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 QuadraticSystems 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 AnalyticEquivalence 100
QuasiSuccessor Function 102
3.3
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
xii Contents
4.2 Linear Recursive Formulas to Compute Period Constants 116
4.3 Isochronous Center for a Class of Quintic System in the Complex Domain 122 123
4.3.1 The Conditions of Isochronous Center Under Condition C1
124
4.3.2 The Conditions of Isochronous Center Under Condition C2
127
4.3.3 The Conditions of Isochronous Center Under Condition C3
. . . 128
4.3.4 Non-Isochronous Center under Condition C4 and C4 .
4.4 TheMethodofTime-AngleDi.erence 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 In.nity of a Class of Systems 138
5.1 De.nition of the Focal Values of In.nity 138
Conversionof Questions 141
5.2
5.3 Method of Formal Series and Singular Point Value of In.nity . 144
5.4 The Algebraic Construction of Singular Point Values of In.nity . 156
5.5 Singular Point Values at In.nity and Integrable Conditions for a Class of Cubic System 161
5.6 Bifurcation of Limit Cycles at In.nity 168
5.7 Isochronous Centers at In.nity 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 aClass of MultipleSingularPoints . 180
6.1 Succession Function and Focal Values for a Class of Multiple Singular Points 180 Conversion of the Questions 182
6.2
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
Contents xiii
a Class of Analytic System 202
7 OnQuasi Analytic Systems 205
7.1 Preliminary 205
Reduction of the Problems 208
7.2
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
219
the Condition of C1
7.6.2 The Problem of Complex Isochronous Centers Under
222
the Condition of C2
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 Zq-Equivariant Hamiltonian Vector Fields 232
8.1 Zq-Equivariant Planar Vector Fields and an Example 232
8.2 The Method of Detection Functions: Rough Perturbations of Zq-Equivariant Hamiltonian Vector Fields 242
8.3 Bifurcations of Limit Cycles of a Z2-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(2k .1) Given in [Christopher and Lloyd, 1995] 262
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