing bifurcation (a0 < 0). LS: lower-saddle, US: upper-saddle, SI: sink, SO: source. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

Bi Bj i, j 2 f1, 2, . . . , sg; i 6¼ j, Δi > Δiþ1 ði ¼ 1, 2, . . . , s; s  n < mÞ, Δi ¼ 0 with kpi k < kpiþ1 k:

ð6:161Þ

The two series of the fish-scale switching bifurcations in Fig. 6.8a, c for a0 < 0 have the following detailed structures: 8 ( ð2r 1  1Þth SO, > > th > ð ð  1 Þ Þ US ! 2 2r > 1 > > > ð2r 1  1Þth SI; > > > ( > > > ð2r 2 Þth LS, < ð2ð2r 2 ÞÞth LS ! ð2r 2 Þth LS; > > > > > ⋮ > ( > > > ð2r n  1Þth SO, > > th > > : ð2ð2r n  1ÞÞ US ! ð2r n  1Þth SI; and

ð6:162Þ

6 (2m)th-Degree Polynomial Systems

274

a

b

c

d Fig. 6.8 The fish-scale appearing bifurcation patterns in a (2m)th-degree polynomial system: (a) (2 (2r1  1))thUS ‐ (2(2r2))thLS ‐    (a0 > 0), (b) (2(2r1  1))thLS ‐ (2(2r2))thUS ‐    (a0 < 0), (c) (2 (2r1))thUS ‐ (2(2r2))thUS ‐    (a0 > 0), (d) (2(2r1))thLS ‐ (2(2r2  1))thLS ‐    (a0 < 0). LS: lower saddle, US: upper saddle, SI: sink, SO: source. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

6.3 Higher Order Equilibrium Bifurcations

8 ( ð2r 1 Þth US, > > th > ð 2 ð 2r Þ Þ US ! > 1 > > > ð2r 1 Þth US; > > > ( > > > ð2r 2  1Þth SO, < th ð2ð2r 2  1ÞÞ US ! ð2r 2  1Þth SI; > > > > > ⋮ > ( > > > ð2r n  1Þth SO, > > th > > : ð2ð2r n  1ÞÞ US ! ð2r n Þth SI:

275

ð6:163Þ

Two series of fish-scale appearing bifurcations in Fig. 6.8b, d for a0 < 0 have the following structures as 8 ( ð2r 1  1Þth SI, > > th > ð ð  1 Þ Þ LS ! 2 2r > 1 > > > ð2r 1  1Þth SO; > > > ( > > > ð2r 2 Þth US, < th ð2ð2r 2 ÞÞ US ! ð2r 2 Þth US; > > > > > ⋮ > ( > > > ð2r n  1Þth SI, > > th > ð ð  1 Þ Þ LS ! 2 2r > n : ð2r n  1Þth SO;

ð6:164Þ

8 ( ð2r 1 Þth LS, > > th > > ð2ð2r 1 ÞÞ LS ! > > > ð2r 1 Þth LS; > > > ( > > > ð2r 2  1Þth SI, < th ð2ð2r 2  1ÞÞ LS ! ð2r 2  1Þth SO; > > > > > ⋮ > ( > > > ð2r n  1Þth SI, > > th > ð ð  1 Þ Þ LS ! 2 2r > n : ð2r n Þth SO:

ð6:165Þ

and

The four fish-scale appearing bifurcation patterns for higher order equilibriums are different from the fish-scale appearing bifurcation patterns for simple equilibriums.

6 (2m)th-Degree Polynomial Systems

276

6.3.2

Switching Bifurcations α

Consider the roots of ðx2 þ Bi x þ C i Þ i ¼ 0 as ðiÞ

ðiÞ

ðiÞ

ðiÞ

Bi ¼ ðb1 þ b2 Þ, Δi ¼ ðb1  b2 Þ2 0, ðiÞ

ðiÞ

ðiÞ

ðiÞ

x1,2 ¼ b1,2 , Δi > 0 if b1 6¼ b2 ði ¼ 1, 2, . . . , nÞ; ) Bi 6¼ Bj ði, j ¼ 1, 2, . . . , n; i 6¼ jÞ at bifurcation: ðiÞ ðiÞ Δi ¼ 0 at b1 ¼ b2 ði ¼ 1, 2, . . . , nÞ

ð6:166Þ

The αth i -order singularity bifurcation is for the switching of a pair of higher order equilibriums (i.e., sinks, sources, upper saddles, and lower saddles). There are six switching bifurcations for i 2 {1, 2, . . ., n}: ði Þ

ðiÞ

li ¼r1 þr 2 1

ð2li Þth order US ! switching bifurcation 8 th ð i Þ < ð2r  1Þ order SO # SI, for bðiÞ ¼ a2i # a2i1 , 2 2 :

ðiÞ

th

ð6:167Þ

ðiÞ

ð2r 1  1Þ order SI " SO, for b1 ¼ a2i1 " a2i ; ði Þ

ði Þ

li ¼r1 þr2 1

ð2li Þth order LS ! switching bifurcation 8 th < ð2r ðiÞ  1Þ order SI # SO, for bðiÞ ¼ a2i # a2i1 , 2 2 :

ðiÞ

th

ð6:168Þ

ðiÞ

ð2r 1  1Þ order SO " SI, for b1 ¼ a2i1 " a2i ; ði Þ

ðiÞ

li ¼r1 þr 2

ð2li Þth order US ! switching bifurcation 8 th ð i Þ < ð2r Þ order US # US, for bðiÞ ¼ a2i # a2i1 , 2 2 :

ðiÞ th

ð6:169Þ

ðiÞ

ð2r 1 Þ order US " US for b1 ¼ a2i1 " a2i ; ði Þ

ði Þ

li ¼r1 þr2

ð2li Þth order LS ! switching bifurcation 8 th < ð2r ðiÞ Þ order LS # LS, for bðiÞ ¼ a2i # a2i1 , 2 2 :

ðiÞ th

ðiÞ

ð2r 1 Þ order LS " LS for b1 ¼ a2i1 " a2i ;

ð6:170Þ

6.3 Higher Order Equilibrium Bifurcations

277

ðiÞ

ði Þ

li ¼r 1 þr2

ð2li  1Þth order SO ! switching bifurcation 8 th ð i Þ < ð2r  1Þ order SO # SO, for bðiÞ ¼ a2i # a2i1 , 2 2 :

ðiÞ th

ð6:171Þ

ðiÞ

ð2r 1 Þ order LS " US for b1 ¼ a2i1 " a2i ; ði Þ

ði Þ

li ¼r1 þr2

ð2li  1Þth order SI ! switching bifurcation 8 th < ð2r ðiÞ  1Þ order SI # SI, for bðiÞ ¼ a2i # a2i1 , 2 2 :

ðiÞ th

ð6:172Þ

ðiÞ

ð2r 1 Þ order US " LS for b1 ¼ a2i1 " a2i :

A set of n-paralleled higher order XX switching bifurcations is called the ðða1 Þth XX : ða2 Þth XX : . . . : ðan Þth XXÞ parallel switching bifurcation in the (2m)thdegree polynomial nonlinear system. Such a bifurcation is also called the nðða1 Þth XX : ða2 Þth XX : . . . : ðan Þth XXÞ antenna switching bifurcation. αi 2 {2li, 2li  1} ðiÞ ðiÞ and XX 2 {SO, SI, US, LS}. For non-switching points, Δi > 0 at b1 6¼ b2 (i ¼ 1, ðiÞ ðiÞ 2, . . . , n). At the bifurcation point, Δi ¼ 0 at b1 ¼ b2 (i ¼ 1, 2, . . . , n). The nðða1 Þth XX : ða2 Þth XX : . . . : ðan Þth XXÞ parallel antenna switching bifurcation is 8 8 < ðr ðnÞ Þth XXðnÞ > th > n bifurcation > 2 2 th > αn XXn ! > > > switching : ðnÞ th ðnÞ > ðr 1 Þ XX1 > > > > > ⋮ 8 > < < ðr ð2Þ Þth XXð2Þ 2nd bifurcation 2 2 th α XX ! 2 > > 2 switching : ð2Þ th ð2Þ > > > 8 ðr 1 Þ XX1 > > > < ðr ð1Þ Þth XXð1Þ > > 1st bifurcation > 2 2 th > XX α > 1 1 ! > : switching : ð1Þ th ð1Þ ðr 1 Þ XX1

ðnÞ

ðnÞ

ðnÞ

ðnÞ

ð2Þ

ð2Þ

# YY1 , for b2 ¼ a2n # a2n1 , " YY2 , for b1 ¼ a2n1 " a2n ; # YY1 , for b2 ¼ a4 # a3 , #

ð2Þ YY2 , ð1Þ YY1 ,

"

ð1Þ YY2 ,

"

¼ a3 " a4 ;

for

ð2Þ b1 ð1Þ b2

for

ð1Þ b1

¼ a1 " a2 :

for

ð6:173Þ

¼ a2 # a1 ,

Such eight sets of parallel switching bifurcations of ðða1 Þth XX : ða2 Þth XX : . . . : ðan Þth XXÞ are presented in Fig. 6.9(a, c, e, g) and (b, d, f, h) for a0 > 0 and a0 < 0, respectively. The eight switching bifurcation structures are as follows: (a) (b) (c) (d) (e) (f) (g) (h)

((2l1)thUS :    : (2ln  1  1)thSO : (2ln)thUS) for a0 > 0, ((2l1)thLS :    : (2ln  1  1)thSI : (2ln)thLS) for a0 < 0, ((2l1)thLS :    : (2ln  1  1)thSI : (2ln  1)thSO) for a0 > 0, ((2l1)thUS :    : (2ln  1  1)thSO : (2ln  1)thSI) for a0 < 0, ((2l1)thLS :    : (2ln  1  1)thSI : (2ln  1)thSO) for a0 > 0, ((2l1)thUS :    : (2ln  1  1)thSI : (2ln  1)thSI) for a0 < 0, ((2l1)thUS :    : (2ln  1  1)thSO : (2ln)thUS) for a0 > 0, ((2l1)thLS :    : (2ln  1  1)thSI : (2ln)thLS) for a0 < 0.

6 (2m)th-Degree Polynomial Systems

278

a

b

c

d

Fig. 6.9 Parallel antenna switching bifurcations for high-order equilibriums in a (2m)th-degree polynomial system. ðða1 Þth XX : ða2 Þth XX : . . . : ðan Þth XXÞ: (a, c, e, g) for a0 > 0. (b, d, f, h) for a0 < 0. LS: lower saddle, US: upper saddle, SI: sink, SO: source. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

6.3 Higher Order Equilibrium Bifurcations

279

e

f

g

h

Fig. 6.9 (continued)

6 (2m)th-Degree Polynomial Systems

280

The same switching bifurcations with different higher order equilibriums are illustrated, which is different from the m-USN and m-LSN for simple sinks and sources. Consider a switching bifurcation for a cluster of higher order equilibriums with the following conditions: ðiÞ

ðiÞ

ðiÞ

ðiÞ

Bi ¼ ðb1 þ b2 Þ, Δi ¼ ðb1  b2 Þ2 0, ðiÞ

ðiÞ

ðiÞ

ðiÞ

x1,2 ¼ b1,2 , Δi > 0 if b1 6¼ b2 ði ¼ 1, 2, . . . , nÞ; ) Bi ¼ Bj ði, j 2 f1, 2, . . . , ng; i 6¼ jÞ at bifurcation: ðiÞ ðiÞ Δi ¼ 0 at b1 ¼ b2 ði ¼ 1, 2, . . . , nÞ

ð6:174Þ

Thus, the (αi)th-order switching bifurcation can be for a cluster of higher order equilibriums. The (αi)th-order switching bifurcations for i 2 {1, 2, . . ., s} are αi ¼

th

Pli

ði Þ r j¼1 j

ðαi Þ order XX ! switching bifurcation 8 th ðiÞ ðiÞ ðiÞ ðiÞ ð i Þ > ðr s Þ order XXli # YYli , for bli # ali , > > > > > > ⋮ > > < ðiÞ th ðiÞ ðiÞ ðiÞ ðiÞ ðr j Þ order XXj # YYj , for bj # aj , > > > > > ⋮ > > > > : ðiÞ th ðiÞ ðiÞ ðiÞ ðiÞ ðr 1 Þ order XX1 " YY1 , for b1 # a1 ,

ð6:175Þ

where ðiÞ

ðiÞ

ðiÞ

ðiÞ

fa1 ; a2 ; . . . ; ali1 ; ali g ðiÞ

ðiÞ

ðiÞ

ðiÞ

fb1 ; b2 ; . . . ; bli1 ; bli g

ð1Þ

ð1Þ

ðnÞ

ðnÞ



sortfb1 ; b2 ; . . . ; b1 ; b2 g,



sortfb1 ; b2 ; . . . ; b1 ; b2 g:

before bifurcation After bifurcation

ð1Þ

ð1Þ

ðnÞ

ðnÞ

ð6:176Þ

A set of paralleled, different, higher order upper-saddle-node switching bifurcations with multiplicity is the ((α1)thXX : (α2)thXX :    : (αs)thXX) parallel switching bifurcation in the (2m)th-degree polynomial system. At the straw-bundle switching bifurcation, Δi ¼ 0 (i ¼ 1, 2, . . . , n) and Bi ¼ Bj ( i, j 2 {1, 2, . . ., n}; i 6¼ j). The parallel straw-bundle switching bifurcation for higher order equilibriums is

6.3 Higher Order Equilibrium Bifurcations

281

ððα1 Þth XX : ðα2 Þth XX :    : ðαs Þth XXÞ‐switching 8 ðαs Þth order XX switching, > > > > > ðα2 Þth order XX switching, > > > > : ðα1 Þth order XX switching,

ð6:177Þ

αi 2 f2li , 2li  1g and XX 2 fUS, LS, SO, SIg:

ð6:178Þ

where


 th th Eight parallel straw-bundle switching bifurcations of αth 1 XX : α2 XX : . . . : αn XX are presented in Figs. 6.10 and 6.11 for a0 > 0 and a0 < 0, respectively.

6.3.3

Appearing and Switching Bifurcations

Consider a (2m)th-degree polynomial system in a form of x_ ¼ a0 QðxÞ

2n1 Y i¼1

ðx  ci Þαi

n2 Y


x2 þ B j x þ C j

αj

:

ð6:179Þ

j¼1

Without loss of generality, a function of Q(x) > 0 is either a polynomial function or a non-polynomial function. The roots of x2 + Bjx + Cj ¼ 0 are 1 1 pffiffiffiffiffi ð jÞ Δj , Δj ¼ B2j  4Cj 0 ð j ¼ 1, 2, . . . , n2 Þ; b1,2 ¼  Bj  2 2

ð6:180Þ

either     fa 1 , a2 , . . . , a2n1 g ¼ sortfc1 ; c2 ; . . . ; c2n1 g, as  asþ1 before bifurcation ð1Þ

ð1Þ

ðn Þ

ðn Þ

2 2 þ þ faþ 1 , a2 , . . . , a2n3 g ¼ sortfc1 ; . . . ; c2n1 ; b1 ; b2 ; . . . ; b1 ; b2 g,

þ aþ s  asþ1 , n3 ¼ n1 þ n2 after bifurcation;

ð6:181Þ

or ð1Þ

ð1Þ

ðn Þ

ðn Þ

2 2   fa 1 , a2 , . . . , a2n3 g ¼ sortfc1 ; c2 ; . . . ; c2n1 ; b1 ; b2 ; . . . ; b1 ; b2 g,

ð6:182Þ

 a s  asþ1 , n3 ¼ n1 þ n2 before bifurcation; þ faþ 1 , a2 ,

. . . , aþ 2n1 g

¼ sortfc1 ; . . . ; c2n1 g, aþ s 

aþ sþ1

after bifurcation;

6 (2m)th-Degree Polynomial Systems

282 a0 > 0 (2r2 n −2 −1) th SO (2ln ) th US

(2r2 n −1 ) th LS (2r2 n −1) th SI

a2n

(2r2 n −1) th SO

a2 n−1

(2r2 n −1 ) th LS

a0 > 0 (2r2 n −2 −1) th SO

a2 n−2

(2r2 n −2 −1) th SI

a2 n−3

(2r2 n −3 ) th US

(2ln −1) th SO

(2r2 n −1 ) th LS (2r2 n ) th LS

a2 n−3 •

 

•

(2r2 n −3 ) th US

(2r2i −2 ) th US th

(2r2i −3 −1) SO

 

•

(2r1 ) th LS

a3 a2

(2l1 −1) th SI

(2r2 −1) th SI

a2 n−3

a1

• •

 

(2r2i ) th US

•



Δ iq > 0

a3 a2

(2l1 −1) th SO (2r2 −1) th SO

x*

(2r2 ) th US

Δ iq > 0 Δ iq = 0

(2r2i −2 ) th LS (2r2i −3 −1) th SI

(2r2 −1) th SI (2r1 ) US

(2r2i −1 −1) th SO

(2li ) th US

(2r1 ) th US

th

(2r2 n −4 −1) th SI

(2r2i ) th US

•

(2r2i −1 −1) th SI

(2r3 ) th LS

a1

(2r3 ) th US (2r2 −1) th SO (2r1 ) th LS

(2r2 ) th LS

Δ iq > 0

Δ iq > 0 Δ iq = 0

|| p ||

b

a a0 > 0 (2r2 n −2 ) th US (2ln −1) th SO

(2r2 n −1 ) th US

a2n

(2r2 n −1) th SO

a2 n−1

(2r2 n −1 ) th LS

a2 n−2

th

(2r2 n −1) SO

a2 n−3

a0 > 0 (2r2 n −2 ) th US

th

(2r2 n −2 ) LS

(2ln −1) th SO

(2r2 n −1 −1) th SO th

th

(2r2 n ) LS

(2r2 n −3 ) LS

(2ln −1 −1) th SO

a2 n−3 

 (2r2i −1 −1) th SO

(2r2 n −4 −1) th SI

• •

(2r2i ) th US

•

(2r2i −1 −1) th SO

(2r2 n −3 ) US

 

(2r2i −2 ) th LS

th

(2r2i −1 −1) SI

 (2r2i −1 −1) th SO

(2r2 n −2 ) th LS

a2 n−3

(2r2 n −3 ) th LS

(2r2 n −4 −1) th SI

(2r2i −3 −1) th SI

 

•

th

(2r1 ) US

(2l1 −1) th SI

a3 a2 a1

th

(2r2 ) LS

Δ iq > 0 Δ iq = 0

• •

(2r2i ) th US

•

(2r2i −1 −1) th SO

Δ iq > 0

(2li ) US

(2r2i −2 ) th LS (2r2i −1 −1) SI

(2r2i ) th US

(2r3 ) th US

(2r2i −3 −1) th SI •

 

• •



th

(2r1 ) US (2r2 −1) th SO

(2r2i −2 ) th LS

th

• •



|| p ||

a2 n−2

th

(2li ) LS

(2r2i −2 ) th LS

x*

(2r2 n −1 −1) th SO

a2 n−3

th

(2r2 −1) th SO

a2 n−1

th



(2r2i ) th US

(2r2 n ) th US

(2ln −1 −1) th SI

th

(2r2 n −3 ) US

a2n

(2r2 n −4 −1) th SI

(2r2 n −4 −1) th SI

c

(2r2 n −2 −1) th SO (2r2 n −3 ) th LS

•

(2r2i −2 ) th LS

• •



|| p ||

a2 n−2

•



  (2r2i −1 −1) th SO

(2r2i −1 −1) SI

(2r2i −1 −1) th SO

x*

(2r2 n −1 ) th US

a2 n−3

th

(2li ) th LS

(2r2i −2 ) th US

(2r2 n −4 −1) th SO

(2r2i ) th LS

•



(2r2i ) th LS

a2 n−1

(2r2 n −4 −1) th SI

(2r2 n −4 −1) th SO

(2r2i −1 −1) th SI

(2r2 n ) th US

(2ln −1 −1) th SI

(2ln −1 −1) th SO

(2r2 n −3 ) th LS

a2n

(2r2 −1) th SO

(2l1 −1) th SO

(2r1 ) th LS

x*

a3 a2 a1

(2r3 ) th US (2r2 −1) th SO (2r1 ) th LS

th

(2r2 ) LS

|| p ||

Δ iq > 0 Δ iq = 0

Δ iq > 0

d

Fig. 6.10 (a–d) Four types of (r1th XX : r2th XX :    : rmth XX) parallel switching bifurcation for a0 > 0 in the (2m)th-degree polynomial system. LS: lower saddle, US: upper saddle, SI: sink, SO: source. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

6.3 Higher Order Equilibrium Bifurcations a0 < 0

a2n

(2r2 n −2 −1) th SI

a2 n−1

(2ln ) th LS

(2r2 n −1 ) th US (2r2 n −1) th SO

283 a0 < 0

(2r2 n −1) th SI

a2n

(2r2 n −2 −1) th SI

a2 n−2 a2 n−3

(2r2 n −3 ) th LS

(2r2 n −1 ) th US (2r2 n ) th US

th

(2ln −1 −1) SI

(2r2 n −4 −1) th SI

(2r2 n −3 ) th US

a2 n−3 (2r2 n −3 ) th LS

(2r2i ) th US

•

•



th

(2r2i −1 −1) SO

(2r2i −1 −1) th SI

•



(2r2i −2 ) th LS

(2r2i −1 −1) th SI • • •



a3

(2r1 ) th US

a2

(2l1 −1) th SO

th

(2r2 −1) SO

a1

q

• • •

  

a3

(2r1 ) th LS

a2

(2l1 −1) th SI

th

(2r2 −1) SO (2r2 −1) th SI

th

(2r1 ) LS

x*

Δi > 0

a1

a

(2r3 ) th LS (2r2 −1) th SI (2r1 ) th US

(2r2 ) th US

Δi > 0

Δi > 0 Δi = 0

|| p ||

q

q

(2r2i −3 −1) th SO

(2r2i ) th LS

(2r3 ) th US

(2r2 ) th LS

Δi > 0 Δi = 0

(2r2i −2 ) th US

(2r2i −1 −1) th SO

(2r2i −3 −1) SI  

(2li ) LS

(2r2i −2 ) th US

th

(2r2i ) th US

(2r2i −1 −1) th SI

th

(2li ) US

(2r2i −2 ) th LS

(2r2 n −4 −1) th SO

(2r2i ) th LS

•



• th

q

q

q

b a0 < 0

a2n

(2r2 n −2 ) th LS

a2 n−1

(2ln −1) th SI

(2r2 n −1 ) th LS

a2 n−2

th

(2r2 n −1) SI

a2 n−3

a0 < 0

(2r2 n −1) th SI

(2r2 n −2 ) th LS (2ln −1) th SI

th

(2r2 n −1 ) US (2r2 n −1 −1) th SI

th

(2r2 n −2 ) US

th

th

(2r2 n ) US

(2r2 n −3 ) US

a2 n−4 •

  

(2r2 n −3 ) LS

th

(2r2i −1 −1) SI

(2li ) th LS

(2r2i −1 −1) th SI

(2r2i −2 ) th US

th

(2r2i −1 −1) SO

a2 n−3

(2r2 n −3 ) th US

(2r2 n −4 −1) th SO

a3 a2

th

(2l1 −1) SI

(2r2 −1) th SI

a1

q

q

(2r2i −1 −1) th SI

(2li ) th LS

(2r2i −2 ) th US

(2r2i −1 −1) SO

(2r2i ) th LS

(2r3 ) th LS

(2r2i −3 −1) th SO • • •

 

a3

th

(2r1 ) LS

(2l1 −1) SI

(2r2 −1) th SI

(2r2 −1) th SI

a2

th

(2r1 ) th US

x*

(2r2 ) th US

Δi > 0 Δi = 0





• •



•

th

(2r2i −3 −1) th SO



(2r2i ) th LS

•

(2r2i −2 ) th US

•



|| p ||

(2r2 n −2 ) th US

•

 

•

(2r2i −2 ) th US

x*

a2 n−2

a2 n−3

(2r2i ) th LS

•

(2r2i −1 −1) th SI

(2r1 ) LS

(2r2 n −1 −1) th SI

th

(2r2 n −3 ) LS

th

(2r2 n ) th LS

(2r2 n −4 −1) th SO

(2r2 n −4 −1) th SO

th

(2r2i ) th LS

a2n a2 n−1

(2ln −1 −1) th SI

(2ln −1 −1) th SO (2r2 n −4 −1) th SO

c

a2 n−3 (2ln −1 −1) SO

•

 

 (2r2i −1 −1) th SO

|| p ||

a2 n−2

(2r2 n −4 −1) th SO

a2 n−3

x*

(2r2 n −1 ) th LS (2r2 n −2 −1) th SI

th

(2r2 n −4 −1) th SI

(2r2 n −3 ) th US

a2 n−1

(2ln −1) th SI

(2r2 n −1 ) th US (2r2 n −2 −1) th SO

(2r2 n ) th LS

Δi > 0

a1

(2r2 −1) th SI (2r1 ) th LS

(2r2 ) th US

|| p ||

q

(2r3 ) th LS

Δi > 0 Δi = 0 q

q

Δi > 0 q

d

Fig. 6.11 (a–d) Four types of (r1th XX : r2th XX :    : rmth XX) parallel switching bifurcation for a0 < 0 in the (2m)th-degree polynomial system. LS: lower saddle, US: upper saddle, SI: sink, SO: source. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

6 (2m)th-Degree Polynomial Systems

284

and
9 Bj1 ¼ Bj2 ¼    ¼ Bjs jk1 2 f1, 2, . . . , ng; jk1 6¼ jk2 > > > > > > ðk1 , k2 2 f1, 2, . . . , sg; k1 6¼ k2 Þ = Δj ¼ 0 ðj 2 U f1, 2, . . . , n2 g 1 ci 6¼  Bj ði ¼ 1, 2, . . . , 2n1 , j ¼ 1, 2, . . . , n2 Þ 2

> > > > > > ;

at bifurcation:

ð6:183Þ

th  th  th    Consider a just before bifurcation of ððβ 1 Þ XX1 : ðβ2 Þ XX2 : . . . : ðβ s1 Þ XXs1 Þ     for higher order equilibriums. For βi ¼ 2li  1, XXi 2 fSO, SIg and for αi ¼ 2l i , XX 2 US, LS (i ¼ 1, 2, . . . , s ). The detailed structures are as follows: f g 1 i

9 th ðiÞ ðr ðsii Þ Þ order XXðsii Þ , x ¼ aks , > > i > > > > > ⋮ > > = βi ¼Pksi rðj iÞ j¼1 th ðiÞ ðiÞ th ðiÞ  ðiÞ : ! ðβ ðr j Þ order XXj , x ¼ aj i Þ order XX switching bifurcation > > > > > ⋮ > > > > th ; ðiÞ ðiÞ  ðiÞ ðr 1 Þ order XX1 , x ¼ aj ð6:184Þ
th 
 th  th  XX1 : β2 XX2 : . . . : ðβ The bifurcation set of ð β 1 s1 Þ XXs1 Þ at the same parameter point is called a left-parallel-straw-bundle switching bifurcation.
th þ
þ th þ th þ XX1 : β2 XX2 : . . . : ðβþ Consider a just after bifurcation of ð βþ 1 s2 Þ XXs2 Þ þ þ for simple sources and sinks. For βþ i ¼ 2li  1, XXi 2 fSO, SIg and for þ þ þ βi ¼ 2li , XXi 2 fUS, LSg. The detailed structures are as follows: 8 ðiÞþ ðiÞþ ðiÞþ th  > > > ðr si Þ order XXsi , x ¼ aki , > > > > ⋮ > P s > ð i Þþ i < β ¼ r i j th j¼1 th ðiÞþ ðβþ ! ðr ðj iÞþ Þ order XXðj iÞþ , x ¼ aðj iÞþ i Þ order XX switching bifurcation > > > > > ⋮ > > > > : ðiÞþ th ðiÞþ ðiÞþ ðr 1 Þ order XX1 , x ¼ aj : ð6:185Þ
th þ
þ th þ th þ The bifurcation set of ð βþ XX1 : β2 XX2 : . . . : ðβþ 1 s2 Þ XXs2 Þ at the same parameter point is called a right-parallel-straw-bundle switching bifurcation.

6.3 Higher Order Equilibrium Bifurcations

285

(i) For the just before and after bifurcation structure, if there exists a relation of th th þ th þ  þ   ðβ i Þ XXi ¼ ðβ j Þ XX j ¼ ðβj Þ XX, for x ¼ ai ¼ a j

ði; j 2 f1; 2; . . . ; kgÞ, XX 2 fUS; LS; SO; SIg

ð6:186Þ

then the bifurcation is a (βj)thXXi switching bifurcation for higher order equilibriums. (ii) Just for the just before bifurcation structure, if there exists a relation of th  th   
ð2li Þ XXi ¼ ð2li Þ XX, for x ¼ ai ¼ ai i 2 f1; 2; . . . ; s1 g, XX 2 fUS; LSg

ð6:187Þ

then the bifurcation is a (2l)thXX left-appearing (or right-vanishing) bifurcation for higher order equilibriums. (iii) Just for the just after bifurcation structure, if there exists a relation of th þ þ  ð2lþ i Þ XXi ¼ ð2li Þ XX, for x ¼ ai ¼ ai ði 2 f1; 2; . . . ; s1 gÞ, XX 2 fUS; LSg th

ð6:188Þ

then the bifurcation is a (2l)thXX right-appearing (or left-vanishing) bifurcation for higher order equilibriums. (iv) For the just before and after bifurcation structure, if there exists a relation of th þ þ  þ   ðβ i Þ XXi 6¼ ðβ j Þ XX j for x ¼ ai ¼ a j th

ð6:189Þ

þ XX i , XX j 2 fUS; LS; SO; SIg

ði 2 f1; 2; . . . ; s1 g; j 2 f1; 2; . . . ; s2 gÞ, then two flower-bundle switching bifurcations of higher order equilibriums are as follows. þ (iv1) For βj ¼ βi + 2l, the bifurcation is called a ðβþ j Þ XXj right flower-bundle th

th þ  þ switching bifurcation for the ðβ i Þ XXi to ðβj Þ XXj switching of higher order equilibriums with the appearance (or birth) of (2l)thXX right-appearing (or left-vanishing) bifurcation. th  (iv2) For βj ¼ βi  2l, the bifurcation is called a ðβ i Þ XXi left flower-bundle th

th þ  þ switching bifurcation for the ðβ i Þ XXi to ðβj Þ XXj switching of higher order equilibriums with the vanishing ( or death) of (2l)thXX leftappearing (or right-vanishing) bifurcation. th

6 (2m)th-Degree Polynomial Systems

286

A general parallel switching bifurcation is switching

th   th   th  ððβ 1 Þ XX1 : ðβ2 Þ XX2 : . . . : ðβs1 Þ XXs1 Þ ! bifucation

th þ ððβþ 1 Þ XX1

:

th þ ðβþ 2 Þ XX2

: ... :

th þ ðβþ s2 Þ XXs2 Þ:

ð6:190Þ

Such a general, parallel switching bifurcation consists of the left and right parallel-bundle switching bifurcations for higher order equilibriums. If the left and right parallel-bundle switching bifurcations are same in a parallel flower-bundle switching bifurcation, i.e., th þ th  þ ðβ i Þ XXi ¼ ðβ i Þ XXi ¼ β XX, th

þ for x ¼ a i ¼ ai ði ¼ 1; 2; . . . ; sg

ð6:191Þ

then the parallel flower-bundle switching bifurcation becomes a parallel strawbundle switching bifurcation of ((α1)thXX : (β2)thXX :    : (βs)thXX). If the left and right parallel-bundle switching bifurcations are different in a parallel flower-bundle switching bifurcation, i.e., th   th þ þ þ ðα i Þ XXi ¼ ð2lj Þ XX, ðαj Þ XXj ¼ ð2lj Þ YY, th

th

þ for x ¼ a i 6¼ aj ði ¼ 1, 2, . . . , s1 ; j ¼ 1, 2, . . . , s2 Þ, XX 2 fUS, LSg, YY 2 fUS, LSg,

ð6:192Þ

then the parallel flower-bundle switching bifurcation for higher order equilibriums becomes a combination of two independent left and right parallel appearing bifurcations: th  th  th    (i) a ðð2l 1 Þ XX1 : ð2l2 Þ XX2 :    : ð2ls1 Þ XXs1 Þ left parallel sprinklerspraying-appearing (or right-vanishing) bifurcation and th þ th þ th þ þ þ (ii) a ðð2lþ 1 Þ XX1 : ð2l2 Þ XX2 :    : ð2ls2 Þ XXs2 Þ right parallel sprinklerspraying-appearing (or left-vanishing) bifurcation.

The parallel switching and appearing bifurcations for higher order equilibriums are presented in Fig. 6.12a–d. The waterfall appearing bifurcations and the flowerbundle switching bifurcations for higher order equilibriums are presented.

6.3 Higher Order Equilibrium Bifurcations

287

a

c

b

d

Fig. 6.12 (r1th XX : r2th XX :    : rnth XX) parallel switching/appearing bifurcations (a0 > 0): (a) without switching, and (b) with switching. The (r1th XX : r2th XX :    : rnth XX) parallel switching/appearing bifurcation (a0 < 0): (c) without switching, and (d) with switching. LS: lower saddle, US: upper saddle, SI: sink, SO: source. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

288

6 (2m)th-Degree Polynomial Systems

Reference Luo, A.C.J., 2020, The stability and bifurcation of the (2m)th degree polynomial systems, Journal of Vibration Testing and System Dynamics, 4(1), pp. 1–42.

Chapter 7

(2m+1)th-Degree Polynomial Systems

In this chapter, the global stability and bifurcations of equilibriums in the (2m+1)th-degree polynomial systems are discussed for a better understanding of the complexity of bifurcations and stability of equilibriums in such a (2m+1)thdegree polynomial system. The appearing and switching bifurcations are presented for simple equilibriums and higher-order equilibriums. The broom-appearing bifurcations, broom-spraying-appearing and broom-sprinkler-spraying-appearing bifurcations for simple and higher-order equilibriums are presented. The antennaswitching bifurcations for simple and higher-order equilibriums are discussed and the parallel straw-bundle-switching and flower-bundle-switching bifurcations for simple and higher order equilibriums are also presented.

