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Chapter 1 Limit Cycle and Its Perturbations 1

1.1 Basic notations and facts 1

1.2 Further discussion on property of limit cycles 7

1.3 Perturbations of a limit cycle 13

Chapter 2 Focus Values and Hopf Bifurcation 20

2.1 Poincarémap and focus value 20

2.2 Normal form and Poincaré-Lyapunov technique 28

2.3 Hopf bifurcation near a focus and systems with symmetry 44

2.4 Degenerate Hopf bifurcation near a center 58

2.5 Hopf bifurcation for Liénard systems 72

2.6 Hopf bifurcation for some polynomial systems 92

Chapter 3 Perturbations of Hamiltonian Systems 106

3.1 General theory 106

3.2 Limit cycles near homoclinic and heteroclinic loops 124

3.3 Finding more limit cycles by Melnikov functions 163

3.4 Limit cycle bifurcations near a nilpotent center 183

3.5 Limit cycle bifurcations with a nilpotent cusp 200

3.6 Limit cycle bifurcations with a nilpotent saddle 214

Chapter 4 Stability of Homoclinic Loops and Limit Cycle Bifurcations 254

4.1 Local behavior near a saddle 254

4.2 Stability of a homoclinic loop and bifurcation near it 269

4.3 Homoclinic and heteroclinic bifurcations in near-Hamiltonian systems 290

Chapter 5 The Number of Limit Cycles of Polynomial Systems 310

5.1 Introduction 310

5.2 Some fundamental results 312

5.3 Further study for general polynomial systems 322

Bibliography 333

Index 347