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# Differential Equations, Dynamical Systems, and an Introduction to Chaos

**by**Hirsch, Morris W.; Smale, Stephen; Devaney, Robert L.

3rd

### 9780123820105

0123820103

Hardcover

3/12/2012

Academic Pr

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## Summary

Hirsch, Devaney, and Smale's classic Differential Equations, Dynamical Systems, and an Introduction to Chaos has been used by professors as the primary text for undergraduate and graduate level courses covering differential equations. It provides a theoretical approach to dynamical systems and chaos written for a diverse student population among the fields of mathematics, science, and engineering. Prominent experts provide everything students need to know about dynamical systems as students seek to develop sufficient mathematical skills to analyze the types of differential equations that arise in their area of study. The authors provide rigorous exercises and examples clearly and easily by slowly introducing linear systems of differential equations. Calculus is required as specialized advanced topics not usually found in elementary differential equations courses are included, such as exploring the world of discrete dynamical systems and describing chaotic systems. Classic text by three of the world's most prominent mathematicians Continues the tradition of expository excellence Contains updated material and expanded applications for use in applied studies

## Table of Contents

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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