7.1

Global Stability and Bifurcations

In a similar fashion of low-degree polynomial systems, the global stability and bifurcation of equilibriums in the (2m+1)th-degree polynomial nonlinear systems are discussed as in Luo (2019). The stability and bifurcation of each individual equilibrium are analyzed from the local analysis in Chapters 1 and 2. Definition 7.1 Consider a (2m+1)th-degree polynomial nonlinear system x_ ¼ A0 ðpÞx2mþ1 þ A1 ðpÞx2m þ . . . þ A2m1 ðpÞx2 þ A2m x þ A2mþ1 ðpÞ

 ¼ a0 ðpÞðx  aðpÞÞ x2 þ B1 ðpÞx þ C 1 ðpÞ . . . x2 þ Bm ðpÞx þ C m ðpÞ

ð7:1Þ

where A0(p) 6¼ 0, and p ¼ ð p1 , p2 , . . . , pm Þ T :

© Springer Nature Switzerland AG 2019 A. C. J. Luo, Bifurcation and Stability in Nonlinear Dynamical Systems, Nonlinear Systems and Complexity 28, https://doi.org/10.1007/978-3-030-22910-8_7

ð7:2Þ

289

7 (2m+1)th-Degree Polynomial Systems

290

(i) If Δi ¼ B2i  4C i < 0 for i ¼ 1, 2, . . . , m,

ð7:3Þ

the (2m+1)th-degree polynomial system has one equilibrium of x ¼ a, and the corresponding standard form is 1 2 1 1 1 2 x_ ¼ a0 ðx  aÞ½ðx þ B1 Þ þ ðΔ1 Þ . . . ½ðx þ Bm Þ þ ðΔm Þ: 2 4 2 4

ð7:4Þ

The flow of such a system with one equilibrium is called a single-equilibrium flow. (a) If a0 > 0, the equilibrium flow with x ¼ a is called a source flow. (b) If a0 < 0, the equilibrium flow with x ¼ a is called a sink flow. (ii) If Δi ¼ B2i  4C i > 0, i ¼ i1 , i2 , . . . , il 2 f1, 2, . . . , mg, Δj ¼ B2j  4C j < 0, j ¼ ilþ1 , ilþ2 , . . . , im 2 f1, 2, . . . , mg with l 2 f0, 1, . . . , mg,

ð7:5Þ

the (2m+1)th-degree polynomial nonlinear system has (2l + 1)-equilibriums as pffiffiffiffiffi pffiffiffiffiffi 1 1 ðiÞ ðiÞ x ¼ b1 ¼  ðBi þ Δi Þ, x ¼ b2 ¼  ðBi  Δi Þ 2 2 i 2 fi1 , i2 , . . . , il g  f1, 2, . . . , mg:

ð7:6Þ

ðjÞ bðiÞ r 6¼ bs for r,s 2 f1; 2g;i,j ¼ 1,2, . . . , l ð1Þ ð1Þ ðlÞ ðlÞ fa1 ; a2 ; . . . ; a2l g ¼ sortfa, b1 , b2 ,   , b1 , b2 g, as < asþ1 ,

ð7:7Þ

(ii1) If

then the corresponding standard form is x_ ¼ a0

2Y lþ1 i1 ¼1

ðx  ai1 Þ

m Y

1 1 2 ½ðx þ Bik Þ þ ðΔik Þ: 2 4 k¼lþ1

ð7:8Þ

(a) If a0 > 0, the simple equilibrium separatrix flow is called a (SO : SI : . . .: SO : SI : . . .SI : SO)-flow.

7.1 Global Stability and Bifurcations

291

(b) If a0 < 0, the simple equilibrium separatrix flow is called a (SI : SO : . . .: SI : SO : . . .SO : SI)-flow. (ii2) If ð1Þ

ð1Þ

ðlÞ

ðlÞ

fa1 ; a2 ; . . . ; a2lþ1 g ¼ sortfa; b1 ; b2 ; . . . ; b1 ; b2 g, ai 1  a1 ¼ . . . ¼ al 1 , ai2  al1 þ1 ¼ . . . ¼ al1 þl2 , ⋮

ð7:9Þ

air  aΣr1 ¼ . . . ¼ aΣr1 ¼ a2lþ1 i¼1 li þ1 i¼1 li þlr

r with Σs¼1 ls ¼ 2l þ 1,

then the corresponding standard form is x_ ¼ a0

r Y

ðx  ais Þls

s¼1

m Y

1 2 1 ½ðx þ Bik Þ þ ðΔik Þ: 2 4 k¼lþ1

ð7:10Þ

The equilibrium separatrix flow is called a (l1th XX : l2th XX : . . . : lrth XX)flow. (a) For a0 > 0 and p ¼ 1, 2, . . . , r,

lp th XX ¼

f

 

2r p  1

th th

order source, for αp ¼ 2M p  1, lp ¼ 2r p  1;

2r p  1 order sink, for αp ¼ 2M p , lp ¼ 2r p  1; th 2r p order lower‐saddle, for αp ¼ 2M p  1, lp ¼ 2r p ;  th 2r p order upper‐saddle, for αp ¼ 2M p , lp ¼ 2r p , 

ð7:11Þ

where αp ¼

Xr

l: s¼p s

ð7:12Þ

(b) For a0 < 0 and p ¼ 1, 2,   , r,

lp th XX ¼

f

 

2r p  1

th th

order sink, for αp ¼ 2M p  1, lp ¼ 2r p  1;

2r p  1 order source, for αp ¼ 2M p , lp ¼ 2r p  1; th 2r p order upper‐saddle, for αp ¼ 2M p  1, lp ¼ 2r p ;  th 2r p order lower‐saddle, for αp ¼ 2M p , lp ¼ 2r p : 

ð7:13Þ

7 (2m+1)th-Degree Polynomial Systems

292

(c) The equilibrium of x ¼ aip for (lp > 1)-repeated equilibrium switching is called an lpth XX bifurcation of ðlp1 th XX : lp2 th XX : . . . : lpβ th XXÞ equilibrium switching at a point p ¼ p1 2 ∂Ω12, and the bifurcation condition is aip  aΣp1 li þ1 ¼ . . . ¼ aΣp1 li þlp , i¼1 i¼1 Xβ   l : aΣp1 l þ1 6¼ . . . 6¼ aΣp1 l þl ; lp ¼ i¼1 pi i¼1 i

i¼1 i

ð7:14Þ

p

(iii) If Δi ¼ B2i  4C i ¼ 0, i 2 fi11 ; i12 ; . . . ; i1s g  fi1 ; i2 ; . . . ; il g  f1; 2; . . . ; mg, Δk ¼ B2k  4C k > 0, k 2 fi21 ; i22 ; . . . ; i2r g  fi1 ; i2 ; . . . ; il g  f1; 2; . . . ; mg, Δ j ¼ B2j  4C j < 0, j 2 filþ1 ; ilþ2 ; . . . ; im g  f1; 2; . . . ; mg, ð7:15Þ the (2m + 1)th-degree polynomial nonlinear system has (2l + 1)-equilibriums as 1 1 ðiÞ ðiÞ x ¼ b1 ¼  Bi , x ¼ b2 ¼  Bi for i 2 fi11 , i12 , . . . , i1s g, 2 2 pffiffiffiffiffiffi  pffiffiffiffiffiffi 1 1 ðk Þ ðk Þ  x ¼ b1 ¼  ðBk þ Δk Þ, x ¼ b2 ¼  ðBk  Δk Þ 2 2 for k 2 fi21 , i22 , . . . , i2r g:

ð7:16Þ

If n o ð1Þ ð1Þ ðlÞ ðlÞ fa1 , a2 , . . . , a2lþ1 g ¼ sort a, b1 , b2 , . . . , b1 , b2 , ai 1  a 1 ¼ . . . ¼ al 1 , ai2  al1 þ1 ¼ . . . ¼ al1 þl2 , ⋮ air  aΣr1 ¼ . . . ¼ aΣr1 ¼ a2lþ1 i¼1 li þ1 i¼1 li þlr

ð7:17Þ

with Σrs¼1 ls ¼ 2l þ 1, then the corresponding standard form is x_ ¼ a0

r Y s¼1

ðx  ais Þls

m Y

1 2 1 ½ðx þ Bik Þ þ ðΔik Þ: 2 4 k¼lþ1

ð7:18Þ

7.1 Global Stability and Bifurcations

293

The equilibrium separatrix flow is called a (l1th XX : l2th XX : . . . : lrth XX)flow. (a) The equilibrium of x ¼ aip for (lp > 1)-repeated equilibriums appearing or vanishing is called an lpth XX bifurcation of equilibrium at a point p ¼ p1 2 ∂Ω12, and the bifurcation condition is 1 aip  aΣp1 li þ1 ¼ . . . ¼ aΣp1 li þlp ¼  Biq , i¼1 i¼1 2 with Δiq ¼ B2iq  4C iq ¼ 0 ðiq 2 fi1 ; i2 ; . . . ; il gÞ,

ð7:19Þ

6¼ . . . 6¼ aþ or a 6¼ . . . 6¼ a : aþ Σp1 l þ1 Σp1 l þl Σp1 l þ1 Σp1 l þl i¼1 i

i¼1 i

i¼1 i

p

i¼1 i

p

(b) The equilibrium of x ¼ aiq for (lq > 1)-repeated equilibriums switching is called an lqth XX bifurcation of ðlq1 th XX : lq2 th XX : . . . : lqβ th XXÞ equilibrium switching at a point p ¼ p1 2 ∂Ω12, and the switching bifurcation condition is aiq  aΣq1 li þ1 ¼ . . . ¼ aΣq1 li þlp , i¼1

a l þ1 Σq1 i¼1 i

ð7:20Þ

i¼1

6¼ . . . 6¼

a ;l l þlq q Σq1 i¼1 i

β ¼ Σi¼1 lqi :

  (c) The equilibrium of x ¼ aip for l 1 -repeated equilibriums appearp 1  ance/vanishing and lp2 2 repeated equilibriums switching of ðlp21 th XX : lp22 th XX : . . . : lp2β th XXÞ is called an lpth XX bifurcation of equilibrium at a point p = p1 2 ∂Ω12, and the bifurcation condition is aip  aΣp1 li þ1 ¼    ¼ aΣp1 li þlp i¼1

i¼1

with Δiq ¼ B2iq  4Ciq ¼ 0 ðiq 2 fi1 ; i2 ; . . . ; il gÞ 6¼    6¼ aþ or ap1 1 aþ Σp1 l þj Σp1 l þj i¼1 i

i¼1 i

1

p1

Σi¼1 li þj1

6¼    6¼ ap1 1

Σi¼1 li þjp 1

for fj1 ; j2 ; . . . ; jp1 g  f1; 2; . . . ; lp g, a Σp1 l þk 1 i¼1 i

6¼    6¼

, ð7:21Þ

a Σp1 l þk p 2 i¼1 i

for fk1 ; k2 ; . . . ; k p2 g  f1; 2; . . . ; lp g, β with lp1 þ lp2 ¼ lp ; lp2 ¼ Σi¼1 lp2i

(iv) If Δi ¼ B2i  4C i > 0 for i ¼ 1,2, . . . , m

ð7:22Þ

the (2m + 1)th-degree polynomial nonlinear system has (2m+1)-equilibriums as

7 (2m+1)th-Degree Polynomial Systems

294

pffiffiffiffiffi pffiffiffiffiffi 1 1 ðiÞ ðiÞ x ¼ b1 ¼  ðBi þ Δi Þ, x ¼ b2 ¼  ðBi  Δi Þ 2 2

ð7:23Þ

for i ¼ 1, 2, . . . , m: (iv1) If bðriÞ 6¼ bðs jÞ for r, s 2 f1, 2g; i, j ¼ 1, 2, . . . , m n o ð1Þ ð1Þ ðmÞ ðmÞ ðak < akþ1 Þ, fa1 , a2 , . . . , a2m g ¼ sort a, b1 , b2 , . . . , b1 , b2

ð7:24Þ

then the corresponding standard form is x_ ¼ a0 ðx  a1 Þðx  a2 Þðx  a3 Þðx  a4 Þ . . . ðx  a2m Þðx  a2mþ1 Þ:

ð7:25Þ

This flow is formed with all the simple equilibriums. (a) If a0 > 0, the separatrix flow with (2m + 1) equilibriums is called a (SO : SI : . . .: SO : SI : . . .SI : SO)-flow. (b) If a0 < 0, the separatrix flow with (2m + 1) equilibriums is called a (SI : SO : . . .: SI : SO : . . .SO : SI)-flow. (iv2) If ð1Þ

ð1Þ

ðmÞ

ðmÞ

fa1 ; a2 . . . ; a2mþ1 g ¼ sortfa; b1 ; b2 ; . . . ; b1 ; b2 g, ai 1  a1 ¼ . . . ¼ al 1 , ai2  al1 þ1 ¼ . . . ¼ al1 þl2 , ⋮ air  aΣr1 ¼ . . . ¼ aΣr1 ¼ a2mþ1 i¼1 li þ1 i¼1 li þlr

ð7:26Þ

r with Σs¼1 ls ¼ 2m þ 1,

then the corresponding standard form is x_ ¼ a0

r Y

ðx  ais Þls :

ð7:27Þ

s¼1

The equilibrium separatrix flow is called a (l1th XX : l2th XX : . . . : lrth XX)flow. The equilibrium of x ¼ aip for lp-repeated equilibriums switching is called an lpth XX bifurcation of ðlp1 th XX : lp2 th XX : . . . : lpβ th XXÞ equilibrium switching at a point p ¼ p1 2 ∂Ω12, and the switching bifurcation condition is

7.1 Global Stability and Bifurcations

295

aip  aΣp1 li þ1 ¼ . . . ¼ aΣp1 li þlp , i¼1 i¼1 Xβ   aΣp1 l þ1 6¼ . . . 6¼ aΣp1 l þl ; lp ¼ l : i¼1 pi i¼1 i

i¼1 i

ð7:28Þ

p

Definition 7.2 Consider a (2m+1)th-degree polynomial nonlinear system as x_ ¼ A0 ðpÞx2mþ1 þ A1 ðpÞx2m þ . . . þ A2m1 ðpÞx2 þ A2m x þ A2mþ1 ðpÞ n Y q ¼ a0 ðpÞðx  aðpÞÞ ½x2 þ Bi ðpÞx þ C i ðpÞ i

ð7:29Þ

i¼1

where A0(p) 6¼ 0, and p ¼ ðp1 ; p2 ; . . . ; pm ÞT , m ¼

Xn

qi :

ð7:30Þ

Δi ¼ B2i  4Ci < 0 for i ¼ 1,2, . . . , n

ð7:31Þ

i¼1

(i) If

the (2m+1)th-degree polynomial nonlinear system has one equilibrium of x ¼ a, and the corresponding standard form is x_ ¼ a0 ðx  aÞ

n Y

qi 1 1 ½ðx þ Bi Þ2 þ ðΔi Þ : 2 4 i¼1

ð7:32Þ

The flow of such a system with one equilibrium is called a single equilibrium flow. (a) If a0 > 0, the equilibrium flow of x ¼ a is called a source flow. (b) If a0 < 0, the equilibrium flow of x ¼ a is called a sink flow. (ii) If Δi ¼ B2i  4C i > 0, i 2 fi1 ; i2 ; . . . ; il g  f1; 2; . . . ; ng, Δ j ¼ B2j  4C j < 0, j 2 filþ1 ; ilþ2 ; . . . ; in g  f1; 2; . . . ; ng

ð7:33Þ

the (2m + 1)th-degree polynomial nonlinear system has (2l + 1)-equilibriums as pffiffiffiffiffi pffiffiffiffiffi 1 1 ðiÞ ðiÞ x ¼ b1 ¼  ðBi þ Δi Þ, x ¼ b2 ¼  ðBi  Δi Þ 2 2 i 2 fi1 , i2 , . . . , il g  f1, 2, . . . , ng: (ii1) If

ð7:34Þ

7 (2m+1)th-Degree Polynomial Systems

296

bðriÞ 6¼ bðs jÞ for r, s 2 f1, 2g; i, j ¼ 1, 2, . . . , l ð1Þ

ð1Þ

ðr Þ

ðr Þ

fa1 , a2 , . . . , a2lþ1 g ¼ sortfa, b1 , b2 , . . . , b1 , b2 g, as asþ1 , |fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl} q1 sets

ð7:35Þ

qr sets

then the corresponding standard form is x_ ¼ a0

2Y lþ1

n Y

s¼1

k¼lþ1

ðx  as Þls

½ðx þ 12Bik Þ2 þ 14ðΔik Þ

qik

ð7:36Þ

with ls 2 fqi1 ; qi2 ;   ; qil ; 1g: The equilibrium separatrix flow is called a (l1thXX : l2thXX :. . .: l2l+1th XX)flow. (a) For a0 > 0 and p ¼ 1, 2, . . . , 2l + 1,

lp th XX ¼

f

 

2r p  1

th th

order source, for αp ¼ 2M p  1, lp ¼ 2r p  1;

2r p  1 order sink, for αp ¼ 2M p , lp ¼ 2r p  1; th 2r p order lower‐saddle, for αp ¼ 2M p  1, lp ¼ 2r p ;  th 2r p order upper‐saddle, for αp ¼ 2M p , lp ¼ 2r p , 

ð7:37Þ

where αp ¼

X2lþ1 s¼p

ls :

ð7:38Þ

(b) For a0 < 0 and p ¼ 1, 2, . . . , 2l + 1,

lp th XX ¼

(ii2) If

f

 

2r p  1

th th

order sink, for αp ¼ 2M p  1, lp ¼ 2r p  1;

2r p  1 order source, for αp ¼ 2M p , lp ¼ 2r p  1; th 2r p order upper‐saddle, for αp ¼ 2M p  1, lp ¼ 2r p ;  th 2r p order lower‐saddle, for αp ¼ 2M p , lp ¼ 2r p : 

ð7:39Þ

7.1 Global Stability and Bifurcations

297 ð1Þ

ð1Þ

ðr Þ

ðr Þ

fa1 , a2 , . . . , a2lþ1 g ¼ sortfa, b1 , b2 , . . . , b1 , b2 g, |fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl} q1 sets

ai1  a1 ¼ . . . ¼ al1 , ai2  al1 þ1 ¼ . . . ¼ al1 þl2 ,

qr sets

ð7:40Þ

⋮ air  aΣr1 ¼ . . . ¼ aΣr1 ¼ a2lþ1 i¼1 li þ1 i¼1 li þlr with Σrs¼1 ls ¼ 2l þ 1, then the corresponding standard form is x_ ¼ a0

r Y

ðx  ais Þls

s¼1

n Y

qi 1 1 2 ½ðx þ Bik Þ þ ðΔik Þ k : 2 4 k¼lþ1

ð7:41Þ

The equilibrium separatrix flow is called an (l1th XX : l2th XX : . . . : lrth XX)flow. (a) For a0 > 0 and s ¼ 1, 2, . . . , r,

lp th XX ¼

f

 

2r p  1

th th

order source, for αp ¼ 2M p  1, lp ¼ 2r p  1;

2r p  1 order sink, for αp ¼ 2M p , lp ¼ 2r p  1; th 2r p order lower‐saddle, for αp ¼ 2M p  1, lp ¼ 2r p ;  th 2r p order upper‐saddle, for αp ¼ 2M p , lp ¼ 2r p , 

ð7:42Þ

where αp ¼

Xr

l: s¼p s

ð7:43Þ

(b) For a0 < 0 and p ¼ 1, 2, . . . , r,

lp th XX ¼

f

 

2r p  1

th th

order sink, for αp ¼ 2M p  1, lp ¼ 2r p  1;

2r p  1 order source, for αp ¼ 2M p , lp ¼ 2r p  1; th 2r p order upper‐saddle, for αp ¼ 2M p  1, lp ¼ 2r p ;  th 2r p order lower‐saddle, for αp ¼ 2M p , lp ¼ 2r p : 

ð7:44Þ

(c) The equilibrium of x ¼ aip for (lp > 1)-repeated equilibriums switching is called an lpth XX bifurcation of ðlp1 th XX : lp2 th XX : . . . : lpβ th XXÞ equilibrium switching at a point p ¼ p1 2 ∂Ω12, and the switching bifurcation condition is

7 (2m+1)th-Degree Polynomial Systems

298

aip  aΣp1 li þ1 ¼ . . . ¼ aΣp1 li þlp , i¼1 i¼1 Xβ   aΣp1 l þ1 6¼ . . . 6¼ aΣp1 l þl ; lp ¼ l : i¼1 pi i¼1 i

i¼1 i

ð7:45Þ

p

(iii) If Δi ¼ B2i  4Ci ¼ 0, i 2 fi11 ; i12 ; . . . ; i1s g  fi1 ; i2 ; . . . ; il g  f1; 2; . . . ; ng, Δk ¼ B2k  4Ck > 0, k 2 fi21 ; i22 ; . . . ; i2r g  fi1 ; i2 ; . . . ; il g  f1; 2; . . . ; ng, ð7:46Þ Δ j ¼ B2j  4C j < 0, j 2 filþ1 ; ilþ2 ; . . . ; in g  f1; 2; . . . ; ng with i 6¼ j 6¼ k, the (2m+1)th-degree polynomial nonlinear system has (2l + 1)-equilibriums as 1 1 ðiÞ ðiÞ x ¼ b1 ¼  Bi , x ¼ b2 ¼  Bi for i 2 fi11 ; i12 ; . . . ; i1s g, 2 2 pffiffiffiffiffiffi pffiffiffiffiffiffi 1 1 ðkÞ ðkÞ x ¼ b1 ¼  ðBk þ Δk Þ, x ¼ b2 ¼  ðBk  Δk Þ 2 2 for k 2 fi21 ; i22 ; . . . ; i2r g:

ð7:47Þ

If ð1Þ

ð1Þ

ðlÞ

ðlÞ

fa1 ; a2 ; . . . ; a2lþ1 g ¼ sortfa; b1 ; b2 ; . . . ; b1 ; b2 g, ai 1  a1 ¼ . . . ¼ al 1 , ai2  al1 þ1 ¼ . . . ¼ al1 þl2 , ⋮

ð7:48Þ

air  aΣr1 ¼ . . . ¼ aΣr1 ¼ a2lþ1 i¼1 li þ1 i¼1 li þlr

r with Σs¼1 ls ¼ 2l þ 1,

then the corresponding standard form is x_ ¼ a0

r Y s¼1

ðx  ais Þls

n Y

qi 1 2 1 ½ðx þ Bik Þ þ ðΔik Þ k : 2 4 k¼lþ1

ð7:49Þ

The equilibrium separatrix flow is called an (l1th XX : l2th XX : . . . : lrth XX)flow. (a) The equilibrium of x ¼ aip for (lp > 1)-repeated equilibriums appearing or vanishing is called an lpth XX bifurcation of equilibrium at a point p ¼ p1 2 ∂Ω12, and the bifurcation condition is

7.1 Global Stability and Bifurcations

299

1 aip  aΣp1 li þ1 ¼ . . . ¼ aΣp1 li þlp ¼  Biq i¼1 i¼1 2   with Δiq ¼ B2iq  4C iq ¼ 0 iq 2 fi1 , i2 , . . . , il g , aþ Σp1 l þ1 i¼1 i

6¼ . . . 6¼

aþ or Σp1 l þlp i¼1 i

a Σp1 l þ1 i¼1 i

6¼ . . . 6¼

ð7:50Þ

a : Σp1 l þlp i¼1 i

(b) The equilibrium of x ¼ aiq for (lq > 1)-repeated equilibriums switching is called an lqth XX bifurcation of ðlq1 th XX : lq2 th XX : . . . : lqβ th XXÞ equilibrium switching at a point p ¼ p1 2 ∂Ω12, and the bifurcation condition is aiq  aΣq1 li þ1 ¼ . . . ¼ aΣq1 li þlp , i¼1

a Σq1 l þ1 i¼1 i

i¼1

6¼ . . . 6¼

a ; lp Σq1 l þlq i¼1 i

¼

Xβ

l : i¼1 pi

ð7:51Þ

(iv) If Δi ¼ B2i  4Ci > 0 for i ¼ 1,2, . . . , n

ð7:52Þ

the (2m + 1)th-degree polynomial nonlinear system has (2n + 1)-equilibriums as pffiffiffiffiffi pffiffiffiffiffi 1 1 ðiÞ ðiÞ x ¼ b1 ¼  ðBi þ Δi Þ, x ¼ b2 ¼  ðBi  Δi Þ 2 2 for i ¼ 1, 2, . . . , n:

ð7:53Þ

(iv1) If bðriÞ 6¼ bðs jÞ for r, s 2 f1, 2g, ði, j ¼ 1, 2, . . . , nÞ; ð1Þ

ð1Þ

ðnÞ

ðnÞ

fa1 , a2 , . . . , a2nþ1 g ¼ sortfa, b1 , b2 , . . . , b1 , b2 g ðas asþ1 Þ, |fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl} q1 sets

ð7:54Þ

qn sets

then the corresponding standard form is x_ ¼ a0

2nþ1 Y

ðx  as Þls with ls 2 fqi1 ; qi2 ; . . . ; qin ; 1g:

ð7:55Þ

s¼1

The equilibrium separatrix flow is called an (l1thXX : l2thXX :. . .: l2n+1th XX)-flow.

7 (2m+1)th-Degree Polynomial Systems

300

(a) For a0 > 0 and p ¼ 1, 2, . . . , 2n + 1,

lp th XX ¼

f

 

2r p  1

th th

order source, for αp ¼ 2M p  1, lp ¼ 2r p  1;

2r p  1 order sink, for αp ¼ 2M p , lp ¼ 2r p  1; th 2r p order lower‐saddle, for αp ¼ 2M p  1, lp ¼ 2r p ;  th 2r p order upper‐saddle, for αp ¼ 2M p , lp ¼ 2r p , 

ð7:56Þ

where αp ¼

X2nþ1 s¼p

ls :

ð7:57Þ

(b) For a0 < 0 and p ¼ 1, 2, . . . , 2n + 1,

lp th XX ¼

f

 

2r p  1

th th

order sink, for αp ¼ 2M p  1, lp ¼ 2r p  1;

2r p  1 order source, for αp ¼ 2M p , lp ¼ 2r p  1; th 2r p order upper‐saddle, for αp ¼ 2M p  1, lp ¼ 2r p ;  th 2r p order lower‐saddle, for αp ¼ 2M p , lp ¼ 2r p : 

ð7:58Þ

(iv2) If ð1Þ

ð1Þ

ðnÞ

ðnÞ

fa1 , a2 , . . . , a2nþ1 g ¼ sortfa, b1 , b2 , . . . , b1 , b2 g, |fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl} q1 sets

ai 1  a1 ¼ . . . ¼ al 1 , ai2  al1 þ1 ¼ . . . ¼ al1 þl2 ,

qn sets

ð7:59Þ

⋮ air  aΣr1 ¼ . . . ¼ aΣr1 ¼ a2nþ1 , i¼1 li þ1 i¼1 li þlr with Σrs¼1 ls ¼ 2n þ 1, then the corresponding standard form is x_ ¼ a0

r Y

ðx  ais Þls :

ð7:60Þ

s¼1

The equilibrium separatrix flow is called an (l1th XX : l2th XX : . . . : lrth XX)flow. The equilibrium of x ¼ aip for lp-repeated equilibriums switching is  called an lpth XX switching bifurcation of lp1 th XX : lp2 th XX : . . . : lpβ th XXÞ equilibrium at a point p ¼ p1 2 ∂Ω12, and the switching bifurcation condition is

7.1 Global Stability and Bifurcations

301

aip  aΣp1 li þ1 ¼ . . . ¼ aΣp1 li þlp , i¼1 i¼1 Xβ   aΣp1 l þ1 6¼ . . . 6¼ aΣp1 l þl ; lp ¼ l : i¼1 pi i¼1 i

i¼1 i

ð7:61Þ

p

Definition 7.3 Consider a 1-dimensional, (2m + 1)th-degree polynomial nonlinear dynamical system x_ ¼ A0 ðpÞx2mþ1 þ A1 ðpÞx2m þ . . . þ A2m1 ðpÞx2 þ A2m x þ A2mþ1 ðpÞ r n Y Y q ¼ a0 ðpÞ ðx  cis ðpÞÞls ½x2 þ Bi ðpÞx þ C i ðpÞ i s¼1

ð7:62Þ

i¼rþ1

where A0(p) 6¼ 0, and Xr

¼ 2l þ 1,

l s¼1 s

Xn i¼rþ1

qi ¼ ðm  lÞ, p ¼ ðp1 ; p2 ; . . . ; pm ÞT :

ð7:63Þ

(i) If Δi ¼ B2i  4C i < 0 for i ¼ r þ 1,r þ 2, . . . , n, fa1 ; a2 ; . . . ; ar g ¼ sortfc1 ; c2 ; . . . ; cr g with ai < aiþ1

ð7:64Þ

the (2m+1)th-degree polynomial system has equilibriums of x ¼ ais ðpÞ (s ¼ 1, 2, . . . , r), and the corresponding standard form is x_ ¼ a0 ðpÞ

r Y j¼1

ðx  ai j Þl j

n Y

1 1 ½ðx þ Bi Þ2 þ ðΔi Þli : 2 4 i¼rþ1

ð7:65Þ

The equilibrium separatrix flow is called an (l1th XX : l2th XX : . . . : lrth XX)flow. (a) For a0 > 0 and s ¼ 1, 2, . . . , r,

lp th XX ¼

f

 

2r p  1

th th

order source, for αp ¼ 2M p  1, lp ¼ 2r p  1;

2r p  1 order sink, for αp ¼ 2M p , lp ¼ 2r p  1; th 2r p order lower‐saddle, for αp ¼ 2M p  1, lp ¼ 2r p ;  th 2r p order upper‐saddle, for αp ¼ 2M p , lp ¼ 2r p , 

ð7:66Þ

7 (2m+1)th-Degree Polynomial Systems

302

where αp ¼

Xr

l: s¼p s

ð7:67Þ

(b) For a0 < 0 and p ¼ 1, 2, . . . , r,

lp th XX ¼

f

 

2r p  1

th th

order sink, for αp ¼ 2M p  1, lp ¼ 2r p  1;

2r p  1 order source, for αp ¼ 2M p , lp ¼ 2r p  1; th 2r p order upper‐saddle, for αp ¼ 2M p  1, lp ¼ 2r p ;  th 2r p order lower‐saddle, for αp ¼ 2M p , lp ¼ 2r p :

ð7:68Þ



(ii) If Δi ¼ B2i  4Ci > 0, i ¼ j1 , j2 , . . . , js 2 fl þ 1; l þ 2; . . . ; ng, ð7:69Þ

Δ j ¼ B2j  4C j < 0, j ¼ jsþ1 , jsþ2 , . . . , jn 2 fl þ 1; l þ 2; . . . ; ng with s 2 f1; . . . ; n  lg,

the (2m+1)th-degree polynomial nonlinear system has 2n2-equilibriums as pffiffiffiffiffi pffiffiffiffiffi 1 1 ðiÞ ðiÞ x ¼ b1 ¼  ðBi þ Δi Þ, x ¼ b2 ¼  ðBi  Δi Þ 2 2  i 2 j1 , j2 , . . . , jn1  fl þ 1, l þ 2, . . . , ng:

ð7:70Þ

If ðrþ1Þ

ðrþ1Þ

ðn Þ

ðn Þ

fa1 , a2 , . . . , a2n2 þ1 g ¼ sortfc1 , c2 , . . . , c2lþ1 , b1 , b2 , . . . , b1 1 , b2 1 g, |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} |fflfflfflfflfflffl{zfflfflfflfflfflffl} qrþ1 sets

ai1  a1 ¼ . . . ¼ al1 , ai2  al1 þ1 ¼ . . . ¼ al1 þl2 ,

qn1 sets

⋮ ain1  aΣn1 1 li þ1 ¼ . . . ¼ aΣn1 1 li þln ¼ a2n2 þ1 i¼1

i¼1

1

1 with Σns¼1 ls ¼ 2n2 þ 1,

ð7:71Þ

7.1 Global Stability and Bifurcations

303

then the corresponding standard form is x_ ¼ a0

n1 Y

ðx  ais Þls

s¼1

n Y

qi 1 2 1 ½ðx þ Bi Þ þ ðΔi Þ : 2 4 i¼n þ1

ð7:72Þ

2

The equilibrium separatrix flow is called an ðl1 th XX : l2 th XX : . . . : ln1 th XXÞflow. (a) For a0 > 0 and p ¼ 1, 2, . . . , r, r + 1, . . ., n1,

lp th XX ¼

f

 

2r p  1

th th

order source, for αp ¼ 2M p  1, lp ¼ 2r p  1;

2r p  1 order sink, for αp ¼ 2M p , lp ¼ 2r p  1; th 2r p order lower‐saddle, for αp ¼ 2M p  1, lp ¼ 2r p ;  th 2r p order upper‐saddle, for αp ¼ 2M p , lp ¼ 2r p , 

ð7:73Þ

where αp ¼

Xn1

l: s¼p s

ð7:74Þ

(b) For a0 < 0 and p ¼ 1, 2, . . . , r, r + 1, . . ., n1,

lp th XX ¼

f

 

2r p  1

th th

order sink, for αp ¼ 2M p  1, lp ¼ 2r p  1;

2r p  1 order source, for αp ¼ 2M p , lp ¼ 2r p  1; th 2r p order upper‐saddle, for αp ¼ 2M p  1, lp ¼ 2r p ;  th 2r p order lower‐saddle, for αp ¼ 2M p , lp ¼ 2r p : 

ð7:75Þ

(c) The equilibrium of x ¼ aip for (lp > 1)-repeated  equilibriums switching is called an lpth XX switching bifurcation of lp1 th XX : lp2 th XX : . . . : lpβ th XXÞ equilibrium at a point p ¼ p1 2 ∂Ω12, and the bifurcation condition is aip  aΣp1 li þ1 ¼ . . . ¼ aΣp1 li þlp , i¼1 i¼1 Xβ   aΣp1 l þ1 6¼ . . . 6¼ aΣp1 l þl ; lp ¼ l : i¼1 pi i¼1 i

i¼1 i

p

ð7:76Þ

7 (2m+1)th-Degree Polynomial Systems

304

(iii) If Δi ¼ B2i  4C i ¼ 0, for i 2 fi11 , i12 , . . . , i1s g  filþ1 , ilþ2 , . . . , in2 g  fl þ 1, l þ 2, . . . , ng, Δk ¼ B2k  4Ck > 0, for k 2 fi21 , i22 , . . . , i2r g  filþ1 , ilþ2 , . . . , in2 g  fl þ 1, l þ 2, . . . , ng,

ð7:77Þ

Δj ¼ B2j  4C j < 0, for j 2 fin2 þ1 , in2 þ2 , . . . , in g  fl þ 1, l þ 2, . . . , ng, the (2m + 1)th-degree polynomial nonlinear system has (2n2+1)-equilibriums as 1 1 ðiÞ ðiÞ x ¼ b1 ¼  Bi , x ¼ b2 ¼  Bi for i 2 fi11 , i12 , . . . , i1s g, 2 2 ffiffiffiffiffi ffi p pffiffiffiffiffiffi 1 1 ð k Þ ðk Þ x ¼ b1 ¼  ðBk þ Δk Þ, x ¼ b2 ¼  ðBk  Δk Þ 2 2 for i 2 fi21 , i22 , . . . , i2r g:

ð7:78Þ

If ðr Þ

ðr Þ

ðn Þ

ðn Þ

fa1 ,a2 , ...,a2n2 þ1 g ¼ sortfa,c1 ,c2 , ...,c2l ,b1 ,b2 , ...,b1 1 ,b2 1 g, |fflfflfflffl{zfflfflfflffl} |fflfflfflfflffl{zfflfflfflfflffl} qr sets

ai1  a1 ¼ ... ¼ al1 ,

qn1 sets

ai2  al1 þ1 ¼ ... ¼ al1 þl2 ,

ð7:79Þ

⋮ ain1  aΣn1 1 li þ1 ¼ ... ¼ aΣn1 1 li þln ¼ a2n2 þ1 i¼1

i¼1

1

1 with Σns¼1 ls ¼ 2n2 þ1,

then the corresponding standard form is x_ ¼ a0

n1 Y s¼1

ðx  ais Þls

n Y

qi 1 2 1 ½ðx þ Bi Þ þ ðΔi Þ : 2 4 i¼n þ1

ð7:80Þ

2

The equilibrium separatrix flow is called an l1 th XX : l2 th XX : . . . : ln1 th XXÞflow. (a) The equilibrium of x ¼ aip for (qp > 1)-repeated equilibriums appearing or vanishing is called an lpth XX bifurcation of equilibrium at a point p ¼ p1 2 ∂Ω12, and the bifurcation condition is

7.1 Global Stability and Bifurcations

305

1 aip  aΣp1 li þ1 ¼ . . . ¼ aΣp1 li þlp ¼  Biq i¼1 i¼1 2   with Δiq ¼ B2iq  4Ciq ¼ 0 iq 2 fi1 , i2 , . . . , il g aþ Σp1 q þ1 i¼1 i

6¼ . . . 6¼

aþ or Σp1 q þqp i¼1 i

a Σp1 q þ1 i¼1 i

6¼ . . . 6¼

ð7:81Þ

a : Σp1 q þqp i¼1 i

(b) The equilibrium of x ¼ aiq for (lp > 1)-repeated equilibriums switching is called an lpth XX bifurcation of ðlp1 th XX : lp2 th XX : . . . : lpβ th XXÞ equilibrium switching at a point p ¼ p1 2 ∂Ω12, and the bifurcation condition is aiq  aΣq1 li þ1 ¼ . . . ¼ aΣq1 li þlp , i¼1

a Σq1 l þ1 i¼1 i

6¼ . . . 6¼

i¼1

a , Σq1 l þlq i¼1 i

lp ¼

Xβ

l : i¼1 pi

ð7:82Þ

  (c) The equilibrium of x ¼ aip for lp1 1 -repeated equilibriums appearing/ vanishing and lp2 2 -repeated equilibriums switching of lp21 th XX : lp22 th XX : . . . : lp2β th XXÞ is called an qpth XX bifurcation of equilibrium at a point p ¼ p1 2 ∂Ω12, and the bifurcation condition is aip  aΣp1 q þ1 ¼ . . . ¼ aΣp1 q þq p i¼1 i i¼1 i   with Δiq ¼ B2iq  4C iq ¼ 0 iq 2 fi1 , i2 , . . . , il g 6¼ . . . 6¼ aþ or a 6¼ . . . 6¼ a , aþ p1 1 p1 1 q þj1 q þjp1 Σp1 Σp1 Σi¼1 Σi¼1 qi þj1 qi þjp1 i¼1 i i¼1 i  for fj1 j2 . . . jp1 g  1, 2, . . . , qp ,

ð7:83Þ

6¼ . . . 6¼ a a Σp1 Σp1 qi þk p q þk 1 i¼1 i  i¼1  2 for k 1 , k 2 , . . . , kp2  1, 2, . . . , qp , with lp1 þ lp2 ¼ lp : (iv) If Δi ¼ B2i  4C i > 0 for i ¼ l þ 1,l þ 2, . . . , n

ð7:84Þ

the (2m+1)th-degree polynomial nonlinear system has (2m + 1)-equilibriums as pffiffiffiffiffi pffiffiffiffiffi 1 1 ðiÞ ðiÞ x ¼ b1 ¼  ðBi þ Δi Þ, x ¼ b2 ¼  ðBi  Δi Þ 2 2 for i ¼ l þ 1, l þ 2, . . . , n:

ð7:85Þ

7 (2m+1)th-Degree Polynomial Systems

306

If ðrþ1Þ

ðrþ1Þ

ðnÞ

ðnÞ

fa1 , a2 , . . . , a2mþ1 g ¼ sortfc1 , c2 , . . . , c2lþ1 , b1 , b2 , . . . , b1 , b2 g, |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} |fflfflfflffl{zfflfflfflffl} qrþ1 sets

ai1  a1 ¼ . . . ¼ al1 , ai2  al1 þ1 ¼ . . . ¼ al1 þl2 ,

qn sets

ð7:86Þ

⋮ ain  aΣn1 ¼ . . . ¼ aΣn1 ¼ a2mþ1 i¼1 li þ1 i¼1 li þlr with Σns¼1 ls ¼ 2m þ 1, then the corresponding standard form is x_ ¼ a0

r Y

ðx  ais Þls :

ð7:87Þ

s¼1

The equilibrium separatrix flow is called an (l1th XX : l2th XX : . . . : lrth XX)flow. The equilibrium of x ¼ aip for lp-repeated equilibriums switching  is called an lpth XX switching bifurcation of lp1 th XX : lp2 th XX : . . . : lpβ th XXÞ equilibrium at a point p ¼ p1 2 ∂Ω12, and the bifurcation condition is aip  aΣp1 li þ1 ¼ . . . ¼ aΣp1 li þlp , i¼1 i¼1 Xβ   aΣp1 l þ1 6¼ . . . 6¼ aΣp1 l þl ; lp ¼ l : i¼1 pi i¼1 i

7.2

i¼1 i

ð7:88Þ

p

Simple Equilibrium Bifurcations

To illustrate the bifurcations in the (2m+1)th-degree polynomial system as in Luo (2019), the detailed discussion with graphical illustrations will be presented as follows.

