Differential Equations, Dynamical Systems, and an Introduction to Chaos

Differential Equations, Dynamical Systems, and an Introduction to Chaos

3rd Edition - March 12, 2012

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  • Authors: Morris W. Hirsch, Stephen Smale, Robert L. Devaney
  • Hardcover ISBN: 9780123820105
  • eBook ISBN: 9780123820112

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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 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


Product details

  • No. of pages: 432
  • Language: English
  • Copyright: © Academic Press 2012
  • Published: March 12, 2012
  • Imprint: Academic Press
  • Hardcover ISBN: 9780123820105
  • eBook ISBN: 9780123820112

About the Authors

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

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