Differential Equations, Dynamical Systems, and an Introduction to Chaos

3rd Edition

Authors: Morris W. Hirsch Stephen Smale Robert L. Devaney
Hardcover ISBN: 9780123820105
eBook ISBN: 9780123820112
Imprint: Academic Press
Published Date: 12th March 2012
Page Count: 432
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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.

Key Features

  • 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


Advanced Undergraduate and Graduate students studying mathematics, biology, chemistry, economics, physical sciences, physics, computer science and engineering

Table of Contents

Preface to Third Edition


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

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

3. Phase Portraits for Planar Systems

3.1 Real Distinct Eigenvalues

3.2 Complex Eigenvalues

3.3 Repeated Eigenvalues

3.4 Changing Coordinates

4. Classification of Planar Systems

4.1 The Trace–Determinant Plane

4.2 Dynamical Classification

4.3 Exploration: A 3D Parameter Space

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

6. Higher-Dimensional Linear Systems

6.1 Distinct Eigenvalues

6.2 Harmonic Oscillators

6.3 Repeated Eigenvalues

6.4 The Exponential of a Matrix

6.5 Nonautonomous Linear Systems

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

7.6 Exploration: Numerical Methods and Chaos

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

9. Global Nonlinear Techniques

9.1 Nullcl


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About the Author

Morris W. Hirsch

Affiliations and Expertise

University of Wisconsin, Madison, USA

Stephen Smale

Affiliations and Expertise

University of California, Berkeley, USA

Robert L. Devaney

Affiliations and Expertise

Boston University, MA, USA