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 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
8.4.2 A Correction to the Lower Bounds of H(2k .1) Given in [Christopher and Lloyd, 1995] ...................................262 8.4.3 A New Lower Bound for H(2k .1) ................................265
8.4.4 Lower Bound for H(3 × 2k.1 .1) .................................267
9 Center-Focus Problem and Bifurcations of Limit Cycles xiv Contents for a Z2-Equivariant Cubic System .. .. ... .. .. .. .. .. .. .. ... .. .. .. .. .. . 272 9.1 Standard Form of a Class of System (EZ2 ) . . . . . . . . . . . . . ... .. .. .. .. .. .. 272 3 9.2 Liapunov Constants, Invariant Integrals and the Necessary and Su.cient 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 (EZ2 ) System With 13 Limit Cycles ....................... 290
3 Proofs of Lemma 9.4.1 and Theorem 9.4.1 ............................ 294
10 Center-Focus Problem and Bifurcations of Limit Cycles forThree-Multiple NilpotentSingularPoints....................... 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 Classi.cation of Three-Multiple Nilpotent Singular Points and Inverse Integral Factor .......................................... 321 10.5 Quasi-Lyapunov Constants For the Three-Multiple Nilpotent Singular Point ............................................. 326 Proof of Theorem 10.5.2 ............................................. 329 10.6 10.7 On the Computation of Quasi-Lyapunov Constants .. .. .. ... .. .. .. .. . 333 10.8 Bifurcations of Limit Cycles Created from a Three-Multiple Nilpotent Singular Point of a Cubic System ........................... 336 Bibliography ................................................................ 341 Index ........................................................................ 368
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