7.2.1

Appearing Bifurcations

Consider a (2m+1)th-degree polynomial system as x_ ¼ a0 QðxÞðx  aÞ

n Y i¼1

ðx2 þ Bi x þ Ci Þ

ð7:89Þ

7.2 Simple Equilibrium Bifurcations

307

where without loss of generality, a function of Q(x) > 0 is either a polynomial function or a non-polynomial function. The roots of x2 + Bix + Ci ¼ 0 are 1 1pffiffiffiffiffi ðiÞ Δi , Δi ¼ B2i  4Ci 0 ði ¼ 1; 2; . . . ; nÞ; b1,2 ¼  Bi  2 2 ð1Þ ð1Þ ð2Þ ð2Þ ðnÞ ðnÞ fa1 ; a2 ; . . . ; a2l g  sortfb1 , b2 , b1 , b2 , . . . , b1 , b2 g, as asþ1 ; Bi 6¼ B j ði; j ¼ 1; 2; . . . ; n; i 6¼ jÞ at bifurcation: Δi ¼ 0 ði ¼ 1; 2; . . . ; nÞ

ð7:90Þ

g

The second-order singularity bifurcation is for the birth of a pair of simple sink and source. There are two appearing bifurcations for i 2 {1, 2, . . ., n}: ith quadratic factor



2nd order US







! appearing bifurcation ith quadratic factor

2 order LS







! nd

appearing bifurcation



SO, for x ¼ a2i , SI, for x ¼ a2i1 :

ð7:91Þ

SI, for x ¼ a2i , SO, for x ¼ a2i1 :

ð7:92Þ

If x ¼ a 6¼  12 Bi (i 2 {1, 2, . . ., m}), the equilibrium of x ¼ a breaks a cluster of teeth comb appearing bifurcations to two parts. The teeth comb appearing bifurcation generated by the m-pairs of quadratics becomes a broom appearing bifurcation. The two broom appearing bifurcations are l1 þl2 ¼m

SO ð x ¼ aÞ







! ðl1 ‐LSN : SO : l2 ‐USNÞ appearing bifurcation 8 l2 ‐USN, for x 2 fa2j , a2jþ1 , i ¼ l1 þ 1 . . . l1 þ l2 g, > > < ¼ SO, for x ¼ a ¼ a2ðl1 þ1Þ1 > > : l1 ‐LSN, for x 2 fa2i1 , a2i , i ¼ 1, 2, . . . , l1 g

ð7:93Þ

and l1 þl2 ¼m

SI ð x ¼ aÞ







! ðl1 ‐USN : SO : l2 ‐LSNÞ m‐appearing bifurcation 8 l2 ‐LSN, for x 2 fa2j , a2jþ1 : j ¼ l1 þ 1 . . . l1 þ l2 g, > > < ¼ SI, for x ¼ a ¼ a2ðl1 þ1Þ1 > > : l1 ‐USN, for x 2 fa2i1 , a2i : i ¼ 1, 2, . . . , l1 g where the lj-USN and lj-LSN ( j ¼ 1,2) are

ð7:94Þ

308

7 (2m+1)th-Degree Polynomial Systems

8 8 > < SO, for x ¼ a2ðs þl Þþδ2 , th > ðl þs Þ bifurcation j j > j j j > > US







! > > appearing : SI, for x ¼ a > > 2ðsj þlj Þ1þδ2j ; < lj ‐USN ⋮ 8 > > > < SO, for x ¼ a2sj þδ2j , > th > ðs Þ bifurcation j > > US





! > > appearing : SI, for x ¼ a2s 1þδ2 : j j

ð7:95Þ

8 8 > < SI, for x ¼ a2ðsj þlj Þþδ2j , th > ðlj þsj Þ bifurcation > > > LS





! > > appearing : SO, for x ¼ a2ðs þl Þ1þδ2 ; > > j j j < lj ‐LSN ⋮ 8 > > > < SI, for x ¼ a2sj þδ2j , > th > ðsj Þ bifurcation > > LS





! > > appearing : SO, for x ¼ a2s 1þδ2 : j j

ð7:96Þ

for sj 2 {0, 1, 2, . . ., m} and 0 lj m with 0 lj m. Four special broom appearing bifurcations are 8 SO ! SO, for x ¼ a ¼ a2mþ1 , > > 8 ( > > SI, for x ¼ a2m , > th > > m bifurcation > > > >

! > LS





> > > > appearing < > SO, for x ¼ a2m1 ; > <  SO ðx ¼ aÞ ! > m‐LSN ⋮ > > > > ( > > > > > SI, for x ¼ a2 , > st > > 1 bifurcation > > > >

! > : : LS





appearing SO, for x ¼ a1 : 8 SI ! SI, for x ¼ a ¼ a2mþ1 , > > 8 ( > > SO, for x ¼ a2m , > > > mth bifurcation > > > > US





! > > > > > appearing < > SI, for x ¼ a2m1 ; > <  SI ðx ¼ aÞ ! > m‐USN ⋮ > > > > ( > > > > > SO, for x ¼ a2 , > > > 1st bifurcation > > > >

! > : : US





appearing SI, for x ¼ a1 : and

ð7:97Þ

ð7:98Þ

7.2 Simple Equilibrium Bifurcations

309

8 ( 8 SO, for x ¼ a2mþ1 , th > > m bifurcation > > > > US





! > > > > > appearing > > > SI, for x ¼ a2m ; > > < > > < m‐USN ⋮ > SO ðx ¼ aÞ ! ( > > > > SO, for x ¼ a3 , > st > > 1 bifurcation > > > >

! > > : US





> appearing > > SI, for x ¼ a2 ; > : SO ! SO, for x ¼ a ¼ a1 :

ð7:99Þ

8 ( 8 SI, for x ¼ a2mþ1 , th > > m bifurcation > > > > LS





! > > > > > appearing > > > SO, for x ¼ a2m ; > > < > > < m‐LSN ⋮ > SI ðx ¼ aÞ ! ( > > > > SI, for x ¼ a3 , > > > 1st bifurcation > > > > > > : LS



! > appearing > > SO, for x ¼ a2 ; > :  SI ! SI,for x ¼ a ¼ a1 :

ð7:100Þ

If x ¼ a ¼  12 Bi (i 2 {1, 2, . . ., m}), the equilibrium of x ¼ a possesses a thirdorder SI or SO switching bifurcation (or pitchfork bifurcation). The teeth comb appearing bifurcation generated by the m-pairs of quadratics becomes a broom appearing bifurcation. The two broom appearing bifurcations are m¼l1 þl2 þ1

SO ðx ¼ aÞ







! ðl1 ‐LSN : 3rd SO : l2 ‐USNÞ appearing bifurcation 8 ‐USN, for x 2 fa2j , a2jþ1 , j ¼ l1 þ 2 . . . l1 þ l2 g, l > 2 > > > 8 >  > > > > > SO, for x ¼ a2ðl1 þ2Þ1 > > < < ¼ 3rd SO ! SI, for x ¼ a ¼ a2ðl1 þ1Þ > > > > > > > : > SO, for x ¼ a2ðl1 þ1Þ1 > > > > : l1 ‐LSN, for x 2 fa2i1 , a2i , i ¼ 1, 2, . . . , l1 g

ð7:101Þ

and m¼l1 þl2 þ1

SI ð x ¼ aÞ







! ðl1 ‐USN : 3rd SI : l2 ‐LSNÞ appearing bifurcation 8 l2 ‐LSN, for x 2 fa2j , a2jþ1 , j ¼ l1 þ 2 . . . l1 þ l2 g, > > > > 8 > > SI, for x ¼ a2ðl1 þ2Þ1 > > > > > > < < ¼ 3rd SI ! SO, for x ¼ a ¼ a2ðl1 þ1Þ > > > > > > > : > SI, for x ¼ a2ðl1 þ1Þ1 > > > > : l1 ‐USN, for x 2 fa2i1 , a2i , i ¼ 1, 2, . . . , l1 g:

ð7:102Þ

7 (2m+1)th-Degree Polynomial Systems

310

Consider an appearing bifurcation for a cluster of sink and source equilibriums with the following conditions:

g

Bi ¼ B j ði; j 2 f1; 2; . . . ; ng; i 6¼ jÞ at bifurcation: Δ j ¼ 0 ði ¼ 1; 2; . . . ; nÞ

ð7:103Þ

Thus, the (2l)th-order appearing bifurcation is for a cluster of simple sinks and sources. Two (2l)th-order appearing bifurcations for l 2 {1, 2, . . ., s} are 8 SO, for x ¼ a2sl , > > > >  > > < SI, for x ¼ a2sl 1 , cluster of l‐quadratics th ð2lÞ order USN







! ⋮ appearing bifurcation > > > SO, for x ¼ a2s , > 1 > > : SI, for x ¼ a2s1 1

ð7:104Þ

8 SI, for x ¼ a2sl , > > > >  > > < SO, for x ¼ a2sl 1 , cluster of l‐quadratics th ð2lÞ order LSN







! ⋮ appearing bifurcation > > > SI, for x ¼ a2s , > 1 > > : SO, for x ¼ a2s1 1 :

ð7:105Þ

If x ¼ a 6¼  12 Bi (i 2 {1, 2, . . ., n}), the equilibrium of x ¼ a breaks a cluster of sprinkler-spraying appearing bifurcations to two parts. The sprinkler-spraying appearing bifurcation generated by the m-pairs of quadratics becomes a broomsprinkler-spraying appearing bifurcation. The two broom-sprinkler-spraying appearing bifurcations are m¼m1 þm2

SO ð x ¼ aÞ





! ðr 1 ‐LSG : SO : r 2 ‐USGÞ appearing bifurcation 8 8 th > > > ð2lðr22 Þ Þ USN ðx ¼ ar1 þr2 þ1 Þ, > > > > > < > > > > r ‐USG ! ⋮ > 2 > > > > > > > > > : ð2Þ th > > ð2l1 Þ USN ðx ¼ ar1 þ2 Þ, > > < ¼ SO ða ¼ ar1 þ1 Þ ! SO ða ¼ a2ðm1 þ1Þ1 Þ, > > 8 > > th > > > > ð2lðr11 Þ Þ LSN ðx ¼ ar1 Þ, > > > > > < > > > > r 1 ‐LSG ! ⋮ > > > > > > > > > : : ð1Þ th ð2l1 Þ LSN ðx ¼ a1 Þ

ð7:106Þ

7.2 Simple Equilibrium Bifurcations

311

and

f

m¼m1 þm2

SI ð x ¼ aÞ







! ðr 1 ‐USG : SI : r 2 ‐LSGÞ

¼

appearing bifurcation

r 2 ‐LSG !

f

th

ð2lðr22 Þ Þ LSN ðx ¼ ar1 þr2 þ1 Þ, ⋮ ð2Þ th

ð2l1 Þ LSN ðx ¼ ar1 þ2 Þ,   SI ða ¼ ar1 þ1 Þ ! SI a ¼ a2ðm1 þ1Þ1 ,

r 1 ‐USG !

f

ð7:107Þ

th

ð2lðr11 Þ Þ USN ðx ¼ ar1 Þ, ⋮ ð1Þ th

ð2l1 Þ USN ðx ¼ a1 Þ

P 1 ð1Þ P 2 ð2Þ li , m2 ¼ rj¼1 lj ; and the acronyms USG and LSG are the upperfor m1 ¼ ri¼1 saddle-node and lower-saddle-node bifurcation groups, respectively. Four special broom-sprinkler-spraying appearing bifurcations are 8 SO ða ¼ a2rþ1 Þ ! SO ða ¼ a2mþ1 Þ, > > 8 > Pr > < > ð2lr Þth LSN ðx ¼ ar Þ, m¼ li > i¼1 <  SO ð x ¼ aÞ







! appearing bifurcation > r‐LSG ! ⋮ > > > > > : : ð2l1 Þth LSN ðx ¼ a1 Þ,

ð7:108Þ

8 SI ða ¼ a2rþ1 Þ ! SI ða ¼ a2mþ1 Þ, > > 8 > Pr > < > ð2lr Þth USN ðx ¼ ar Þ, m¼ li > i¼1 <  SI ð x ¼ aÞ







! appearing bifurcation > r‐USG ! ⋮ > > > > > : : ð2l1 Þth USN ðx ¼ a1 Þ,

ð7:109Þ

and 8 8 > ð2lr Þth USN ðx ¼ arþ1 Þ, > > > < > Pr > < r‐USG ! m¼ l ⋮ i¼1 i  > SO ð x ¼ aÞ







! > : appearing bifurcation > > ð2l1 Þth USN ðx ¼ a2 Þ, > > : SO ða ¼ a1 Þ ! SO ða ¼ a1 Þ,

ð7:110Þ

7 (2m+1)th-Degree Polynomial Systems

312

8 8 > ð2lr Þth LSN ðx ¼ arþ1 Þ, > > > < > Pr > < r‐LSG ! m¼ l ⋮ i¼1 i > SI ð x ¼ aÞ







! > : appearing bifurcation > > ð2l1 Þth LSN ðx ¼ a2 Þ, > > : SI ða ¼ a1 Þ ! SO ða ¼ a1 Þ

ð7:111Þ

If x ¼ a ¼  12 Bi (i 2 {1, 2, . . ., l}), the equilibrium of x ¼ a possesses a (2l + 1)th-order SI or SO switching bifurcation (or broom-switching bifurcation). The sprinkler-spraying appearing bifurcation generated by the m-pairs of quadratics becomes a broom-sprinkler-spraying switching bifurcation. The two broomswitching bifurcations are m¼m1 þm2 þl

SO ð x ¼ aÞ







! ðr 1 ‐LSG : ð2l þ 1Þrd SO : r 2 ‐USGÞ switching bifurcation 8 8 th > > > ð2lðr22 Þ Þ USN ðx ¼ ar2 þr1 þ1 Þ, > > > > > < > > > > ‐USG ! r ⋮ 2 > > > > > > > > > > : ð2Þ th > > ð2l1 Þ USN ðx ¼ ar1 þ2 Þ, > > < ¼ ð2l þ 1Þrd SO ða ¼ ar1 þ1 Þ; > > 8 > > th > > > > ð2lðr11 Þ Þ LSN ðx ¼ ar1 Þ, > > > > > < > > > > ‐LSG ! r ⋮ 1 > > > > > > > > > : : ð1Þ th ð2l1 Þ LSN ðx ¼ a1 Þ

ð7:112Þ

and m¼m1 þm2 þl

SI ð x ¼ aÞ





! ðr 1 ‐USG : ð2l þ 1Þrd SI : r 2 ‐LSGÞ switching bifurcation 8 8 th > > > ð2lðr22 Þ Þ LSN ðx ¼ ar2 þr1 þ1 Þ, > > > > > < > > > > ‐LSG ! r ⋮ 2 > > > > > > > > > > : ð2Þ th > > ð2l1 Þ LSN ðx ¼ ar1 þ2 Þ, > > < ¼ ð2l þ 1Þrd SI ða ¼ ar1 þ1 Þ; > > 8 > > th > > > > ð2lðr11 Þ Þ USN ðx ¼ ar1 Þ, > > > > > < > > > > r 1 ‐USG ! ⋮ > > > > > > > > > : : ð1Þ th ð2l1 Þ USN ðx ¼ a1 Þ,

ð7:113Þ

7.2 Simple Equilibrium Bifurcations

313

where

cluster of l‐quadratics

ð2l þ 1Þth order SO ðx ¼ aÞ







!

8 SO, for x ¼ a2sl þ1 , > > > >  > > < SI, for x ¼ a2sl ,

⋮ > > > SI, for x ¼ a2s1 , > > > : SO, for x ¼ a2s1 1 8 SI, for x ¼ a2sl þ1 , > > > >  > > < SO, for x ¼ a2sl , cluster of l‐quadratics ð2l þ 1Þth order SI ðx ¼ aÞ







! ⋮ appearing bifurcation > > > > SO, for x ¼ a2s1 , > > : SI, for x ¼ a2s1 1 appearing bifurcation

ð7:114Þ

ð7:115Þ

where x ¼ a 2 fa2s1 1 , . . . , a2sl , a2sl þ1 g. In Fig. 7.1a, b, the simple switching with two teeth comb appearing bifurcations is presented for a0 > 0 and a0 < 0, respectively. The two bifurcation structures are (a) SO ! (l1 ‐ LSN : SO : l2 ‐ USN), (b) SI ! (l1 ‐ USN : SI : l2 ‐ LSN) with l1 + l2 ¼ m. In Fig. 7.1c, d, the third-order pitchfork switching bifurcation with two teeth comb appearing bifurcations is presented for a0 > 0 and a0 < 0, respectively. The two bifurcation structures are (c) SO ! (l1 ‐ LSN : 3rdSO : l2 ‐ USN), (d) SI ! (l1 ‐ USN : 3rdSI : l2 ‐ LSN) with l1 + l2 ¼ m  1. In Fig. 7.2a, b, the simple switching with two sprinkler-spraying-appearing bifurcations is presented for a0 > 0 and a0 < 0, respectively. The two bifurcation structures are (a) SO ! (r1 ‐ LSG : SO : r2 ‐ USG), (b) SI ! (r1 ‐ USG : SI : r2 ‐ LSG)

P 1 ð1Þ P 2 ð2Þ li , m2 ¼ rj¼1 lj . In with r1 + r2 + 1 ¼ n and m1 + m2 ¼ m where m1 ¼ ri¼1 Fig. 7.2c, d, the (2l + 1)th-order broom-switching with two sprinkler-sprayingappearing bifurcations is presented for a0 > 0 and a0 < 0, respectively. The two bifurcation structures are (c) SO ! (r1 ‐ LSG : (2l + 1)thSO : r2 ‐ USG), (d) SI ! (r1 ‐ USG : (2l + 1)thSI : r2 ‐ LSG)

P 1 ð1Þ P 2 ð2Þ li , m2 ¼ rj¼1 lj . with r1 + r2 + 1 ¼ n and m1 + m2 + l ¼ m where m1 ¼ ri¼1 For a cluster of m-quadratics, Bi ¼ Bj (i, j 2 {1, 2, . . ., m}; i 6¼ j) and Δi ¼ 0 (i 2 {1, 2, . . ., m}). The (2m)th-order upper-saddle-node appearing bifurcation for mpairs of sink and source equilibriums is

7 (2m+1)th-Degree Polynomial Systems

314

a

b

c

d

Fig. 7.1 Simple broom-switching bifurcations: (a) (US : . . . : US : SO : LS : . . . : LS) (a0 > 0), (b) (LS : . . . : LS : SI : US : . . . : US) (a0 < 0), (c) (US : . . . : US : 3rdSO : LS : . . . : LS) (a0 > 0), (d) (LS : . . . : LS : 3rdSI : US : . . . : US) (a0 < 0) in a (2m + 1)th-degree polynomial system. LS: lower saddle, US: upper saddle, SI: sink, SO: source. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

7.2 Simple Equilibrium Bifurcations

315

a

b

c

d

Fig. 7.2 Broom appearing bifurcation: (a) (r1 ‐ LSN : SO : r2 ‐ USN) (a0 > 0). (b) (r1 ‐ USN : SI : r2 ‐ LSN) (a0 < 0); broom-sprinkler-spraying-switching bifurcation. (c) (r1 ‐ LSG : (2lk + 1)thSO : r2 ‐ USG) (a0 > 0). (d) (r1 ‐ LSG : (2lk + 1)thSO : r2 ‐ USG) (a0 < 0) in a (2m + 1)th-degree polynomial system. LS: lower saddle, US: upper saddle, SI: sink, SO: source. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

316

7 (2m+1)th-Degree Polynomial Systems

8 SO, for x ¼ a2m , > > > >  > > < SI, for x ¼ a2m1 ; cluster of m‐quadratics th ð2mÞ order US





! ⋮ appearing bifurcation > > > SO, for x ¼ a2 , > > > : SI, for x ¼ a1 :

ð7:116Þ

The (2m)th-order lower-saddle-node appearing bifurcation for m-pairs of sink and source equilibriums is 8 SI, for x ¼ a2m , > > > >  > > < SO, for x ¼ a2m1 ; cluster of m‐quadratics th ð2mÞ order LS







! ⋮ appearing bifurcation > > > SI, for x ¼ a2 , > > > : SO, for x ¼ a1 :

ð7:117Þ

There are four simple switching and (2m)th-order saddle-node appearing bifurcations: The two switching bifurcations of SO ! ((2m)thUS : SO) and SI ! ((2m)thLS : SI) with two (2m)th-order USN and LSN spraying-appearing bifurcations are 8 SO ! SO, for x ¼ a ¼ a2mþ1 > > 8 > > SI, for x ¼ a2m , > > > > > > > > > > > > < SO, for x ¼ a2m1 ; > > <  SO ðx ¼ aÞ ! > ð2mÞth order LSN ! ⋮ > > > > > > > > > > SI, for x ¼ a2 , > > > > > > > : : SO, for x ¼ a1 8 SI ! SI, for x ¼ a ¼ a2mþ1 > > 8 > > SO, for x ¼ a2m , > > > > > > > > > > > > < SI, for x ¼ a2m1 ; > > <  SI ðx ¼ aÞ ! > ð2mÞth order USN ! ⋮ > > > > > > > > > > SO, for x ¼ a2 , > > > > > > > : : SI, for x ¼ a1

ð7:118Þ

ð7:119Þ

and the two switching bifurcations of SO ! (SO : (2m)thUS) and SI ! (SI: (2m)thLS) with two (2m)th-order USN and LSN spraying-appearing bifurcations are

7.2 Simple Equilibrium Bifurcations

8 8 SO, for x ¼ a2mþ1 , > > > > > > > > > > > > SI, for x ¼ a2m ; > > > > < > > < ð2mÞth order USN ! ⋮ > SO ðx ¼ aÞ ! > > > > > SO, for x ¼ a3 , > > > > > > > > : > > SI, for x ¼ a2 ; > > :  SO ! SO, for x ¼ a ¼ a1 8 8 SI, for x ¼ a2mþ1 , > > > > > > > > > > > > SO, for x ¼ a2m ; > > > > < > > < ð2mÞth order LSN ! ⋮ > SI ðx ¼ aÞ ! > > > > > SI, for x ¼ a3 , > > > > > > > > : > > SO, for x ¼ a2 ; > > :  SI ! SI, for x ¼ a ¼ a1 :

317

ð7:120Þ

ð7:121Þ

The (2m+1)th-order source broom-switching bifurcation is 8 SO, for x ¼ a2mþ1 , > > > >  > > < SI, for x ¼ a2m , switching SOðx ¼ aÞ



! ð2m þ 1Þth order SO ⋮ > > > > SI, for x ¼ a2 , > > : SO, for x ¼ a1 :

ð7:122Þ

The (2m+1)th-order sink broom-switching bifurcation is 8 SI, for x ¼ a2mþ1 , > > > >  > > < SO, for x ¼ a2m , switching SI ðx ¼ a1 Þ



! ð2m þ 1Þth order SI ⋮ > > > > SO, for x ¼ a2 , > > : SI, for x ¼ a1 :

ð7:123Þ

The switching bifurcation consists of a simple switching and the (2m)th-order saddle-node appearing bifurcation with m-pairs of source and sinks. The (2m)thorder saddle-node appearing bifurcation is a sprinkler-spraying cluster of the mpairs of sources and sinks. Thus, the four switching bifurcations of SO ! (SO : (2m)thUS) for a0 > 0, SI ! (SI : (2m)thLS) for a0 < 0, SO ! ((2m)thUS : SO) for a0 > 0, SI ! ((2m)thLS : SI) for a0 < 0

318

7 (2m+1)th-Degree Polynomial Systems

Fig. 7.3 (a) (SO : (2m)thLS)-switching bifurcation (a0 > 0), (b) (SI : (2m)thSO)-switching bifurcation (a0 < 0), (c) ((2m)thUS : SO)-switching bifurcation (a0 > 0), (d) ((2m)thLS : SI)-switching bifurcation (a0 < 0), (e) (2m + 1)th SO broom appearing bifurcation (a0 > 0), (f) (2m + 1)th SI broom appearing bifurcation (a0 < 0) in the (2m + 1)th-degree polynomial system. LS: lower saddle, US: upper saddle, SI: sink, SO: source. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

are presented in Fig. 7.3a–d, respectively. The (2m + 1)th-order source switching bifurcation is named the (2m+1)thSO broom-switching bifurcation and the (2m + 1)th-order sink switching bifurcation is named the (2m + 1)thSI broomswitching bifurcation. Such a (2m + 1)thXX broom-switching bifurcation is from

7.2 Simple Equilibrium Bifurcations

319

simple equilibrium to a (2m + 1)thXX broom-switching bifurcation. The two broomswitching bifurcations of SO ! (2m + 1)thSO for a0 > 0, SI ! (2m + 1)thSI for a0 < 0 are presented in Fig. 7.3e, f, respectively. A series of the third-order source and sink bifurcations is aligned up with varying parameters. Such a special pattern is from m-quadratics in the (2m+1)th-order polynomial systems, and the following conditions should be satisfied:   1 1 aðpi Þ ¼  Bi and a pj ¼  Bj 2 2 Bi Bj i, j 2 f1, 2, . . . , ng; i 6¼ j, Δi > Δiþ1 ði ¼ 1, 2, . . . , n; n mÞ, Δi ¼ 0 with kpi k < kpiþ1 k:

ð7:124Þ

Thus, a series of m ‐ (3rdSO ‐ 3rdSI ‐ 3rdSO ‐ . . .) switching bifurcations (a0 > 0) and a series of m ‐ (3rdSI ‐ 3rdSO ‐ 3rdSI ‐ . . .) switching bifurcations (a0 < 0) are presented in Fig. 7.4a, b. The bifurcation scenario is formed by the swapping pattern of third SI and third SO switching bifurcations. Such a bifurcation scenario is like the fish bone. Thus, such a bifurcation swapping pattern of third SI and third SO switching bifurcations is called the fish-bone switching bifurcation in the (2m+1)thdegree polynomial nonlinear system. There are two swaps of the third SI and third SO bifurcations: (a) the third SO-third SI fish-bone switching bifurcation and (b) the third SI-third SO fish-bone switching bifurcation.

7.2.2

Switching Bifurcations

In the (2m + 1)th-order polynomial system, among the possible (2m + 1) roots, there are two roots to satisfy x2 + Bix + Ci ¼ 0 with ðiÞ

ðiÞ

ðiÞ

ðiÞ 2

Bi ¼ ðb1 þ b2 Þ, Δi ¼ ðb1  b2 Þ 0, ðiÞ

ðiÞ

ðiÞ

ðiÞ

x1,2 ¼ b1,2 , Δi > 0 if b1 6¼ b2 ði ¼ 1, 2, . . . , nÞ; ) Bi 6¼ Bj ði, j ¼ 1, 2, . . . , n; i 6¼ jÞ at bifurcation: ðiÞ ðiÞ Δi ¼ 0 at b1 ¼ b2 ði ¼ 1, 2, . . . , nÞ

ð7:125Þ

The second-order singularity bifurcation is for the switching of a pair of simple sink and source. There are two switching bifurcations for i 2 {1, 2, . . ., n}:

7 (2m+1)th-Degree Polynomial Systems

320

a

b Fig. 7.4 (a) m ‐ (3rdSO ‐ 3rdSI ‐ 3rdSO. . .) series bifurcation (a0 > 0), (b) m ‐ (3rdSI ‐ 3rdSO ‐ 3rdSI. . .) series switching bifurcation (a0 < 0) in the (2m + 1)th-degree polynomial system. SI: sink, SO: source. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

(

ðiÞ

SO, for a2i ¼ b2 ! b1 ,

switching bifurcation

SI, for a2i1 ¼ b1 ! b2

( ith quadratic factor

2 order LS







! nd

ðiÞ

ith quadratic factor

2 order US







! nd

switching bifurcation

ðiÞ

ðiÞ

ðiÞ

ðiÞ

SI, for a2i ¼ b2 ! b1 , ðiÞ

ðiÞ

SO, for a2i1 ¼ b1 ! b2 : ðiÞ

ðiÞ

ð7:126Þ

ð7:127Þ

For non-switching point, Δi > 0 at b1 6¼ b2 (i ¼ 1, 2, . . ., n). At the bifurcation ðiÞ ðiÞ point, Δi ¼ 0 at b1 ¼ b2 (i ¼ 1, 2, . . ., n). The l-USN parallel switching bifurcation for si 2 {0, 1, . . ., m} (i ¼ 1, 2, . . ., l) is

7.2 Simple Equilibrium Bifurcations

8 8 > < SO # SI, th > s bifurcation > l > > US





! > > switching : SI " SO, > > < l‐USN ⋮ 8 > > > < SO # SI, > th > s bifurcation l > > US





! > > switching : : SI " SO,

321 ðs Þ

for b2 l ¼ a2sl # a2sl 1 , ðs Þ

for b1 l ¼ a2sl 1 " a2sl ; ð7:128Þ for

ðs Þ b2 1

¼ a2s1 # a2s1 1 ,

ðs Þ

for b1 1 ¼ a2s1 1 " a2s1 :

The l-LSN antenna-switching bifurcation for si 2 {0, 1, . . ., m} (i ¼ 1, 2, . . ., l) is 8 8 > < SI # SO, th > s bifurcation > l > > LS





! > > switching : SO " SI, > > < l‐LSN ⋮ 8 > > > < SI # SO, > th > s bifurcation l > > LS





! > > switching : : SO " SI,

ðs Þ

for b2 l ¼ a2sl # a2sl 1 , ðs Þ

for b1 l ¼ a2sl 1 " a2sl ; ð7:129Þ for

ðs Þ b2 1

¼ a2s1 # a2s1 1 ,

ðs Þ

for b1 1 ¼ a2s1 1 " a2s1 :

Two antenna-switching bifurcation structures exist for the (2m + 1)th-order polynomial system. The (l1 ‐ LSN : SO : l2 ‐ USN)-switching bifurcation for a0 > 0 is 8 > < l2 ‐USN l1 þl2 ¼m ðl1 ‐LSN : SO : l2 ‐USNÞ



! SO ! SO, > : l1 ‐LSN

ð7:130Þ

and the (l1 ‐ USN : SI : l2 ‐ LSN)-switching bifurcation for a0 < 0 is 8 > < l2 ‐LSN, l1 þl2 ¼m ðl1 ‐USN : SI : l2 ‐LSNÞ



! SI ! SI, > : l1 ‐USN:

ð7:131Þ

As in the (2m + 1)th-order polynomial system, consider a switching bifurcation for a bundle of sink and source equilibriums with the following conditions: Bi ¼ Bj ði, j 2 f1, 2, . . . , ng; i 6¼ jÞ ðiÞ

ðiÞ

Δi ¼ 0 at b1 ¼ b2 ði ¼ 1, 2, . . . , nÞ

) at bifurcation:

Two (2l)th-order switching bifurcations for l 2 {1, 2, . . ., s} are

ð7:132Þ

7 (2m+1)th-Degree Polynomial Systems

322

8 SO, for a2sl ! b2sl , > > > > > > < SI, for a2sl 1 ! b2sl 1 , a bundle of ð2lÞ‐equilibriums th ð2lÞ order US









! ⋮ switching bifurcation > > > SO, for a2s ! b2s , > 1 1 > > : SI, for a2s1 1 ! b2s1 1 8 SI, for a2sl ! b2sl , > > > > > > < SO, for a2sl 1 ! b2sl 1 , a bundle of ð2lÞ‐equilibriums th ð2lÞ order LS









! ⋮ switching bifurcation > > > > > SI, for a2s1 ! b2s1 , > : SO, for a2s1 1 ! b2s1 1

ð7:133Þ

ð7:134Þ

where Δij ¼ (ai  aj)2 ¼ (bi  bj)2 ¼ 0 with Bi ¼ Bj (i, j ¼ 2s1  1, 2s1, . . ., 2sl  1, 2sl) and fa2s1 1 ; a2s1 ; . . . ; a2sl 1 ; a2sl g fb2s1 1 ; b2s1 ; . . . ; b2sl 1 ; b2sl g



