Differential Equations, Dynamical Systems, and an Introduction to Chaos

Differential Equations, Dynamical Systems, and an Introduction to Chaos, 3rd Edition

Differential Equations, Dynamical Systems, and an Introduction to Chaos, 3rd Edition,Morris W. Hirsch,Stephen Smale,Robert L. Devaney,ISBN9780123820105

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Academic Press




234 X 155

This classic book presents the mathematical rigor, theory, and applications needed to understand dynamical systems in an attempt to solve problems in engineering, science, and biology.

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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


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.


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

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

Differential Equations, Dynamical Systems, and an Introduction to Chaos, 3rd Edition

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 Nullclines

9.2 Stability of Equilibria

9.3 Gradient Systems

9.4 Hamiltonian Systems

9.5 Exploration: The Pendulum with Constant Forcing

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

11. Applications in Biology

11.1 Infectious Diseases

11.2 Predator–Prey Systems

11.3 Competitive Species

11.4 Exploration: Competition and Harvesting

11.5 Exploration: Adding Zombies to the SIR Model

12. Applications in Circuit Theory

12.1 An RLC Circuit

12.2 The Liénard Equation

12.3 The van der Pol Equation

12.4 A Hopf Bifurcation

12.5 Exploration: Neurodynamics

13. Applications in Mechanics

13.1 Newton’s Second Law

13.2 Conservative Systems

13.3 Central Force Fields

13.4 The Newtonian Central Force System

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

13.10 Exploration: Motion of a Glider

14. The Lorenz System

14.1 Introduction

14.2 Elementary Properties of the Lorenz System

14.3 The Lorenz Attractor

14.4 A Model for the Lorenz Attractor

14.5 The Chaotic Attractor

14.6 Exploration: The Rössler Attractor

15. Discrete Dynamical Systems

15.1 Introduction

15.2 Bifurcations

15.3 The Discrete Logistic Model

15.4 Chaos

15.5 Symbolic Dynamics

15.6 The Shift Map

15.7 The Cantor Middle-Thirds Set

15.8 Exploration: Cubic Chaos

15.9 Exploration: The Orbit Diagram

16. Homoclinic Phenomena

16.1 The Shilnikov System

16.2 The Horseshoe Map

16.3 The Double Scroll Attractor

16.4 Homoclinic Bifurcations

16.5 Exploration: The Chua Circuit

17. Existence and Uniqueness Revisited

17.1 The Existence and Uniqueness Theorem

17.2 Proof of Existence and Uniqueness

17.3 Continuous Dependence on Initial Conditions

17.4 Extending Solutions

17.5 Nonautonomous Systems

17.6 Differentiability of the Flow


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