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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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