Planar Dynamical Systems

In 2008, November 23-28, the workshop of Classical Problems on Planar Polynomial Vector Fields was held in the Banff International Research Station, Canada. Called "classical problems", it was concerned with the following:

(1) Problems on integrability of planar polynomial vector fields.

(2) The problem of the center stated by Poincaré for real polynomial differential systems, which asks us to recognize when a planar vector field defined by polynomials of degree at most n possesses a singularity which is a center.

(3) Global geometry of specific classes of planar polynomial vector fields.

(4) Hilbert's 16th problem.

These problems had been posed more than 110 years ago. Therefore, they are called "classical problems" in the studies of the theory of dynamical systems. The qualitative theory and stability theory of differential equations, created by Poincaré and Lyapunov at the end of the 19th century, had major developments as two branches of the theory of dynamical systems during the 20th century. As a part of the basic theory of nonlinear science, it is one of the very active areas in the new millennium.

This book presents in an elementary way the recent significant developments in the qualitative theory of planar dynamical systems. The subjects are covered as follows: the studies of center and isochronous center problems, multiple Hopf bifurcations and local and global bifurcations of the equivariant planar vector fields which concern with Hilbert's 16th problem.

The book is intended for graduate students, post-doctors and researchers in dynamical systems. For all engineers who are interested in the theory of dynamical systems, it is also a reasonable reference. It requires a minimum background of a one-year course on nonlinear differential equations.

Autorentext

Yirong Liu, Jibin Li and Wentao Huang, Zhejiang Normal University, Jinhua, Zhejiang, P. R. China.

Klappentext
In 2008, November 23-28, the workshop of ?Classical Problems on Planar Polynomial Vector Fields ? was held in the Banff International Research Station, Canada. Called "classical problems", it was concerned with the following: (1) Problems on integrability of planar polynomial vector fields. (2) The problem of the center stated by Poincaré for real polynomial differential systems, which asks us to recognize when a planar vector field defined by polynomials of degree at most n possesses a singularity which is a center. (3) Global geometry of specific classes of planar polynomial vector fields. (4) Hilbert's 16th problem. These problems had been posed more than 110 years ago. Therefore, they are called "classical problems" in the studies of the theory of dynamical systems. The qualitative theory and stability theory of differential equations, created by Poincaré and Lyapunov at the end of the 19th century, had major developments as two branches of the theory of dynamical systems during the 20th century. As a part of the basic theory of nonlinear science, it is one of the very active areas in the new millennium. This book presents in an elementary way the recent significant developments in the qualitative theory of planar dynamical systems. The subjects are covered as follows: the studies of center and isochronous center problems, multiple Hopf bifurcations and local and global bifurcations of the equivariant planar vector fields which concern with Hilbert's 16th problem. The book is intended for graduate students, post-doctors and researchers in dynamical systems. For all engineers who are interested in the theory of dynamical systems, it is also a reasonable reference. It requires a minimum background of a one-year course on nonlinear differential equations.

Inhalt

Preface i
1 Basic Concept and Linearized Problem of Systems 1
1.1 Basic Concept and Variable Transformation . . . . . . . . . . 1
1.2 Resultant of the Weierstrass Polynomial and Multiplicity of
a Singular Point . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Quasi-Algebraic Integrals of Polynomial Systems . . . . . . . 12
1.4 Cauchy Majorant and Analytic Properties in a Neighborhood
of an Ordinary Point . . . . . . . . . . . . . . . . . . . . . . 17
1.5 Classification of Elementary Singular Points and Linearized
Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.6 Node Value and Linearized problem of the Integer-Ratio Node 33
1.7 Linearized Problem of the Degenerate Node . . . . . . . . . . 39
1.8 Integrability and Linearized Problem of Weak Critical Singular
Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
1.9 Integrability and Linearized Problem of the Resonant Singular
Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2 Focal Values, Saddle Values and Singular Point Values 77
2.1 Successor Functions and Properties of Focal Values . . . . . . 77
2.2 Poincar e Formal Series and Algebraic Equivalence . . . . . . 83
2.3 Linear Recursive Formulas for the Computation of Singular
Point Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
2.4 The Algebraic Construction of Singular Values . . . . . . . . 92
2.5 Elementary Generalized Rotation Invariants of the Cubic Systems
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
2.6 Singular Point Values and Integrability Condition of the
Quadratic Systems . . . . . . . . . . . . . . . . . . . . . . . . 100
2.6.1 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . 102

2.7 Singular Point Values and Integrability Condition of the Cubic
Systems Having Homogeneous Nonlinearities . . . . . . . 103
2.7.1 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . 104
3 Multiple Hopf Bifurcations 107
3.1 The Zeros of Successor Functions in the Polar Coordinates . . 107
3.2 Analytic Equivalence . . . . . . . . . . . . . . . . . . . . . . . 111
3.3 Quasi Successor Function . . . . . . . . . . . . . . . . . . . . 113
3.4 Bifurcations of Limit Circle of a Class of Quadratic Systems . 119
4 Isochronous Center In Complex Domain 123
4.1 Isochronous Centers and Period Constants . . . . . . . . . . 123
4.2 Linear Recursive Formulas to Compute Period Constants . . 129
4.3 Isochronous Center for a Class of Quintic System in the Complex
Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
4.3.1 The Conditions of Isochronous Center under Condition
C1 . . . . . . . . . . . . . . . . . . . . . . . . . . 136
4.3.2 The Conditions of Isochronous Center under Condition
C2 . . . . . . . . . . . . . . . . . . . . . . . . . . 137
4.3.3 The Conditions of Isochronous Center under Condition
C3 . . . . . . . . . . . . . . . . . . . . . . . . . . 141
4.3.4 Non-Isochronous Center under Condition C4 and C.4 . 142
4.4 The Method of Time-Angle Difference . . . . . . . . . . . . . 142
4.5 The Conditions of Isochronous Center of the Origin for a Cubic
System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
5 Theory of Center-Focus and Bifurcation of Limit Cycles at
Infinity of a Class of System 153
5.1 Definition of the Focal Values of Infinity . . . . . . . . . . . . 153
5.2 Conversion of Questions . . . . . . . . . . . . . . . . . . . . . 156
5.3 Method of Formal Series and Singular Point Value of Infinity 159
5.4 The Algebraic Construction of Singular Point Values of Infinity173
5.5 Singular Point Values at Infinity and Integrable Conditions
for a Class of Cubic System . . . . . . . . . . . . . . . . . . . 178
5.6 Bifurcation of Limit Cycles at Infinity . . . . . . . . . . . . . 185
5.7 Isochronous Centers at Infinity of a Polynomial Systems . . . 190
5.7.1 Conditions of Complex Center for System (5.7.6) . . . 191
5.7.2 Conditions of Complex Isochronous Center for System
(5.7.6) . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

6 Theory of Center-Focus and Bifurcations of Limit Cycles
For a Class of Multi…