Differential Equations, Dynamical Systems, and an Introduction to Chaos
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Table of contents
- Cover
- Copyright Pageiv
- Contentsv
- Prefacex
- Chapter 1. First-Order Equations1
- 1.1 The Simplest Example1
- 1.2 The Logistic Population Model4
- 1.3 Constant Harvesting and Bifurcations7
- 1.4 Periodic Harvesting and Periodic Solutions9
- 1.5 Computing the Poincaré Map12
- 1.6 Exploration: A Two-Parameter Family15
- Chapter 2. Planar Linear Systems21
- 2.1 Second-Order Differential Equations23
- 2.2 Planar Systems24
- 2.3 Preliminaries from Algebra26
- 2.4 Planar Linear Systems29
- 2.5 Eigenvalues and Eigenvectors30
- 2.6 Solving Linear Systems33
- 2.7 The Linearity Principle36
- Chapter 3. Phase Portraits for Planar Systems39
- 3.1 Real Distinct Eigenvalues39
- 3.2 Complex Eigenvalues44
- 3.3 Repeated Eigenvalues47
- 3.4 Changing Coordinates49
- Chapter 4. Classification of Planar Systems61
- 4.1 The Trace-Determinant Plane61
- 4.2 Dynamical Classification64
- 4.3 Exploration: A 3D Parameter Space71
- Chapter 5. Higher Dimensional Linear Algebra75
- 5.1 Preliminaries from Linear Algebra75
- 5.2 Eigenvalues and Eigenvectors83
- 5.3 Complex Eigenvalues86
- 5.4 Bases and Subspaces89
- 5.5 Repeated Eigenvalues95
- 5.6 Genericity101
- Chapter 6. Higher Dimensional Linear Systems107
- 6.1 Distinct Eigenvalues107
- 6.2 Harmonic Oscillators114
- 6.3 Repeated Eigenvalues119
- 6.4 The Exponential of a Matrix123
- 6.5 Nonautonomous Linear Systems130
- Chapter 7. Nonlinear Systems139
- 7.1 Dynamical Systems140
- 7.2 The Existence and Uniqueness Theorem142
- 7.3 Continuous Dependence of Solutions147
- 7.4 The Variational Equation149
- 7.5 Exploration: Numerical Methods153
- Chapter 8. Equilibria in Nonlinear Systems159
- 8.1 Some Illustrative Examples159
- 8.2 Nonlinear Sinks and Sources165
- 8.3 Saddles168
- 8.4 Stability174
- 8.5 Bifurcations176
- 8.6 Exploration: Complex Vector Fields182
- Chapter 9. Global Nonlinear Techniques189
- 9.1 Nullclines189
- 9.2 Stability of Equilibria194
- 9.3 Gradient Systems203
- 9.4 Hamiltonian Systems207
- 9.5 Exploration: The Pendulum with Constant Forcing210
- Chapter 10. Closed Orbits and Limit Sets215
- 10.1 Limit Sets215
- 10.2 Local Sections and Flow Boxes218
- 10.3 The Poincaré Map220
- 10.4 Monotone Sequences in Planar Dynamical Systems222
- 10.5 The Poincaré-Bendixson Theorem225
- 10.6 Applications of Poincaré-Bendixson227
- 10.7 Exploration: Chemical Reactions That Oscillate230
- Chapter 11. Applications in Biology235
- 11.1 Infectious Diseases235
- 11.2 Predator/Prey Systems239
- 11.3 Competitive Species246
- 11.4 Exploration: Competition and Harvesting252
- Chapter 12. Applications in Circuit Theory257
- 12.1 An RLC Circuit257
- 12.2 The Lienard Equation261
- 12.3 The van der Pol Equation262
- 12.4 A Hopf Bifurcation270
- 12.5 Exploration: Neurodynamics272
- Chapter 13. Applications in Mechanics277
- 13.1 Newton’s Second Law277
- 13.2 Conservative Systems280
- 13.3 Central Force Fields281
- 13.4 The Newtonian Central Force System285
- 13.5 Kepler’s First Law289
- 13.6 The Two-Body Problem292
- 13.7 Blowing Up the Singularity293
- 13.8 Exploration: Other Central Force Problems297
- 13.9 Exploration: Classical Limits of Quantum Mechanical Systems298
- Chapter 14. The Lorenz System303
- 14.1 Introduction to the Lorenz System304
- 14.2 Elementary Properties of the Lorenz System306
- 14.3 The Lorenz Attractor310
- 14.4 A Model for the Lorenz Attractor314
- 14.5 The Chaotic Attractor319
- 14.6 Exploration: The Rössler Attractor324
- Chapter 15. Discrete Dynamical Systems327
- 15.1 Introduction to Discrete Dynamical Systems327
- 15.2 Bifurcations332
- 15.3 The Discrete Logistic Model335
- 15.4 Chaos337
- 15.5 Symbolic Dynamics342
- 15.6 The Shift Map347
- 15.7 The Cantor Middle-Thirds Set349
- 15.8 Exploration: Cubic Chaos352
- 15.9 Exploration: The Orbit Diagram353
- Chapter 16. Homoclinic Phenomena359
- 16.1 The Shil’nikov System359
- 16.2 The Horseshoe Map366
- 16.3 The Double Scroll Attractor372
- 16.4 Homoclinic Bifurcations375
- 16.5 Exploration: The Chua Circuit379
- Chapter 17. Existence and Uniqueness Revisited383
- 17.1 The Existence and Uniqueness Theorem383
- 17.2 Proof of Existence and Uniqueness385
- 17.3 Continuous Dependence on Initial Conditions392
- 17.4 Extending Solutions395
- 17.5 Nonautonomous Systems398
- 17.6 Differentiability of the Flow400
- Bibliography407
- Index411
Thirty years in the making, this revised text by three of the world's leading mathematicians covers the dynamical aspects of ordinary differential equations. it explores the relations between dynamical systems and certain fields outside pure mathematics, and has become the standard textbook for graduate courses in this area. The Second Edition now brings students to the brink of contemporary research, starting from a background that includes only calculus and elementary linear algebra.
The authors are tops in the field of advanced mathematics, including Steve Smale who is a recipient of the Field's Medal for his work in dynamical systems.
* Developed by award-winning researchers and authors
* Provides a rigorous yet accessible introduction to differential equations and dynamical systems
* Includes bifurcation theory throughout
* Contains numerous explorations for students to embark upon
NEW IN THIS EDITION
* New contemporary material and updated applications
* Revisions throughout the text, including simplification of many theorem hypotheses
* Many new figures and illustrations
* Simplified treatment of linear algebra
* Detailed discussion of the chaotic behavior in the Lorenz attractor, the Shil'nikov systems, and the double scroll attractor
* Increased coverage of discrete dynamical systems
The authors are tops in the field of advanced mathematics, including Steve Smale who is a recipient of the Field's Medal for his work in dynamical systems.
* Developed by award-winning researchers and authors
* Provides a rigorous yet accessible introduction to differential equations and dynamical systems
* Includes bifurcation theory throughout
* Contains numerous explorations for students to embark upon
NEW IN THIS EDITION
* New contemporary material and updated applications
* Revisions throughout the text, including simplification of many theorem hypotheses
* Many new figures and illustrations
* Simplified treatment of linear algebra
* Detailed discussion of the chaotic behavior in the Lorenz attractor, the Shil'nikov systems, and the double scroll attractor
* Increased coverage of discrete dynamical systems