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. This book is intended for graduate students, post-doctors and researchers in the area of theories and applications of dynamical systems. For all engineers who are interested the theory of dynamical systems, it is also a reasonable reference. It requires a minimum background of an one-year course on nonlinear differential equations.
Preface v
1 Basic Concept and Linearized Problem of 1 (68)
Systems
1.1 Basic Concept and Variable 1 (2)
Transformation
1.2 Resultant of the Weierstrass 3 (7)
Polynomial and Multiplicity of a Singular
Point
1.3 Quasi-Algebraic Integrals of 10 (5)
Polynomial Systems
1.4 Cauchy Majorant and Analytic 15 (9)
Properties in a Neighborhood of an
Ordinary Point
1.5 Classification of Elementary Singular 24 (6)
Points and Linearized Problem
1.6 Node Value and Linearized Problem of 30 (5)
the Integer-Ratio Node
1.7 Linearized Problem of the Degenerate 35 (4)
Node
1.8 Integrability and Linearized Problem 39 (19)
of Weak Critical Singular Point
1.9 Integrability and Linearized Problem 58 (11)
of the Resonant Singular Point
2 Focal Values, Saddle Values and Singular 69 (28)
Point Values
2.1 Successor Functions and Properties of 69 (5)
Focal Values
2.2 Poincare Formal Series and Algebraic 74 (4)
Equivalence
2.3 Linear Recursive Formulas for the 78 (5)
Computation of Singular Point Values
2.4 The Algebraic Construction of 83 (5)
Singular Values
2.5 Elementary Generalized Rotation 88 (2)
Invariants of the Cubic Systems
2.6 Singular Point Values and 90 (3)
Integrability Condition of the Quadratic
Systems
2.7 Singular Point Values and 93 (4)
Integrability Condition of the Cubic
Systems Having Homogeneous Nonlinearities
3 Multiple Hopf Bifurcations 97 (14)
3.1 The Zeros of Successor Functions in 97 (3)
the Polar Coordinates
3.2 Analytic Equivalence 100(2)
3.3 Quasi Successor Function 102(6)
3.4 Bifurcations of Limit Circle of a 108(3)
Class of Quadratic Systems
4 Isochronous Center In Complex Domain 111(27)
4.1 Isochronous Centers and Period 111(5)
Constants
4.2 Linear Recursive Formulas to Compute 116(6)
Period Constants
4.3 Isochronous Center for a Class of 122(6)
Quintic System in the Complex Domain
4.3.1 The Conditions of Isochronous 123(1)
Center Under Condition C1
4.3.2 The Conditions of Isochronous 124(3)
Center Under Condition C2
4.3.3 The Conditions of Isochronous 127(1)
Center Under Condition C3
4.3.4 Non-Isochronous Center under 128(1)
Condition C4 and C4
4.4 The Method of Time-Angle Difference 128(6)
4.5 The Conditions of Isochronous Center 134(4)
of the Origin for a Cubic System
5 Theory of Center-Focus and Bifurcation of 138(42)
Limit Cycles at Infinity of a Class of
Systems
5.1 Definition of the Focal Values of 138(3)
Infinity
5.2 Conversion of Questions 141(3)
5.3 Method of Formal Series and Singular 144(12)
Point Value of Infinity
5.4 The Algebraic Construction of 156(5)
Singular Point Values of Infinity
5.5 Singular Point Values at Infinity and 161(7)
Integrable Conditions for a Class of
Cubic System
5.6 Bifurcation of Limit Cycles at 168(4)
Infinity
5.7 Isochronous Centers at Infinity of a 172(8)
Polynomial Systems
5.7.1 Conditions of Complex Center for 173(3)
System (5.7.6)
5.7.2 Conditions of Complex Isochronous 176(4)
Center for System (5.7.6)
6 Theory of Center-Focus and Bifurcations 180(25)
of Limit Cycles for a Class of Multiple
Singular Points
6.1 Succession Function and Focal Values 180(2)
for a Class of Multiple Singular Points
