Preface to the Third Edition | p. ix |

Preface | p. xi |

First-Order Equations | p. 1 |

The Simplest Example | p. 1 |

The Logistic Population Model | p. 4 |

Constant Harvesting and Bifurcations | p. 7 |

Periodic Harvesting and Periodic Solutions | p. 10 |

Computing the Poincaré Map | p. 11 |

Exploration: A Two-Parameter Family | p. 15 |

Planar Linear Systems | p. 21 |

Second-Order Differential Equations | p. 23 |

Planar Systems | p. 24 |

Preliminaries from Algebra | p. 26 |

Planar Linear Systems | p. 29 |

Eigenvalues and Eigenvectors | p. 30 |

Solving Linear Systems | p. 33 |

The Linearity Principle | p. 36 |

Phase Portraits for Planar Systems | p. 39 |

Real Distinct Eigenvalues | p. 39 |

Complex Eigenvalues | p. 44 |

Repeated Eigenvalues | p. 47 |

Changing Coordinates | p. 49 |

Classification of Planar Systems | p. 61 |

The Trace-Determinant Plane | p. 61 |

Dynamical Classification | p. 64 |

Exploration: A 3D Parameter Space | p. 71 |

Higher-Dimensional Linear Algebra | p. 73 |

Preliminaries from Linear Algebra | p. 73 |

Eigenvalues and Eigenvectors | p. 82 |

Complex Eigenvalues | p. 85 |

Bases and Subspaces | p. 88 |

Repeated Eigenvalues | p. 93 |

Genericity | p. 100 |

Higher-Dimensional Linear Systems | p. 107 |

Distinct Eigenvalues | p. 107 |

Harmonic Oscillators | p. 114 |

Repeated Eigenvalues | p. 120 |

The Exponential of a Matrix | p. 123 |

Nonautonomous Linear Systems | p. 130 |

Nonlinear Systems | p. 139 |

Dynamical Systems | p. 140 |

The Existence and Uniqueness Theorem | p. 142 |

Continuous Dependence of Solutions | p. 147 |

The Variational Equation | p. 149 |

Exploration: Numerical Methods | p. 153 |

Exploration: Numerical Methods and Chaos | p. 156 |

Equilibria in Nonlinear Systems | p. 159 |

Some Illustrative Examples | p. 159 |

Nonlinear Sinks and Sources | p. 165 |

Saddles | p. 168 |

Stability | p. 174 |

Bifurcations | p. 175 |

Exploration: Complex Vector Fields | p. 182 |

Global Nonlinear Techniques | p. 187 |

Nullclines | p. 187 |

Stability of Equilibria | p. 192 |

Gradient Systems | p. 202 |

Hamiltonian Systems | p. 206 |

Exploration: The Pendulum with Constant Forcing | p. 209 |

Closed Orbits and Limit Sets | p. 213 |

Limit Sets | p. 213 |

Local Sections and Flow Boxes | p. 216 |

The Poincaré Map | p. 218 |

Monotone Sequences in Planar Dynamical Systems | p. 220 |

The Poincaré-Bendixson Theorem | p. 222 |

Applications of Poincaré-Bendixson | p. 225 |

Exploration: Chemical Reactions that Oscillate | p. 228 |

Applications in Biology | p. 233 |

Infectious Diseases | p. 233 |

Predator-Prey Systems | p. 237 |

Competitive Species | p. 244 |

Exploration: Competition and Harvesting | p. 250 |

Exploration: Adding Zombies to the SIR Model | p. 251 |

Applications in Circuit Theory | p. 257 |

An RLC Circuit | p. 257 |

The Liénard Equation | p. 261 |

The van der Pol Equation | p. 263 |

A Hopf Bifurcation | p. 270 |

Exploration: Neurodynamics | p. 272 |

Applications in Mechanics | p. 277 |

Newton's Second Law | p. 277 |

Conservative Systems | p. 280 |

Central Force Fields | p. 282 |

The Newtonian Central Force System | p. 285 |

Kepler's First Law | p. 290 |

The Two-Body Problem | p. 293 |

Blowing Up the Singularity | p. 294 |

Exploration: Other Central Force Problems | p. 298 |

Exploration: Classical Limits of Quantum Mechanical Systems | p. 299 |

Exploration: Motion of a Glider | p. 301 |

The Lorenz System | p. 305 |

Introduction | p. 306 |

Elementary Properties of the Lorenz System | p. 308 |

The Lorenz Attractor | p. 312 |

A Model for the Lorenz Attractor | p. 316 |

The Chaotic Attractor | p. 321 |

Exploration: The Rössler Attractor | p. 326 |

Discrete Dynamical Systems | p. 329 |

Introduction | p. 329 |

Bifurcations | p. 334 |

The Discrete Logistic Model | p. 337 |

Chaos | p. 340 |

Symbolic Dynamics | p. 344 |

The Shift Map | p. 349 |

The Cantor Middle-Thirds Set | p. 351 |

Exploration: Cubic Chaos | p. 354 |

Exploration: The Orbit Diagram | p. 355 |

Homoclinic Phenomena | p. 361 |

The Shilnikov System | p. 361 |

The Horseshoe Map | p. 368 |

The Double Scroll Attractor | p. 375 |

Homoclinic Bifurcations | p. 377 |

Exploration: The Chua Circuit | p. 381 |

Existence and Uniqueness Revisited | p. 385 |

The Existence and Uniqueness Theorem | p. 385 |

Proof of Existence and Uniqueness | p. 387 |

Continuous Dependence on Initial Conditions | p. 394 |

Extending Solutions | p. 397 |

Nonautonomous Systems | p. 401 |

Differentiability of the Flow | p. 404 |

Bibliography | p. 411 |

Index | p. 415 |

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