非线性振动.动力学系统和矢量场的分叉

CHAPTER 1 Introduction： Differential Equations and Dynamical Systems 1.1 Existence and Uniqueness of Solutions 1.1 The Linear System x = Ax 1.2 Flows and Invariant Subspaces 1.3 The Nonlinear System x = f （x） 1.4 Linear and Nonlinear Maps 1.5 Closed Orbits， Poincare Maps.and Forced Oscillations 1.6 Asymptotic Behavior 1.7 Equivalence Relations and Structural Stability 1.8 Two-Dimensional Flows 1.9 Peixoto's Theorem for Two-Dimensional Flows CHAPTER 2 An Introduction to Chaos： Four Examples 2.1 Van der Pol's Equation 2.2 Duffing's Equaiion 2.3 The Lorenz Equations 2.4 The Dynamics of a Bouncing Ball 2.5 Conclusions： The Moral of the Tales CHAPTER 3 Local Bifurcations 3.1 BiFurcation Problems 3.2 Center Manifolds 3.3 Normal Forms 3.4 Codimension One Bifurcations of Equilibria 3.5 Codimension One Bifurcations of Maps and Periodic Orbits CHAPTER 4 Averaging and Perturbation from a Geometric Viewpoint 4.1 Averaging and Poincare Maps 4.2 Examples of Averaging 4.3 Averaging and Local Bifurcations 4.4 Averaging， Hamikonian Systems， and Global Behavior： Cautionary Notes 4.5 Melnikov's Method： Perturbations of Planar Homoclinic Orbits 4.6 Melnikov's Method： Perturbations of Hamiltonian Systems and Subharmonic Orbits 4.7 Stability or Subharmonic Orbits 4.8 Two Degree of Freedom Hamiltonians and Area Preserving Maps of the Plane CHAPTER 5 Hyperbolic Sets， Symbolic Dynamics， and Strange Attractors 5.0 Introduction 5.1 The Smale Horseshoe： An Example of a Hyperbolic Limit Set 5.2 Invariant Sets and Hyperbolicity 5.3 Markov Partitions and Symbolic Dynamics 5.4 Strange Auractors and the Stability Dogma 5.5 Structurally Stable Attractors 5.6 One-Dimensional Evidence for Strange Attractors 5.7 The Geometric Lorenz Attractor 5.8 Statistical Properties： Dimension， Entropy， and Liapunov Exponents CHAPTER 6 Global Bifurcations 6.1 Saddle Connections 6.2 Rotation Numbers 6.3 Bifurcations or One-Dimensional Maps 6.4 The Lorenz Bifurcations 6.5 Homoclinic Orbits in Three-Dimensional Flows： Silnikov's Example 6.6 Homoclinic aifurcations of Periodic Orbits 6.7 Wild Hyperbolic Sets 6.8 Renormalization and Universality CHAPTER 7 Local Codimension Two Bifurcations of Flows 7.1 Degeneracy in Higher-Order Terms 7.2 A Note on k-Jets and Determinacy 7.3 The Double Zero Eigenvalue 7.4 A Pure Imaginary Pair and a Simple Zero Eigenvalue 7.5 Two Pure Imaginary Pairs of Eigenvalues without Resonance 7.6 Applicaiions to Large Systems APPENDIX Suggestions for Further Reading Postscript Added at Second Printing Glossary References Index