Differential Equations, Dynamical Systems, and an Introduction to Chaos

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  • Edition: 3rd
  • Format: Hardcover
  • Copyright: 2012-03-12
  • Publisher: Elsevier Science Ltd

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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 Editionp. ix
Prefacep. xi
First-Order Equationsp. 1
The Simplest Examplep. 1
The Logistic Population Modelp. 4
Constant Harvesting and Bifurcationsp. 7
Periodic Harvesting and Periodic Solutionsp. 10
Computing the Poincaré Mapp. 11
Exploration: A Two-Parameter Familyp. 15
Planar Linear Systemsp. 21
Second-Order Differential Equationsp. 23
Planar Systemsp. 24
Preliminaries from Algebrap. 26
Planar Linear Systemsp. 29
Eigenvalues and Eigenvectorsp. 30
Solving Linear Systemsp. 33
The Linearity Principlep. 36
Phase Portraits for Planar Systemsp. 39
Real Distinct Eigenvaluesp. 39
Complex Eigenvaluesp. 44
Repeated Eigenvaluesp. 47
Changing Coordinatesp. 49
Classification of Planar Systemsp. 61
The Trace-Determinant Planep. 61
Dynamical Classificationp. 64
Exploration: A 3D Parameter Spacep. 71
Higher-Dimensional Linear Algebrap. 73
Preliminaries from Linear Algebrap. 73
Eigenvalues and Eigenvectorsp. 82
Complex Eigenvaluesp. 85
Bases and Subspacesp. 88
Repeated Eigenvaluesp. 93
Genericityp. 100
Higher-Dimensional Linear Systemsp. 107
Distinct Eigenvaluesp. 107
Harmonic Oscillatorsp. 114
Repeated Eigenvaluesp. 120
The Exponential of a Matrixp. 123
Nonautonomous Linear Systemsp. 130
Nonlinear Systemsp. 139
Dynamical Systemsp. 140
The Existence and Uniqueness Theoremp. 142
Continuous Dependence of Solutionsp. 147
The Variational Equationp. 149
Exploration: Numerical Methodsp. 153
Exploration: Numerical Methods and Chaosp. 156
Equilibria in Nonlinear Systemsp. 159
Some Illustrative Examplesp. 159
Nonlinear Sinks and Sourcesp. 165
Saddlesp. 168
Stabilityp. 174
Bifurcationsp. 175
Exploration: Complex Vector Fieldsp. 182
Global Nonlinear Techniquesp. 187
Nullclinesp. 187
Stability of Equilibriap. 192
Gradient Systemsp. 202
Hamiltonian Systemsp. 206
Exploration: The Pendulum with Constant Forcingp. 209
Closed Orbits and Limit Setsp. 213
Limit Setsp. 213
Local Sections and Flow Boxesp. 216
The Poincaré Mapp. 218
Monotone Sequences in Planar Dynamical Systemsp. 220
The Poincaré-Bendixson Theoremp. 222
Applications of Poincaré-Bendixsonp. 225
Exploration: Chemical Reactions that Oscillatep. 228
Applications in Biologyp. 233
Infectious Diseasesp. 233
Predator-Prey Systemsp. 237
Competitive Speciesp. 244
Exploration: Competition and Harvestingp. 250
Exploration: Adding Zombies to the SIR Modelp. 251
Applications in Circuit Theoryp. 257
An RLC Circuitp. 257
The Liénard Equationp. 261
The van der Pol Equationp. 263
A Hopf Bifurcationp. 270
Exploration: Neurodynamicsp. 272
Applications in Mechanicsp. 277
Newton's Second Lawp. 277
Conservative Systemsp. 280
Central Force Fieldsp. 282
The Newtonian Central Force Systemp. 285
Kepler's First Lawp. 290
The Two-Body Problemp. 293
Blowing Up the Singularityp. 294
Exploration: Other Central Force Problemsp. 298
Exploration: Classical Limits of Quantum Mechanical Systemsp. 299
Exploration: Motion of a Gliderp. 301
The Lorenz Systemp. 305
Introductionp. 306
Elementary Properties of the Lorenz Systemp. 308
The Lorenz Attractorp. 312
A Model for the Lorenz Attractorp. 316
The Chaotic Attractorp. 321
Exploration: The Rössler Attractorp. 326
Discrete Dynamical Systemsp. 329
Introductionp. 329
Bifurcationsp. 334
The Discrete Logistic Modelp. 337
Chaosp. 340
Symbolic Dynamicsp. 344
The Shift Mapp. 349
The Cantor Middle-Thirds Setp. 351
Exploration: Cubic Chaosp. 354
Exploration: The Orbit Diagramp. 355
Homoclinic Phenomenap. 361
The Shilnikov Systemp. 361
The Horseshoe Mapp. 368
The Double Scroll Attractorp. 375
Homoclinic Bifurcationsp. 377
Exploration: The Chua Circuitp. 381
Existence and Uniqueness Revisitedp. 385
The Existence and Uniqueness Theoremp. 385
Proof of Existence and Uniquenessp. 387
Continuous Dependence on Initial Conditionsp. 394
Extending Solutionsp. 397
Nonautonomous Systemsp. 401
Differentiability of the Flowp. 404
Bibliographyp. 411
Indexp. 415
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