ð1Þ

before bifurcation



after bifurcation

ð1Þ

ðnÞ

ðnÞ

sortfb1 ; b2 ; . . . ; b1 ; b2 ; ag, ð1Þ

ð1Þ

ðnÞ

ðnÞ

sortfb1 ; b2 ; . . . ; b1 ; b2 ; ag: ð7:135Þ

Two (2l + 1)th-order switching bifurcations for l 2 {1, 2, . . ., s} are 8 SO, for a2sl þ1 ! b2sl þ1 , > > > < ⋮ a bundle of ð2lþ1Þ‐equilibriums ð7:136Þ ð2l þ 1Þth order SO











! > switching bifurcation SI, for a2s1 ! b2s1 , > > : SO, for a2s1 1 ! b2s1 1 8 SI, for a2sl þ1 ! b2sl þ1 , > > > < ⋮ a bundle of ð2lþ1Þ‐equilibriums ð2l þ 1Þth order SI









! ð7:137Þ > switching bifurcation SO, for a ! b , 2s 2s > 1 1 > : SI, for a2s1 1 ! b2s1 1 where Δij ¼ (ai  aj)2 ¼ (bi  bj)2 ¼ 0 with Bi ¼ Bj (i, j ¼ 2s1  1, 2s1, . . ., 2sl þ 1) and fa2s1 1 ; a2s1 ; . . . ; a2sl þ1 g fb2s1 1 ; b2s1 ; . . . ; b2sl þ1 g

ð1Þ

ð1Þ

ðnÞ

ðnÞ



sortfb1 ; b2 ; . . . ; b1 ; b2 ; ag,



sortfb1 ; b2 ; . . . ; b1 ; b2 ; ag:

before bifurcation After bifurcation

ð1Þ

ð1Þ

ðnÞ

ðnÞ

ð7:138Þ

A set of paralleled, different, higher order upper-saddle-node switching bifurcations with multiplicity is the ((α1)thXX : (α2)thXX : . . . : (αs)thXX) parallel switching bifurcation in the (2m+1)th-degree polynomial system. At the straw-bundle

7.2 Simple Equilibrium Bifurcations

323

switching bifurcation, Δi ¼ 0 (i ¼ 1, 2, . . ., n) and Bi ¼ Bj (i, j 2 {1, 2, . . ., n}; i 6¼ j). Thus, the parallel straw-bundle switching bifurcation is ððα1 Þth XX : ðα2 Þth XX : . . . : ðαs Þth XXÞ‐switching 8 ðαs Þth order XX switching, > > > > >

> ðα2 Þth order XX switching, > > > : ðα1 Þth order XX switching

ð7:139Þ

where αi 2 f2li , 2li  1g with

Xs

α i¼1 i

¼ 2m þ 1;

and XX 2 fUS, LS, SO, SIg: ð jÞ

ð7:140Þ

th

ð jÞ

The ð2l1 : 2l2 : . . . : 2lðs jÞ Þ USN parallel switching bifurcation is called the

ð jÞ

th

ð jÞ

ð2l1 : 2l2 : . . . : 2lðs jÞ Þ USN parallel straw-bundle switching bifurcation: ð jÞ

ð jÞ

th

sj ‐USG ¼ ð2l1 : 2l2 : . . . : 2lðsjjÞ Þ USN switching 8 th > ð2lðs jÞ Þ order USN switching, > > > > >

> ð2l2 Þ order USN switching, > > > > : ð jÞ th ð2l1 Þ order USN switching: ð jÞ

ð7:141Þ

th

ð jÞ

The ð2l1 : 2l2 : . . . : 2lðs jÞ Þ LSN parallel switching bifurcation is called the

ð jÞ

ð jÞ

th

ð2l1 : 2l2 : . . . : 2lðs jÞ Þ LSN parallel straw-bundle switching bifurcation: ð jÞ

ð jÞ

th

sj ‐LSG ¼ ð2l1 : 2l2 : . . . : 2lðsjjÞ Þ LSN switching 8 th > 2lðs jÞ order LSN switching, > > > > >

> ð2l2 Þ order LSN switching, > > > > : ð jÞ th ð2l1 Þ order LSN switching:

ð7:142Þ

7 (2m+1)th-Degree Polynomial Systems

324 th

ð2Þ

The ðs1 ‐LSG : ð2l1 þ 1Þ SO : s3 ‐USGÞ-switching bifurcation for a0 > 0 is

th

ð 2Þ

ðs1 ‐LSG : ð2l1 þ 1Þ SO : s3 ‐USGÞ ¼

8 th ð3Þ > ð2l1 : . . . : 2lðs32 Þ Þ ‐USN, > > < ð2Þ

th

ð2l1 þ 1Þ SO, > > > th : ð1Þ ð2l1 : . . . : 2lðs11 Þ Þ ‐LSN

ð7:143Þ

th

ð2Þ

and the ðs1 ‐USG : ð2l1 þ 1Þ SI : s3 ‐LSGÞ-switching bifurcation for a0 < 0 is

ð2Þ

th

ðs1 ‐USG : ð2l1 þ 1Þ SI : s3 ‐LSGÞ ¼

8 th ð3Þ > ð2l1 : . . . : 2lðs33 Þ Þ ‐LSN, > > < ð2Þ

th

ð2l1 þ 1Þ SI, > > > th : ð1Þ ð2l1 : . . . : 2lðs11 Þ Þ ‐USN:

ð7:144Þ

The two (l1 ‐ USN : SO : l2 ‐ LSN) and (l1 ‐ LSN : SI : l2 ‐ USN) parallel switching bifurcations (l1 + l2 ¼ m) are presented in Fig. 7.5a, b for a0 > 0 and a < 0, respectively. A set of (3rdSO : . . . : SI : 3rdSO) parallel, switching bifurcations for SI and SO equilibriums is presented in Fig. 7.5c for a0 > 0. However, for a0 < 0, the set of (3rdSI :    : SO : 3rdSI) switching bifurcations for sources and sinks is presented in Fig. 7.5d.

7.2.3

Switching and Appearing Bifurcations

Consider a (2m + 1)th-degree polynomial system in a form of x_ ¼ a0 QðxÞ

2n 1 þ1 Y i¼1

ðx  ci Þ

n2 Y

ðx2 þ B j x þ C j Þ:

ð7:145Þ

j¼1

Without loss of generality, a function of Q(x) > 0 is either a polynomial function or a non-polynomial function. The roots of x2 + Bjx + Cj ¼ 0 are 1 1 pffiffiffiffiffiffi ðjÞ Δ j ,Δ j ¼ B2j  4C j 0 ðj ¼ 1; 2; . . . ; n2 Þ; b1,2 ¼  B j  2 2

ð7:146Þ

7.2 Simple Equilibrium Bifurcations

325

a

b

c

d

Fig. 7.5 Parallel switching bifurcations: (a) (l1 ‐ USG : SO : l2 ‐ USG) (a0 > 0), (b) (l1 ‐ USG : SI : l2 ‐ LSG) (a0 < 0), (c) (3rdSI : . . . : USN : 3rdSO) (a0 > 0), (d) (3rdSO : . . . : LSN : 3rdSI) (a0 < 0) in the (2m + 1)th-degree polynomial nonlinear system. LSN: lower saddle node, USN: upper saddle node, SI: sink, SO: source. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

7 (2m+1)th-Degree Polynomial Systems

326

either   fa 1 , a2 , . . . , a2n1 þ1 g ¼ sortfc1 ; c2 ; . . . ; c2n1 þ1 g,  a s asþ1 before bifurcation; ð1Þ

ð1Þ

ðn Þ

ðn Þ

2 2 þ þ faþ 1 , a2 , . . . , a2n3 þ1 g ¼ sortfc1 ; . . . ; c2n1 þ1 ; b1 ; b2 ; . . . ; b1 ; b2 g,

ð7:147Þ

þ aþ s asþ1 , n3 ¼ n1 þ n2 after bifurcation

or ð1Þ

ð1Þ

ðn Þ

ðn Þ

2 2   fa 1 , a2 , . . . , a2n3 þ1 g ¼ sortfc1 ; c2 ; . . . ; c2n1 ; b1 ; b2 ; . . . ; b1 ; b2 ; ag,

 a s asþ1 , n3 ¼ n1 þ n2 before bifurcation; þ þ faþ 1 , a2 , . . . , a2n1 þ1 g ¼ sortfc1 ; . . . ; c2n1 ; ag, þ aþ s asþ1 after bifurcation

ð7:148Þ and   Bj1 ¼ Bj2 ¼ . . . ¼ Bjs jk1 2 f1, 2, . . . , ng; jk1 6¼ jk2 ðk1 , k2 2 f1, 2, . . . , sg; k1 6¼ k 2 Þ Δj ¼ 0 ðj 2 U  f1, 2, . . . , n2 g

g

at bifurcation:

ð7:149Þ

1 ci 6¼  Bj ði ¼ 1, 2, . . . , 2n1 , j ¼ 1, 2, . . . , n2 Þ 2  th    th   th  Consider a just before bifurcation of α XX1 : α2 XX2 : . . . :ðα 1 s1 Þ XXs1 Ps1 with i¼1 αi ¼ 2m1 þ 1 for simple sources and sinks in the (2m +1)th-degree poly     nomial system. For α i ¼ 2li  1, XXi 2 fSO, SIg and for αi ¼ 2li , XXi 2 fUS, LSg (i ¼ 1, 2, . . ., s1). The detailed structures are as follows:

7.2 Simple Equilibrium Bifurcations

327

9 9 SI > SO > > > > > > > > > > SO > SI > > > > = =   th  th ! 2li  1 SI, and ⋮ ! 2l ⋮ i  1 SO; > > > > > > > SO > SI > > > > > > > > ; ; SI SO 9 9 SO > SI > > > > > > > > > > SI > SO > > > > = =   th  th ! 2li US, and ⋮ ! 2l ⋮ LS: i > > > > > > > > SO > SI > > > > > > > ; ; SI SO

ð7:150Þ

th     th  th The bifurcation set of ððα 1 Þ XX1 : ðα2 Þ XX2 : . . . : ðαs1 Þ XXs1 Þ at the same parameter point is called a left-parallel-bundle switching bifurcation. th þ þ th þ þ þ th Consider a just after bifurcation of ððαþ 1 Þ XX1 : ðα2 Þ XX2 : . . . : ðαs2 Þ XXs2 Þ Ps 2 þ th with i¼1 αi ¼ 2m2 þ 1 for simple sources and sinks in the (2m + 1) -degree þ þ  polynomial system. XXþ i 2 fSO, SIg for αi ¼ 2li  1, and XXi 2 fUS, LSg for þ þ αi ¼ 2li . The four detailed structures are as follows:

8 8 SI SO > > > > > > > > > > > > SO SI > > > > < <  þ th  þ th 2li  1 SI ! ⋮ , and 2li  1 SO ! ⋮ ; > > > > > > > > SO SI > > > > > > > > : : SI SO 8 8 SO SI > > > > > > > > > > > > SI SO > > > > < <  þ th  þ th 2li US ! ⋮ , and 2li LS ! ⋮ : > > > > > > > > SO SI > > > > > > > > : : SI SO

ð7:151Þ

þ þ þ þ þ The bifurcation set of ððαþ 1 Þ XX1 : ðα2 Þ XX2 : . . . : ðαs2 Þ XXs2 Þ at the same parameter point is called a right-parallel:bundle switching bifurcation. th

th

th

7 (2m+1)th-Degree Polynomial Systems

328

(i) For the just before and after bifurcation structure, if there exists a relation of th þ th þ  þ th   ðα i Þ XXi ¼ ðα j Þ XX j ¼ α XX, for x ¼ ai ¼ a j

ði 2 f1; 2; . . . ; s1 g; j 2 f1; 2; . . . ; s2 gÞ, XX 2 fUS; LS; SO; SIg

ð7:152Þ

then the bifurcation is a αthXX switching bifurcation for simple equilibriums. (ii) Just for the just before bifurcation structure, if there exists a relation of th   ð2l Þth XX i ¼ ð2lÞ XX, for x ¼ ai ¼ ai  i i 2 f1; 2; . . . ; s1 g, XX 2 fUS; LSg

ð7:153Þ

then the bifurcation is a(2l)thXX left-appearing (or right-vanishing) bifurcation for simple equilibriums. (iii) Just for the just after bifurcation structure, if there exists a relation of th þ þ  ð2lþ i Þ XXi ¼ ð2lÞ XX, for x ¼ ai ¼ ai ði 2 f1; 2; . . . ; s1 gÞ, XX 2 fUS; LSg th

ð7:154Þ

then the bifurcation is a (2l)thXX right-appearing (or left-vanishing) bifurcation for simple equilibriums. (iv) For the just before and after bifurcation structure, if there exists a relation of th þ th þ  þ   ðα i Þ XXi 6¼ ðα j Þ XX j for x ¼ ai ¼ a j

ð7:155Þ

þ XX i , XX j 2 fUS; LS; SO; SIg

ði 2 f1; 2; . . . ; s1 g; j 2 f1; 2; . . . ; s2 gÞ, then there are two flower-bundle switching bifurcations of simple equilibriums: þ (iv1) For αj ¼ αi + 2l, the bifurcation is called a ðαþ i Þ XXi right flower-bundle switching bifurcation for αi to αj-simple equilibriums with the appearance (birth) of 2l-simple equilibriums.  th  (iv2) For αj ¼ αi  2l, the bifurcation is called a α XXi left flower-bundle i switching bifurcation for αi to αj-simple equilibriums with the vanishing (death) of 2l-simple equilibriums. th

A general parallel switching bifurcation is switching

th   th   th  ððα 1 Þ XX1 : ðα2 Þ XX2 : . . . : ðαs1 Þ XXs1 Þ



! bifucation

þ þ þ þ þ ððαþ 1 Þ XX1 : ðα2 Þ XX2 : . . . : ðαs2 Þ XXs2 Þ: th

th

th

ð7:156Þ

7.2 Simple Equilibrium Bifurcations

329

Such a general, parallel switching bifurcation consists of the left and right parallel-bundle switching bifurcations. If the left and right parallel-bundle switching bifurcations are same in a parallel flower-bundle switching bifurcation, i.e., th th þ  þ ðα i Þ XXi ¼ ðαi Þ XXi ¼ ðαi Þ XXi þ for x ¼ a i ¼ ai ði ¼ 1; 2; . . . ; sg th

ð7:157Þ

then the parallel flower-bundle switching bifurcation becomes a parallel strawbundle switching bifurcation of ((α1)thXX : (α2)thXX : . . . : (αs)thXX). If the left and right parallel-bundle switching bifurcations are different in a parallel flower-bundle switching bifurcation, i.e., th  th þ th þ th  þ ðα i Þ XXi ¼ ð2li Þ XX, ðα j Þ XX j ¼ ð2l j Þ YY, þ for x ¼ a i 6¼ ai ði ¼ 1; 2; . . . ; sg

ð7:158Þ

XX 2 fUS; LSg, YY 2 fUS; LSg then the parallel flower-bundle switching bifurcation becomes a combination of two independent left and right parallel appearing bifurcations:   th    th   th  (i) A 2l1 XX1 : 2l2 XX2 : . . . : ð2l s1 Þ XXs1 -left parallel sprinklerspraying-appearing (or right vanishing) bifurcation   þ th þ  þ th þ th þ (ii) A 2l1 XX1 : 2l2 XX2 : . . . : ð2lþ s1 Þ XXs2 -right parallel sprinklerspraying-appearing (or left vanishing) bifurcation The (4thLS : . . . : SO : 6thUS) parallel appearing bifurcation for a0 > 0 is presented in Fig. 7.6a. The (4thUS : . . . : SI : 6thLS) parallel appearing bifurcation for a0 < 0 is shown in Fig. 7.6b. Such a kind of bifurcation is also like a waterfall appearing bifurcation. The (5thSO : . . . : 6thUS : 6thUS) parallel, flowerbundle switching bifurcation for SI and SO equilibriums is presented in Fig. 7.6c for a0 > 0. Such a parallel flower-bundle switching bifurcation is from (SO : SI : SO) to (5thSO : . . . : 6thUS : 6thUS) with a waterfall appearance. The set of (5thSI : . . . : 6thLS : 6thLS) flower-bundle switching bifurcation for SI and SO equilibriums is presented in Fig. 7.6d for a0 < 0. Such a parallel flower-bundle switching bifurcation is from (SI : SO : SI) to (5thSI : . . . : 6thLS : 6thLS) with a waterfall appearance. After the bifurcation, the waterfall equilibrium birth can be observed. The equilibriums before such a bifurcation are much less than after the bifurcation.

7 (2m+1)th-Degree Polynomial Systems

330

a

b

c

d

Fig. 7.6 Switching and appearing bifurcations. Simple switching: (a) (4thLSN : . . . : SO : 6thUSN) (a0 > 0), (b) (4thUSN : . . . : SI : 6thLSN) (a0 < 0). Higher order switching: (c) (5thSI : . . . : 6thUSN : 6thUSN) (a0 > 0), (d) (5thSO : . . . : 6thLSN : 6thLSN) (a0 < 0) in the (2m + 1)th-degree polynomial nonlinear system. LSN: lower saddle node, USN: upper saddle node, SI: sink, SO: source. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

7.3 Higher Order Equilibrium Bifurcations

7.3

331

Higher Order Equilibrium Bifurcations

The afore-discussed appearing and switching bifurcations in the (2m +1)th-degree polynomial system are relative to the simple sources and sinks. As similar to the (2m)th-degree polynomial system, the higher order singularity bifurcations in the (2m +1)th-degree polynomial system can be for higher order sinks, sources, upper saddles, and lower saddles as in Luo (2019).

7.3.1

Higher Order Equilibrium Bifurcations

Consider a (2m +1)th-degree polynomial system as x_ ¼ a0 QðxÞðx  aÞ

s Y

αi

ðx2 þ Bi x þ C i Þ ,

ð7:159Þ

i¼1

where αi 2 {2li  1, 2li}. Without loss of generality, a function of Q(x) > 0 is either a polynomial function or a non-polynomial function. The roots of x2 + Bix + Ci ¼ 0 are 1 1pffiffiffiffiffi ðiÞ Δi ,Δi ¼ B2i  4C i 0; b1,2 ¼  Bi  2 2 ð1Þ ð1Þ ðsÞ ðsÞ fa1 ; a2 ; . . . ; a2s1 ; a2s ; a2sþ1 g ¼ sortfb1 ; b2 ; . . . ; b1 ; b2 ; ag, a j a jþ1 : ð7:160Þ For a 6¼  12 Bi (i ¼ 1, 2, . . ., s), there are four higher order bifurcations as follows: ð2li 1Þth order quadratics

ð2ð2li  1ÞÞth order US







! appearing bifurcation 8 th < ð2li  1Þ order SO, x ¼ bðiÞ , 2 :

ð7:161Þ

ðiÞ

ð2li  1Þth order SI, x ¼ b1 ; ð2li 1Þth order quadratics

ð2ð2li  1ÞÞth order LS







! appearing bifurcation 8 th < ð2li  1Þ order SI, x ¼ bðiÞ , 2 :

ðiÞ

ð2li  1Þth order SO, x ¼ b1 ;

ð7:162Þ

7 (2m+1)th-Degree Polynomial Systems

332

ð2li Þth ‐order quadratics

ð2ð2li ÞÞth order US







! appearing bifurcation 8 < ð2li Þth order US, x ¼ bðiÞ , 2 :

ð7:163Þ

ðiÞ

ð2li Þth order US, x ¼ b1 ; ð2li Þth ‐order quadratics

ð2ð2li ÞÞth order LS







! appearing bifurcation 8 th < ð2li Þ order LS, x ¼ bðiÞ , 2

ð7:164Þ

: ð2l Þth order LS, x ¼ bðiÞ : i 1 (i) For αi ¼ 2li  1, the (2(2li  1))th-order upper-saddle (US) appearing bifurðiÞ cation is for the onset of the (2li  1)th-order source (SO) ðx ¼ b2 Þ and the ðiÞ ðiÞ ðiÞ th  (2li  1) -order sink (SI) ðx ¼ b1 Þ with b2 > b1 . (ii) For αi ¼ 2li  1, the (2(2li  1))th-order lower-saddle (LS) appearing bifurcaðiÞ tion is for the onset of the (2li  1)th-order sink (SI) (x ¼ b2 ) and the ðiÞ

ðiÞ

ðiÞ

(2li  1)th-order source (SI) (x ¼ b1 ) with b2 > b1 . (iii) For αi ¼ 2li, the (2(2li))th-order upper-saddle (US) appearing bifurcation is for ðiÞ ðiÞ ðiÞ ðiÞ the onset of two (2li)th-order upper saddles (US) (x ¼ b1 , b2 ) with b2 > b1 . (iv) For αi ¼ 2li, the (2(2li))th order lower-saddle (LS) appearing bifurcation is for ðiÞ ðiÞ ðiÞ ðiÞ the onset of two (2li)th-order lower saddles (LS) (x ¼ b1 , b2 ) with b2 > b1 . The equilibrium of x ¼ a 6¼  12 Bi (i ¼ 1, 2, . . ., s) breaks a cluster of teeth comb appearing bifurcations of higher order equilibrium to two parts. The teeth comb appearing bifurcation generated by the s-pairs of quadratics becomes a broom appearing bifurcation for higher order equilibriums. The two broom appearing bifurcations for higher order equilibriums are 8 th ð2Þ th > ðð2α1 Þ US : . . . : ð2αðs22 Þ Þ USÞ, > < j¼1 i¼1 SO ð x ¼ aÞ







! SO, for x ¼ a ¼ a2ðs1 þ1Þ1 , > appearing bifurcation > : th ð1Þ th ðð2α1 Þ LS : . . . : ð2αðs11 Þ Þ LSÞ

ð7:165Þ

8 th ð2Þ th > ðð2α1 Þ LS : . . . : ð2αðs22 Þ Þ LSÞ, > < j¼1 i¼1 SI ð x ¼ aÞ







! SI, for x ¼ a ¼ a2ðs1 þ1Þ1 , > appearing bifurcation > : th ð1Þ th ðð2α1 Þ US : . . . : ð2αðs11 Þ Þ USÞ

ð7:166Þ

sj 2 P P

ð jÞ

αi ¼m

and sj 2 P P

ð jÞ

αi ¼m

7.3 Higher Order Equilibrium Bifurcations

333

where 8 8 th < ðαðsjjÞ Þ XX, > > th > ð j Þ > ð2αsj Þ US ! > > > : ð jÞ th > ðαsj Þ XX; > < th th ð jÞ ð jÞ ðð2α1 Þ US : . . . : ð2αsj Þ USÞ ¼ ⋮ > 8 > th > > < ðαð jÞ Þ XX, > th > 1 ð j Þ > > ð2α1 Þ US ! > : : ð jÞ th ðα1 Þ XX;

ð7:167Þ

8 8 th < ðαðsjjÞ Þ XX, > > th > > ð2αðsjjÞ Þ LS ! > > > : ð jÞ th > ðαsj Þ XX; > < th th ð jÞ ðð2α1 Þ LS : . . . : ð2αðsjjÞ Þ LSÞ ¼ ⋮ > 8 > th > > < ðαð jÞ Þ XX, > th > 1 ð jÞ > > > ð2α1 Þ LS ! : : ð jÞ th ðα1 Þ XX

ð7:168Þ

for j ¼ 1,2. Four special broom appearing bifurcations for higher order equilibriums are Ps



α ¼m i¼1 i



SO ðx ¼ aÞ







!

ðð2α1 Þth LS : . . . : ð2αs Þth LSÞ

appearing bifurcation

Ps 

α ¼m i¼1 i

SO, for x ¼ a ¼ a2sþ1 ,



SI ðx ¼ aÞ







! appearing bifurcation

SO, for x ¼ a ¼ a2sþ1 , ðð2α1 Þth US : . . . : ð2αs Þth USÞ

ð7:169Þ

ð7:170Þ

and Ps 

α ¼m i¼1 i

(

SO ðx ¼ aÞ







! appearing bifurcation

Ps 

α ¼m i¼1 i

SI ðx ¼ aÞ







! appearing bifurcation

(

ðð2α1 Þth US : . . . : ð2αs Þth USÞ SO, for x ¼ a ¼ a1 ðð2α1 Þth LS : . . . : ð2αs Þth LSÞ SI, for x ¼ a ¼ a1 :

ð7:171Þ

ð7:172Þ

7 (2m+1)th-Degree Polynomial Systems

334

For a ¼  12 Bi (i 2 {1, 2, . . ., s}), there are four higher order bifurcations as follows: 8 ðiÞ > ð2l 1Þth orderSO, x ¼b2 , > < i SO ðx ¼aÞ! ð2ð2li 1Þþ1Þth SO¼ SI,x ¼a, > > : ðiÞ ð2li 1Þth orderSO,x ¼b1 ; 8 ðiÞ th  > < ð2li  1Þ order SI, x ¼ b2 , th  SI ðx ¼ aÞ ! ð2ð2li  1Þ þ 1Þ SI ¼ SO, x ¼ a, > : ðiÞ ð2li  1Þth order SI, x ¼ b1 ; 8 ðiÞ th  > < ð2li Þ order US, x ¼ b2 , SI ðx ¼ aÞ ! ð2ð2li Þ þ 1Þth SO ¼ SO, x ¼ a, > : ðiÞ ð2li Þth order LS, x ¼ b1 ; 8 ðiÞ th  > < ð2li Þ order LS, x ¼ b2 , SI ðx ¼ aÞ ! ð2ð2li Þ þ 1Þth SI ¼ SI, x ¼ a, > : ðiÞ ð2li Þth order US, x ¼ b1 :

ð7:173Þ

ð7:174Þ

ð7:175Þ

ð7:176Þ

(i) For αi ¼2li – 1, the (2(2li  1) + 1)th-order source (SO) switching bifurcation is ðiÞ with the (2li – 1)th-order source (SO) ðx ¼ b2 Þ and the (2l – 1)th-order sink ðiÞ ðiÞ ðiÞ  (SI) ðx ¼ b1 Þ with b2 > a > b1 . (ii) For αi ¼ 2li – 1, the (2(2li  1) + 1)th-order sink (SI) switching bifurcation is ðiÞ with the (2li – 1)th-order sink (SI) ðx ¼ b2 Þ and the (2li  1)th-order source ðiÞ

ðiÞ

ðiÞ

ðiÞ

ðiÞ

ðiÞ

ðiÞ

ðiÞ

ðiÞ

(SO) ðx ¼ b1 Þ with b2 > a > b1 . (iii) For αi ¼ 2li, the (2(2li) + 1)th-order source (SO) switching bifurcation is with the ðiÞ (2li)th-order upper saddle (US) ðx ¼ b2 Þ and the (2li)th-order upper saddles (LS) ðx ¼ b1 Þ with b2 > a > b1 . (iv) For αi ¼ 2li, the (2(2li) + 1)th-order sink (SI) switching bifurcation is with the ðiÞ (2li)th-order upper saddle (LS) ðx ¼ b2 Þ and the (2li)th-order upper saddles (US) ðx ¼ b1 Þ with b2 > a > b1 . If x ¼ a ¼  12 Bi (i 2 {1, 2, . . ., m}), the equilibrium of x ¼ a possesses a (2(2li  1) + 1)th and (2(2li) + 1)th-order SI or SO switching bifurcations (or pitchfork bifurcations) for higher order equilibriums. The teeth comb appearing bifurcation generated by the m-pairs of quadratics becomes a broom-switching bifurcation. Such a broom-switching bifurcation consists of a pitchfork-switching bifurcation and two

7.3 Higher Order Equilibrium Bifurcations

335

teeth comb appearing bifurcations in the (2m + 1)th-degree polynomial system. Four broom-switching bifurcations for higher order equilibriums are 8 ð2Þ th ð2Þ th > > > ðð2α1 Þ US : . . . : ð2αs2 Þ USÞ, > > 8 > > sj > 2 P P > > ð2ls1 þ1 Þth US, ð jÞ > αi þ2ls1 þ1 ¼m > > > < < j¼1 i¼1 SO ð x ¼ aÞ









! ð2ð2ls1 þ1 Þ þ 1Þth SO SO, x ¼ a, appearing bifurcation > > > > > > > : > > ð2ls1 þ1 Þth LS, > > > > > th : ð1Þ th ðð2α1 Þ LS : . . . : ð2αðs11 Þ Þ LSÞ; ð7:177Þ 8 th ð2Þ th > > ðð2α1 Þ US : . . . : ð2αðs22 Þ Þ USÞ, > > > 8 > > sj > 2 P P > > ð2ls1 þ1  1Þth SO, ð jÞ > αi þ2ls1 þ1 1¼m > > > < < j¼1 i¼1 SO ð x ¼ aÞ









! ð2ð2ls1 þ1  1Þ þ 1Þth SO SI, x ¼ a, appearing bifurcation > > > > > > > : > > ð2ls1 þ1  1Þth SO, > > > > > th : ð1Þ th ðð2α1 Þ LS : . . . : ð2αðs11 Þ Þ LSÞ; ð7:178Þ and 8 ð2Þ th ð2Þ th > > > ðð2α1 Þ LS : . . . : ð2αs2 Þ LSÞ, > 8 > s > j 2 P P > ð jÞ > ð2ls1 þ1 Þth LS, αi þ2ls1 þ1 ¼m > > > < < j¼1 i¼1 SI ð x ¼ aÞ









! ð2ð2ls1 þ1 Þ þ 1Þth SI SI, x ¼ a, ð7:179Þ appearing bifurcation > > > > > : th > > ð2ls1 þ1 Þ US, > > > > th th : ð1Þ ðð2α1 Þ US : . . . : ð2αðs11 Þ Þ USÞ; 8 th ð2Þ th > > ðð2α1 Þ LS : . . . : ð2αðs22 Þ Þ LSÞ, > > > 8 > > sj > 2 P P > > ð2ls1 þ1  1Þth SI, ð jÞ > > αi þ2ls1 þ1 1¼m > > < < j¼1 i¼1 SI ð x ¼ aÞ









! ð2ð2ls1 þ1  1Þ þ 1Þth SI SO, x ¼ a, > > appearing bifurcation > > > > > : > > ð2ls1 þ1  1Þth SI; > > > > > th : ð1Þ th ðð2α1 Þ US : . . . : ð2αðs11 Þ Þ USÞ: ð7:180Þ

7 (2m+1)th-Degree Polynomial Systems

336

Consider a (2m + 1)th-degree polynomial system as x_ ¼ a0 QðxÞðx  aÞ

n Y

αi

ðx2 þ Bi x þ Ci Þ

ð7:181Þ

i¼1

where αi 2 {2ri  1, 2ri} (i ¼ 1, 2, . . ., n). Without loss of generality, a function of Q(x) > 0 is either a polynomial function or a non-polynomial function. The roots of x2 + Bix + Ci ¼ 0 are 1 1pffiffiffiffiffi ðiÞ Δi ,Δi ¼ B2i  4Ci 0; x1,2 ¼  Bi  2 2 Bi ¼ B j ði; j 2 f1; 2; . . . ; ng; i 6¼ jÞ ð1Þ

ð1Þ

ð2Þ

ð2Þ

ðnÞ

ðnÞ

fa1 ; a2 ; . . . ; a2nþ1 g  sortfx1 ; x2 ; x1 ; x2 ; . . . ; x1 ; x2 ; ag, ai aiþ1 : ð7:182Þ The higher order singularity bifurcation can be for a cluster of higher order equilibriums. There are four higher order bifurcations as follows: (i) The (2(2l  1))th-order upper-saddle (US) spraying-appearing bifurcation for a cluster of higher order sinks, sources, upper saddles, and lower saddles is ð2βÞth US ¼ ð2ð2l  1ÞÞth order US 8 ðα2n Þth order XX for x ¼ a2n , > > > > > th < ðα  a cluster of 2n‐XX 2n1 Þ order XX for x ¼ a2n1 ,







! appearing bifurcation > >⋮ > > > : ðα1 Þth order XX for x ¼ a1 ; where 2ð2l  1Þ ¼

Pn

i¼1 αi

( th

ðα2n Þ order XX ¼ ( th

ðα1 Þ order XX ¼

ð7:183Þ

and

ð2r 2n Þth order US, for α2n ¼ 2r n , ð2r 2n  1Þth order SO, for α2n ¼ 2r n  1;

ð2r 1 Þth order US, for α1 ¼ 2r 1 ,

ð7:184Þ

ð2r 1  1Þth order SO, for α1 ¼ 2r 1  1:

(ii) The (2(2l))th-order upper-saddle (US) spraying-appearing bifurcation for a cluster of higher order sinks, sources, upper saddles, and lower saddles is

7.3 Higher Order Equilibrium Bifurcations

ð2βÞth US ¼ ð2ð2lÞÞth order US 8 ðα2n Þth order XX for x ¼ a2n , > > > > > th < ðα  a cluster of 2n‐XX 2n1 Þ order XX for x ¼ a2n1 ,







! appearing bifurcation > >⋮ > > > : ðα1 Þth order XX for x ¼ a1 where 2ð2lÞ ¼

Pn

i¼1 αi

( th

ðα2n Þ order XX ¼ ( th

ðα1 Þ order XX ¼

337

ð7:185Þ

and

ð2r 2n Þth order US, for α2n ¼ 2r n , ð2r 2n  1Þth order SO, for α2n ¼ 2r n  1;

ð2r 1 Þth order US, for α1 ¼ 2r 1 ,

ð7:186Þ

ð2r 1  1Þth order SI, for α1 ¼ 2r 1  1:

For the higher order lower-saddle bifurcation, the cluster of the higher order equilibriums is given by the following two cases. (iii) The (2(2l  1))th-order lower-saddle (LS) spraying-appearing bifurcation for a cluster of higher order sinks, sources, upper saddles, and lower saddles is ð2βÞth LS ¼ ð2ð2l  1ÞÞth order LS 8 ðα2n Þth order XX, for x ¼ a2n , > > > > > th < ðα  a cluster of 2n‐XX 2n1 Þ order XX, for x ¼ a2n1 ,







! appearing bifurcation > >⋮ > > > : ðα1 Þth order XX, for x ¼ a1 where 2ð2l  1Þ ¼

Pn

i¼1 αi

( th

ðα2n Þ order XX ¼ ( th

ðα1 Þ order XX ¼

ð7:187Þ

and

ð2r 2n Þth order LS, for α2n ¼ 2r n , ð2r 2n  1Þth order SI, for α2n ¼ 2r n  1;

ð2r 1 Þth order LS, for α1 ¼ 2r 1 ,

ð7:188Þ

ð2r 1  1Þth order SI, for α1 ¼ 2r 1  1:

(iv) The (2(2l))th-order lower-order spraying-appearing bifurcation for a cluster of higher order sinks, sources, upper saddles, and lower saddles is

7 (2m+1)th-Degree Polynomial Systems

338

ð2βÞth LS ¼ ð2ð2lÞÞth order LS 8 ðα2n Þth order XX, for x ¼ a2n , > > > > > th < ðα  a cluster of 2n‐XX 2n1 Þ order XX, for x ¼ a2n1 ,









! appearing bifurcation > >⋮ > > > : ðα1 Þth order XX, for x ¼ a1 where 2ð2lÞ ¼

Pn

i¼1 αi

( th

ðα2n Þ order XX ¼ ( th

ðα1 Þ order XX ¼

ð7:189Þ

and

ð2r 2n Þth order LS, for α2n ¼ 2r n , ð2r 2n  1Þth order SI, for α2n ¼ 2r n  1;

ð2r 1 Þth order LS, for α1 ¼ 2r 1 ,

ð7:190Þ

ð2r 1  1Þth order SO, for α1 ¼ 2r 1  1:

If x ¼ a 6¼  12 Bi (i 2 {1, 2, . . ., n}), the equilibrium of x ¼ a breaks a cluster of sprinkler-spraying appearing bifurcations for higher order equilibriums to two parts. The sprinkler-spraying appearing bifurcation generated by the m-pairs of quadratics becomes a broom-sprinkler-spraying-appearing bifurcation. The two broom-sprinkler-spraying-appearing bifurcations in the (2m + 1)th-degree polynomial system are 8 th ð2Þ th > > ðð2β1 Þ US : . . . : ð2βðr22 Þ Þ USÞ, > > <   m¼m1 þm2 SO ð x ¼ aÞ







! SO ða ¼ ar1 þ1 Þ ! SO a ¼ a2ðm1 þ1Þ1 , appearing bifurcation > > > > : ðð2βð1Þ Þth LS : . . . : ð2βð1Þ Þth LSÞ 1 r1 ð7:191Þ and 8 ð2Þ th ð2Þ th > > ðð2β > 1 Þ LS : . . . : ð2β r2 Þ LSÞ, > <   m¼m1 þm2 SI ð x ¼ aÞ







! SI ða ¼ ar1 þ1 Þ ! SO a ¼ a2ðm1 þ1Þ1 , appearing bifurcation > > > > : ðð2βð1Þ Þth US : . . . : ð2βð1Þ Þth USÞ r1 1 ð7:192Þ P 1 ð1Þ P 2 ð2Þ for m1 ¼ ri¼1 βi , m2 ¼ rj¼1 βj ; and the acronyms USG and LSG are the uppersaddle-node bifurcation group and lower-saddle-node bifurcation group, respectively.