6.2 Conversion of the Questions 182(2)
6.3 Formal Series, Integral Factors and 184(12)
Singular Point Values for a Class of
Multiple Singular Points
6.4 The Algebraic Structure of Singular 196(2)
Point Values of a Class of Multiple
Singular Points
6.5 Bifurcation of Limit Cycles From a 198(1)
Class of Multiple Singular Points
6.6 Bifurcation of Limit Cycles Created 199(3)
from a Multiple Singular Point for a
Class of Quartic System
6.7 Quasi Isochronous Center of Multiple 202(3)
Singular Point for a Class of Analytic
System
7 On Quasi Analytic Systems 205(27)
7.1 Preliminary 205(3)
7.2 Reduction of the Problems 208(2)
7.3 Focal Values, Periodic Constants and 210(4)
First Integrals of (7.2.3)
7.4 Singular Point Values and 214(3)
Bifurcations of Limit Cycles of
Quasi-Quadratic Systems
7.5 Integrability of Quasi-Quadratic 217(2)
Systems
7.6 Isochronous Center of Quasi-Quadratic 219(9)
Systems
7.6.1 The Problem of Complex 219(3)
Isochronous Centers Under the Condition
of C1
7.6.2 The Problem of Complex 222(3)
Isochronous Centers Under the Condition
of C2
7.6.3 The Problem of Complex 225(3)
Isochronous Centers Under the Other
Conditions
7.7 Singular Point Values and Center 228(4)
Conditions for a Class of Quasi-Cubic
Systems
8 Local and Non-Local Bifurcations of 232(40)
Perturbed Zq-Equivariant Hamiltonian Vector
Fields
8.1 Zq-Equivariant Planar Vector Fields 232(10)
and an Example
8.2 The Method of Detection Functions: 242(2)
Rough Perturbations of Zq- Equivariant
Hamiltonian Vector Fields
8.3 Bifurcations of Limit Cycles of a Z2- 244(14)
Equivariant Perturbed Hamiltonian Vector
Fields
8.3.1 Hopf Bifurcation Parameter Values 246(1)
8.3.2 Bifurcations From Heteroclinic or 247(5)
Homoclinic Loops
8.3.3 The Values of Bifurcation 252(3)
Directions of Heteroclinic and
Homoclinic Loops
8.3.4 Analysis and Conclusions 255(3)
8.4 The Rate of Growth of Hilbert Number 258(14)
H(n) with n
8.4.1 Preliminary Lemmas 259(3)
8.4.2 A Correction to the Lower Bounds 262(3)
of h(2k -- 1) Given in [Christopher and
Lloyd, 1995]
8.4.3 A New Lower Bound for h(2k -- 1) 265(2)
8.4.4 Lower Bound for H(3 X 2k-1 -- 1) 267(5)
9 Center-Focus Problem and Bifurcations of 272(36)
Limit Cycles for a Z2-Equivariant Cubic
System
9.1 Standard Form of a Class of System (E) 272(2)
9.2 Liapunov Constants, Invariant 274(12)
Integrals and the Necessary and
Sufficient Conditions of the Existence
for the Bi-Center
9.3 The Conditions of Six-Order Weak 286(4)
Focus and Bifurcations of Limit Cycles
9.4 A Class of (E) System With 13 Limit 290(4)
Cycles
9.5 Proofs of Lemma 9.4.1 and Theorem 294(6)
9.4.1
9.6 The Proofs of Lemma 9.4.2 and Lemma 300(8)
9.4.3
10 Center-Focus Problem and Bifurcations of 308(34)
Limit Cycles for Three-Multiple Nilpotent
Singular Points
10.1 Criteria of Center-Focus for a 308(3)
Nilpotent Singular Point
10.2 Successor Functions and Focus Value 311(3)
of Three-Multiple Nilpotent Singular Point
10.3 Bifurcation of Limit Cycles Created 314(8)
from Three-Multiple Nilpotent Singular
Point
10.4 The Classification of Three-Multiple 322(4)
Nilpotent Singular Points and Inverse
Integral Factor
10.5 Quasi-Lyapunov Constants For the 326(3)
Three-Multiple Nilpotent Singular Point
10.6 Proof of Theorem 10.5.2 329(5)
10.7 On the Computation of Quasi-Lyapunov 334(2)
Constants
10.8 Bifurcations of Limit Cycles Created 336(6)
from a Three-Multiple Nilpotent Singular
Point of a Cubic System
Bibliography 342(27)
Index 369
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