Title Page -- Copyright Page -- Preface -- I. Center-focus problem -- II. Small-amplitude limit cycles created by multiple Hopf bifurcations -- III. Local and non-local bifurcations of Zq-equivariant perturbed planarHamiltonian vector fields -- IV. Isochronous center problem and periodic map -- Table of Contents -- Chapter 1 - Basic Concept and Linearized Problem of Systems -- 1.1 Basic Concept and Variable Transformation -- 1.2 Resultant of the Weierstrass Polynomial and Multiplicity of a Singular Point -- 1.3 Quasi-Algebraic Integrals of Polynomial Systems -- 1.4 Cauchy Majorant and Analytic Properties in a Neighborhood of an Ordinary Point -- 1.5 Classification of Elementary Singular Points and Linea-rized Problem -- 1.6 Node Value and Linearized Problem of the Integer-Ratio Node -- 1.7 Linearized Problem of the Degenerate Node -- 1.8 Integrability and Linearized Problem of Weak Critical Singular Point -- 1.9 Integrability and Linearized Problem of the Resonant Singular Point -- Chapter 2 - Focal Values, Saddle Values and Singular Point Values -- 2.1 Successor Functions and Properties of Focal Values -- 2.2 Poincaré Formal Series and Algebraic Equivalence -- 2.3 Linear Recursive Formulas for the Computation of Singular Point Values -- 2.4 The Algebraic Construction of Singular Values -- 2.5 Elementary Generalized Rotation Invariants of the Cubic Systems -- 2.6 Singular Point Values and Integrability Condition of the Quadratic Systems -- 2.7 Singular Point Values and Integrability Condition of theCubic Systems Having Homogeneous Nonlinearities -- Chapter 3 - Multiple Hopf Bifurcations -- 3.1 The Zeros of Successor Functions in the Polar Coordinates -- 3.2 Analytic Equivalence -- 3.3 Quasi Successor Function -- 3.4 Bifurcations of Limit Circle of a Class of Quadratic Systems -- Chapter 4 - Isochronous Center In Complex Domain.

4.1 Isochronous Centers and Period Constants -- 4.2 Linear Recursive Formulas to Compute Period Constants -- 4.3 Isochronous Center for a Class of Quintic System in the Complex Domain -- 4.3.1 The Conditions of Isochronous Center Under Condition C1 -- 4.3.2 The Conditions of Isochronous Center Under Condition C2 -- 4.3.3 The Conditions of Isochronous Center Under Condition C3 -- 4.3.4 Non-Isochronous Center under Condition C4 and -- 4.4 The Method of Time-Angle Difference -- 4.5 The Conditions of Isochronous Center of the Origin for a Cubic System -- Chapter 5 - Theory of Center-Focus and Bifurcation of Limit Cycles at Infinity of a Class of Systems -- 5.1 Definition of the Focal Values of Infinity -- 5.2 Conversion of Questions -- 5.3 Method of Formal Series and Singular Point Value of Infinity -- 5.4 The Algebraic Construction of Singular Point Values of Infinity -- 5.5 Singular Point Values at Infinity and Integrable Conditions for a Class of Cubic System -- 5.6 Bifurcation of Limit Cycles at Infinity -- 5.7 Isochronous Centers at Infinity of a Polynomial Systems -- 5.7.1 Conditions of Complex Center for System (5.7.6) -- 5.7.2 Conditions of Complex Isochronous Center for System (5.7.6) -- Chapter 6 - Theory of Center-Focus and Bifurcations of Limit Cycles for a Class of Multiple Singular Points -- 6.1 Succession Function and Focal Values for a Class of Multiple Singular Points -- 6.2 Conversion of the Questions -- 6.3 Formal Series, Integral Factors and Singular Point Values for a Class of Multiple Singular Points -- 6.4 The Algebraic Structure of Singular Point Values of a Class of Multiple Singular Points -- 6.5 Bifurcation of Limit Cycles From a Class of Multiple Singular Points -- 6.6 Bifurcation of Limit Cycles Created from a Multiple Singular Point for a Class of Quartic System.

6.7 Quasi Isochronous Center of Multiple Singular Point for a Class of Analytic System -- Chapter 7 - On Quasi Analytic Systems -- 7.1 Preliminary -- 7.2 Reduction of the Problems -- 7.3 Focal Values, Periodic Constants and First Integrals of (7.2.3) -- 7.4 Singular Point Values and Bifurcations of Limit Cycles of Quasi-Quadratic Systems -- 7.5 Integrability of Quasi-Quadratic Systems -- 7.6 Isochronous Center of Quasi-Quadratic Systems -- 7.6.1 The Problem of Complex Isochronous Centers Under the Condition of C1 -- 7.6.2 The Problem of Complex Isochronous Centers Under the Condition of C2 -- 7.6.3 The Problem of Complex Isochronous Centers Under the Other Conditions -- 7.7 Singular Point Values and Center Conditions for a Class of Quasi-Cubic Systems -- Chapter 8 - Local and Non-Local Bifurcations of Perturbed Zq-Equivariant Hamiltonian Vector Fields -- 8.1 Zq-Equivariant Planar Vector Fields and an Example -- 8.2 The Method of Detection Functions: Rough Perturbations of Zq-Equivariant Hamiltonian Vector Fields -- 8.3 Bifurcations of Limit Cycles of a Z2-Equivariant Perturbed Hamiltonian Vector Fields -- 8.3.1 Hopf Bifurcation Parameter Values -- 8.3.2 Bifurcations from Heteroclinic or Homoclinic Loops -- 8.3.3 The Values of Bifurcation Directions of Heteroclinic and HomoclinicLoops -- 8.3.4 Analysis and Conclusions -- 8.4 The Rate of Growth of Hilbert Number H(n) with n -- 8.4.1 Preliminary Lemmas -- 8.4.2 A Correction to the Lower Bounds of H (2k − 1) Given in[Christopher and Lloyd, 1995] -- 8.4.3 A New Lower Bound for H (2k − 1) -- 8.4.4 Lower Bound for H (3 × 2k−1 − 1) -- Chapter 9 - Center-Focus Problem and Bifurcations of Limit Cycles for a Z2-Equivariant Cubic System -- 9.1 Standard Form of a Class of System () -- 9.2 Liapunov Constants, Invariant Integrals and the Necessary and Sufficient Conditions of the Existence for the Bi-Center.

9.3 The Conditions of Six-Order Weak Focus and Bifurcations of Limit Cycles -- 9.4 A Class of ( ) System With 13 Limit Cycles -- 9.5 Proofs of Lemma 9.4.1 and Theorem 9.4.1 -- 9.6 The Proofs of Lemma 9.4.2 and Lemma 9.4.3 -- 9.7 Appendix -- Chapter 10 - Center-Focus Problem and Bifurcations of Limit Cycles for Three-Multiple Nilpotent Singular Points -- 10.1 Criteria of Center-Focus for a Nilpotent Singular Point -- 10.2 Successor Functions and Focus Value of Three-Multiple Nilpotent Singular Point -- 10.3 Bifurcation of Limit Cycles Created from Three-Multiple Nilpotent Singular Point -- 10.4 The Classification of Three-Multiple Nilpotent SingularPoints and Inverse Integral Factor -- 10.5 Quasi-Lyapunov Constants For the Three-Multiple Nilpotent Singular Point -- 10.6 Proof of Theorem 10.5.2 -- 10.7 On the Computation of Quasi-Lyapunov Constants -- 10.8 Bifurcations of Limit Cycles Created from a Three-Multiple Nilpotent Singular Point of a Cubic System -- Bibliography -- Index.