7.3 Higher Order Equilibrium Bifurcations

339

Four special broom-sprinkler-spraying-appearing bifurcations of the (2m + 1)thdegree polynomial system are m¼



(

Pr

β i¼1 i

SO ð x ¼ aÞ







! appearing bifurcation m¼



Pr

β i¼1 i

SI ð x ¼ aÞ







!

SO ða ¼ a2mþ1 Þ ! SO ða ¼ a2mþ1 Þ,

ðð2β1 Þth LS : . . . : ð2βr Þth LSÞ; ( SI ða ¼ a2mþ1 Þ ! SI ða ¼ a2mþ1 Þ,

appearing bifurcation

ðð2β1 Þth US : . . . : ð2βr Þth USÞ;

ð7:193Þ

ð7:194Þ

and m¼

Pr

β i¼1 i

(

SO ð x ¼ aÞ







! appearing bifurcation m¼



Pr

β i¼1 i

SI ð x ¼ aÞ







! appearing bifurcation

(

ðð2β1 Þth US : . . . : ð2βr Þth USÞ, SO ða ¼ a1 Þ ! SO ða ¼ a1 Þ; ðð2β1 Þth LS : . . . : ð2βr Þth LSÞ, SI ða ¼ a1 Þ ! SI ða ¼ a1 Þ:

ð7:195Þ

ð7:196Þ

If x ¼ a ¼  12 Bi (i 2 {1, 2, . . ., l}), the equilibrium of x ¼ a possesses a (2l + 1)th-order SI or SO switching bifurcation (or broom-switching bifurcation) for higher order equilibriums. The sprinkler-spraying-appearing bifurcation generated by the m-pairs of quadratics becomes a broom-sprinkler-spraying-switching bifurcation. The two broom-switching bifurcations in the (2m + 1)th-degree polynomial system are m¼m1 þm2 þβ

SO ð x ¼ aÞ







! switching bifurcation 8 8 th > > > ð2βðr22 Þ Þ US ðx ¼ ar1 þr2 þ1 Þ, > > P > r > 2 ð 2Þ < > > m2 ¼ β th > j¼1 j > > ð2βð12Þ : . . . : 2βðr22 Þ Þ US





! ⋮ > > appearing > > > > > > > > : ð2Þ th > > ð2β1 Þ US ðx ¼ ar1 þ2 Þ; > > < th

ð2β þ 1Þ SO ða ¼ ar1 þ1 Þ; > > 8 > > > > ð2βð1Þ Þth LS ðx ¼ ar Þ, > > 1 > r1 > P r 1 ð 1Þ > > > < > m1 ¼ β th > i¼1 i ð1Þ > ð 1 Þ > ð2β1 : . . . : 2βr1 Þ LS





! ⋮ > > appearing > > > > > > > : : ð1Þ th ð2β1 Þ LS ðx ¼ a1 Þ; and

ð7:197Þ

7 (2m+1)th-Degree Polynomial Systems

340 m¼m1 þm2 þβ

SI ð x ¼ aÞ







! switching bifurcation 8 8 > ð2βð2Þ Þth LS ðx ¼ ar þr þ1 Þ, > > 1 2 > r2 P r 2 ð 2Þ > > > > < > m2 ¼ β th > j¼1 j ð 2Þ ð 2 Þ > > ð2β1 : . . . : 2βr2 Þ LS





! ⋮ > > appearing > > > > > > > > : ð2Þ th > > ð2β1 Þ LS ðx ¼ ar1 þ2 Þ; > >
> 8 > > > > ð2βð1Þ Þth US ðx ¼ ar Þ, > > 1 > r1 > Pr1 ð1Þ > > > < > m1 ¼ βi th > ð 1 Þ i¼1 > > ð2β1 : . . . : 2βðr12 Þ Þ US





! ⋮ > > appearing > > > > > > > : : ð1Þ th ð2β1 Þ US ðx ¼ a1 Þ where cluster of l‐quadratics

ð2β þ 1Þth order SOðx ¼ aÞ







! appearing bifurcation

8 ðα2sl þ1 Þth XX, for x ¼ a2sl þ1 , > > > > > > > > ðα2sl Þth XX, for x ¼ a2sl , > > > < ⋮ > > > > > > ðα2s1 Þth XX, for x ¼ a2s1 , > > > > > : ðα2s1 1 Þth XX, for x ¼ a2s1 1 :

ð7:199Þ

cluster of l‐quadratics

ð2β þ 1Þth order SI ðx ¼ aÞ







! appearing bifurcation

8 ðα2sl þ1 Þth XX, for x ¼ a2sl þ1 , > > > > > > > th > > ðα2sl Þ XX, for x ¼ a2sl , > > < ⋮ > > > > > > ðα2s1 Þth XX, for x ¼ a2s1 , > > > > > : ðα2s1 1 Þth XX, for x ¼ a2s1 1 where x ¼ a 2 fa2s1 1 , . . . , a2sl , a2sl þ1 g and 2β þ 1 ¼

ð7:200Þ

Pl

i¼1 α2si 1

þ α2si þ α2sl þ1 .

7.3 Higher Order Equilibrium Bifurcations

341

The two appearing bifurcations for the higher order singularity of equilibriums are (i) SO ! ((2α1)thLS : . . . : (2αi)thLS : SO : . . . : (2αn  1)thUS : (2αn)thUS), (ii) SO ! ((2α1)thUS : . . . : (2αi)thUS : SO : . . . : (2αn  1)thLS : (2αn)thLS), as presented in Figs. 7.7a, b for a0 > 0 and a0 < 0, respectively. The broom appearing bifurcation for the higher order equilibriums are illustrated. The components of the broom appearing bifurcation are 8  < 2r j th US 2αj US



!   ð j ¼ i, n  1, . . .Þ, th appearing : 2r j US ( ð2r k  1Þth SO αk ¼2r k 1 th ð2αk Þ US



! ðk ¼ 1, n, . . .Þ; appearing ð2r k  1Þth SI

ð7:201Þ

8  < 2r j th LS ð j ¼ i, n  1, . . .Þ, 2αj LS



! th appearing :  2r j LS ( ð2r k  1Þth SI αk ¼2r k 1 th ð2αk Þ LS



ðk ¼ 1, n, . . .Þ:

! appearing ð2r k  1Þth SO

ð7:202Þ



th

αj ¼2r j

and 

th

αj ¼2r j

The simple equilibrium does not interact with the bifurcation points. There are four switching and appearing bifurcations of the higher-order equilibriums. The two broom-sprinkler-spraying-switching bifurcations for the higher order singularity of equilibriums are (iii) (iv) (v) (vi)

((2(2r1  1)thLS : . . . : (2(2ri) + 1)SO : . . . : (2(2rn  1))thUS : (2(2rn  1))thUS), ((2(2r1  1)thUS : . . . : (2(2ri) + 1)SI : . . . : (2(2rn  1))thLS : (2(2rn  1))thLS), ((2(2r1  1)thLS : . . . : (2(2ri))LS : . . . : (2(2rn  1))thLS : (2(2rn  1) + 1)thSO), ((2(2r1  1)thUS : . . . : (2(2ri))US : . . . : (2(2rn  1))thUS : (2(2rn  1) + 1)thSI),

as presented in Figs. 7.7(c, e) and (d, g) for a0 > 0 and a0 < 0, respectively. The (2(2ri) + 1)thSO and (2(2ri) + 1)thSI switching bifurcations are

7 (2m+1)th-Degree Polynomial Systems

342

a

b

c

d

Fig. 7.7 Six bifurcations in a (2m + 1)th-degree polynomial system. (a, b) Two broom appearing bifurcations. (c–f) Broom-switching bifurcations. LS: lower saddle, US: upper saddle, SI: sink, SO: source. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

7.3 Higher Order Equilibrium Bifurcations

e

343

f

Fig. 7.7 (continued)

8 > ð2r i Þth US, > < αj ¼2rj ð2ð2r i Þ þ 1Þth SO



! SO, appearing > > : ð2r i Þth LS, 8 > ð2r Þth LS, > < i α ¼2r j j ð2ð2r i Þ þ 1Þth SI



! SI appearing > > : ð2r i Þth US,

ð7:203Þ

and the (2(2rn  1) + 1)thSO and (2(2rn  1) + 1)thSI switching bifurcations are 8 > ð2r n  1Þth SO, > < αj ¼2rj ð2ð2r n  1Þ þ 1Þth SO



! SI, appearing > > : ð2r n  1Þth SO, 8 > ð2r n  1Þth SI, > < αj ¼2r j ð2ð2r n  1Þ þ 1Þth SI



! SO appearing > > : ð2r n  1Þth SI:

ð7:204Þ

7 (2m+1)th-Degree Polynomial Systems

344

In Fig. 7.8a, b, the simple switching with two sprinkler-spraying-appearing bifurcations are presented for a0 > 0 and a0 < 0, respectively. The two bifurcation structures are (i) SO ! ((2β1)thLS : . . . : (2βi)thLS : SO : . . . : (2βn  1)thUS : (2βn)thUS), (ii) SI ! ((2β1)thUS : . . . : (2βi)thUS : SI : . . . : (2βn  1)thLS : (2βn)thLS), P where m ¼ ni¼1 βi , β1 ¼ (2l1  1), . . ., βi ¼ 2li, . . ., βn  1 ¼ 2ln  1, βn ¼ 2ln. In Fig. 7.8 (c, e) and (d, f), the (2l + 1)th-order broom-switching with two sprinklerspraying-appearing bifurcations are presented for a0 > 0 and a0 < 0, respectively. The two bifurcation structures are: (iii) (iv) (v) (vi)

SO ! ((2β1)thUS : . . . : (2βi + 1)thSO : . . . : (2βn  1)thUS : (2βn)thUS), SI ! ((2β1)thLS : . . . : (2βi + 1)thSI : . . . : (2βn  1)thLS : (2βn)thLS), SO ! ((2β1 + 1)thSO : . . . : (2βi)thUS : . . . : (2βn  1)thUS : (2βn)thUS), SI ! ((2β1 + 1)thSI : . . . : (2βi)thLS : . . . : (2βn  1)thLS : (2βn)thLS).

For a cluster of m-quadratics, Bi ¼ Bj (i, j 2 {1, 2, . . ., m}; i 6¼ j) and Δi ¼0 (i 2 {1, 2, . . ., m}). The (2m)th-order upper-saddle-node appearing bifurcation for higher-order equilibriums is 8 > ðα2s Þth XX, for x ¼ a2s , > > > < cluster of s‐quadratics ðα2s1 Þth XX, for x ¼ a2s1 , ð2mÞth order US







! appearing bifurcation > ⋮ > > > : ðα1 Þth XX, for x ¼ a1 , where 2m ¼

P2s

j¼1 αj

ð7:205Þ

and m ¼ (2l  1), (2l): ( th

ðα1 Þ XX ¼ ðα2s Þth XX ¼

ð2l1 Þth LS, for α1 ¼ 2l1 ,

ð2l1  1Þth SI, for α1 ¼ 2l1  1; ( ð2l2s Þth US, for α2s ¼ 2l2s ,

ð7:206Þ

ð2l2s  1Þth SI, for α2s ¼ 2l2s  1:

The (2m)th-order lower-saddle-node appearing bifurcation for higher-order equilibriums is 8 > ðα2s Þth XX, for x ¼ a2s , > > > < cluster of s‐quadratics ðα2s1 Þth XX, for x ¼ a2s1 ; ð2mÞth order LS







! appearing bifurcation > ⋮ > > > : ðα1 Þth XX, for x ¼ a1

ð7:207Þ

7.3 Higher Order Equilibrium Bifurcations

345

a

b

c

d

Fig. 7.8 Six types of bifurcations in a (2m + 1)th-degree polynomial system. (a, b) Broomsprinkler-spraying-appearing bifurcations, (c–f) broom-spraying-switching bifurcations with equilibrium clusters. LS: lower saddle, US: upper saddle, SI: sink, SO: source. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

7 (2m+1)th-Degree Polynomial Systems

346

e

f

Fig. 7.8 (continued)

where ( th

ðα1 Þ XX ¼ ðα2s Þth XX ¼

ð2l1 Þth US, for α1 ¼ 2l1

ð2l1  1Þth SO, for α1 ¼ 2l1  1 ( ð2l2s Þth LS, for α2s ¼ 2l2s

ð7:208Þ

ð2l2s  1Þth SI, for α2s ¼ 2l2s  1

There are four simple switching and (2m)th-order saddle-node appearing bifurcations for higher order equilibriums: The two switching bifurcations of SO ! ((2m)thUS : SO) and SI ! ((2m)thLS : SI) with two (2m)th-order USN and LSN spraying-appearing bifurcations in the (2m + 1)th-degree polynomial system are SO ðx ¼ aÞ ! 

SI ðx ¼ aÞ !



SO ! SO, for x ¼ a ¼ a2mþ1 , ð2mÞth order LS



SI ! SI, for x ¼ a ¼ a2mþ1 , ð2mÞth order US

ð7:209Þ

ð7:210Þ

7.3 Higher Order Equilibrium Bifurcations

347

and the two switching bifurcations of SO ! (SO : (2m)thUS) and SI ! (SI : (2m)thLS) with two (2m)th-order USN and LSN spraying-appearing bifurcations in the (2m + 1)th-degree polynomial system are SO ðx ¼ aÞ ! 



ð2mÞth order US, SO ! SO, for x ¼ a ¼ a1 ;



SI ðx ¼ aÞ !

ð2mÞth order LS, SI ! SI, for x ¼ a ¼ a1 :

ð7:211Þ ð7:212Þ

The (2m + 1)th-order source broom-switching bifurcation for higher order equilibrium is 8 > ðα2sþ1 Þth XX, for x ¼ a2sþ1 , > > > < switching ðα2s Þth XX, for x ¼ a2s , SOðx ¼ aÞ



! ð2m þ 1Þth order SO > ⋮ > > > : ðα1 Þth XX, for x ¼ a1 ð7:213Þ where 2m þ 1 ¼

P2sþ1 j¼1

( th

ðα1 Þ XX ¼ th

ðα2sþ1 Þ XX ¼

αj , m ¼ (2l  1), (2l) and

ð2l1 Þth LS, for α1 ¼ 2l1 , ð2l1  1Þth SI, for α1 ¼ 2l1  1; ( ð2l2sþ1 Þth US, for α2sþ1 ¼ 2l2sþ1 ,

ð7:214Þ

ð2l2sþ1  1Þth SO, for α2sþ1 ¼ 2l2sþ1  1:

The (2m + 1)th-order sink broom-switching bifurcation is 8 th  > > > ðα2sþ1 Þ XX, for x ¼ a2sþ1 , > < th switching ðα2s Þ XX, for x ¼ a2s , th SIðx ¼ aÞ



! ð2m þ 1Þ order SI >⋮ > > > : ðα1 Þth XX, for x ¼ a1 ð7:215Þ where

7 (2m+1)th-Degree Polynomial Systems

348

( th

ðα1 Þ XX ¼ ( th

ðα2sþ1 Þ XX ¼

ð2l1 Þth US, for α1 ¼ 2l1 , ð2l1  1Þth SO, for α1 ¼ 2l1  1; ð2l2sþ1 Þth LS, for α2sþ1 ¼ 2l2sþ1 ,

ð7:216Þ

ð2l2sþ1  1Þth SI, for α2sþ1 ¼ 2l2sþ1  1:

The switching bifurcation consists of a simple switching and the (2m)th-order saddle-node appearing bifurcation with m-pairs of source and sinks. The (2m)thorder saddle-node appearing bifurcation is a sprinkler-spraying cluster of the mpairs of sources and sinks. Thus, the four switching bifurcations of SO ! (SO : (2m)thUS) for higher order equilibriums for a0 > 0, SI ! (SI : (2m)thLS) for higher order equilibriums for a0 < 0, SO ! ((2m)thUS : SO) for higher order equilibriums for a0 > 0, SI ! ((2m)thLS : SI) for higher order equilibrium for a0 < 0 are presented in Fig. 7.9a–d, respectively. The (2m + 1)th-order source switching bifurcation is named the (2m + 1)thSO broom-sprinkle-spraying-switching bifurcation, and the (2m + 1)th-order sink switching bifurcation is named the (2m + 1)thSI broom-switching bifurcation. Such a (2m + 1)thXX broom-switching bifurcation is from simple equilibrium to a (2m + 1)thXX broom-switching bifurcation. The two broom-switching bifurcations for higher order equilibriums of SO ! (2m + 1)thSO for higher order equilibrium for a0 > 0, SI ! (2m + 1))thSI for higher order equilibrium for a0 < 0 are presented in Fig. 7.9e, f, respectively. A series of the (2αi + 1)th-order source and sink bifurcations are aligned up with varying parameters, which is formed in a special pattern. Such a special pattern is from m-quadratics in the (2m + 1)th-degree polynomial system; the following conditions should be satisfied: 1 1 aðpi Þ ¼  Bi and aðp j Þ ¼ ‐ B j 2 2 Bi B j i, j 2 f1; 2; . . . ; ng;i 6¼ j, Δi > Δiþ1 ði ¼ 1; 2; . . . ; n; n mÞ,

ð7:217Þ

Δi ¼ 0 with kpi k < kpiþ1 k: Four series of switching bifurcations in the (2m + 1)th-degree polynomial nonlinear system are (i) (ii) (iii) (iv)

(2(2r1  1) + 1)thSO ‐ (2(2r2) + 1)thSI ‐ . . . ‐ (2(2rn  1) + 1)thSO), (2(2r1  1) + 1)thSI ‐ (2(2r2) + 1)thSO ‐ . . . ‐ (2(2rn  1) + 1)thSI), (2(2r1) + 1)thSO ‐ (2(2r2  1) + 1)thSO ‐ . . . ‐ (2(2rn  1))thSO), (2(2r1) + 1)thSI ‐ (2(2r2  1) + 1)thSI ‐ . . . ‐ (2(2rn  1) + 1)thSI),

7.3 Higher Order Equilibrium Bifurcations

349

a

b

c

d

e

f

Fig. 7.9 Broom-switching bifurcations of equilibriums in (2m + 1)th polynomial system: (a) (SO : (2m)thUS)-appearing bifurcation (a0 > 0), (b) (SI : (2m)thUS)-appearing bifurcation (a0 < 0), (c) ((2m)thUS, SO)-appearing bifurcation (a0 > 0), (d) ((2m)thLS : SI)-appearing bifurcation (a0 < 0). (e) (2m + 1)thSO-switching-appearing bifurcation (a0 > 0), (f) (2m + 1)thSO-switchingappearing bifurcation (a0 < 0). LS: lower saddle, US: upper saddle, SI: sink, SO: source. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

as presented in Fig. 7.10 (a, c)–(b, f) for (a0 > 0) and (a0 < 0), respectively. The swapping pattern of higher order sinks and sources switching bifurcations cannot be observed. Such a bifurcation scenario is like the fish bone for the higher order switching bifurcations for higher order equilibriums.

a

b

c

d Fig. 7.10 Four series of switching bifurcations of equilibriums in a (2m + 1)th polynomial system: (a, c) for a0 > 0, (b, d) for a0 < 0. LS: lower saddle, US: upper saddle, SI: sink, SO: source. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

7.3 Higher Order Equilibrium Bifurcations

7.3.2

351

Switching Bifurcations α

Consider the roots of ðx2 þ Bi x þ C i Þ i ¼ 0 as ðiÞ

ðiÞ

ðiÞ 2

ðiÞ

Bi ¼ ðb1 þ b2 Þ, Δi ¼ ðb1  b2 Þ 0, ðiÞ

ðiÞ

ðiÞ

ðiÞ

x1,2 ¼ b1,2 , Δi > 0 if b1 6¼ b2 ði ¼ 1, 2, . . . , nÞ; ) Bi 6¼ Bj ði, j ¼ 1, 2, . . . , n; i 6¼ jÞ at bifurcation: ðiÞ ðiÞ Δi ¼ 0 at b1 ¼ b2 ði ¼ 1, 2, . . . , nÞ

ð7:218Þ

The ðαi Þth -order singularity bifurcation is for the switching of a pair of higher order equilibriums (i.e., sinks, sources, upper saddles, and lower saddles). There are six switching bifurcations for i 2 {1, 2, . . ., n}: ðiÞ

ði Þ

li ¼r 1 þr2 1

ð2li Þth order US







! switching bifurcation 8 th < ð2r ðiÞ  1Þ order SO # SI, for bðiÞ ¼ a2i # a2i1 , 2 2 :

ðiÞ

th

ð7:219Þ

ðiÞ

ð2r 1  1Þ order SI " SO, for b1 ¼ a2i1 " a2i ; ði Þ

ði Þ

li ¼r1 þr2 1

ð2li Þth order LS







! switching bifurcation 8 th ð i Þ < ð2r  1Þ order SI # SO, for bðiÞ ¼ a2i # a2i1 , 2 2 :

ðiÞ

th

ð7:220Þ

ðiÞ

ð2r 1  1Þ order SO " SI, for b1 ¼ a2i1 " a2i ; ði Þ

ðiÞ

li ¼r1 þr 2

ð2li Þth order US







! switching bifurcation 8 th < ð2r ðiÞ Þ order US # US, for bðiÞ ¼ a2i # a2i1 , 2 2 :

ðiÞ th

ð7:221Þ

ðiÞ

ð2r 1 Þ order US " US, for b1 ¼ a2i1 " a2i ; ði Þ

ði Þ

li ¼r1 þr2

ð2li Þth order LS







! switching bifurcation 8 th ð i Þ < ð2r Þ order LS # LS, for bðiÞ ¼ a2i # a2i1 , 2 2 :

ðiÞ th

ðiÞ

ð2r 1 Þ order LS " LS, for b1 ¼ a2i1 " a2i ;

ð7:222Þ

7 (2m+1)th-Degree Polynomial Systems

352 ðiÞ

ði Þ

li ¼r 1 þr2

ð2li  1Þth order SO







! switching bifurcation 8 th ð i Þ < ð2r  1Þ order SO # SO, for bðiÞ ¼ a2i # a2i1 , 2 2 :

ðiÞ th

ð7:223Þ

ðiÞ

ð2r 1 Þ order LS " US for b1 ¼ a2i1 " a2i ; ðiÞ

ði Þ

li ¼r 1 þr2

ð2li  1Þth order SI







! switching bifurcation 8 th < ð2r ðiÞ  1Þ order SI # SI, for bðiÞ ¼ a2i # a2i1 , 2 2 :

ðiÞ th

ð7:224Þ

ðiÞ

ð2r 1 Þ order US " LS for b1 ¼ a2i1 " a2i :

A set of n-paralleled higher order XX switching bifurcations is called the ððα1 Þth XX : ðα2 Þth XX : . . . : ðαn Þth XXÞ parallel switching bifurcation in the (2m + 1)th-degree polynomial nonlinear system. Such a bifurcation is also called the ððα1 Þth XX : ðα2 Þth XX : . . . : ðαn Þth XXÞ antenna-switching bifurcation. αi 2 {2li, 2li  1} and XX 2 {SO, SI, US, LS}. For non-switching points, ðiÞ ðiÞ ðiÞ ðiÞ Δi > 0 at b1 6¼ b2 (i ¼ 1, 2, . . ., n). At the bifurcation point, Δi ¼ 0 at b1 ¼ b2 (i ¼ 1, 2, . . ., n). The parallel antenna-switching bifurcation for higher order equilibriums in the (2m + 1)th-degree polynomial system is 8 ð2Þ th ð2Þ th > > < ððα1 Þ XX1 : . . . : ðαl2 Þ XXl2 Þ SI ðor SOÞ, for x ¼ a ð7:225Þ > > th th : ð1Þ ð1Þ ððα1 Þ XX1 : . . . : ðαl1 Þ XXl1 Þ where th

si th bifurcation

ðαðsii Þ Þ XXsi





! switching 8 th > < ðr ð2si Þ Þ XXð2si Þ # YYð1si Þ , for bð2si Þ ¼ að2siÞ # að2siÞ 1 , i i

ð7:226Þ

> : ðr ðsi Þ Þth XXðsi Þ " YYðsi Þ , for bðsi Þ ¼ aðiÞ " aðiÞ ; 1 1 2 1 2si 1 2si ðsi ¼ 1, 2, . . . , li , i ¼ 1, 2Þ: Such eight sets of parallel switching bifurcations for higher order equilibrium are presented in Fig. 7.11(a, c, e, g) and (b, d, f, h) for a0 > 0 and a0 < 0, respectively. The eight switching bifurcation structures are as follows:

7.3 Higher Order Equilibrium Bifurcations

353

a

b

c

d

Fig. 7.11 Antenna parallel switching bifurcation of equilibriums for a (2m + 1)th-degree polynomial nonlinear dynamical system. (a, c, e, g) Four parallel bifurcations for a0 > 0. (b, d, f, h) Four parallel bifurcations for a0 < 0. LS: lower saddle, US: upper saddle, SI: sink, SO: source. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

7 (2m+1)th-Degree Polynomial Systems

354

e

f

g

h

Fig. 7.11 (continued)

7.3 Higher Order Equilibrium Bifurcations

355

((2l1)thUS : . . . : SI : . . . : (2ln1  1)thSO : (2ln)thUS) for a0 > 0, ((2l1)thLS : . . . : SO : . . . : (2ln1  1)thSI : (2ln)thLS) for a0 < 0, ((2l1)thLS : . . . : SO : . . . : (2ln1  1)thSI : (2ln  1)thSO) for a0 > 0, ((2l1)thUS : . . . : SI : . . . : (2ln1  1)thSO : (2ln  1)thSI) for a0 < 0, ((2l1)thLS : . . . : SO : . . . : (2ln1  1)thSI : (2ln  1)thSO) for a0 > 0, ((2l1)thUS : . . . : SI : . . . : (2ln1  1)thSI : (2ln  1)thSI) for a0 < 0, ((2l1)thUS : . . . : SI : . . . : (2ln1  1)thSO : (2ln)thUS) for a0 > 0, ((2l1)thLS : . . . : SO : . . . : (2ln1  1)thSI : (2ln)thLS) for a0 < 0.

(i) (ii) (iii) (iv) (v) (vi) (vii) (viii)

The switching bifurcations with different higher order equilibriums are similar to the (l1 ‐ LSN : SO : l2 ‐ USN) and (l1 ‐ USN : SI : l2 ‐ LSN) switching bifurcations for simple sinks and sources. Consider a switching bifurcation for a cluster of higher order equilibriums with the following conditions: Bi ¼ B j ði; j 2 f1; 2; . . . ; ng; i 6¼ jÞ g at bifurcation: ðiÞ ðiÞ Δi ¼ 0 at b1 ¼ b2 ði ¼ 1; 2; . . . ; nÞ

ð7:227Þ

Thus, the (2l)th-order switching bifurcation can be for a cluster of higher order equilibriums. Two (2l)th-order switching bifurcations for l 2 {1, 2, . . ., s} are th

αi ¼

Ps

ði Þ r j¼1 j

ðαi Þ order XX







! switching bifurcation 8  th ð i Þ > rs order XXðsiÞ # YYðsiÞ , for bðsiÞ # aðsiÞ > > > > > > ⋮ > > < ðiÞ th ðiÞ ðiÞ ðiÞ ðr j Þ order XXj # YYj , for bj # aðsiÞ > > > > >⋮ > > > > : ðiÞ th ðiÞ ðiÞ ðiÞ ðr 1 Þ order XX1 " YY1 , for b1 # aðsiÞ

ð7:228Þ

where fa2s1 1 ; a2s1 ; . . . ; a2sl 1 ; a2sl g fb2s1 1 ; b2s1 ; . . . ; b2sl 1 ; b2sl g

¼

ðs Þ

before bifurcation

¼

after bifurcation

ðs Þ

ðs Þ

ðs Þ

sortfb1 1 ; b2 1 ; . . . ; b1 l ; b2 l g, ðs Þ

ðs Þ

ðs Þ

ðs Þ

sortfb1 1 ; b2 1 ; . . . ; b1 l ; b2 l g: ð7:229Þ

A set of paralleled, different, higher order upper-saddle-node switching bifurcations with multiplicity is the ((α1)thXX : (α2)thXX : . . . : (αs)thXX) parallel switching bifurcation in the (2m+1)th-degree polynomial system. At the straw-bundle switching bifurcation, Δi ¼0 (i ¼ 1, 2, . . ., n) and Bi ¼ Bj (i, j 2 {1, 2, . . ., n}; i 6¼ j). The parallel straw-bundle switching bifurcation for higher order equilibriums is

7 (2m+1)th-Degree Polynomial Systems

356

ððα1 Þth XX : ðα2 Þth XX : . . . : ðαs Þth XXÞ switching 8 ðαs Þth order XX switching, > > > > > ðα2 Þth order XX switching, > > > > : ðα1 Þth order XX switching

ð7:230Þ

αi 2 f2li ; 2li  1g and XX 2 fUS; LS; SO; SIg:

ð7:231Þ

8 ð2Þ th ð2Þ th ð2Þ th > > < ððα1 Þ XX : ðα2 Þ XX : . . . : ðαs2 Þ XXÞ SI ðor SOÞ > > : ð1Þ th th ð1Þ th ððα1 Þ XX : ðα2 Þ XX : . . . : ðαðs11 Þ Þ XXÞ:

ð7:232Þ

where

Thus,

Eight parallel straw-bundle switching bifurcations of ððα1 Þth XX : ðα2 Þth XX : . . . : ðαn Þth XXÞ are presented in Figs. 7.12 and 7.13 for a0 > 0 and a0 < 0, respectively.

7.3.3

Switching and Appearing Bifurcations

Consider a (2m + 1)th-degree 1-dimensional polynomial system in a form of x_ ¼ a0 QðxÞ

n1 Y i¼1

ðx  ci Þαi

n2 Y

αj

ðx2 þ B j x þ C j Þ :

ð7:233Þ

j¼1

P1 where ni¼1 αi ¼ 2s1 þ 1. Without loss of generality, a function of Q(x) > 0 is either a polynomial function or a non-polynomial function. The roots of x2 + Bjx + Cj ¼ 0 are 1 1 pffiffiffiffiffiffi ðjÞ Δ j ,Δ j ¼ B2j  4C j 0 ðj ¼ 1; 2; . . . ; n2 Þ; b 1 ,2 ¼  B j  2 2 either

ð7:234Þ

7.3 Higher Order Equilibrium Bifurcations

357

a

b

c

d

Fig. 7.12 (a–d) Four types of (r1th XX : r2th XX : . . . : rnth XX) parallel switching bifurcation for a0 > 0 in the (2m + 1)th-degree polynomial system. LS: lower saddle, US: upper saddle, SI: sink, SO: source. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

7 (2m+1)th-Degree Polynomial Systems

358

a

b

c

d

Fig. 7.13 (a–d) Four types of (r1th XX : r2th XX : . . . : rnth XX) parallel switching bifurcation for a0 < 0 in the (2m + 1)th-degree polynomial system. LS: lower saddle, US: upper saddle, SI: sink, SO: source. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

7.3 Higher Order Equilibrium Bifurcations

359

    fa 1 , a2 , . . . , a2n1 þ1 g ¼ sortfc1 ; c2 . . . ; c2n1 ; ag, as asþ1 before bifurcation ð1Þ

ð1Þ

ðn Þ

ðn Þ

2 2 þ þ faþ 1 , a2 , . . . , a2n3 þ1 g ¼ sortfc1 ; . . . ; c2n1 ; a; b1 ; b2 ; . . . ; b1 ; b2 g,

þ aþ s asþ1 , n3 ¼ n1 þ n2 after bifurcation

ð7:235Þ or ð1Þ

ð1Þ

ðn Þ

ðn Þ

2 2   fa 1 , a2 , . . . , a2n3 þ1 g ¼ sortfc1 ; c2 . . . ; c2n1 ; a; b1 ; b2 ; . . . ; b1 ; b2 g,  a s asþ1 , n3 ¼ n1 þ n2 before bifurcation;

þ þ þ þ faþ 1 , a2 , . . . , a2n1 þ1 g ¼ sortfc1 ; . . . ; c2n1 ; ag, as asþ1 after bifurcation;

ð7:236Þ and  9 Bj1 ¼ Bj2 ¼ . . . ¼ Bjs jk1 2 f1, 2, . . . , ng; jk1 6¼ jk2 > > > > > = ðk1 , k2 2 f1, 2, . . . , sg; k1 6¼ k2 Þ Δj ¼ 0 ð j 2 U  f1, 2, . . . , n2 gÞ 1 ci 6¼  Bj ði ¼ 1, 2, . . . , 2n1 , j ¼ 1, 2, . . . , n2 Þ 2

> > > > > ;

at bifurcation:

ð7:237Þ

  th  th th   Consider a just before bifurcation of ððβ XX2 : . . . : ðβ 1 Þ XX1 : β 2 s1 Þ XXs1 Þ     for higher order equilibriums. For β i ¼ 2li  1, XXi 2 fSO, SIg and for αi ¼ 2li ,  XXi 2 fUS, LSg (i ¼ 1, 2, . . ., s1). The detailed structures are as follows: 9 th ðiÞ ðr ðsiÞ Þ order XXðsiÞ , x ¼ aki , > > > > > > > ⋮ > > = βi ¼Pki rðj iÞ   th j¼1 th ðiÞ ðiÞ  ðiÞ order XXðiÞ :







! β ðr j Þ order XXj , x ¼ aj i switching bifurcation > > > > > ⋮ > > > > ðiÞ th ðiÞ  ðiÞ ; ðr 1 Þ order XX1 , x ¼ aj ð7:238Þ   th  th th   The bifurcation set of ððβ XX2 : . . . : ðβ 1 Þ XX1 : β 2 s1 Þ XXs1 Þ at the same parameter point is called a left-parallel-straw-bundle switching bifurcation. th þ th þ th þ þ þ Consider a just after bifurcation of ððβþ 1 Þ XX1 , ðβ2 Þ XX2 : . . . : ðβs2 Þ XXs2 Þ þ þ þ þ for simple sources and sinks. For βi ¼ 2li  1, XXi 2 fSO, SIg and for βi ¼ 2lþ i , XXþ 2 US, LS The four detailed structures are as follows: f g i

7 (2m+1)th-Degree Polynomial Systems

360

8 th ðiÞþ > ðr ðsii Þþ Þ order XXðsii Þþ , x ¼ aki , > > > > >⋮ > Psi ðiÞþ > >  < βi ¼ r  þ th th j¼1 j ðiÞþ βi order XX







! ðr ðj iÞþ Þ order XXðj iÞþ , x ¼ aðj iÞþ switching bifurcation > > > >⋮ > > > > > : ðiÞþ th ðiÞþ ðiÞþ ðr 1 Þ order XX1 , x ¼ aj : ð7:239Þ  þ th þ th th þ þ The bifurcation set of ððβþ XX2 : . . . : ðβþ 1 Þ XX1 : β 2 s2 Þ XXs2 Þ at the same parameter point is called a right-parallel-straw-bundle switching bifurcation. (i) For the just before and after bifurcation structure, if there exists a relation of   th  th þ þ th   þ βi XXi ¼ ðβþ j Þ XXj ¼ ðβ j Þ XX, for x ¼ ai ¼ aj ði, j 2 f1, 2, . . . , k gÞ, XX 2 fUS, LS, SO, SIg

ð7:240Þ

then the bifurcation is a ðβj Þth switching bifurcation for higher order equilibriums. (ii) Just for the just before bifurcation structure, if there exists a relation of 

2l i

th

th   XX i ¼ ð2lÞ XX, for x ¼ ai ¼ ai

ði 2 f1, 2, . . . , s1 g, XX 2 fUS, LSgÞ

ð7:241Þ

then the bifurcation is a (2l)thXX left-appearing (or right-vanishing) bifurcation for higher order equilibriums. (iii) Just for the just after bifurcation structure, if there exists a relation of 

2lþ i

th

th  þ XXþ i ¼ ð2lÞ XX, for x ¼ ai ¼ ai

ði 2 f1, 2, . . . , s1 gÞ, XX 2 fUS, LSg

ð7:242Þ

then the bifurcation is a (2l)thXX right-appearing (or left-vanishing) bifurcation for higher order equilibriums. (iv) For the just before and after bifurcation structure, if there exists a relation of 

β i

th

þ þ   þ XX i 6¼ ðβj Þ XXj for x ¼ ai ¼ aj th

þ XX i , XXj 2 fUS, LS, SO, SIg

ð7:243Þ

ði 2 f1, 2, . . . , s1 g, j 2 f1, 2, . . . , s2 gÞ, then two flower-bundle switching bifurcations of higher order equilibriums are as follows.

