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

Preface v

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 3

1.3 Quasi-Algebraic Integrals of Polynomial Systems 10

1.4 Cauchy Majorant and Analytic Properties in a Neighborhood of an Ordinary Point 15

1.5 Classification of Elementary Singular Points and Linearized Problem 24

1.6 Node Value and Linearized Problem of the Integer-Ratio Node 30

1.7 Linearized Problem of the Degenerate Node 35

1.8 Integrability and Linearized Problem of Weak Critical Singular Point 39

1.9 Integrability and Linearized Problem of the Resonant Singular Point 58

2 Focal Values, Saddle Values and Singular Point Values 69

2.1 Successor Functions and Properties of Focal Values 69

2.2 Poincar’e Formal Series and Algebraic Equivalence 74

2.3 Linear Recursive Formulas for the Computation of Singular Point Values 78

2.4 The Algebraic Construction of Singular Values 83

2.5 Elementary Generalized Rotation Invariants of the Cubic Systems 88

2.6 Singular Point Values and Integrability Condition of the Quadratic Systems 90

2.7 Singular Point Values and Integrability Condition of the Cubic Systems Having Homogeneous Nonlinearities 93

3 Multiple Hopf Bifurcations 97

3.1 The Zeros of Successor Functions in the Polar Coordinates 97

3.2 Analytic Equivalence 100

3.3 Quasi Successor Function 102

3.4 Bifurcations of Limit Circle of a Class of Quadratic Systems 108

4 Isochronous Center In Complex Domain 111

4.1 Isochronous Centers and Period Constants 111

4.2 Linear Recursive Formulas to Compute Period Constants 116

4.3 Isochronous Center for a Class of Quintic System in the Complex Domain 122

      4.3.1 The Conditions of Isochronous Center Under Condition C 1 123

      4.3.2 The Conditions of Isochronous Center Under Condition C 2 124

      4.3.3 The Conditions of Isochronous Center Under Condition C 3 127

      4.3.4 Non-Isochronous Center under Condition C_4 and C~_4 128

4.4 The Method of Time-Angle Difference 128

4.5 The Conditions of Isochronous Center of the Origin for a Cubic System 134

5 Theory of Center-Focus and Bifurcation of Limit Cycles at Infinity of a Class of Systems 138

5.1 Definition of the Focal Values of Infinity 138

5.2 Conversion of Questions 141

5.3 Method of Formal Series and Singular Point Value of Infinity 144

5.4 The Algebraic Construction of Singular Point Values of Infinity 156

5.5 Singular Point Values at Infinity and Integrable Conditions for a Class of Cubic System 161

5.6 Bifurcation of Limit Cycles at Infinity 168

5.7 Isochronous Centers at Infinity of a Polynomial Systems 172

      5.7.1 Conditions of Complex Center for System (5.7.6) 173

      5.7.2 Conditions of Complex Isochronous Center for System (5.7.6) 176

6 Theory of Center-Focus and Bifurcations of Limit Cycles for a Class of Multiple Singular Points 180

6.1 Succession Function and Focal Values for a Class of Multiple Singular Points 180

6.2 Conversion of the Questions 182

6.3 Formal Series, Integral Factors and Singular Point Values for a Class of Multiple Singular Points 184

6.4 The Algebraic Structure of Singular Point Values of a Class of Multiple Singular Points 196

6.5 Bifurcation of Limit Cycles From a Class of Multiple Singular Points 198

6.6 Bifurcation of Limit Cycles Created from a Multiple Singular Point for a Class of Quartic System 199

6.7 Quasi Isochronous Center of Multiple Singular Point for a Class of Analytic System 202

7 On Quasi Analytic Systems 205

7.1 Preliminary 205

7.2 Reduction of the Problems 208

7.3 Focal Values, Periodic Constants and First Integrals of (7.2.3) 210

7.4 Singular Point Values and Bifurcations of Limit Cycles of Quasi-Quadratic Systems 214

7.5 Integrability of Quasi-Quadratic Systems 217

7.6 Isochronous Center of Quasi-Quadratic Systems 219

      7.6.1 The Problem of Complex Isochronous Centers Under the Condition of C 1 219

      7.6.2 The Problem of Complex Isochronous Centers Under the Condition of C 2 222

      7.6.3 The Problem of Complex Isochronous Centers Under the Other Conditions 225

      7.7 Singular Point Values and Center Conditions for a Class of Quasi-Cubic Systems 228

8 Local and Non-Local Bifurcations of Perturbed Z q -Equivariant Hamiltonian Vector Fields 232

8.1 Z_q -Equivariant Planar Vector Fields and an Example 232

8.2 The Method of Detection Functions: Rough Perturbations of Z_q - Equivariant Hamiltonian Vector Fields 242

8.3 Bifurcations of Limit Cycles of a Z_2 - Equivariant Perturbed Hamiltonian Vector Fields 244

      8.3.1 Hopf Bifurcation Parameter Values 246

      8.3.2 Bifurcations From Heteroclinic or Homoclinic Loops 247

      8.3.3 The Values of Bifurcation Directions of Heteroclinic and Homoclinic Loops 252

      8.3.4 Analysis and Conclusions 255

8.4 The Rate of Growth of Hilbert Number H(n) with n 258

      8.4.1 Preliminary Lemmas 259

      8.4.2 A Correction to the Lower Bounds of H(2~k − 1) Given in [Christopher and Lloyd, 1995] 262

      8.4.3 A New Lower Bound for H(2~k − 1) 265

      8.4.4 Lower Bound for H(3 ×2~( k−1) − 1) 267

9 Center-Focus Problem and Bifurcations of Limit Cycles for a Z 2 -Equivariant Cubic System 272

9.1 Standard Form of a Class of System (E~(Z_2)_3) 272

9.2 Liapunov Constants, Invariant Integrals and the Necessary and Sufficient Conditions of the Existence for the Bi-Center 274

9.3 The Conditions of Six-Order Weak Focus and Bifurcations of Limit Cycles 286

9.4 A Class of (E~(Z_2)_3) System With 13 Limit Cycles 290

9.5 Proofs of Lemma 9.4.1 and Theorem 9.4.1 294

9.6 The Proofs of Lemma 9.4.2 and Lemma 9.4.3 300

10 Center-Focus Problem and Bifurcations of Limit Cycles for Three-Multiple Nilpotent Singular Points 308

10.1 Criteria of Center-Focus for a Nilpotent Singular Point 308

10.2 Successor Functions and Focus Value of Three-Multiple Nilpotent Singular Point 311

10.3 Bifurcation of Limit Cycles Created from Three-Multiple Nilpotent Singular Point 314

10.4 The Classification of Three-Multiple Nilpotent Singular Points and Inverse Integral Factor 322

10.5 Quasi-Lyapunov Constants For the Three-Multiple Nilpotent Singular Point 326

10.6 Proof of Theorem 10.5.2 329

10.7 On the Computation of Quasi-Lyapunov Constants 334

10.8 Bifurcations of Limit Cycles Created from a Three-Multiple Nilpotent Singular Point of a Cubic System 336

Bibliography 342

Index 369

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