平面动力系统的若干经典问题

Rough PerturbAtions of Zq-EquivAriAnt HAmiltoniAn Vector Fields.242

8.3 BifurcAtions of Limit Cycles of A Z2-EquivAriAnt Perturbed HAmiltoniAn Vector Fields244

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 n258

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

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

9.5

9.6 TheProofsofLemmA9.4.2AndLemmA9.4.3 . 300

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

　　浅水波，非线性光学、电磁学、等离子物理、凝聚态物理、生物及化学、通讯等领域均存在非线性波运动。对其数学模型--波方程的解研究有重要价值。上世纪90年代，数学家发现了行波方程的非光滑的孤粒子解（peakon）、有限支集解（compacton）和圈解（loopsolution）等，为理解这些解，特别是非光滑解的出现，导致用动力系统的分支理论及方法对奇行波方程进行研究的新方向。《数学专著丛书：平面动力系统的若干经典问题（英文版）》介绍两类奇行波方程的研究的动力系统方法，及对大量数学物理问题的应用。