7.3 Higher Order Equilibrium Bifurcations

361

þ (iv1) For βj ¼ βi + 2l, the bifurcation is called a ðβþ j Þ XXj right flower-bundle th

th þ  þ switching bifurcation for the ðβ i Þ XXi to ðβj Þ XXj switching of higher order equilibriums with the appearance (or birth) of (2l)thXX right-appearing (or left-vanishing) bifurcation. th  (iv2) For βj ¼ βi  2l, the bifurcation is called a ðβ i Þ XXi left flower-bundle th

th þ  þ switching bifurcation for the ðβ i Þ XXi to ðβj Þ XXj switching of higher order equilibriums with the vanishing ( or death) of (2l)thXX leftappearing (or right-vanishing) bifurcation. th

A general parallel switching bifurcation is   th  switching th th   ððβ XX2 : . . . : ðβ

! 1 Þ XX1 : β2 s1 Þ XXs1 Þ



bifucation  þ th þ th th þ þ ððβþ XX2 : . . . : ðβþ 1 Þ XX1 : β 2 s2 Þ XXs2 Þ:

ð7:244Þ

Such a general, parallel switching bifurcation consists of the left and right parallel-bundle switching bifurcations for higher order equilibriums. If the left and right parallel-bundle switching bifurcations are same in a parallel flower-bundle switching bifurcation, i.e., th þ th  þ ðβ i Þ XXi ¼ ðβ i Þ XXi ¼ β XX, th

þ for x ¼ a i ¼ ai ði ¼ 1; 2; . . . ; sÞ

ð7:245Þ

then the parallel flower-bundle switching bifurcation becomes a parallel strawbundle switching bifurcation of ((α1)thXX : (β2)thXX : . . . : (βs)thXX). If the left and right parallel-bundle switching bifurcations are different in a parallel flower-bundle switching bifurcation, i.e., th  th þ þ  þ ðα i Þ XXi ¼ ð2li Þ XX, ðα j Þ XX j ¼ ð2l j Þ YY, th

th

þ for x ¼ a i 6¼ a j ði ¼ 1; 2; . . . ; s1 ; j ¼ 1; 2; . . . ; s2 Þ,

ð7:246Þ

XX 2 fUS; LSg, YY 2 fUS; LSg then the parallel flower-bundle switching bifurcation for higher order equilibriums becomes a combination of two independent left and right parallel appearing bifurcations:  th    th  th  (i) ð 2l XX1 : 2l2 XX2 : . . . : ð2l 1 s1 Þ XXs1 Þ-left parallel sprinkler-sprayingappearing (or right-vanishing) bifurcation and  th þ  þ th þ th þ (ii) ð 2lþ XX1 : 2l2 XX2 : . . . : ð2lþ 1 s2 Þ XXs2 Þ-right parallel sprinkler-sprayingappearing (or left-vanishing) bifurcation. The parallel switching and appearing bifurcations for higher order equilibriums are presented in Fig. 7.14a–d. The waterfall appearing bifurcations and the flowerbundle switching bifurcations for higher order equilibriums are presented.

7 (2m+1)th-Degree Polynomial Systems

362

a

c

b

d

Fig. 7.14 (r1th XX : r2th XX : . . . : rnth XX) parallel bifurcation (a0 > 0): (a) without switching, and (b) with switching. The (r1th XX : r2th XX : . . . : rnth XX) parallel bifurcation (a0 < 0): (c) without switching, and (d) with switching. LS: lower saddle, US: upper saddle, SI sink, SO: source. Stable and unstable equilibriums are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

Reference

363

Reference Luo, A.C.J., 2019, The stability and bifurcation of the (2m + 1)th-degree polynomial systems, Journal of Vibration Testing and System Dynamics, in press.

Chapter 8

Infinite-Equilibrium Systems

In this chapter, dynamical systems with infinite equilibriums are discussed through the local analysis. A method for equilibriums in nonlinear dynamical systems is developed. The generalized normal forms of nonlinear dynamical systems at equilibriums are presented for a better understanding of singularity in nonlinear dynamical systems. The dynamics of infinite-equilibrium dynamical systems is discussed for the complexity and singularity of nonlinear dynamical systems. A few examples are presented for complexity and singularity in infinite-equilibrium systems.

8.1

Equilibrium Computations

In this section, a method for equilibriums in nonlinear dynamical systems is presented, which is the extension of the Newton-Raphson method. Using the local analysis, the existence of equilibriums is discussed as in Luo (2019). Definition 8.1 Consider an n-dimensional, dynamical system x_ ¼ fðx, pÞ:

ð8:1Þ

For a given point x0 with parameter p0, there is a neighborhood U(x0). Suppose for parameter p0, the dynamical system in Eq. (8.1) has an equilibrium x. The vector field in U(x0) is expanded as fðx , p0 Þ  fðx0 , p0 Þ þ Dfðx0 , p0 Þðx  x0 Þ ¼ 0:

© Springer Nature Switzerland AG 2019 A. C. J. Luo, Bifurcation and Stability in Nonlinear Dynamical Systems, Nonlinear Systems and Complexity 28, https://doi.org/10.1007/978-3-030-22910-8_8

ð8:2Þ

365

366

8 Infinite-Equilibrium Systems

For the Jacobian matrix of the vector field at the given point (x0, p0), the corresponding distinct eigenvalues are given by j Dfðx0 , p0 Þ  λI j¼ 0,

ð8:3Þ

and the eigenvectors vk (k ¼ 1, 2, . . . , n) for simple eigenvalues are computed by ðDfðx0 , p0 Þ  λk IÞvk ¼ 0:

ð8:4Þ

The covariant matrix for the Jacobian matrix Df(x0, p0) is Q ¼ ðv1 , v2 , . . . , vn Þ ¼ ðaij Þnn ,

ð8:5Þ

and the contravariant matrix of the Jacobian matrix Df(x0, p0) is P ¼ ðv1 , v2 , . . . , vn Þ ¼ ðaij Þnn ¼ Q1 :

ð8:6Þ

Suppose a new transform is x ¼ ck vk , x0 ¼ ck0 vk ,

ð8:7Þ

and the covariant component is ck ¼ ðvk ÞT  x ¼ ðvk ÞT  cj vj ¼ cj δkj :

ð8:8Þ

Thus, the component of the vector field on the covariant direction of vk is expanded by f k ¼ðvk ÞT  fðx, pÞ ¼ðvk ÞT  fðx0 , p0 Þ þ Gkðj1 Þ zj1 þ þ

mk X 1

q! q¼2

ðqÞ

Gkðj

1 j2 ...jq Þ

ðzj1 zj2 . . . zjq Þ

1 ðm þ1Þ ðθ zj1 Þðθ2 zj2 Þ . . . ðθmk þ1 zjm k þ 1 Þ G k ðmk þ 1Þ! kðj1 j2 ...jmk þ1 Þ 1

ð8:9Þ

8.1 Equilibrium Computations

367

where T

z j ¼ ðv j Þ  y ¼ c j  c0j ðj ¼ 1; 2; . . . ; nÞ, T

Gkðj1 Þ ¼ ðvk Þ  ∂cj1 fðx; pÞjðx0 ;p0 Þ T

¼ ðvk Þ  ∂x fðx; pÞjðx0 ;p0 Þ vj1 , ðrÞ

Gkðj

T

1 j2 jr Þ

0

k T

¼ ðv Þ 

0Þ

ðrÞ ∂x fðx; pÞjðx0 ;p0 Þ vj1 vj2

ðrÞ

¼ Gkðj

ð8:10Þ

ðrÞ

¼ ðvk Þ  ∂cj1 cj2 ...cjr fðx; pÞjðx ;p

1 j2 ...jr Þ

. . . vj r

ðx0 ; p0 Þ:

Definition 8.2 Consider an n-dimensional, dynamical system in Eq. (8.1). For a given point x0 with parameter p0, there is a neighborhood U(x0). Suppose for parameter p0, the dynamical system in Eq. (8.1) has an equilibrium x. The Jacobian matrix of the vector field f(x, p) in U(x0) has distinct eigenvalues λk (k ¼ 1, 2, . . . , n) and the corresponding covariant and contravariant eigenvectors are vk and vk (k ¼ 1, 2, . . . , n), respectively. The covariant component of the vector field on the covariant direction of vk is f k ¼ ðvk ÞT  fðx, pÞ:

ð8:11Þ

For a specific p ¼ p0, if λk 6¼ 0 ðk ¼ 1, 2, . . . , nÞ ð2Þ

j Gkðj

1 j2 Þ

ðx0 , p0 Þzj1 zj2 j 0 for i 2 fi1 , . . . , ir1 g f1, 2, . . . , mk g, Δj ¼ B2j  4Cj < 0 for j 2 fj1 , . . . , jr2 g f1, 2, . . . , mk g,

ð8:26Þ

r 1 þ r 2 ¼ mk , then Eq. (8.22) has r1-pairs of solutions in the neighborhood U(x0). Thus, there are 2r1-approximate solutions of equilibriums for a specific p ¼ p0,

370

8 Infinite-Equilibrium Systems

pffiffiffiffiffi 1 kðiÞ z1, 2 ¼  ðBi  Δi Þ for i 2 fi1 , . . . , ir1 g f1, 2, . . . , mk g, 2 ðkÞ

ðkÞ

ðkÞ

kði1 Þ

fz1 , z2 , . . . , z2r1 g ¼ sortfz1

kði1 Þ

, z2

kðir1 Þ

, . . . , z1

kðir2 Þ

, z2

ð8:27Þ

g

ðkÞ

with zðkÞ s  zsþ1 , and ðiÞ

ði1Þ

ði1Þ ði1Þ zj , ði

xj ¼ xj

þ Qj

ð0Þ

 xj0 , zj

with xj ði1Þ

Qj

ð0Þ

ði1Þ

¼ ðvj1

ði1Þ

ðDfðxj

¼ 1, 2, . . . ;j ¼ 1, 2, . . . , 2r 1 Þ,

 zj0 , Qð0Þ ¼ Q for i ¼ 1,

ði1Þ

, vj2

ði1Þ

, p0 Þ  λjk

ði1Þ

, . . . , vjn ði1Þ

IÞvjk

ð8:28Þ

Þ for

¼0

ðk ¼ 1, 2, . . . , nÞ: If ðiÞ

kfðxj , p0 Þk  ε,

ð8:29Þ

ðiÞ

then xj  xj ( j ¼ 1, 2, . . . , 2r1) is called a set of approximate solutions of equilibrium for f(x, p0)  0 in the sense of ε. (c) If Δi ¼ B2i  4Ci ¼ 0 for i 2 fi1 , . . . , ir1 g f1, 2, . . . , mk g, Δj ¼ B2j  4Cj > 0 for j 2 fj1 , . . . , jr2 g f1, 2, . . . , mk g,

ð8:30Þ

Δl ¼ B2l  4Cl < 0 for l 2 fl1 , . . . , lr3 g f1, 2, . . . , mk g, r 1 þ r 2 þ r 3 ¼ mk ,

then Eq. (8.22) has r1-pairs of repeated solutions and r2-pairs of simple solutions in the neighborhood U(x0). Thus, there are (r1 + 2r2)-approximate solutions of equilibriums for a specific p ¼ p0, 1 kðiÞ z1, 2 ¼  Bi , Δi ¼ 0 for i 2 fi1 , . . . , ir1 g f1, 2, . . . , mk g 2 pffiffiffiffiffi 1 kðjÞ z1, 2 ¼  ðBj  Δj Þ for j 2 fj1 , . . . , jr2 g f1, 2, . . . , mk g, 2 ðkÞ

ðkÞ

ðkÞ

kði1 Þ

fz1 , z2 , . . . , z2ðr1 þr2 Þ g ¼ sortfz1

kði1 Þ

, z2

kðir1 Þ

, . . . , z2

kðj1 Þ

;z1

kðj1 Þ

, z2

kðjr2 Þ

, . . . , z2

g

ðkÞ

with zðkÞ s  zsþ1 , ð8:31Þ

8.1 Equilibrium Computations

371

and ðiÞ

ði1Þ

xj ¼ xj

ð0Þ

with xj ði1Þ

Qj

ð0Þ

 xj0 , zj ði1Þ

¼ ðvj1

ði1Þ

ðDfðxj

ði1Þ

þ Qði1Þ zj

, ði ¼ 1, 2, . . . ;j ¼ 1, 2, . . . , r 1 þ 2r 2 Þ

 zj0 , Qð0Þ ¼ Q for i ¼ 1,

ði1Þ

, vj2

ði1Þ

, p0 Þ  λjk

ði1Þ

, . . . , vjn ði1Þ

IÞvjk

Þ for

ð8:32Þ

¼0

ðk ¼ 1, 2, . . . , nÞ: If ðiÞ

kfðxj , p0 Þk  ε,

ð8:33Þ

ðiÞ

then xj  xj ( j ¼ 1, 2, . . . , r1 + 2r2) is called a set of approximate solutions of equilibrium for f(x, p0)  0 in the sense of ε. (ii2) Equation (8.22) is equivalent to f k  A0 ðzk Þ2mk þ1 þ A1 ðzk Þ2mk þ    þ A2mk zk þ A2mk þ1 ¼ 0 Ai ¼ Ai ðz1 , z2 , . . . , zk1 , zk , . . . , zn Þ ði ¼ 0, 1, 2, . . . , 2mk þ 1Þ,

ð8:34Þ

and the standard form for simple solutions is f k  a0 ðzk  aÞ

mk Y

½ðzk Þ2 þ Bi zk þ C i ¼ 0:

ð8:35Þ

i¼1

(a) If Δi ¼ B2i  4C i < 0 for i ¼ 1, 2, . . . , mk ,

ð8:36Þ

then Eq. (8.22) has a solution. Thus, the equilibrium has one solution of equilibrium in the neighborhood U(x0). (b) If Δi ¼ B2i  4Ci > 0 for i 2 fi1 , . . . , ir1 g f1, 2, . . . , mk g, Δj ¼ B2j  4Cj < 0 for j 2 fj1 , . . . , jr2 g f1, 2, . . . , mk g,

ð8:37Þ

r 1 þ r 2 ¼ mk , then Eq. (8.22) has (2r1 + 1)-simple solutions in the neighborhood U(x0). Thus, there exist (2r1 + 1)-approximate solutions of equilibriums for a specific p ¼ p0 as

372

8 Infinite-Equilibrium Systems

pffiffiffiffiffi 1 kðiÞ z1, 2   ðBi  Δi Þ for i 2 fi1 , . . . , ir1 g f1, 2, . . . , mk g, 2 ðkÞ

ðkÞ

ðkÞ

kði1 Þ

fz1 , z2 , . . . , z2r1 þ1 g ¼ sortfa, z1

kði1 Þ

, z2

kðir1 Þ

, . . . , z1

kðir2 Þ

, z2

g

ð8:38Þ

ðkÞ

with zðkÞ s  zsþ1 , and ðiÞ

ði1Þ

xj ¼ xj

ði1Þ

þ Qði1Þ zj

,

ði ¼ 1, 2, . . . ; j ¼ 1, 2, . . . , 2r 1 þ 1Þ with

ð0Þ xj



ð0Þ xj0 , zj

ð8:39Þ

 zj0 for i ¼ 1:

If ðiÞ

kfðxj , p0 Þk  ε,

ð8:40Þ

ðiÞ

then xj  xj ( j ¼ 1, 2, . . . , 2r1 + 1) is called a set of approximate solutions of equilibrium for f(x, p0)  0 in the sense of ε. (c) If Δi ¼ B2i  4Ci ¼ 0 for i 2 fi1 , . . . , ir1 g f1, 2, . . . , mk g, Δj ¼ B2j  4Cj > 0 for j 2 fj1 , . . . , jr2 g f1, 2, . . . , mk g, Δl ¼ B2l  4Cl < 0 for l 2 fj1 , . . . , jr3 g f1, 2, . . . , mk g,

ð8:41Þ

r 1 þ r 2 þ r 3 ¼ mk , then Eq. (8.22) has r1-pairs of repeated solutions and r2-pairs of simple solutions in the neighborhood U(x0). Thus, there are (r1 + 2r2 + 1)approximate solutions of equilibriums for a specific p ¼ p0, 1 kðiÞ z1, 2   Bi , Δi ¼ 0 for i 2 fi1 , . . . , ir1 g f1, 2, . . . , mk g 2 pffiffiffiffiffi 1 kðjÞ z1, 2   ðBj  Δj Þ for j 2 fj1 , . . . , jr2 g f1, 2, . . . , mk g, 2 ðkÞ

ðkÞ

ð8:42Þ

ðkÞ

fz1 , z2 , . . . , z2ðr1 þr2 Þþ1 g kði1 Þ

¼ sortfa, z1

ðkÞ

with zðkÞ s  zsþ1 and

kði1 Þ

, z2

kðir1 Þ

, . . . , z2

kðj1 Þ

;z1

kðj1 Þ

, z2

kðjr2 Þ

, . . . , z2

g

8.2 Normal Forms

373 ðiÞ

ði1Þ

xj ¼ xj

ði1Þ

þ Qði1Þ zj

,

ði ¼ 1, 2, . . . ; j ¼ 1, 2, . . . , r 1 þ 2r 2 þ 1Þ ð0Þ

ð0Þ

with xj

 xj0 , zj

ði1Þ

ði1Þ

Qj

¼ ðvj1

ði1Þ

for ðDfðxj

 zj0 , Qð0Þ ¼ Q for i ¼ 1

ði1Þ

, vj2

ði1Þ

, . . . , vjn ði1Þ

, p0 Þ  λjk

Þ

ði1Þ

IÞvjk

ð8:43Þ

¼ 0:

ðk ¼ 1, 2, . . . , nÞ: If ðiÞ

kfðxj , p0 Þk  ε,

ð8:44Þ

ðiÞ

then xj  xj ( j ¼ 1, 2, . . . , r1 + 2r2 + 1) is called a set of approximate solutions of equilibrium for f(x, 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 equilibriums of nonlinear dynamical 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 NewtonRaphson method. The singularity of nonlinear systems in eigenvector space is employed to determine solution existence and multiplicity.

8.2

Normal Forms

In this section, a general normal form of dynamical systems is presented through the eigenvector space of equilibriums. As in Luo (2019), the eigenvector space is developed, and the corresponding normal forms are developed by the Taylor series expansion. Nonlinear dynamical systems experiencing the simple eigenvalues are discussed first, and the eigenvector space is developed through the following definition. Definition 8.4 Consider an n-dimensional, dynamical system x_ ¼ fðx, pÞ:

ð8:45Þ

There is an equilibrium point x with parameter p0, fðx , p0 Þ ¼ 0:

ð8:46Þ

374

8 Infinite-Equilibrium Systems

In a neighborhood U(x) at parameter p0, the vector field in Eq. (8.45) is linearized by fðx, p0 Þ ¼ Dfðx , p0 Þðx  x Þ:

ð8:47Þ

For the Jacobian matrix, the corresponding eigenvalues are determined by j Dfðx , p0 Þ  λI j¼ 0:

ð8:48Þ

Thus, the foregoing equation gives f ðλÞ ¼ ð1Þn λn þ a1 λn1 þ    þ an1 λ þ an ¼ 0:

ð8:49Þ

(i) For n ¼ 2m, consider simple eigenvalues of Eq. (8.48) as f ðλÞ ¼ λ2m þ a1 λ2n1 þ    þ a2m1 λ þ a2m ¼ 0 equivalent to f ðλÞ ¼

m Y

ð8:50Þ

ðλ þ Bi λ þ C i Þ ¼ 0: 2

i¼1

(i1) If Δi1 ¼ B2i1  4Ci1 > 0 ð1Þ

ð1Þ

ði1 2 fs1 , s2 , . . . , sð1Þ n1 g f1, 2, . . . , 2mgÞ, then

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 ði Þ B2i1  4C i1 , λ1,12 ¼  Bi1  2 2 ði Þ ði Þ ðDfðx , p0 Þ  λj 1 IÞvj 1 ¼ 0 ðj ¼ 1, 2Þ, ð1Þ

ð8:51Þ

ð8:52Þ

ð1Þ

ði1 2 fs1 , s2 , . . . , sð1Þ n1 g f1, 2, . . . , mgÞ: (i2) If Δi2 ¼ B2i2  4Ci2 < 0 ð2Þ

ð2Þ

ði2 2 fs1 , s2 , . . . , sð2Þ n2 g f1, 2, . . . , mgÞ, then

ð8:53Þ

8.2 Normal Forms

375

1 1 ði Þ λ1,22 ¼  Bi2  2 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4Ci2  B2i2 i  αði2 Þ  βði2 Þ i,

ði Þ

ði Þ

ði Þ

Dfðx , p0 Þu1 2 ¼ αði2 Þ u1 2 þ βði2 Þ v1 2 , ði Þ

ði Þ

ði Þ

ð8:54Þ

Dfðx , p0 Þv1 2 ¼ βði2 Þ u1 2 þ αði2 Þ v1 2 , pffiffiffiffiffiffiffi ð2Þ ð2Þ i ¼ 1, ði2 2 fs1 , s2 , . . . , sð2Þ n2 g f1, 2, . . . , mgÞ: (i3) If Δi3 ¼ B2i3  4C i3 ¼ 0, Bi3 6¼ Bj3 ð3Þ

ð3Þ

ði3 , j3 2 fs1 , s2 , . . . , sð3Þ n3 g f1, 2, . . . , mgÞ,

ð8:55Þ

then 1 ði Þ λ1,32 ¼  Bi3 ¼ λði3 Þ , 2 ði Þ

ði Þ

ði Þ

ði Þ

Dfðx , p0 Þv1 3 ¼ λði3 Þ v1 3

ði Þ

Dfðx , p0 Þv2 3 ¼ λði3 Þ v2 3 þ δði3 Þ v1 3 , ði Þ

ðDfðx , p0 Þ  λði3 Þ IÞvj 3 ¼ 0 for j ¼ 1, 2 and δði3 Þ ¼ 0;

ð8:56Þ

2 ði Þ

ðDfðx , p0 Þ  λði3 Þ IÞ vj 3 ¼ 0 for j 2 f1, 2g and δði3 Þ ¼ 1 ð3Þ

ð3Þ

ði3 2 fs1 , s2 , . . . , sð3Þ n3 g f1, 2, . . . , mgÞ: (ii) For n ¼ 2m + 1, consider simple eigenvalues of Eq. (8.48) as f ðλÞ ¼ λ2mþ1 þ a1 λ2m þ    þ a2m λ þ a2mþ1 ¼ 0 equivalent to f ðλÞ ¼ ðλ  λ1 Þ

m Y

ð8:57Þ ðλ þ Bi λ þ Ci Þ ¼ 0: 2

i¼1

(ii1) For λ ¼ λ1 ðDfðx , p0 Þ  λ1 IÞv1 ¼ 0:

ð8:58Þ

(ii2) If Δi1 ¼ B2i1  4Ci1 > 0 ð1Þ

ð1Þ

ði1 2 fs1 , s2 , . . . , sð1Þ n1 g f1, 2, . . . , mgÞ,

then

ð8:59Þ

376

8 Infinite-Equilibrium Systems

1 1 ði Þ λ1,12 ¼  Bi1  2 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B2i1  4C i1 ,

ði Þ

ði Þ

ðDfðx , p0 Þ  λj 1 IÞvj 1 ¼ 0 ðj ¼ 1, 2Þ, ð1Þ

ð8:60Þ

ð1Þ

ði1 2 fs1 , s2 , . . . , sð1Þ n1 g f1, 2, . . . , mgÞ: (ii3) If Δi2 ¼ B2i2  4Ci2 < 0 ð2Þ

ð2Þ

ði2 2 fs1 , s2 , . . . , sð2Þ n2 g f1, 2, . . . , mgÞ, then 1 1 ði Þ λ1,22 ¼  Bi2  2 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4Ci2  B2i2 i  αði2 Þ  βði2 Þ i,

ði Þ

ði Þ

ði Þ

Dfðx , p0 Þu1 2 ¼ αði2 Þ u1 2 þ βði2 Þ v1 2 , 

ði Þ , p0 Þv1 2

ð8:61Þ

ði Þ βði2 Þ u1 2

ði Þ αði2 Þ v1 2 ,

ð8:62Þ

¼ þ Dfðx pffiffiffiffiffiffiffi ð2Þ ð2Þ i ¼ 1, ði2 2 fs1 , s2 , . . . , sð2Þ n2 g f1, 2, . . . , 2mgÞ: (ii4) If Δi3 ¼ B2i3  4C i3 ¼ 0, Bi3 6¼ Bj3 ð3Þ

ð3Þ

ði3 , j3 2 fs1 , s2 , . . . , sð3Þ n3 g f1, 2, . . . , mgÞ,

ð8:63Þ

then 1 ði Þ λ1,32 ¼  Bi3 ¼ λði3 Þ , 2 ði Þ

ði Þ

ði Þ

ði Þ

Dfðx , p0 Þv1 3 ¼ λði3 Þ v1 3

ði Þ

Dfðx , p0 Þv2 3 ¼ λði3 Þ v2 3 þ δði3 Þ v1 3 , ði Þ

ðDfðx , p0 Þ  λði3 Þ IÞvj 3 ¼ 0

ð8:64Þ

for j ¼ 1, 2 and δði3 Þ ¼ 0; ði Þ

ðDfðx , p0 Þ  λði3 Þ IÞ2 vj 3 ¼ 0 for j 2 f1, 2g and δði3 Þ ¼ 1 ð3Þ

ð3Þ

ði3 2 fs1 , s2 , . . . , sð3Þ n3 g f1, 2, . . . , mgÞ: Nonlinear dynamical systems experiencing the repeated eigenvalues are discussed first. The eigenvector space is developed through the following definition.

8.2 Normal Forms

377

Definition 8.5 Consider an n-dimensional, dynamical system of x_ ¼ fðx, pÞ in Eq. (8.45). An equilibrium point x with parameter p0 is given by f(x, p0) ¼ 0 in Eq. (8.46). In a neighborhood U(x) at parameter p0, the vector field in Eq. (8.45) is linearized, i.e., f(x, p0) ¼ Df(x, p0)(x  x) in Eq. (8.47). For the Jacobian matrix of Df(x, p0), the corresponding eigenvalues are determined by Eq. (8.48). Thus, the eigenvalue equation is given in Eq. (8.49). Consider a general case for simple and repeated eigenvalues of f ðλÞ ¼ð1Þn λn þ a1 λn1 þ    þ an1 λ þ an ¼ 0 equivalent to f ðλÞ ¼ð1Þn

li Y i1 ¼1



l3 Y i3 ¼1

ð1Þ

ðλ  λi1 Þ

i2 ¼1

l4 Y

ð3Þ

ðλ  λi3 Þqi3

l2 Y

i4 ¼1

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ðλ  αi2  βi2 iÞðλ  αi2 þ βi2 iÞ ð4Þ

ð4Þ

ð4Þ

ð8:65Þ

ð4Þ

ððλ  αi4  βi4 iÞðλ  αi4 þ βi4 iÞÞqi4

¼ 0: (i) For simple real eigenvalues, if ð1Þ

λ ¼ λi 1

ð1Þ

ð1Þ

ð1Þ

ði1 2 fs1 , s2 , . . . , sl1 g f1, 2, . . . , ngÞ,

ð8:66Þ

then ð1Þ

ðDfðx , p0 Þ  λi1 IÞvði1 Þ ¼ 0 ð1Þ

ð1Þ

ð1Þ

ði1 2 fs1 , s2 , . . . , sl1 g f1, 2, . . . , ngÞ:

ð8:67Þ

(ii) For complex eigenvalues with simple pairs, if ð2Þ

ð2Þ

ð2Þ

ð2Þ

λ1 ¼ αi2 þ βi2 i, λ2 ¼ αi2  βi2 i, pffiffiffiffiffiffiffi ð2Þ ð2Þ ð2Þ i ¼ 1, ði2 2 fs1 , s2 , . . . , sl2 g f1, 2, . . . , ngÞ, then

ð8:68Þ

378

8 Infinite-Equilibrium Systems ði Þ

ð2Þ ði Þ

ð2Þ ði Þ

Dfðx , p0 Þv1 2 ¼ αi2 v1 2 þ βi2 v2 2 , ði Þ

ð2Þ ði Þ

ð2Þ ði Þ

Dfðx , p0 Þv1 2 ¼ βi2 v1 2 þ αi2 v2 2 , ð2Þ

ð2Þ

ð8:69Þ

ð2Þ

ði2 2 fs1 , s2 , . . . , sl2 g f1, 2, . . . , ngÞ: (iii) For repeated real eigenvalues, if ð3Þ

λ ¼ λi3 with qi3 ‐repeated ð3Þ

ð3Þ

ð3Þ

ði Þ

ð3Þ ði Þ

ði Þ

ð3Þ ði Þ

ð8:70Þ

ði3 2 fs1 , s2 , . . . , sl3 g f1, 2, . . . , ngÞ, then

Dfðx , p0 Þv1 3 ¼ λi3 v1 3

ði Þ ði Þ

3 Dfðx , p0 Þvj 3 ¼ λi3 vj 3 þ δj 3 vj1 ,

ð8:71Þ

ði Þ

ðj ¼ 2, 3,   , N i3 Þ, δj 3 2 f0, 1g ð3Þ

ð3Þ

ð3Þ

ði3 2 fs1 , s2 , . . . , sl3 g f1, 2, . . . , ngÞ: (iv) For complex eigenvalues with repeated pairs, if ð4Þ

ð4Þ

ð4Þ

ð4Þ

λ1 ¼ αi4 þ βi4 i, λ2 ¼ αi4  βi4 i, with qi4 ‐repeated: pffiffiffiffiffiffiffi ð4Þ ð4Þ ð4Þ i ¼ 1, ði4 2 fs1 , s2 , . . . , sl4 g f1, 2, . . . , ngÞ,

ð8:72Þ

then ði Þ

ð4Þ ði Þ

ð4Þ ði Þ

Dfðx , p0 Þu1 4 ¼ αi4 u1 4 þ βi4 v1 4 , ði Þ

ð4Þ ði Þ

ð4Þ ði Þ

Dfðx , p0 Þv1 4 ¼ βi4 u1 4 þ αi4 v1 4 , ði Þ

ð4Þ ði Þ

ð4Þ ði Þ

g ði Þ ði Þ

4 Dfðx , p0 Þuj 4 ¼ αi4 uj 4 þ βi4 vj 4 þ δj 4 uj1 ,

ði Þ

ð4Þ ði Þ

ð4Þ ði Þ

ði Þ ði Þ

4 Dfðx , p0 Þvj 4 ¼ βi4 uj 4 þ αi4 vj 4 þ δj 4 vj1 ,

ði Þ

δj 3 2 f0, 1g, ðj ¼ 2, 3, . . . , qi4 Þ ð4Þ

ð4Þ

ð4Þ

ði4 2 fs1 , s2 , . . . , sl4 g f1, 2, . . . , ngÞ: The covariant matrix for the Jacobian matrix Df(x0, p0) is

g

ð8:73Þ

8.2 Normal Forms

379

Q ¼ ðv1 , v2 , . . . , vn Þ ¼ ðaij Þnn , ð1Þ

ð1Þ

ðsl Þ

ðs Þ

fv1 , v2 , . . . , vn g ¼ inorderfv1 1 , . . . , vl1 1 ; ð2Þ

ð2Þ

ð2Þ

ðs Þ

ð2Þ

ðsl Þ

ðs Þ

ðsl Þ

u1 1 , v1 1 , . . . , ul2 2 , vl2 2 ; ði Þ

ð3Þ

ð8:74Þ

ð3Þ

v1 3 , . . . , vðiqi3 Þ ;ði3 ¼ s1 , . . . , sl3 Þ; 3 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} l3 ‐group

ði Þ

ði Þ

ð4Þ

ð4Þ

u1 4 , v1 4 , . . . , uðiqi4 Þ , vðiqi4 Þ ;ði4 ¼ s1 , . . . , sl3 Þg, 4 4 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} l4 ‐group

and the contravariant matrix of the Jacobian matrix Df(x0, p0) is P ¼ ðv1 , v2 , . . . , vn Þ ¼ ðaij Þnn ¼ Q1 :

ð8:75Þ

Thus PDfðx , p0 ÞQ ¼ Q1 Dfðx , p0 ÞQ ¼ diag ðAð1Þ , Að2Þ , Að3Þ , Að4Þ Þ,

ð8:76Þ

where ð1Þ

ð1Þ

ð1Þ

Að1Þ ¼ diag ðλ1 , λ2 , . . . , λl1 Þ; ð2Þ

ð2Þ

ð8:77Þ

ð2Þ

Að2Þ ¼ diag ðA1 , A2 , . . . , Al2 Þ, Að2Þ r

¼

½

ð2Þ

αi2

ð2Þ

βi2

ð2Þ

βi2

ð2Þ

αi2

ð3Þ



ði2 ¼ sð2Þ r ;r ¼ 1, 2, . . . , l2 Þ; ð3Þ

ð3Þ

Að3Þ ¼ diag ðB1 , B2 , . . . , Bl3 Þ,

Bð3Þ r ¼

½

λi 3

ð3Þ

δ1 3

ði Þ



0

0

0

λi 3

ð3Þ



0

0

⋮

⋮

⋱

⋮

⋮

0

0

   λi3

ð3Þ

δqi3 1

0

0



0

ði Þ 3

ð3Þ

λi 3



ð8:78Þ

ð8:79Þ

qi3 qi3

ði3 ¼ sð3Þ r ;r ¼ 1, 2, . . . , l3 Þ, δðis 3 Þ 2 f0, 1g ðs ¼ 1, 2, . . . , qi3  1Þ;

