CHAPTER 1 First-Order Equations 1.1 The Simplest Example 1.2 The Logistic Population Model 1.3 Constant Harvesting and Bifurcations 1.4 Periodic Harvesting and Periodic Solutions 1.5 Computing the Poincard Map 1.6 Exploration：A Two-Parameter Family CHAPTER 2 Planar Linear Systems 2.1 Second-Order Differential Equations 2.2 Planar Systems 2.3 Preliminaries from Algebra 2.4 Planar Linear Systems 2.5 Eigenvalues and Eigenvectors 2.6 Solving Linear Systems 2.7 The Linearity Principle CHAPTER 3 Phase Portraits for Planar Systems 3.1 Real Distinct Eigenvalues 3.2 Complex Eigenvalues 3.3 Repeated Eigenvalues 3.4 Changing Coordinates CHAPTER 4 Classification of Planar Systems 4.1 The Trace-Determinant Plane 4.2 Dynamical Classification 4.3 Exploration：A 3D Parameter Space CHAPTER 5 Higher Dimensional Linear Algebra 5.1 Preliminaries from Linear Algebra 5.2 Eigenvalues and Eigenvectors 5.3 Complex Eigenvalues 5.4 Bases and Subspaces 5.5 Repeated Eigenvalues 5.6 Genericity CHAPTER 6 Higher Dimensional Linear Systems 6.1 Distinct Eigenvalues 6.2 Harmonic Oscillators 6.3 Repeated Eigenvalues 6.4 The Exponential of a Matrix 6.5 Nonautonomous Linear Systems CHAPTER 7 Nonlinear Systems 7.1 Dynamical Systems 7.2 The Existence and Uniqueness Theorem 7.3 Continuous Dependence of Solutions 7.4 The Variational Equation 7.5 Exploration：Numerical Methods CHAPTER 8 Equilibria in Nonlinear Systems 8.1 Some Nustrative Examples 8.2 Nonlinear Sinks and Sources 8.3 Saddles 8.4 Stability 8.5 Bifurcations 8.6 Exploration：Complex Vector Fields CHAPTER 9 Global Nonlinear Techniques 9.1 Nullclines 9.2 Stability of Equilibria 9.3 Gradient Systems 9.4 Hamiltonian Systems 9.5 Exploration：The Pendulum with Constant Forcing CHAPTER 10 Closed Orbits and Limit Sets 10.1 Limit Sets 10.2 Local Sections and Flow Boxes 10.3 The Poincare Map 10.4 Monotone Sequences in Planar Dynamical Systems 10.5 The Poincare-Bendixson Theorem 10.6 Applications of Poincare-Bendixson 10.7 Expl0ration：Chemical Reactions That Oscillate CHAPTER 11 Applications in Biology 11.1 Infectious Diseases 11.2 Predator／Prey Systems 11.3 Competitive Species 11.4 Exploration：Competition and Harvesting CHAPTER 12 Applications in Circuit Theory 12.1 An RLC Circuit 12.2 The Lienard Equation 12.3 The van der Pol Equation 12.4 A Hopf Bifurcation 12.5 Exploration：Neurodynamics CHAPTER 13 Applications in Mechanics 13.1 Newton’S Second Law 13.2 Conservative Systems 13.3 Central Force Fields 13.4 The Newtonian Central Force System 13.5 Kepler’s First Law 13.6 The Two-Body Problem 13.7 Blowing Up the Singularity 13.8 Exploration：Other Central Force Problems 13.9 Exploration：Classical Limits of Quantum Mechanical Systems CHAPTER 14 The Lorenz System 14.1 Introduction to the Lorenz System 14.2 Elementary Properties of the Lorenz System 14.3 The Lorenz Attractor 14.4 A Model for the Lorenz Attractor 14.5 The Chaotic Attractor 14.6 Exploration：The Rossler Attractor CHAPTER 15 Discrete Dynamical Systems 15.1 Introduction to Discrete Dynamical Systems 15.2 Bifurcations 15.3 The Discrete Logistic Model 15.4 Chaos 15.5 Symbolic Dynamics 15.6 The Shift Map 15.7 The Cantor Middle-Thirds Set 15.8 Exploration：Cubic Chaos 15.9 Exploration：The Orbit Diagram CHAPTER 16 Homoclinic Phenomena 16.1 The Shil’nikov System 16.2 The Horseshoe Map 16.3 The Double Scroll Attractor 16.4 Homoclinic Bifurcations 16.5 Exploration：The Chua Circuit CHAPTER 17 Existence and Uniqueness Revisited 17.1 The Existence and Uniqueness Theorem 17.2 Proof of Existence and Uniqueness 17.3 Continuous Dependence on Initial Conditions 17.4 Extending Solutions 17.5 Nonautonomous Systems 17.6 Differentiability of the Flow Bibliography Index
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