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

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 Multiple Singular Points 199
6.1 Succession Function and Focal Values for a Class of Multiple
Singular Points . . . . . . . . . . . . . . . . . . . . . . . . . . 199
6.2 Conversion of the Questions . . . . . . . . . . . . . . . . . . . 201
6.3 Formal Series, Integral Factors and Singular Point Values for
a Class of Multiple Singular Points . . . . . . . . . . . . . . . 203
6.4 The Algebraic Structure of Singular Point Values of a Class
of Multiple Singular Points . . . . . . . . . . . . . . . . . . . 217
6.5 Bifurcation of Limit Cycles From a Class of Multiple Singular
Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
6.6 Bifurcation of Limit Cycles Created from a Multiple Singular
Point for a Class of Quartic System . . . . . . . . . . . . . . 221
6.7 Quasi Isochronous Center of Multiple Singular Point for a
Class of Analytic System . . . . . . . . . . . . . . . . . . . . 224
7 On Quasi Analytic Systems 227
7.1 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
7.2 Reduction of the Problems . . . . . . . . . . . . . . . . . . . . 230
7.3 Focal Values, Periodic Constants and First Integrals of (7.2.3) 232
7.4 Singular Point Values and Bifurcations of Limit Cycles of
Quasi-Quadratic Systems . . . . . . . . . . . . . . . . . . . . 236
7.5 Integrability of Quasi-Quadratic Systems . . . . . . . . . . . . 240
7.6 Isochronous Center of Quasi-Quadratic Systems . . . . . . . . 243
7.6.1 The Problem of Complex Isochronous Centers Under
the Condition of C1 . . . . . . . . . . . . . . . . . . . 243
7.6.2 The Problem of Complex Isochronous Centers Under
the Condition of C2 . . . . . . . . . . . . . . . . . . . 246
7.6.3 The Problem of Complex Isochronous Centers Under
the Other Conditions . . . . . . . . . . . . . . . . . . 250
7.7 Singular Point Values and Center Conditions for a Class of
Quasi-Cubic Systems . . . . . . . . . . . . . . . . . . . . . . . 253
8 Local and Non-Local Bifurcations of Perturbed Zq- Equiv-
ariant Hamiltonian Vector Fields 257
8.1 Zq-Equivariant Planar Vector Fields and an Example . . . . . 258
8.2 The Method of Detection Functions: Rough Perturbations of
Zq-Equivariant Hamiltonian Vector Fields . . . . . . . . . . 267
8.3 Bifurcations of Limit Cycles of a Z2- Equivariant Perturbed
Hamiltonian Vector Fields . . . . . . . . . . . . . . . . . . . 269

8.3.1 Hopf Bifurcation Parameter Values . . . . . . . . . . 271
8.3.2 Bifurcations From Heteroclinic or Homoclinic Loops . 273
8.3.3 The Values of Bifurcation Directions of Heteroclinic
and Homoclinic Loops . . . . . . . . . . . . . . . . . . 278
8.3.4 Analysis and Conclusions . . . . . . . . . . . . . . . . 281
8.4 The Rate of Growth of Hilbert Number H(n) with n . . . . . 285
8.4.1 Preliminary Lemmas . . . . . . . . . . . . . . . . . . . 285
8.4.2 A Correction to the Lower Bounds of H(2k -1) Given
in [Christopher and Lloyd, 1995] . . . . . . . . . . . . 289
8.4.3 A New Lower Bound for H(2k - 1) . . . . . . . . . . . 291
8.4.4 Lower Bound for H(3 × 2k-1 - 1) . . . . . . . . . . . 293
9 Center-Focus Problem and Bifurcations of Limit Cycles for
a Z2-Equivariant Cubic System 299
9.1 Standard Form of a Class of System EZ2
3 . . . . . . . . . . . . 299
9.2 Liapunov Constants, Invariant Integrals and the Necessary
and Sufficient Conditions of the Existence for the Bi-Center . 301
9.3 The Conditions of Six-Order Weak Focus and Bifurcations of
Limit Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
9.4 A Class of EZ2
3 System With 13 Limit Cycles . . . . . . . . . 318
9.5 Proofs of Lemma 9.4.1 and Theorem 9.4.1 . . . . . . . . . . . 322
9.6 The Proofs of Lemma 9.4.2 and Lemma 9.4.3 . . . . . . . . . 330
9.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
10 Center-Focus Problem and Bifurcations of Limit Cycles for
Three-Multiple Nilpotent Singular Points 339
10.1 Criteria of Center-Focus for a Nilpotent Singular Point . . . . 339
10.2 Successor Functions and Focus Value of Three-Multiple
Nilpotent Singular Point . . . . . . . . . . . . . . . . . . . . . 342
10.3 Bifurcation of Limit Cycles Created from Three-Multiple
Nilpotent Singular Point . . . . . . . . . . . . . . . . . . . . . 346
10.4 The Classification of Three-Multiple Nilpotent Singular
points and Inverse Integral Factor . . . . . . . . . . . . . . . 354
10.5 Quasi-Lyapunov Constants For the Three-Multiple Nilpotent
Singular Point . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
10.6 Proof of Theorem 10.5.2 . . . . . . . . . . . . . . . . . . . . . 363
10.7 On the Computation of Quasi-Lyapunov Constants . . . . . . 368
10.8 Bifurcations of Limit Cycles Created From a Three-Multiple
Nilpotent Singular Point of a Cubic System . . . . . . . . . . 371

Bibliography 377

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.