Contents
CHAPTER1
Introduction: Differential Equations and Dynamical Systems
1.0. Existence and Uniqueness ofSolutions
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 Equation
2.3. The Lorenz Equations
2.4. The Dynamics ofa Bouncing Ball
2.5. Conclusions: The Moral ofthe 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 ofMaps 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 Bifureations
4.4. Averaging, Hamiltonian Systems, and Global Behavior:
Cautionary Notes
4.5. Melnikov's Method: Perturbations ofPlanar Homoclinic Orbits
4.6. Melnikov's Method: Perturbations of Hamiltonian Systems and
Subharmonic Orbits
4.7. Stability of 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 ofa Hyperbolic Limit Set
5.2. Invariant Sets and Hyperbolicity
5.3. Markov Partitions and Symbolic Dynamics
5.4. Strange Attractors 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 Propenies: Dimension. Entropy and Liapunov Exponents
CHAPTER 6
Global Bifurcations
6.1. Saddle Connections
6.2. Rotation Numbers
6.3. Bifurcations of One-Dimensional Maps
6.4. The Lorenz Bifurcations
6.5. Homoclinic Orbits in Three-Dimensional Flows: Silnikov's Example
6.6. Homoclinic Bifurcations of Periodic Orbits
6.7. Wild Hyperbolic Sets
6.8. Renormalization and Universality
CHAPTER7
Local Codimension Two Bifurcations of Flows
7.1. Degeneracy in Higher-Order Terms
7.2. A Note on k-Sels 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. Applications to Large Systems
APPENDIX
Suggestions for Further Reading
Postscript Added at Second Printing
Glossary
References
Index
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