380

8 Infinite-Equilibrium Systems ð4Þ

ð4Þ

ð4Þ

Að4Þ ¼ diag ðB1 , B2 , . . . , Bl4 Þ,

Bð4Þ r ¼

½

Ai4

ð4Þ

δ1 4

ði Þ



0

0

0

Ai4

ð4Þ



0

0

⋮

⋮

⋱

⋮

⋮

0

0

   Ai4

0

ð4Þ

0



ði Þ

δqi4 1 4

ð4Þ Ai4

0



ð8:80Þ ð2qi4 2qi4 Þ

ði4 ¼ sð4Þ r , r ¼ 1, 2, . . . , l4 Þ, ð4Þ Ai4

¼

½

ð4Þ

αi4

ð4Þ

βi4

ð4Þ

β i4

ð4Þ

αi4



, δðis 4 Þ 22

¼

½

δðis 4 Þ

0

0

δðis 4 Þ



and 22

δðis 4 Þ 2 f0, 1g ðs ¼ 1, 2, . . . , qi4  1Þ: From the eigenvector space of the linearized Jacobian matrix of equilibriums, the corresponding normal forms are developed through the following definition. Definition 8.6 Consider an n-dimensional, dynamical system of x_ ¼ fðx, pÞ in Eq. (8.45). An equilibrium point x with parameter p0 is given by f(x, p0) ¼ 0 in Eq. (8.46). In a neighborhood U(x) at parameter p0, the vector field in Eq. (8.45) is linearized, i.e., f(x, p0) ¼ Df(x, p0)(x  x) in Eq. (8.47). For the Jacobian matrix of Df(x, p0), the corresponding eigenvalues are determined by Eq. (8.48). Thus, the eigenvalue equation is given in Eq. (8.49). The covariant matrix for the Jacobian matrix Df(x0, p0) is Q ¼ ðv1 ; v2 ; . . . ; vn Þ ¼ ðaij Þnn ,

ð8:81Þ

and the contravariant matrix of the Jacobian matrix Df(x0, p0) is P ¼ ðv1 , v2 , . . . , vn Þ ¼ ðaij Þnn ¼ Q1 :

ð8:82Þ

Suppose a new transform is x ¼ ck vk , x ¼ ck vk ,

ð8:83Þ

and the covariant component is ck ¼ ðvk ÞT  x ¼ ðvk ÞT  cj vj ¼ cj δkj :

ð8:84Þ

8.2 Normal Forms

381

Thus, the component of the vector field on the covariant direction of vk is expanded by f k ¼ðvk ÞT  fðx, pÞ ¼ðvk ÞT  fðx , p0 Þ þ Gkðj1 Þ zj1 mk X 1 ðqÞ Gkðj j ...j Þ ðzj1 zj2 . . . zjq Þ þ 1 2 q q! q¼2 þ

ð8:85Þ

1 ðm þ1Þ ðθ zj1 Þðθ2 zj2 Þ . . . ðθmk þ1 zjmk þ1 Þ, G k ðmk þ 1Þ! kðj1 j2 jmk þ1 Þ 1

where z j ¼ ðv j Þ  y ¼ c j  cj ðj ¼ 1; 2; . . . ; nÞ, T

T

Gkðj1 Þ ¼ ðvk Þ  ∂cj1 fðx; pÞjðx ;p0 Þ T

ðrÞ

Gkðj

1 j2

¼ ðvk Þ  ∂x fðx; pÞjðx ;p0 Þ vj1 ,  ðrÞ k T j Þ ¼ ðv Þ  ∂cj1 cj2 cjr fðx; p jðx ;p r

ð8:86Þ 0Þ

ðrÞ

T

¼ ðvk Þ  ∂x fðx; pÞjðx ;p0 Þ vj1 vj2 . . . vjr ðrÞ

¼ Gkðj

1 j2 ...jr Þ

ðx ; p0 Þ:

(i) For simple real eigenvalues, consider ð1Þ

λ ¼ λi1

ð1Þ

ð1Þ

ð1Þ

ði1 2 fs1 , s2 , . . . , sl1 g f1, 2, . . . , ngÞ

ð8:87Þ

for a specific p ¼ p0 on the covariant direction of vi1 . If ð1Þ

ð1Þ

ð1Þ

ð1Þ

λk 6¼ 0 ðk ¼ i1 2 fs1 , s2 , . . . , sl1 g f1, 2, . . . , ngÞ ðs Þ ðx0 , p0 Þzj1 zj2 1 j2 ...jsi Þ

j Gkðjk

. . . zjsk j

1

ðs þ1Þ 1 j2 ...jsi

>>j Gkðjk

1

þ1

jsk þ1 j1 j2 j ε Þ ðx0 , p0 Þz z . . . z

ðj1 , j2 , . . . jsk þ1 ¼ 1, 2, . . . , nÞ, then the normal form on the covariant direction of vi1 is

ð8:88Þ

382

8 Infinite-Equilibrium Systems

ð1Þ

z_i1  λk zi1 þ ð1Þ

si1 X 1 ðqÞ Gi1 ðj j ...j Þ ðzj1 zj2 . . . zjq Þ 1 2 q q! q¼2

ð1Þ

ð8:89Þ

ð1Þ

ði1 2 fs1 , s2 , . . . , sl1 g f1, 2, . . . , ngÞ: (ii) For complex eigenvalues with simple pairs, consider ði Þ

ð2Þ

ð2Þ

ði Þ

ð2Þ

ð2Þ

λ1 2 ¼ αi2 þ βi2 i, λ2 2 ¼ αi2 þ βi2 i, ð2Þ

ð2Þ

ð8:90Þ

ð2Þ

ði2 2 fs1 , s2 , . . . , sl2 g f1, 2, . . . , ngÞ

ði Þ

ði Þ

for a specific p ¼ p0 on the covariant eigenvector plane of u1 2 and v1 2 . If ðsðkÞ1Þ 1 j2 ...js

j Gkðj

Þ ðx0 , p0 Þz

ðkÞ1

ðs þ1Þ >>j Gkðjk j ...j 1 2 s ðs Þ 1 j2 ...js

j Gkðjk

ðkÞ2

ðkÞ1 þ1

Þ ðx0 , p0 Þz

Þ ðx0 , p0 Þz

ðs þ1Þ >>j Gkðjk j ...j 1 2 s

ðkÞ2 þ1

j

z . . . z sðkÞ1 j

j1 j2

j

z . . . z sðkÞ1 þ1 j¼ ε,

j1 j2 j

z . . . z sðkÞ2 j

j1 j2

Þ ðx0 , p0 Þz

z ...z

j1 j2

ð8:91Þ js

ðkÞ2 þ1

j¼ ε,

ðj1 , j2 , . . . , jq ¼ 1, 2, . . . , nÞ, ð2Þ

ð2Þ

ðk ¼ i2 2 fs1 , s2 , . . . , sð2Þ n2 g f1, 2, . . . , ngÞ, ði Þ

ði Þ

then the normal form on the covariant eigenvector plane of u1 2 and v1 2 is ð2Þ

ð2Þ

ð2Þ

ð2Þ

z_ði2 Þ1  αi2 zði2 Þ1  βi2 zði2 Þ2 þ z_ði2 Þ2  βi2 zði2 Þ1 þ αi2 zði2 Þ2 þ ð2Þ

ð2Þ

X 1 ðqÞ Gði2 Þ ðj j ...j Þ ðzj1 zj2 . . . zjq Þ, q 1 1 2 q! q¼2

sði2 Þ1

X 1 ðqÞ Gði2 Þ ðj j ...j Þ ðzj1 zj2 . . . zjq Þ, q 2 1 2 q! q¼2

sði2 Þ2

ð2Þ

with ði2 2 fs1 , s2 , . . . , sl2 g f1, 2, . . . , ngÞ, ðj1 , j2 , . . . , jq ¼ 1, 2, . . . , nÞ: (iii) For repeated real eigenvalues, consider

ð8:92Þ

8.2 Normal Forms

383 ð3Þ

λ ¼ λi3 ði3 2

with qi3 ‐repeated

ð3Þ ð3Þ fs1 , s2 ,

ð3Þ

. . . , sl3 g f1, 2, . . . , ngÞ

ð8:93Þ

ði Þ

for a specific p ¼ p0 on the covariant space of vj 3 (j ¼ 2, 3, . . . , qi3 ). If ð3Þ

ð3Þ

ð3Þ

ð3Þ

λi3 6¼ 0 ðk ¼ i3 2 fs1 , s2 , . . . , sl3 g f1, 2, . . . , ngÞ ðsðkÞ Þ

j GðkÞ ðjj j

1 j2 ...js

ðkÞj

Þ ðx0 , p0 Þz

ðsðkÞ þ1Þ

>>j GðkÞ ðjj j

1 j2 ...jsðkÞ þ1 j

z ...z

j1 j2

js

ðkÞj

j

j1 j2 Þ ðx0 , p0 Þz z . . . z

js

ð8:94Þ ðkÞj þ1

j ε

ðj1 , j2 , . . . , jsðkÞ þ1 ¼ 1, 2, . . . , nÞ, j

ði Þ

then the normal form on the covariant space of vj 3 (j ¼ 1, 2, 3, . . . , qi3 ) is ð3Þ

z_ðkÞ1  λi3 zðkÞ1 þ ði Þ

sk 1 X 1 ðqÞ GðkÞ ðj j j Þ ðzj1 zj2 . . . zjq Þ q 1 1 2 q! q¼2

g

ð3Þ

z_ðkÞj  δj 3 zðkÞj1 þ λi3 zðkÞj

sðkÞj ðj ¼ 2, 3, . . . , qi3 Þ, X 1 ðqÞ GðkÞ ðj j j Þ ðzj1 zj2 . . . zjq Þ þ q j 1 2 q! q¼2

ði Þ

ð3Þ

ð3Þ

ð8:95Þ

ð3Þ

δj 3 2 f0, 1g ðk ¼ i3 2 fs1 , s2 , . . . , sl3 g f1, 2, . . . , ngÞ: (iv) For complex eigenvalues with repeated pairs, consider ð4Þ

ð4Þ

ð4Þ

ð4Þ

λ1 ¼ αi4 þ βi4 i, λ2 ¼ αi4  βi4 i, with qi4 ‐repeated ð4Þ

ð4Þ

ð4Þ

ði4 2 fs1 , s2 , . . . , sl4 g f1, 2, . . . , ngÞ ði Þ

ði Þ

ð8:96Þ

on the covariant eigenvector space of uj 2 and vj 2 (j ¼ 1, 2, 3, . . . , qi4). If

384

8 Infinite-Equilibrium Systems ðsðkÞ Þ

j GðkÞ j1ðj j1

1 j2 ...jsðkÞ

Þ ðx0 , p0 Þz

ðsðkÞ þ1Þ 1 j2 ...jsðkÞ þ1 Þ j1

ðsðkÞ Þ

j GðkÞ j2ðj j2

1 j2 ...jsðkÞ

ðkÞj1

j

ðx0 , p0 Þzj1 zj2 . . . z

Þ ðx0 , p0 Þz

z ...z

j1 j2

js

ðkÞj2

js

ðkÞj1 þ1

j ε;

js

ðkÞj2 þ1

j ε;

j

j2

ðsðkÞ þ1Þ

>>j GðkÞ j2ðj j2

js

j1

>>j GðkÞ j1ðj j1

z ...z

j1 j2

1 j2 ...jsðkÞ þ1 j2

j1 j2 Þ ðx0 , p0 Þz z . . . z

ð8:97Þ

ðj1 , j2 ,   jq ¼ 1, 2, . . . , nÞ, ð4Þ

ð4Þ

ð4Þ

ðk ¼ i4 2 fs1 , s2 , . . . , sl2 g f1, 2, . . . , ngÞ, ðj ¼ 1, 2, . . . , qi4 Þ then ð4Þ

ð4Þ

z_ðkÞ11  αi4 zðkÞ11  βi4 zðkÞ12 þ

X 1 ðqÞ GðkÞ ðj j ...j Þ ðzj1 zj2 . . . zjq Þ, q 11 1 2 q! q¼2

sðkÞ11

ð2Þ

ð4Þ

z_ðkÞ12  βi4 zðkÞ11 þ αi4 zðkÞ12

g g

X 1 ðqÞ GðkÞ ðj j ...j Þ ðzj1 zj2 . . . zjq Þ; þ q 12 1 2 q! q¼2 sðkÞ12

ði Þ

ð4Þ

ð4Þ

z_ðkÞj1  δj 4 zðkÞðj1Þ1 þ αi4 zðkÞj1  βi4 zðkÞj2 s

þ

ðkÞj1 X 1

q! q¼2

ðqÞ

GðkÞ

j1 ðj1 j2 ...jq Þ

ði Þ

ðzj1 zj2 . . . zjq Þ,

ð2Þ

ð4Þ

z_ðkÞ12  δj 4 zðkÞðj1Þ2 þ βi4 zðkÞj1 þ αi4 zðkÞj2 s

þ

ðkÞj2 X 1

q! q¼2

ðqÞ

GðkÞ

j2 ðj1 j2 ...jq Þ

ðzj1 zj2 . . . zjq Þ;

ðj ¼ 2, 3, . . . , qi4 Þ, ð4Þ

ð4Þ

ð4Þ

with ðk ¼ i4 2 fs1 , s2 , . . . , sl4 g f1, 2, . . . , ngÞ ðj1 , j2 , . . . , jq ¼ 1, 2, . . . , nÞ:

ð8:98Þ

8.3 Infinite-Equilibrium Systems

385

The normal forms of nonlinear dynamical systems presented herein are based on the eigenvector space of the corresponding linearized systems at equilibriums in general. From simple and repeated eigenvalues of the linearized systems at equilibriums, the corresponding normal forms of the original nonlinear systems are developed through the differential geometry representation. The infinite-equilibrium systems in the following section can be introduced from the normal forms of nonlinear dynamical systems at bifurcation points via the local analysis.

8.3

Infinite-Equilibrium Systems

In this section, using singularity analysis, the stability of equilibriums in the infiniteequilibrium system will be discussed as in Luo (2019). Definition 8.7 Consider an autonomous dynamical system as x_i ¼ gi1 ðx, pi1 Þgi2 ðx, pi2 Þ . . . giri ðx, piri Þ ði ¼ 1, 2, . . . , nÞ, r i 1

ð8:99Þ

where gi ðx, pÞ  gi1 ðx, pi1 Þgi2 ðx, pi2 Þ . . . giri ðx, piri Þ:

ð8:100Þ

Such a system is called an infinite-equilibrium system if the following conditions exist gi ðx, pÞ ¼ 0ði ¼ 1, 2, . . . , nÞ, gi1 j1 ðx, pi1 j1 Þ  gi2 j2 ðx, pi2 j2 Þ ¼ φðx, pÞ ¼ 0 for i1 , i2 2 f1, 2, . . . , ng, i1 6¼ i2 ;

ð8:101Þ

j1 2 f1, 2, . . . , r i1 g, j2 2 f1, 2, . . . , r i2 g: The corresponding surface of φ(x, p) ¼ 0 is called an infinite-equilibrium surface in such a dynamical system. Definition 8.8 Consider an autonomous dynamical system as x_i ¼ φðx, p1 Þgi ðx, pÞ ði ¼ 1, 2, . . . , nÞ:

ð8:102Þ

(i) An (n  1)-dimensional surface of φ(x, p1) ¼ 0 is called an (n  1)-dimensional infinite-equilibrium surface if equilibrium x in Eq. (8.102) satisfies the following condition:

386

8 Infinite-Equilibrium Systems

φðx , p1 Þ ¼ 0:

ð8:103Þ

(ii) The equilibrium x is called a simple equilibrium if x 6¼ xα for x , xα 2 E and α ¼ 1, 2, . . .

with E ¼ fx jgi ðx , pÞ ¼ 0, i ¼ 1, 2, . . . , ng:

ð8:104Þ

(iii) The equilibrium x is called a r-repeated equilibrium (or rth-order equilibrium) if x ¼ x1 ¼ x2 ¼    ¼ xr1

for x , xα 2 E and α ¼ 1, 2, . . . , r  1 

ð8:105Þ



with E ¼ fx jgi ðx , pÞ ¼ 0, i ¼ 1, 2, . . . , ng: (iv) The equilibrium x is called an intersection equilibrium on the infiniteequilibrium surface if φðx , p1 Þ ¼ 0,

g

gi ðx , pÞ ¼ 0, i ¼ i1 , i2 , . . . , in1 iα 2 f1, 2, . . . , ng and α ¼ 1, 2, . . . , n  1:

8.3.1

ð8:106Þ

One-Infinite-Equilibrium Systems

Consider a simple dynamical system with one infinite equilibrium as x_1 ¼ x2 , x_2 ¼ x1 x2 :

ð8:107Þ

x1 2 ð1, 1Þ, x2 ¼ 0; x1 ¼ 0, x2 ¼ 0:

ð8:108Þ

The equilibriums are

The first integral manifold of Eq. (8.107) is 1 x2 ¼ x21 þ C: 2 Consider the first integral manifolds going through the following points:

ð8:109Þ

8.3 Infinite-Equilibrium Systems

387

1 x2 ¼ x21 for ðx1 , x2 Þ ¼ ð0, 0Þ, 2 1 x2 ¼ ðx21  a2 Þ for ðx1 , x2 Þ ¼ ða, 0Þ, 2 1 1 x2 ¼ ðx21 þ b2 Þ for ðx1 , x2 Þ ¼ ð0, b2 Þ 2 2

ð8:110Þ

where a > 0. From Eq. (8.107), the variational equations at the equilibriums are ð1Þ

Δx_1 ¼ Δx2 ¼ G1ðj Þ zj1 , 1

Δx_2 ¼ x2 Δx1 þ x1 Δx2 þ Δx1 Δx2 1 ð2Þ ð1Þ ¼ G2ðj Þ zj1 þ A2ðj j Þ zj1 zj2 1 1 2 2!

ð8:111Þ

where zj ¼ Δxj ðj ¼ 1, 2Þ, ð1Þ

ð1Þ

ð1Þ

ð1Þ

G1ð1Þ ¼ 0, G1ð2Þ ¼ 1;G2ð1Þ ¼ x2 , G1ð2Þ ¼ x1 , ð2Þ G2ð11Þ

¼

ð2Þ G2ð22Þ

¼

ð2Þ 0, G2ð12Þ

¼

ð2Þ G2ð21Þ

ð8:112Þ

¼ 1:

For equilibrium of ðx1 , x2 Þ ¼ ð0, 0Þ, the variational equation is Δx_1 ¼ Δx2 and Δx_2 ¼ Δx1 Δx2 :

ð8:113Þ

On the x1-direction, Δx_1 ¼ Δx2 > 0 if Δx2 > 0. The equilibrium of (0, 0) is the firstorder upper saddle. However, Δx_1 ¼ Δx2 < 0 if Δx2 < 0. So the equilibrium of (0, 0) is the first-order lower saddle. On the x2-direction, for Δx2 > 0, Δx_2 ¼ Δx1 Δx2 < 0 if Δx1 < 0, and Δx_2 ¼ Δx1 Δx2 < 0 if Δx1 > 0. Thus, the equilibrium of (0, 0) is the second-order upper saddle for Δx2 > 0 on the x2-direction. For Δx2 < 0, Δx_2 ¼ Δx1 Δx2 > 0 if Δx1 < 0, and Δx_2 ¼ Δx1 Δx2 < 0 if Δx1 > 0. Thus, the equilibrium of (0, 0) is the second-order lower saddle for Δx2 > 0 on the x2-direction. Therefore, the equilibrium of (0, 0) on the first integral manifold is the second-order upper saddle. For ðx1 , x2 Þ ¼ ða, 0Þ, the variational equation is Δx_1 ¼ Δx2 and Δx_2 ¼ aΔx2 :

ð8:114Þ

On the x1-direction, Δx_1 ¼ Δx2 > 0 if Δx2 > 0 and Δx_1 ¼ Δx2 < 0 if Δx2 < 0. The equilibrium of (a, 0) is the first-order upper saddle for Δx2 > 0 and the first-order lower saddle for Δx2 < 0 on the x1-direction. On the x2-direction, for a > 0, Δx_2 ¼ aΔx2 < 0 if Δx2 < 0 and Δx_2 ¼ aΔx2 > 0 if Δx2 > 0. Thus, the equilibrium of (a, 0) is the second-order source on the x2-direction. Therefore, the equilibrium of (a, 0) is a source.

388

8 Infinite-Equilibrium Systems

x2

x2

SI

US

SO

US

x1 SI

a

SO

x1

b

Fig. 8.1 Equilibrium stability for an infinite-equilibrium system: (a) local analysis for equilibriums, (b) the first integral manifolds. The solid and dashed thick lines are for sink and source infinite equilibriums, respectively. The parabolic curves are for the first integral manifolds. The filled circular symbol is for the upper saddle (US). The hollow circular symbols are for sink (SI) and source (SO) on the infinite-equilibrium line

For ðx1 , x2 Þ ¼ ða, 0Þ, the variational equation is Δx_1 ¼ Δx2 and Δx_2 ¼ aΔx2 :

ð8:115Þ

On the x1-direction, Δx_1 ¼ Δx2 > 0 if Δx2 > 0 and Δx_1 ¼ Δx2 < 0 if Δx2 < 0. The equilibrium of (a, 0) is the first-order upper saddle for Δx2 > 0 and the first-order lower saddle for Δx2 < 0 on the x1-direction. On the x2-direction, for a > 0, Δx_2 ¼ aΔx2 > 0 if Δx2 < 0 and Δx_2 ¼ aΔx2 < 0 if Δx2 > 0. Thus, the equilibrium of (a, 0) is the first-order sink on the x2-direction. Therefore, the equilibrium of (a, 0) is a sink. The infinite sink of (a, 0) and the infinite source of (a, 0) for a 2 (0, 1) with the second-order upper-saddle switching point (0, 0) form the entire infinite equilibrium. The phase portrait with the first integral manifolds is presented in Fig. 8.1. The solid and dashed thick lines are for sink and source infinite equilibriums, respectively. The parabolic curves are for the first integral manifolds. The filled circular symbol is for the upper saddle (US) on the first integral manifold. The hollow circular symbols are for sink (SI) and source (SO) on the infinite-equilibrium line of x2 ¼ 0.

8.3.2

Two-Infinite-Equilibrium Systems

Consider a dynamical system with two infinite-equilibriums x_1 ¼ x2 ðx1  a1 Þðx2  b1 Þ, x_2 ¼ x1 ðx1  a1 Þðx2  b1 Þ

ð8:116Þ

where a1 > 0, b1 > 0. The one simple equilibrium and two infinite equilibriums are

8.3 Infinite-Equilibrium Systems

389

x1 ¼ a1 , x2 2 ð1, 1Þ, x1 2 ð1, 1Þ, x2 ¼ b1 ;

g

for two infinite equilbriums,

x1 ¼ a1 , x2 ¼ b1 ; for intersection equilbriums, x1 ¼ 0, x2 ¼ 0 for simple eqilibrium, x1 ¼ 0, x2 ¼ b1 ; x1 ¼ a1 , x2 ¼ 0

g

ð8:117Þ

for simple eqilibriums:

The stability of equilibriums is determined by the local analysis. The corresponding G-functions are as follows: ð1Þ

G1ð1Þ ¼ x2 ðx2  b1 Þ, ð1Þ

G1ð2Þ ¼ ðx1  a1 Þðx2  b1 Þ þ x2 ðx1  a1 Þ; ð1Þ

G2ð1Þ ¼ ðx1  a1 Þðx2  b1 Þ  x1 ðx2  b1 Þ, ð1Þ G2ð2Þ

¼

x1 ðx1

 a1 Þ;

ð2Þ

G1ð11Þ ¼ 0, ð2Þ

ð2Þ

G1ð12Þ ¼ G1ð21Þ ¼ 2½ðx2  b1 Þ þ x2 , ð2Þ

G1ð22Þ ¼ 2ðx1  a1 Þ, ð2Þ

G2ð11Þ ¼ 2ðx2  b1 Þ, ð2Þ

ð2Þ

G2ð12Þ ¼ G2ð21Þ ¼ ½ðx1  a1 Þ þ x1 , ð2Þ

G2ð22Þ ¼ 0:

g

g

for the first order,

ð8:118Þ

for the second order:

Consider the infinite equilibriums of x*1 ¼ a1, x*2 2 (1, 1). The corresponding G-functions and variational equations are given by ð1Þ

G1ð1Þ ¼ x2 ðx2  b1 Þ, ð1Þ

G1ð2Þ ¼ 0; ð1Þ

G2ð1Þ ¼ a1 ðx2  b1 Þ, ð1Þ G2ð2Þ

¼ 0;

g

ð1Þ

Δx_1 ¼ G1ð1Þ Δx1 ¼ x2 ðx2  b1 ÞΔx1 , ð1Þ

Δx_2 ¼ G2ð1Þ Δx1 ¼ a1 ðx2  b1 ÞΔx1 :

ð8:119Þ

390

8 Infinite-Equilibrium Systems

Thus, for x2 2 ðb1 , 1Þ, there exists Δx_1 > 0, Δx_2 < 0 for Δx1 > 0, Δx_1 < 0, Δx_2 > 0 for Δx1 < 0:

g

ð8:120Þ

Such an infinite-equilibrium interval of x2 2 ðb1 , 1Þ is a source of the first order. For x2 2 ð0, b1 Þ, we have Δx_1 < 0, Δx_2 > 0 for Δx1 > 0, Δx_1 > 0, Δx_2 < 0 for Δx1 < 0:

g

ð8:121Þ

Such an infinite-equilibrium interval of x2 2 ð0, b1 Þ is a sink of the first order. For x2 2 ð1, 0Þ, we have Δx_1 > 0, Δx_2 > 0 for Δx1 > 0, Δx_1 < 0, Δx_2 > 0 for Δx1 < 0:

g

ð8:122Þ

Such an infinite-equilibrium interval of x2 2 ð1, 0Þ is a source of the first order. Consider the infinite equilibriums of x1 2 (1, 1), x2 ¼ b1. The corresponding G-functions and variational equations are ð1Þ

G1ð1Þ ¼ 0, ð1Þ

G1ð2Þ ¼ b1 ðx1  a1 Þ; ð1Þ

G2ð1Þ ¼ 0, ð1Þ G2ð2Þ

¼

Δx_1 ¼

x1 ðx1

ð1Þ G1ð2Þ Δx1

 a1 Þ;

g

ð8:123Þ

¼ b1 ðx1  a1 ÞΔx2 ,

ð1Þ

Δx_2 ¼ G2ð2Þ Δx1 ¼ x1 ðx1  a1 ÞΔx2 : Thus, for x1 2 ða1 , 1Þ, we have Δx_1 > 0, Δx_2 < 0 for Δx2 > 0, Δx_1 < 0, Δx_2 > 0 for Δx2 < 0:

g

ð8:124Þ

Such an infinite-equilibrium interval of x1 2 ða1 , 1Þ is a sink of the first order. For x1 2 ða1 , 0Þ, there exists Δx_1 < 0, Δx_2 > 0 for Δx2 > 0, Δx_1 > 0, Δx_2 < 0 for Δx2 < 0:

g

ð8:125Þ

8.3 Infinite-Equilibrium Systems

391

Such an infinite-equilibrium interval of x2 2 ð0, b1 Þ is a source of the first order. For x1 2 ð1, 0Þ, there exists Δx_1 < 0, Δx_2 < 0 for Δx2 > 0, Δx_1 > 0, Δx_2 > 0 for Δx2 < 0:

g

ð8:126Þ

Such an infinite-equilibrium interval of x1 2 ð1, 0Þ is a sink of the first order. For the equilibrium point of x1 ¼ 0, x2 ¼ b1 , the variational equation is ð1Þ

Δx_1 ¼ G1ð2Þ Δx2 ¼ a1 b1 Δx2 , Δx_2 ¼

1 ð2Þ ð2Þ þ G2ð21Þ Δx1 Δx2 ¼ a1 Δx1 Δx2 ; ½G 2! 2ð12Þ

where ð1Þ

ð1Þ

ð1Þ

ð8:127Þ

ð1Þ

G1ð1Þ ¼ 0, G1ð2Þ ¼ a1 b1 ;G2ð1Þ ¼ 0, G2ð2Þ ¼ 0; ð1Þ

ð1Þ

ð1Þ

G2ð11Þ ¼ 0, G2ð12Þ ¼ 2a1 , G2ð22Þ ¼ 0: On the x1-direction, Δx_1 > 0 for Δx2 < 0 and Δx_1 < 0 for Δx2 > 0. Thus, there is a lower saddle of the first order for Δx2 > 0, and there is an upper saddle of the first order for Δx2 < 0. However, on the x2-direction, as Δx1 > 0, we have Δx_2 > 0 for Δx2 < 0 and Δx_2 < 0 for Δx2 > 0. The second-order source exists. As Δx1 < 0, we have Δx_2 < 0 for Δx2 < 0 and Δx_2 > 0 for Δx2 > 0. The second-order sink exists. Thus, the second-order upper saddle is for the equilibrium point of x1 ¼ 0, x2 ¼ b1 . For the equilibrium point of x1 ¼ a1 , x2 ¼ 0, the variational equation is Δx_1 ¼

1 ð1Þ ð1Þ ½G þ G1ð21Þ Δx1 Δx2 ¼ b1 Δx1 Δx2 , 2! 1ð12Þ ð1Þ

Δx_2 ¼ G2ð1Þ Δx1 ¼ a1 b1 Δx1 ; where ð1Þ

ð8:128Þ

ð1Þ

G1ð1Þ ¼ 0, G1ð2Þ ¼ 0; ð1Þ

ð1Þ

ð1Þ

ð1Þ

G1ð11Þ ¼ 0, G1ð12Þ ¼ G1ð21Þ ¼ b1 , G1ð22Þ ¼ 0; ð1Þ

ð1Þ

G2ð1Þ ¼ a1 b1 , G2ð2Þ ¼ 0: On the x1-direction, as Δx2 > 0, we have Δx_1 > 0 for Δx1 < 0 and Δx_1 < 0 for Δx1 > 0. There is the second-order source. As Δx2 < 0, we have Δx_1 < 0 for Δx1 < 0 and Δx_1 > 0 for Δx1 > 0. There is a second-order source. On the x2-direction, Δx_2 > 0 for Δx1 > 0 and Δx_2 < 0 for Δx1 < 0. Thus, there is an upper saddle of the first order for Δx1 > 0, and there is a lower saddle of the first order for Δx1 < 0. Thus, the second-order lower saddle in the two directions is for the equilibrium point of x1 ¼ a1 , x2 ¼ b1 :

392

8 Infinite-Equilibrium Systems

For the equilibrium point of x1 ¼ a1 , x2 ¼ b1 , the variational equation is 1 ð2Þ ð2Þ ½G þ G1ð21Þ Δx1 Δx2 ¼ b1 Δx1 Δx2 , 2! 1ð12Þ 1 ð2Þ ð2Þ Δx_2 ¼ ½G2ð12Þ þ Gað21Þ Δx1 Δx2 ¼ a1 Δx1 Δx2 ; 2! where Δx_1 ¼

ð1Þ

ð1Þ

ð1Þ

ð1Þ

G1ð1Þ ¼ 0, G1ð2Þ ¼ 0;G2ð1Þ ¼ 0, G2ð2Þ ¼ 0; ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð8:129Þ

ð2Þ

G1ð11Þ ¼ 0, G1ð12Þ ¼ G1ð21Þ ¼ b1 , G1ð22Þ ¼ 0; ð2Þ

G2ð11Þ ¼ 0, G2ð12Þ ¼ G2ð21Þ ¼ a1 , G2ð22Þ ¼ 0: On the x1-direction, as Δx2 > 0, we have Δx_1 < 0 for Δx1 < 0 and Δx_1 > 0 for Δx1 > 0. There is the second-order source. As Δx2 < 0, we have Δx_1 > 0 for Δx1 < 0 and Δx_1 < 0 for Δx1 > 0. There is a second-order sink. On the x2-direction, as Δx1 > 0, we have Δx_2 > 0 for Δx2 < 0 and Δx_2 < 0 for Δx2 > 0. The second-order source exists. As Δx1 < 0, we have Δx_2 < 0 for Δx2 < 0 and Δx_2 > 0 for Δx2 > 0. The second-order sink exists. Thus, the second-order upper saddle for the diagonal direction is for the equilibrium point of x1 ¼ a1 , x2 ¼ b1 . For equilibrium of x1 ¼ 0, x2 ¼ 0, we have ð1Þ

Δx_1 ¼ G1ð2Þ Δx2 ¼ a1 b1 Δx2 , ð1Þ

Δx_2 ¼ G2ð1Þ Δx1 ¼ a1 b1 Δx1 ; where ð1Þ ð1Þ G1ð1Þ ¼ 0, G1ð2Þ ¼ a1 b1 ; ð1Þ

ð8:130Þ

ð1Þ

G2ð1Þ ¼ a1 b1 , G2ð2Þ ¼ 0: Thus, this equilibrium is a center. From the foregoing local analysis of equilibriums, the stability of equilibriums is summarized in Table 8.1. The equilibrium of (0, 0) is a center, which is not included. The rich dynamics in the infinite dynamical systems can be obtained. Such equilibriums possess higher order singularity, which cannot be analyzed by the traditional eigenvalue analysis. The first integral manifold of Eq. (8.116) is x21 þ x22 ¼ C:

ð8:131Þ

Consider the manifolds going through the simple and intersection equilibriums and the corresponding manifolds are

8.3 Infinite-Equilibrium Systems

393

Table 8.1 Summarization of equilibrium stability for Eq. (8.116) Infinite Infinite Simple

x1 2 (a1, 1), x2 ¼ b1 SI x1 ¼ a1, x2 2 (b1, 1), SO x1 ¼ a1, x2 ¼ 0, LS

x1 2 (0, a1), x2 ¼ b1 SO x1 ¼ a1, x2 2 (0, b1) SI x1 ¼ 0, x2 ¼ b1 US

x1 2 (1, 0), x2 ¼ b1 SI x1 ¼ a1, x2 2 (1, 0) SO x1 ¼ a1, x2 ¼ b1 US

Infinite Infinite equilibrium, Simple: Simple equilibrium, SI: sink, SO: source, US: upper saddle, LS: lower saddle

Table 8.2 Signs of ðx_1 , x_2 Þ in different domains for Eq. (8.116)

ðx_1 , x_2 Þ x2 2 (b1, 1) x2 2 (0, b1) x2 2 (1, 0)

x1 2 (a1, 1) (+, ) (, +) (+, +)

x1 2 (0, a1) (, +) (+, ) (, )

x1 2 (1, 0) (, ) (+, +) (, +)

x21 þ x22 ¼ a21 for x1 ¼ a1 , x2 ¼ 0; x21 þ x22 ¼ b21 for x1 ¼ 0, x2 ¼ b1 ; x21 þ x22 ¼ a21 þ b21 for x1 ¼ a1 , x2 ¼ b;

ð8:132Þ

x21 þ x22 ¼ 0 for x1 ¼ 0, x2 ¼ 0; x21 þ x22 ¼ C for others: From the differential equation in Eq. (8.116), in the different domains separated by the equilibriums, the signs of ðx_1 , x_2 Þ are presented in Table 8.2, which indicate the directions of the first-order integral manifolds. The illustrations for the signs in phase plane are presented in Fig. 8.2. From the local analysis of equilibrium stability, the complete picture of equilibriums with the first integral manifolds is presented in Fig. 8.3. The solid and dashed thick lines are for sink and source infinite equilibriums, respectively. The circles are for the first integral manifolds. The filled circular symbol is for the upper saddle of the second order in two directions. The hollow circular symbols are for the center and two intersection points between infinite-equilibrium lines and coordinates.

