Differential equations, dynamical systems, and an introduction to chaos / Morris W. Hirsch, Stephen Smale, Robert L. Devaney.

Format:

Book

Author/Creator:

Hirsch, Morris W., 1933-

Contributor:

Smale, Stephen, 1930-

Devaney, Robert L., 1948-

Hirsch, Morris W., 1933-

Series:

Pure and applied mathematics (Academic Press) ; 60.

Pure and applied mathematics; a series of monographs and textbooks ; v. 60

Language:

English

Subjects (All):

Differential equations.

Algebras, Linear.

Chaotic behavior in systems.

Physical Description:

1 online resource (432 p.)

Edition:

2nd ed.

Place of Publication:

San Diego, CA : Academic Press, c2004.

Language Note:

English

Summary:

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

Contents:

Cover; Copyright Page; Contents; Preface; Chapter 1. First-Order Equations; 1.1 The Simplest Example; 1.2 The Logistic Population Model; 1.3 Constant Harvesting and Bifurcations; 1.4 Periodic Harvesting and Periodic Solutions; 1.5 Computing the Poincaré Map; 1.6 Exploration: A Two-Parameter Family; Chapter 2. Planar Linear Systems; 2.1 Second-Order Differential Equations; 2.2 Planar Systems; 2.3 Preliminaries from Algebra; 2.4 Planar Linear Systems; 2.5 Eigenvalues and Eigenvectors; 2.6 Solving Linear Systems; 2.7 The Linearity Principle; Chapter 3. Phase Portraits for Planar Systems

3.1 Real Distinct Eigenvalues3.2 Complex Eigenvalues; 3.3 Repeated Eigenvalues; 3.4 Changing Coordinates; Chapter 4. Classification of Planar Systems; 4.1 The Trace-Determinant Plane; 4.2 Dynamical Classification; 4.3 Exploration: A 3D Parameter Space; Chapter 5. Higher Dimensional Linear Algebra; 5.1 Preliminaries from Linear Algebra; 5.2 Eigenvalues and Eigenvectors; 5.3 Complex Eigenvalues; 5.4 Bases and Subspaces; 5.5 Repeated Eigenvalues; 5.6 Genericity; Chapter 6. Higher Dimensional Linear Systems; 6.1 Distinct Eigenvalues; 6.2 Harmonic Oscillators; 6.3 Repeated Eigenvalues

6.4 The Exponential of a Matrix6.5 Nonautonomous Linear Systems; Chapter 7. Nonlinear Systems; 7.1 Dynamical Systems; 7.2 The Existence and Uniqueness Theorem; 7.3 Continuous Dependence of Solutions; 7.4 The Variational Equation; 7.5 Exploration: Numerical Methods; Chapter 8. Equilibria in Nonlinear Systems; 8.1 Some Illustrative Examples; 8.2 Nonlinear Sinks and Sources; 8.3 Saddles; 8.4 Stability; 8.5 Bifurcations; 8.6 Exploration: Complex Vector Fields; Chapter 9. Global Nonlinear Techniques; 9.1 Nullclines; 9.2 Stability of Equilibria; 9.3 Gradient Systems; 9.4 Hamiltonian Systems

9.5 Exploration: The Pendulum with Constant ForcingChapter 10. Closed Orbits and Limit Sets; 10.1 Limit Sets; 10.2 Local Sections and Flow Boxes; 10.3 The Poincaré Map; 10.4 Monotone Sequences in Planar Dynamical Systems; 10.5 The Poincaré-Bendixson Theorem; 10.6 Applications of Poincaré-Bendixson; 10.7 Exploration: Chemical Reactions That Oscillate; Chapter 11. Applications in Biology; 11.1 Infectious Diseases; 11.2 Predator/Prey Systems; 11.3 Competitive Species; 11.4 Exploration: Competition and Harvesting; Chapter 12. Applications in Circuit Theory; 12.1 An RLC Circuit

12.2 The Lienard Equation12.3 The van der Pol Equation; 12.4 A Hopf Bifurcation; 12.5 Exploration: Neurodynamics; Chapter 13. Applications in Mechanics; 13.1 Newton's Second Law; 13.2 Conservative Systems; 13.3 Central Force Fields; 13.4 The Newtonian Central ForceSystem; 13.5 Kepler's First Law; 13.6 The Two-Body Problem; 13.7 Blowing Up the Singularity; 13.8 Exploration: Other Central Force Problems; 13.9 Exploration: Classical Limits of Quantum Mechanical Systems; Chapter 14. The Lorenz System; 14.1 Introduction to the Lorenz System; 14.2 Elementary Properties of the Lorenz System

14.3 The Lorenz Attractor

Notes:

Rev. ed. of: Differential equations, dynamical systems, and linear algebra / Morris W. Hirsch and Stephen Smale. 1974.

Includes bibliographical references (p. 407-409) and index.

Related Work:

Hirsch, Morris W., 1933- Differential equations, dynamical systems, and linear algebra.

ISBN:

1-281-00525-8

9786611005252

0-08-049114-6

OCLC:

437181599