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Description
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
Table of Contents
Preface to Third Edition
Preface
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
Index