8.3.3

Higher Order Infinite-Equilibrium Systems

Consider a dynamical system with an infinite equilibrium of the second-order singularity as x_1 ¼ x1 x22 , x_2 ¼ x1 x2 sin x1 where a1 > 0, b1 > 0.

ð8:133Þ

394

8 Infinite-Equilibrium Systems

Fig. 8.2 Local analysis of equilibriums for the first integral manifolds. The solid and dashed thick lines are for sink and source infinite equilibriums, respectively. The filled circular symbol is for the intersection equilibrium. The hollow circular symbols are for the center and intersection equilibriums between infinite-equilibriums and coordinates. SI: sink, SO: source, US: upper saddle, LS: lower saddle

x2 SO SI

US

US

SO

SI

x2 = b1

SI

center

x1

LS

SO

x1 = a1

Fig. 8.3 Equilibrium stability for an infiniteequilibrium system in Eq. (8.116). The solid and dashed thick lines are for sink and source infinite equilibriums, respectively. The circles are for the first integral manifolds. The filled circular symbol is for the intersection equilibrium. The hollow circular symbols are for the center and intersection equilibriums between infiniteequilibriums and coordinates. SI: sink, SO: source, US: upper saddle, LS: lower saddle

x2

SO US SI

SI

US LS

x1

center

SO

The simple equilibriums and infinite equilibriums of the nonlinear dynamical system in Eq. (8.133) are x1 ¼ 0, x2 2 ð1, 1Þ; x1 2 ð1, 1Þ, x2 ¼ 0

g

for infintite‐equilibriums,

x1 ¼ 0, x2 ¼ 0 for intersection equilbriums, x1 ¼ kπ ðk ¼ 1, 2, . . .Þ, x2 ¼ 0 for simple equilbriums:

ð8:134Þ

8.3 Infinite-Equilibrium Systems

395

For the infinite equilibriums of x2 ¼ 0, x1 sin x1 6¼ 0, the variational equation is Δx_1 ¼ x1 Δx2 Δx2 , Δx_2 ¼ x1 sin x1 Δx2 :

ð8:135Þ

On the x1-direction, Δx_1 < 0 for x1 > 0 which implies the second-order lower saddle, and Δx_1 > 0 for x1 < 0 which implies the second-order upper saddle. On the x2-direction, the equilibrium is the first-order source for x1 sin x1 > 0, and sink for x1 sin x1 < 0. Thus, the infinite equilibrium of x2 ¼ 0, x1 sin x1 6¼ 0 is named a sink or source. For the infinite equilibrium of x1 ¼ 0, x2 6¼ 0, the variational equation is Δx_1 ¼ ðx2 Þ2 Δx1 , Δx_2 ¼ x2 cos x1 Δx1 Δx1 ¼ x2 Δx1 Δx1 :

ð8:136Þ

On the x1-direction, the equilibrium is the first-order sink. On the x2-direction, Δx_2 > 0 for x2 > 0 which implies the second-order upper-saddle, and Δx_2 < 0 for x2 < 0 which implies the second-order lower saddle. Thus, the infinite equilibrium of x1 ¼ 0, x2 6¼ 0 is named the sink with different type on the x2-direction. For the simple equilibriums of x2 ¼ 0, x1 ¼ mπ (m ¼ 1, 2, . . . ), the variational equation is Δx_1 ¼ x1 Δx2 Δx2 , Δx_2 ¼ x1 cos x1 Δx1 Δx2 :

ð8:137Þ

On the x1-direction, Δx_1 < 0 for x1 > 0 which implies the second-order lower saddle, and Δx_1 > 0 for x1 < 0 which implies the second-order upper saddle. On the x2-direction, (i) (ii) (iii) (iv)

Δx_2 Δx_2 Δx_2 Δx_2

> 0 for x1 cos x1 > 0 with Δx1 > 0, which implies the second-order source. < 0 for x1 cos x1 > 0 with Δx1 < 0, which implies the second-order sink. < 0 for x1 cos x1 < 0 with Δx1 > 0, which implies the second-order sink. > 0 for x1 cos x1 < 0 with Δx1 < 0, which implies the second-order source.

Thus, the equilibrium based on the x1-direction is called the second-order lower saddle or upper saddle. For the equilibrium of x1 ¼ 0, x2 ¼ 0, the variational equation is Δx_1 ¼ Δx1 Δx2 Δx2 , Δx_2 ¼ Δx1 Δx1 Δx2 :

ð8:138Þ

396

8 Infinite-Equilibrium Systems

Table 8.3 Summarization of equilibrium stability for Eq. (8.133) x2 ¼ 0 x2 ¼ 0 x1 ¼ 0 x2 ¼ 0 x2 ¼ 0

x1 2 ð4π, 3πÞ x1 2 ð3π, 2πÞ SI SO x1 2 ðπ, 0Þ x1 2 ð2π, πÞ SO SI x2 2 ð0, 1Þ SI x1 ¼ 2mπ x1 ¼ ð2m  1Þπ Second-order LS Second-order LS x1 ¼ 0 Center with third-order SI and SO

x1 2 ð2π, πÞ SI x1 2 ð3π, 2πÞ SO x2 2 ð1, 0Þ SI x1 ¼ 2mπ Second-order US

x1 2 ðπ, 0Þ SO x1 2 ð4π, 3πÞ SI

x1 ¼ ð2m  1Þπ Second-order US

SI: sink, SO: source, US: upper saddle, LS: lower saddle

On the x1-direction, the equilibrium is the third-order sink. On the x2-direction, the equilibrium is the third-order source. The equilibrium of ðx1 , x2 Þ ¼ ð0, 0Þ is named the third-order center herein. From the foregoing local analysis, the stability of equilibriums is tabulated in Table 8.3. The first integral manifold of Eq. (8.133) is 1 2 x  cos x1 ¼ C: 2 2

ð8:139Þ

If the manifolds are going through the intersection equilibriums, the corresponding manifolds are given by 1 2 x  cos x1 ¼ 1 for x1 ¼ ð2m  1Þπ, x2 ¼ 0; 2 2 1 2 x  cos x1 ¼ 1 for x1 ¼ ð2mÞπ, x2 ¼ 0; 2 2 1 2 x  cos x1 ¼ C for others: 2 2

ð8:140Þ

From the local analysis of equilibrium stability, equilibriums with the first integral manifolds are presented in Fig. 8.4. The solid and dashed thick lines are for sink and source infinite-equilibriums, respectively. The curves are for the first integral manifolds. The hollow circular symbols are for switching saddles. The filled circular symbol is for the center with third-order sink and source.

8.4

Network-Infinite-Equilibrium Systems

In this section, nonlinear dynamical systems with a network of infinite equilibriums are discussed as in Luo (2019).

8.4 Network-Infinite-Equilibrium Systems

397

x2 SI US SI

US

US SO

SI

LS SO

SO

LS

LS SI

SO

SI

x1

SI Fig. 8.4 Equilibrium stability for an infinite-equilibrium system. The solid and dashed thick lines are for sink and source infinite-equilibriums, respectively. The filled circular symbol is for the center with third-order sink and source. The hollow circular symbols are for upper-saddle (US) and lowersaddle (LS) on the infinite-equilibrium line

Definition 8.9 Consider an autonomous dynamical system as x_i ¼ φ1 ðx, p1 Þφ2 ðx, p2 Þ . . . φl ðx, pl Þgi ðx, pÞ ði ¼ 1, 2, . . . , nÞ, r i 1:

ð8:141Þ

(i) An (n  1)-dimensional surface network of φj(x, pj) ¼ 0 ( j ¼ 1, 2, . . . , l ) in phase space is called an infinite-equilibrium network of dynamical system in Eq. (8.141). (ii) An (n  1)-dimensional surface of φj(x, pj) ¼ 0 ( j 2 {1, 2, . . ., l}) is called an (n  1)-dimensional infinite-equilibrium surface if the equilibrium x of Eq. (8.141) satisfies the following condition, i.e., φj ðx , pj Þ ¼ 0 for j 2 f1, 2, . . . , lg:

ð8:142Þ

(iii) An (n  2)-dimensional intersection edge of φj1 ðx, pj1 Þ ¼ 0 and φj2 ðx, pj2 Þ ¼ 0 is called an (n  2)-dimensional infinite-equilibrium edge if the equilibrium x of Eq. (8.141) satisfies the following conditions, i.e., φj1 ðx , pj1 Þ ¼ 0, φj2 ðx , pj2 Þ ¼ 0 φj1 ðx, pj1 Þ 6¼ φj2 ðx, pj2 Þ and Dφj1 ðx, pj1 Þ 6¼ Dφj2 ðx, pj2 Þ for j1 , j2 2 f1, 2, . . . , lg:

ð8:143Þ

398

8 Infinite-Equilibrium Systems

(iv) An (n  r)-dimensional intersection edge of φjα ðx, pjα Þ ¼ 0 (α ¼ 1, 2, . . . , r) is called an (n  r)-dimensional infinite-equilibrium edge if the equilibrium x of Eq. (8.141) satisfies the following condition, i.e., φj1 ðx , pj1 Þ ¼ 0, φj2 ðx , pj2 Þ ¼ 0, . . . , φjr ðx , pjr Þ ¼ 0, φjα ðx, pjα Þ 6¼ φjβ ðx, pjβ Þ and Dφjα ðx, pjα Þ 6¼ Dφjβ ðx, pjβ Þ

ð8:144Þ

for j1 , j2 , . . . , jr 2 f1, 2, . . . , lg; α, β 2 f1, 2, . . . , rg, α 6¼ β: (v) An intersection vertex of φjα ðx, pjα Þ ¼ 0 (α ¼ 1, 2, . . . , n) is called an infiniteequilibrium vertex if the equilibrium x of Eq. (8.141) satisfies the following condition, i.e., φj1 ðx , pj1 Þ ¼ 0, φj2 ðx , pj2 Þ ¼ 0, . . . , φjn ðx , pjn Þ ¼ 0, φjα ðx, pjα Þ 6¼ φjβ ðx, pjβ Þ and Dφjα ðx, pjα Þ 6¼ Dφjβ ðx, pjβ Þ

ð8:145Þ

for j1 , j2 , . . . , jn 2 f1, 2, . . . , lg, α, β 2 f1, 2, . . . , ng, α 6¼ β: (vi) An (n  1)-dimensional surface of φj(x, pj) ¼ 0 ( j 2 {1, 2, . . ., l}) in phase space is called a rth-order, (n  1)-dimensional, infinite equilibrium if φj ðx, pj Þ  φj1 ðx, pj1 Þ ¼    ¼ φjr ðx, pjr Þ for j1 , j2 , . . . , jr 2 f1, 2, . . . , lg:

8.4.1

ð8:146Þ

A Network-Infinite-Equilibrium System

To demonstrate the infinite-equilibrium network systems, consider a dynamical system of four infinite equilibriums, i.e., x_1 ¼ x2 ðx1  a1 Þðx2  b1 Þðx1 þ a1 Þðx2 þ b1 Þ, x_2 ¼ x1 ðx1  a1 Þðx1  b1 Þðx1 þ a1 Þðx2 þ b1 Þ

ð8:147Þ

where a1 > 0, b1 > 0. Such four infinite equilibriums form a network of the infinite equilibrium in phase space. The dynamical behaviors of the dynamical systems are separated in different subdomains. The simple equilibriums and infinite equilibriums are

8.4 Network-Infinite-Equilibrium Systems

399

Table 8.4 Summarization of equilibrium stability for Eq. (8.147) x2 ¼ b1 x2 ¼  b1 x1 ¼ a1 x1 ¼  a1 x1 ¼ a1 x1 ¼ 0 x1 ¼  a1

x1 2 (a1, 1), SI x1 2 (a1, 1), SO x2 2 (b1, 1), SO x2 2 (b1, 1), SI x2 ¼ b1, US x2 ¼ b1, US x2 ¼ b1, LS

x1 2 (0, a1) SO x1 2 (0, a1) SI x2 2 (0, b1) SI x2 2 (0, b1) SO x2 ¼ 0 LS x2 ¼ 0 Center x2 ¼ 0 US

x1 2 (a1, 0), SI x1 2 (a1, 0), SO x2 2 (b1, 0) SO x2 2 (b1, 0) SI x2 ¼  b1 US x2 ¼  b1 LS x2 ¼  b1 LS

x1 2 (1, a1) SO x1 2 (1, a1) SI x2 2 (1, b1) SI x2 2 (1, b1) SO

SI: sink, SO: source, US: upper saddle, LS: lower saddle

x1 ¼ a1 , x2 2 ð1, 1Þ; x1 ¼ a1 , x2 2 ð1, 1Þ; x1 2 ð1, 1Þ, x2 ¼ b1 ; x1 2 ð1, 1Þ, x2 ¼ b1

g

for infintite equilibriums,

x1 ¼ a1 , x2 ¼ b1 ;x1 ¼ a1 , x2 ¼ b1 ; x1 ¼ a1 , x2 ¼ b1 ;x1 ¼ a1 , x2 ¼ b1

g

for intersection equilbriums,

ð8:148Þ

x1 ¼ 0, x2 ¼ 0, for a simple equilbrium, x1 ¼ 0, x2 ¼ b1 ;x1 ¼ 0, x2 ¼ b1 ; x1 ¼ a1 , x2 ¼ 0;x1 ¼ a1 , x2 ¼ 0

g

for simple equilbriums:

The similar local analysis of equilibriums can be completed, and the stability of equilibriums is summarized in Table 8.4. The first integral manifold of Eq. (8.147) is x21 þ x22 ¼ C:

ð8:149Þ

If the first integral manifolds go through the simple and intersection equilibrium, the corresponding first integral manifolds are given by x21 þ x22 ¼ a21 for x1 ¼ a1 , x2 ¼ 0; x21 þ x22 ¼ b21 for x1 ¼ 0, x2 ¼ b1 ; x21 þ x22 ¼ a21 þ b21 for x1 ¼ a1 , x2 ¼ b; x21 þ x22 ¼ 0 for x1 ¼ 0, x2 ¼ 0; x21 þ x22 ¼ C for others:

ð8:150Þ

400

8 Infinite-Equilibrium Systems

x2

SI

SO US

LS

SI

SO US US

LS

center

x1

LS

SI

SO US

LS SO

SI

Fig. 8.5 Equilibrium stability for an infinite-equilibrium system in Eq. (8.147). The solid and dashed thick lines are for sink and source infinite equilibriums, respectively. The circles are for the first integral manifolds. The filled circular symbol is for the intersection equilibrium between two infinite equilibrium lines. The hollow circular symbols are for a simple equilibrium and the intersection points between infinite-equilibrium lines and coordinates. SI: Sink, SO: Source, US: upper saddle, LS: lower saddle

From the local analysis of equilibrium stability, equilibriums with the first integral manifolds are presented in Fig. 8.5. The solid and dashed thick lines are for sink and source infinite equilibriums, respectively. The circles are for the first integral manifolds. The filled circular symbol is for the upper saddle of the second order in two directions. The hollow circular symbols are for lower-saddles and upper-saddles in one direction and the center. The network of four infinite equilibriums has nine (9) subdomains separated by the four infinite equilibriums.

8.4.2

Circular Infinite-Equilibrium Systems

In this section, a few dynamical systems of circular infinite-equilibrium systems will be presented. A. Harmonic motions: Consider a dynamical system of a circular infinite equilibrium with a harmonic motion, i.e., x_1 ¼ ðx21 þ x22  R2 Þm x2 , x_2 ¼ ðx21 þ x22  R2 Þm x1 where R > 0. The simple equilibriums and infinite equilibriums are

ð8:151Þ

8.4 Network-Infinite-Equilibrium Systems

401

2 2 x2 1 þ x2 ¼ R for the infintite equilibrium;

x1 ¼ R, x2 ¼ 0; x1 ¼ 0, x2 ¼ R

g

for intersection equilibrium;

ð8:152Þ

x1 ¼ 0, x2 ¼ 0 for simple equilbrium: Consider x1 ¼ ρ cos θ, x2 ¼ ρ sin θ; x_1 ¼ ρ_ cos θ  ρθ_ sin θ,

ð8:153Þ

x_2 ¼ ρ_ sin θ þ ρθ_ cos θ: Thus, Eq. (8.151) becomes ρ_ ¼ 0 ) ρ ¼ C,

ð8:154Þ

θ_ ¼ ðρ  RÞm ðρ þ RÞm : The infinite equilibrium is at ρ ¼ R. From Eq. (8.154), we have θ_ ¼ 0 for ρ ¼ R, θ_ ¼ ðρ  RÞm ðρ þ RÞm > 0 for ρ < R, θ_ ¼ ðρ  RÞm ðρ þ RÞm < 0 for ρ > R

g ð8:155Þ

for m ¼ 2l  1 with l ¼ 1, 2, . . . ; θ_ ¼ ðρ  RÞm ðρ þ RÞm < 0 for ρ < R, θ_ ¼ ðρ  RÞm ðρ þ RÞm < 0 for ρ > R

g

for m ¼ 2l with l ¼ 1, 2, . . . : The motions of ρ > R and ρ < R for m ¼ 2l  1 in phase space are clockwise and counterclockwise, respectively. With ρ ¼ R, the system is static, which is an infinite equilibrium. With decreasing ρ, the rotation is from negative to positive direction. Thus, the infinite equilibrium of ρ ¼ R is the (2l  1)th-order sink (SI). The motions of ρ > R and ρ < R for m ¼ 2l in phase space are clockwise and counterclockwise, respectively. With ρ ¼ R, the system is static, which is an infinite equilibrium. With decreasing ρ, the rotation is in the negative direction only. Thus, the infinite equilibrium of ρ ¼ R is the (2l)th-order lower saddle (LS). From Eq. (8.155), we have

402

8 Infinite-Equilibrium Systems

θ ¼ C 1 for ρ ¼ R, θ ¼ ðρ2  R2 Þm t þ θ0 for ρ < R,

ð8:156Þ

θ ¼ ðρ2  R2 Þm t þ θ0 for ρ > R: The first integral manifold of Eq. (8.151) is x21 þ x22 ¼ C:

ð8:157Þ

If the first integral manifolds go through the simple and intersection equilibrium, the corresponding first integral manifolds are given by 2 2 x2 1 þ x2 ¼ R for the infinite‐equilibrium;

x21 þ x22 ¼ C 21 for C 1 < R with counterclockwise rotation; x21

þ

x22

¼

C 22

ð8:158Þ

for C 2 > R with clockwise rotation:

From the local analysis of equilibrium stability, equilibriums with the first integral manifolds are presented in Fig. 8.6. The thick circle is for the infinite-equilibrium. The thin circles are for the first integral manifolds. The hollow circular symbols are for the center and intersection points between the infinite-equilibrium circle and coordinates. B. Linear motions: Consider a dynamical system of a circular infinite equilibrium with linear motions, i.e., x_1 ¼ ðx21 þ x22  R2 Þm x1 , x_2 ¼ ðx21 þ x22  R2 Þm x2

ð8:159Þ

where R > 0. The simple equilibriums and infinite equilibriums are 2 2 x2 1 þ x2 ¼ R for the infintite‐equilibrium;

x1 ¼ R, x2 ¼ 0; x1 ¼ 0, x2 ¼ R

g

for intersection equilibrium;

x1 ¼ 0, x2 ¼ 0 for simple equilbrium: Consider

ð8:160Þ

8.4 Network-Infinite-Equilibrium Systems Fig. 8.6 Equilibrium stability for an infiniteequilibrium system in Eq. (8.151). (a) The opposite directions of motion rotation and (b) the same directions of motion rotation on both sides of the infinite equilibrium. The solid thick circle is for lower-saddle infinite equilibriums. The thin circles are for the first integral manifolds. The hollow circular symbols are for the center and the intersection points between infinite-equilibrium circle and coordinates. LS: lower saddle, SI: sink

403

x2

(2l – 1) th SI

x1

center

a

x2

(2l ) th LS

x1

center

b x1 ¼ ρ cos θ, x2 ¼ ρ sin θ; x_1 ¼ ρ_ cos θ  ρθ_ sin θ, x_2 ¼ ρ_ sin θ þ ρθ_ cos θ:

ð8:161Þ

ρ_ ¼ ρðρ2  R2 Þm ¼ ρðρ  RÞm ðρ þ RÞm , θ_ ¼ 0 ) θ ¼ C:

ð8:162Þ

Thus, Eq. (8.159) becomes

The equilibriums are ρ ¼ 0 and ρ ¼ R for a specific θ ¼ C 2 [0, 2π). We have

404

8 Infinite-Equilibrium Systems

ρ_ ¼ 0 for ρ ¼ R, ρ_ ¼ ρðρ  RÞm ðρ þ RÞm < 0 for ρ < R, ρ_ ¼ ρðρ  RÞm ðρ þ RÞm > 0 for ρ > R

g ð8:163Þ

for m ¼ 2l  1 with l ¼ 1, 2, . . . ; m

m

ρ_ ¼ ρðρ  RÞ ðρ þ RÞ > 0 for ρ < R, ρ_ ¼ ρðρ  RÞm ðρ þ RÞm > 0 for ρ > R

g

for m ¼ 2l with l ¼ 1, 2, . . . ; ρ_ ¼ 0 for ρ ¼ 0, Δρ_ ¼ ðR2 Þm Δρ < 0 for Δρ > 0, for m ¼ 2l  1 with l ¼ 1, 2, . . . ;

ð8:164Þ

2 m

Δρ_ ¼ ðR Þ Δρ > 0 for Δρ > 0, for m ¼ 2l with l ¼ 1, 2, . . . : The motions of ρ > R and ρ < R for m ¼ 2l  1 in the radial direction are outwards and inwards the center, respectively. With ρ ¼ R, the system is static, which is an infinite-equilibrium. Thus, the infinite-equilibrium of ρ* ¼ R is the (2l)th-order source (SO) in the radial direction. The equilibrium of (0, 0) is a sink. The motions of ρ > R and ρ < R for m ¼ 2l in the radial direction are outwards the center. With ρ ¼ R, the system is static, which is an infinite equilibrium. Thus, the infinite equilibrium of ρ ¼ R is the (2l)th-order upper saddle (US) in the radial direction. The equilibrium of (0, 0) is a source. The corresponding equilibriums are ρ ¼ 0 for a simple equilibrium, ρ ¼ R for infinite‐equilbrium,

ð8:165Þ

1 3 θ ¼ 0, π, π, π;ρ ¼ R for intersection points: 2 2 

The first integral manifold of Eq. (8.152) is x2 ¼ Cx1 :

ð8:166Þ

If the first integral manifolds go through the infinite equilibrium, the corresponding first integral manifolds are given by x2 ¼ Cx1 with C ¼ x2 =x1 ¼ tan θ , 2 2 x2 1 þ x2 ¼ R for the infinite‐equilibrium:

ð8:167Þ

8.4 Network-Infinite-Equilibrium Systems Fig. 8.7 Equilibrium stability for an infiniteequilibrium system in Eq. (8.159). (a) The sink of (0, 0) and the (2l  1)thorder source of infinite equilibrium, (b) the source of (0, 0) and the (2l )th-order upper saddle of infinite equilibrium. The solid circular curve is for infinite equilibriums. The straight lines are for the first integral manifolds. The hollow circular symbols are for a simple equilibrium and intersection points between the infinite-equilibrium circle and coordinates. SI: sink, SO: source, US: upper saddle

405

a

b From the local analysis of equilibrium stability, equilibriums with the first integral manifolds are presented in Fig. 8.7. The thick circle is for the infinite-equilibrium circle. The thin lines are for the first integral manifolds. The hollow circular symbols are for the simple equilibrium and the intersection points between the infiniteequilibrium circle and coordinates. C. Hyperbolic motion: Consider a dynamical system of a circular infinite equilibrium with hyperbolic motions, i.e., x_1 ¼ ðx21 þ x22  R2 Þm x2 , x_2 ¼ ðx21 þ x22  R2 Þm x1 where R > 0.The simple equilibriums and infinite equilibriums are

ð8:168Þ

406

8 Infinite-Equilibrium Systems 2 2 x2 1 þ x2 ¼ R for the infintite‐equilibrium;

x1 ¼ R, x2 ¼ 0; x1 ¼ 0, x2 ¼ R

g

for intersection equilibrium;

ð8:169Þ

x1 ¼ 0, x2 ¼ 0 for simple equilbrium: Consider x1 ¼ ρ cos θ, x2 ¼ ρ sin θ; x_1 ¼ ρ_ cos θ  ρθ_ sin θ,

ð8:170Þ

x_2 ¼ ρ_ sin θ þ ρθ_ cos θ: Thus, Eq. (8.168) becomes ρ_ ¼ ρðρ  RÞm ðρ þ RÞm sin 2θ, θ_ ¼ ðρ  RÞm ðρ þ RÞm cos 2θ:

ð8:171Þ

The corresponding equilibriums are ρ ¼ 0, for simple equilibrium, ρ ¼ R, for infinite‐equilbrium, θ ¼ 0,

1 3 π, π, π;ρ ¼ R, 2 2

1 3 3 5 θ ¼ π, π, π, π;ρ ¼ R 4 4 4 4 

g

ð8:172Þ for intersection points:

For the simple equilibrium of (0, 0), the linearized equation is Δx_1 ¼ ðR2 Þm Δx2 , Δx_2 ¼ ðR2 Þm Δx1

ð8:173Þ

with the corresponding eigenvalues are λ1 ¼ ðR2 Þm , λ2 ¼ ðR2 Þm :

ð8:174Þ

For m ¼ 2l  1, we have λ1 ¼ R2ð2l1Þ > 0, λ2 ¼ R2ð2l1Þ < 0: For m ¼ 2l, we have

ð8:175Þ

8.4 Network-Infinite-Equilibrium Systems

407

Table 8.5 Summarization of equilibrium stability for Eq. (8.168) ρ ¼ R m ¼ 2l  1 m ¼ 2l ρ ¼ R m ¼ 2l  1 m ¼ 2l ρ ¼ R m ¼ 2l  1 m ¼ 2l

θ 2 ð0, 12 πÞ, (2l  1)thSO (2l )thUS θ ¼ 0

θ 2 ð12 π, πÞ (2l  1)thSI (2l )thLS θ ¼ 12 π (2l )thUS (2l + 1)thUS θ ¼ 34 π (2l  1)thSI (2l )thLS

(2l )thUS (2l + 1)thUS θ ¼ 14 π (2l  1)thSO (2l )thUS

θ 2 ðπ, 32 πÞ (2l  1)thSO (2l )thUS θ ¼ π (2l )thLS (2l + 1)thLS θ ¼ 54 π (2l  1)thSO (2l )thUS

θ 2 ð32 π, 2πÞ (2l  1)thSI (2l )thLS θ ¼ 32 π (2l )thUS (2l + 1)thLS θ ¼ 74 π (2l  1)thSI (2l )thLS

SI: sink, SO: source, US: upper saddle, LS: lower saddle

λ1 ¼ R2ð2lÞ < 0, λ2 ¼ R2ð2lÞ > 0:

ð8:176Þ

Therefore, the equilibrium of (0, 0) is a saddle. From the local analysis, the behaviors of equilibriums are summarized in Table 8.5. The first integral manifold of Eq. (8.168) is x22  x21 ¼ C1 , x21  x22 ¼ C2 :

ð8:177Þ

If the first integral manifolds go through the simple and intersection equilibrium, and the infinite equilibrium, the corresponding first integral manifolds are given by x22  x21 ¼ R2 , x21  x22 ¼ R2

g

for the intersection equilibriums,

2 x22  x21 ¼ C 1  x2 2  x1 , 2 x21  x22 ¼ C 2  x2 1  x2 , 2 2 x2 1 þ x2 ¼ R

x2 ¼ x1 , x2 ¼ x1

g

g

for the infinite equilibrium,

ð8:178Þ

for the simple equilibrium of ð0, 0Þ:

From the local analysis of equilibrium stability, equilibriums with the first integral manifolds are presented in Fig. 8.8. The solid and dashed thick lines are for sink and source infinite equilibriums, respectively. The circles are for the first integral manifolds. The hollow circular symbols are for lower-saddles and uppersaddles in one direction, related to the infinte-equilibrium circle and coordinates. The equilibrium at the center point is a saddle-node point.

408 Fig. 8.8 Equilibrium stability for an infiniteequilibrium system in Eq. (8.159). (a) The saddle node of (0, 0) and (2l  1)th-order sources and sinks, and (b) the saddle node of (0, 0) and (2l )thorder upper saddles and lower saddles. The thick circle is for the infinite equilibrium. The hollow circular symbols are for simple equilibrium and intersection equilibriums between the infiniteequilibrium circle and coordinates. SI: sink, SO: source, US: upper saddle, LS: lower saddle, SN: saddle node

8 Infinite-Equilibrium Systems

a

b

Reference Luo, A.C.J., 2019, On dynamics of infinite-equilibrium systems, International Journal of Dynamics and Control, in press.

Index

A Antenna switching bifurcation, 255, 256, 277, 352 Appearing bifurcation, 248–254, 265–275, 306–319, 332–334, 336–339, 341, 344, 346–348 Asymptotically stable equilibrium, 8, 48 Asymptotically unstable equilibrium, 8, 46 Autonomous system, 2, 124

B Bifurcation, 59–60, 97–104, 231–237, 289–294 Bifurcation point, 60 Bifurcation value, 60 Broom appearing bifurcations, 307–309, 332, 333, 335 Broom-sprinkler-spraying appearing bifurcations, 310

C Center, 14–16 Center manifold, 7 Center subspace, 5 Circular-infinite-equilibrium systems, 400–407 Constant velocity system, 150 Continuous dynamical systems, 1–4 Contravariant component, 9, 366 Covariant component, 9, 366 Cubic nonlinear systems, 164–185

D Decreasing saddle, 20 Degenerate equilibrium, 13, 14, 20, 34 (2m)th -Degree polynomial systems, 231–286 (2m+1)th -Degree polynomial systems, 289–361 Dynamical systems, 1–4

E Eigenvector, 9 Eigenvector contravariant matrix, 9 Eigenvector contravariant vector, 9 Eigenvector covariant matrix, 9 Equi-distance surface, 49 Equilibrium, 4–18 Equilibrium computations, 365–366 Equilibrium stability, 127–130 Equi-measuring function surface, 49

F Finite-equilibrium system, 125 Fish-bone switching bifurcation, 273, 319 Fish-scale appearing bifurcation, 254, 275 Flow, 1 Flower-bundle switching bifurcation, 262, 263, 285, 286

G G-function, 51 Global stability, 231–237, 289–294

© Springer Nature Switzerland AG 2019 A. C. J. Luo, Bifurcation and Stability in Nonlinear Dynamical Systems, Nonlinear Systems and Complexity 28, https://doi.org/10.1007/978-3-030-22910-8

409

410 H Higher-order equilibrium bifurcation, 265–286, 331–361 Homeomorphism, 4 Hopf bifurcation, 69–85, 117–120 Hyperbolic bifurcation, 60–69 Hyperbolic equilibrium, 14, 15

I Increasing saddle, 20 Infinite-equilibrium system, 123, 365–407 Invariant circle, 13 Invariant manifold, 6 Invariant subspace, 5

J Jacobian matrix, 3

L Left-parallel-bundle switching bifurcation, 261, 284, 327, 359 Limit cycle, 77 Linear system, 149–151 Lipschitz condition, 3 Locally decreasing flow, 54 Locally decreasing flow of the (2s)th order, 56 Locally increasing flow, 54 Locally increasing flow of the (2s)th order, 56 Locally tangential flow of the (2s+1)th order, 56 Local stable manifold, 6, 7 Local unstable manifold, 6, 7 Low-degree polynomial system, 149–213 Lower saddle, 20, 88, 153, 157, 158, 160 Lower-saddle bifurcation, 207 Lower-saddle flow, 205, 207 Lower-saddle Hopf bifurcation, 73, 120 Lower-saddle-node bifurcation, 66, 68, 102, 154, 169–171, 191–194 Lower-saddle-node bifurcation of the mkth order, 62, 64 Lower-saddle-node bifurcation of the (2m+1)th order, 100 Lower-saddle-node switching bifurcation, 102 Lower saddle of the mkth order, 27 Lower-saddle of the (2m)th order, 94 Lyapunov functions, 48–57

M Measuring function, 49 Metric tensors, 9

Index N Negative flow, 152, 162, 186, 190, 208, 232, 237 Network-infinite-equilibrium system, 396–407 Non-autonomous system, 2, 124 Non-equilibrium flow, 152 Normal form, 373–385

O One-equilibrium system, 130–131 Operator norm, 2

P Periodic motion, 77 Permanent static system, 150 Pitchfork bifurcation, 69, 104, 109 Pitchfork bifurcation of the (2m+1)th order, 100 Positive flow, 152, 162, 186, 190, 208, 232, 237

Q Quadratic nonlinear system, 151–163 Quartic nonlinear system, 186–213

R Right-parallel-bundle switching bifurcation, 261, 284, 327, 360

S Saddle, 14, 15, 17 Saddle flow, 166–168, 187–189 Saddle-node appearing/vanishing bifurcation, 101 Saddle-node bifurcation, 66 Saddle-node switching bifurcation, 102, 104–106 Simple equilibrium, 165 Simple equilibrium bifurcation, 248–258, 306–319 Singularity, 18–43, 87–97, 110–117 Sink, 14, 15, 19, 88 Sink bifurcation, 170, 172, 193, 206 Sink bifurcation of the mkth order, 63, 64 Sink flow, 167, 168, 189, 290, 295 Sink Hopf bifurcation, 73 Sink of the mkth order, 26 Sink of the (2m+1)th order, 94

Index Source, 14, 15, 20, 88 Source bifurcation, 170, 172, 193, 206 Source bifurcation of the mkth order, 63, 65 Source flow, 167, 168, 189, 290, 295 Source Hopf bifurcation, 73 Source of the mkth order, 27 Source of the (2m+1)th order, 94 Spirally stable equilibrium, 13, 34, 43, 46, 48 Spirally stable equilibrium of the mkth order, 35, 39 Spirally unstable equilibrium, 13, 34, 46 Spirally unstable equilibrium of the mkth order, 34 Spiral saddle, 18 Spiral sink, 18 Spiral source, 18 Spiral stability, 30–43 Spraying appearing bifurcation, 250, 267–270, 272, 336, 337, 339, 344, 346 Sprinkler-spraying-appearing bifurcation, 250, 251, 253, 270–272, 310–313, 339, 344, 345 Stability, 4–43, 48–57, 87–97, 110–117 Stable equilibrium, 8, 11 Stable node, 14, 17, 19 Stable node of the mkth order, 26 Stable subspace, 5 Straw-bundle switching bifurcation, 257, 258, 263, 280, 281, 323, 355 Switching-appearing bifurcation, 260–265, 324–329, 356–361 Switching bifurcation, 66, 67, 255–258, 276–281, 319–324, 351–356 Switching point, 59 Switching value, 60

411 T Teeth comb appearing bifurcation, 249, 267 Three-equilibrium systems, 139–145 Transcritical bifurcation, 67 Transcritical switching bifurcation, 102 Two-equilibrium system, 131–139

U Uniformly decreasing flow, 55 Uniformly increasing flow, 55 Uniformly invariant flow, 55 Unstable equilibrium, 8, 11 Unstable node, 14, 17, 20 Unstable node of the mkth order, 27 Unstable pitchfork bifurcation of (2m+1)th order, 100 Unstable subspace, 5 Upper-saddle, 20, 153, 157, 160 Upper-saddle bifurcation, 205, 207 Upper saddle flow, 191 Upper-saddle Hopf bifurcation, 73, 120 Upper-saddle of the mkth order, 27 Upper-saddle of the (2m)th order, 100 Upper-saddle-node bifurcation, 66, 67, 102, 153, 170, 171, 191–194 Upper-saddle-node bifurcation of the mkth order, 62, 64 Upper-saddle-node switching bifurcation, 102

V Vector field, 1

Z Zero-order G-function, 51