Differential Equations, Dynamical Systems, and an Introduction to Chaos

ISBN-10: 0123497035
ISBN-13: 9780123497031
Edition: 2nd 2003 (Revised)
List price: $115.00
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Description: This text is about the dynamical aspects of ordinary differential equations and the relations between dynamical systems and certain fields outside pure mathematics. It has become the standard textbook for graduate courses in this area.

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Book details

List price: $115.00
Edition: 2nd
Copyright year: 2003
Publisher: Elsevier Science & Technology Books
Publication date: 10/17/2003
Binding: Hardcover
Pages: 425
Size: 6.00" wide x 9.00" long x 1.25" tall
Weight: 1.980
Language: English

This text is about the dynamical aspects of ordinary differential equations and the relations between dynamical systems and certain fields outside pure mathematics. It has become the standard textbook for graduate courses in this area.

Preface
First-Order Equations
The Simplest Example
The Logistic Population Model
Constant Harvesting and Bifurcations
Periodic Harvesting and Periodic Solutions
Computing the Poincare Map
Exploration: A Two-Parameter Family
Planar Linear Systems
Second-Order Differential Equations
Planar Systems
Preliminaries from Algebra
Planar Linear Systems
Eigenvalues and Eigenvectors
Solving Linear Systems
The Linearity Principle
Phase Portraits for Planar Systems
Real Distinct Eigenvalues
Complex Eigenvalues
Repeated Eigenvalues
Changing Coordinates
Classification of Planar Systems
The Trace-Determinant Plane
Dynamical Classification
Exploration: A 3D Parameter Space
Higher Dimensional Linear Algebra
Preliminaries from Linear Algebra
Eigenvalues and Eigenvectors
Complex Eigenvalues
Bases and Subspaces
Repeated Eigenvalues
Genericity
Higher Dimensional Linear Systems
Distinct Eigenvalues
Harmonic Oscillators
Repeated Eigenvalues
The Exponential of a Matrix
Nonautonomous Linear Systems
Nonlinear Systems
Dynamical Systems
The Existence and Uniqueness Theorem
Continuous Dependence of Solutions
The Variational Equation
Exploration: Numerical Methods
Equilibria in Nonlinear Systems
Some Illustrative Examples
Nonlinear Sinks and Sources
Saddles
Stability
Bifurcations
Exploration: Complex Vector Fields
Global Nonlinear Techniques
Nullclines
Stability of Equilibria
Gradient Systems
Hamiltonian Systems
Exploration: The Pendulum with Constant Forcing
Closed Orbits and Limit Sets
Limit Sets
Local Sections and Flow Boxes
The Poincare Map
Monotone Sequences in Planar Dynamical Systems
The Poincare-Bendixson Theorem
Applications of Poincare-Bendixson
Exploration: Chemical Reactions That Oscillate
Applications in Biology
Infectious Diseases
Predator/Prey Systems
Competitive Species
Exploration: Competition and Harvesting
Applications in Circuit Theory
An RLC Circuit
The Lienard Equation
The van der Pol Equation
A Hopf Bifurcation
Exploration: Neurodynamics
Applications in Mechanics
Newton's Second Law
Conservative Systems
Central Force Fields
The Newtonian Central Force System
Kepler's First Law
The Two-Body Problem
Blowing Up the Singularity
Exploration: Other Central Force Problems
Exploration: Classical Limits of Quantum Mechanical Systems
The Lorenz System
Introduction to the Lorenz System
Elementary Properties of the Lorenz System
The Lorenz Attractor
A Model for the Lorenz Attractor
The Chaotic Attractor
Exploration: The Rossler Attractor
Discrete Dynamical Systems
Introduction to Discrete Dynamical Systems
Bifurcations
The Discrete Logistic Model
Chaos
Symbolic Dynamics
The Shift Map
The Cantor Middle-Thirds Set
Exploration: Cubic Chaos
Exploration: The Orbit Diagram
Homoclinic Phenomena
The Shil'nikov System
The Horseshoe Map
The Double Scroll Attractor
Homoclinic Bifurcations
Exploration: The Chua Circuit
Existence and Uniqueness Revisited
The Existence and Uniqueness Theorem
Proof of Existence and Uniqueness
Continuous Dependence on Initial Conditions
Extending Solutions
Nonautonomous Systems
Differentiability of the Flow
Bibliography
Index

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