Preface Historical Prologue Part 1. Systems of Nonlinear Differential Equations Chapter 1. Geometric Approach to Differential Equations Chapter 2. Linear Systems 2.1. Fundamental Set of Solutions Exercises 2.1 2.2. Constant Coefficients： Solutions and Phase Portraits Exercises 2.2 2.3. Nonhomogeneous Systems： Time-dependent Forcing Exercises 2.3 2.4. Applications Exercises 2.4 2.5. Theory and Proofs Chapter 3. The Flow： Solutions of Nonlinear Equations 3.1. Solutions of Nonlinear Equations Exercises 3.1 3.2. Numerical Solutions of Differential Equations Exercises 3.2 3.3. Theory and Proofs Chapter 4. Phase Portraits with Emphasis on Fixed Points 4.1. Limit Sets Exercises 4.1 4.2. Stability of Fixed Points Exercises 4.2 4.3. Scalar Equations Exercises 4.3 4.4. Two Dimensions and Nullclines Exercises 4.4 4.5. Linearized Stability of Fixed Points Exercises 4.5 4.6. Competitive Populations Exercises 4.6 4.7. Applications Exercises 4.7 4.8. Theory and Proofs Chapter 5. Phase Portraits Using Scalar Functions 5.1. Predator-Prey Systems Exercises 5.1 5.2. Undamped Forces Exercises 5.2 5.3. Lyapunov Functions for Damped Systems Exercises 5.3 5.4. Bounding Functions Exercises 5.4 5.5. Gradient Systems Exercises 5.5 5.6. Applications Exercises 5.6 5.7. Theory and Proofs Chapter 6. Periodic Orbits 6.1. Introduction to Periodic Orbits Exercises 6.1 6.2. Poincare-Bendixson Theorem Exercises 6.2 6.3. Self-Excited Oscillator Exercises 6.3 6.4. Andronov-HopfBifurcation Exercises 6.4 6.5. Homoclinic Bifurcation Exercises 6.5 6.6. Rate of Change of Volume Exercises 6.6 6.7. Poincare Map Exercises 6.7 6.8. Applications Exercises 6.8 6.9. Theory and Proofs Chapter 7. Chaotic Attractors 7.1. Attractors Exercises 7.1 7.2. Chaotic Attractors Exercise 7.2 7.3. Lorenz System Exercises 7.3 7.4. RSssler Attractor Exercises 7.4 7.5. Forced Oscillator Exercises 7.5 7.6. Lyapunov Exponents Exercises 7.6 7.7. Test for Chaotic Attractors Exercises 7.7 7.8. Applications 7.9. Theory and Proofs Part 2. Iteration of Functions Chapter 8. Iteration of Functions as Dynamics 8.1. One-Dimensional Maps 8.2. Functions with Several Variables Chapter 9. Periodic Points of One-Dimensional Maps 9.1. Periodic Points Exercises 9.1 9.2. Iteration Using the Graph Exercises 9.2 9.3. Stability of Periodic Points Exercises 9.3 9.4. Critical Points and Basins Exercises 9.4 9.5. Bifurcation of Periodic Points Exercises 9.5 9.6. Conjugacy Exercises 9.6 9.7. Applications Exercises 9.7 9.8. Theory and Proofs Chapter 10. Itineraries for One-Dimensional Maps 10.1. Periodic Points from Transition Graphs Exercises 10.1 10.2. Topological Transitivity Exercises 10.2 10.3. Sequences of Symbols Exercises 10.3 10.4. Sensitive Dependence on Initial Conditions Exercises 10.4 10.5. Cantor Sets Exercises 10.5 10.6. Piecewise Expanding Maps and Subshifts Exercises 10.6 10.7. Applications Exercises 10.7 10.8. Theory and Proofs Chapter 11. Invariant Sets for One-Dimensional Maps 11.1. Limit Sets Exercises 11.1 11.2. Chaotic Attractors Exercises 11.2 11.3. Lyapunov Exponents Exercises 11.3 11.4. Invariant Measures Exercises 11.4 11.5. Applications 11.6. Theory and Proofs Chapter 12. Periodic Points of Higher Dimensional Maps 12.1. Dynamics of Linear Maps Exercises 12.1 12.2. Classification of Periodic Points Exercises 12.2 12.3. Stable Manifolds Exercises 12.3 12.4. Hyperbolic Toral Automorphisms Exercises 12.4 12.5. Applications Exercises 12.5 12.6. Theory and Proofs Chapter 13. Invariant Sets for Higher Dimensional Maps 13.1. Geometric Horseshoe Exercises 13.1 13.2. Symbolic Dynamics Exercises 13.2 13.3. Homoclinic Points and Horseshoes Exercises 13.3 13.4. Attractors Exercises 13.4 13.5. Lyapunov Exponents Exercises 13.5 13.6. Applications 13.7. Theory and Proofs Chapter 14. Fractals 14.1. Box Dimension Exercises 14.1 14.2. Dimension of Orbits Exercises 14.2 14.3. Iterated-Function Systems Exercises 14.3 14.4. Theory and Proofs Appendix A. Background and Terminology A.1. Calculus Background and Notation A.2. Analysis and Topology Terminology A.3. Matrix Algebra Appendix B. Generic Properties Bibliography Index