Differential Equations, Dynamical Systems, and an Introduction to Chaos

ISBN-10: 0123820103
ISBN-13: 9780123820105
Edition: 3rd 2012
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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  More...

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Book details

List price: $99.95
Edition: 3rd
Copyright year: 2012
Publisher: Elsevier Science & Technology Books
Publication date: 3/12/2012
Binding: Hardcover
Pages: 432
Size: 6.13" wide x 9.21" long x 1.25" tall
Weight: 2.002

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.Classic text by three of the world's most prominent mathematiciansContinues the tradition of expository excellenceContains updated material and expanded applications for use in applied studies

Preface to the Third Edition
Preface
First-Order Equations
The Simplest Example
The Logistic Population Model
Constant Harvesting and Bifurcations
Periodic Harvesting and Periodic Solutions
Computing the Poincar� 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
Exploration: Numerical Methods and Chaos
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 Poincar� Map
Monotone Sequences in Planar Dynamical Systems
The Poincar�-Bendixson Theorem
Applications of Poincar�-Bendixson
Exploration: Chemical Reactions that Oscillate
Applications in Biology
Infectious Diseases
Predator-Prey Systems
Competitive Species
Exploration: Competition and Harvesting
Exploration: Adding Zombies to the SIR Model
Applications in Circuit Theory
An RLC Circuit
The Li�nard 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
Exploration: Motion of a Glider
The Lorenz System
Introduction
Elementary Properties of the Lorenz System
The Lorenz Attractor
A Model for the Lorenz Attractor
The Chaotic Attractor
Exploration: The R�ssler Attractor
Discrete Dynamical Systems
Introduction
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 Shilnikov 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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