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