Differential Equations, Dynamical Systems, and an Introduction to Chaos

by Hirsch; Smale; Devaney

• ISBN13: 9780123820105
• ISBN10: 0123820103
• Edition: 3rd
• Format: Hardcover
• Copyright: 2012-03-12
• Publisher: Elsevier Science

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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