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Preface to the Third Edition | |
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Preface | |
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First-Order Equations | |
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The Simplest Example | |
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The Logistic Population Model | |
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Constant Harvesting and Bifurcations | |
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Periodic Harvesting and Periodic Solutions | |
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Computing the Poincar� Map | |
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Exploration: A Two-Parameter Family | |
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Planar Linear Systems | |
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Second-Order Differential Equations | |
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Planar Systems | |
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Preliminaries from Algebra | |
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Planar Linear Systems | |
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Eigenvalues and Eigenvectors | |
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Solving Linear Systems | |
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The Linearity Principle | |
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Phase Portraits for Planar Systems | |
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Real Distinct Eigenvalues | |
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Complex Eigenvalues | |
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Repeated Eigenvalues | |
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Changing Coordinates | |
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Classification of Planar Systems | |
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The Trace-Determinant Plane | |
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Dynamical Classification | |
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Exploration: A 3D Parameter Space | |
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Higher-Dimensional Linear Algebra | |
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Preliminaries from Linear Algebra | |
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Eigenvalues and Eigenvectors | |
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Complex Eigenvalues | |
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Bases and Subspaces | |
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Repeated Eigenvalues | |
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Genericity | |
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Higher-Dimensional Linear Systems | |
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Distinct Eigenvalues | |
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Harmonic Oscillators | |
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Repeated Eigenvalues | |
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The Exponential of a Matrix | |
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Nonautonomous Linear Systems | |
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Nonlinear Systems | |
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Dynamical Systems | |
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The Existence and Uniqueness Theorem | |
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Continuous Dependence of Solutions | |
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The Variational Equation | |
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Exploration: Numerical Methods | |
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Exploration: Numerical Methods and Chaos | |
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Equilibria in Nonlinear Systems | |
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Some Illustrative Examples | |
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Nonlinear Sinks and Sources | |
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Saddles | |
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Stability | |
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Bifurcations | |
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Exploration: Complex Vector Fields | |
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Global Nonlinear Techniques | |
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Nullclines | |
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Stability of Equilibria | |
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Gradient Systems | |
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Hamiltonian Systems | |
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Exploration: The Pendulum with Constant Forcing | |
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Closed Orbits and Limit Sets | |
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Limit Sets | |
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Local Sections and Flow Boxes | |
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The Poincar� Map | |
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Monotone Sequences in Planar Dynamical Systems | |
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The Poincar�-Bendixson Theorem | |
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Applications of Poincar�-Bendixson | |
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Exploration: Chemical Reactions that Oscillate | |
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Applications in Biology | |
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Infectious Diseases | |
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Predator-Prey Systems | |
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Competitive Species | |
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Exploration: Competition and Harvesting | |
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Exploration: Adding Zombies to the SIR Model | |
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Applications in Circuit Theory | |
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An RLC Circuit | |
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The Li�nard Equation | |
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The van der Pol Equation | |
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A Hopf Bifurcation | |
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Exploration: Neurodynamics | |
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Applications in Mechanics | |
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Newton's Second Law | |
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Conservative Systems | |
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Central Force Fields | |
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The Newtonian Central Force System | |
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Kepler's First Law | |
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The Two-Body Problem | |
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Blowing Up the Singularity | |
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Exploration: Other Central Force Problems | |
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Exploration: Classical Limits of Quantum Mechanical Systems | |
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Exploration: Motion of a Glider | |
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The Lorenz System | |
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Introduction | |
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Elementary Properties of the Lorenz System | |
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The Lorenz Attractor | |
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A Model for the Lorenz Attractor | |
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The Chaotic Attractor | |
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Exploration: The R�ssler Attractor | |
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Discrete Dynamical Systems | |
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Introduction | |
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Bifurcations | |
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The Discrete Logistic Model | |
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Chaos | |
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Symbolic Dynamics | |
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The Shift Map | |
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The Cantor Middle-Thirds Set | |
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Exploration: Cubic Chaos | |
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Exploration: The Orbit Diagram | |
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Homoclinic Phenomena | |
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The Shilnikov System | |
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The Horseshoe Map | |
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The Double Scroll Attractor | |
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Homoclinic Bifurcations | |
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Exploration: The Chua Circuit | |
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Existence and Uniqueness Revisited | |
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The Existence and Uniqueness Theorem | |
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Proof of Existence and Uniqueness | |
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Continuous Dependence on Initial Conditions | |
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Extending Solutions | |
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Nonautonomous Systems | |
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Differentiability of the Flow | |
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Bibliography | |
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Index | |