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More About This Textbook
Overview
The study of nonlinear dynamical systems has exploded in the past 25 years, and Robert L. Devaney has made these advanced research developments accessible to undergraduate and graduate mathematics students as well as researchers in other disciplines with the introduction of this widely praised book. In this second edition of his best-selling text, Devaney includes new material on the orbit diagram fro maps of the interval and the Mandelbrot set, as well as striking color photos illustrating both Julia and Mandelbrot sets. This book assumes no prior acquaintance with advanced mathematical topics such as measure theory, topology, and differential geometry, Assuming only a knowledge of calculus, Devaney introduces many of the basic concepts of modern dynamical systems theory and leads the reader to the point of current research in several areas.
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Meet the Author
Professor Robert L. Devaney received his A.B. from Holy Cross College and his Ph.D. from the University of California at Berkeley in 1973. He taught at Northwestern University, Tufts University, and the University of Maryland before coming to Boston University in 1980. He served there as chairman of the Department of Mathematics from 1983 to 1986. His main area of research is dynamical systems, including Hamiltonian systems, complex analytic dynamics, and computer experiments in dynamics. He is the author of An Introduction to Chaotic Dynamical Systems, and Chaos, Fractals, and Dynamics: Computer Experiments in Modern Mathematics, which aims to explain the beauty of chaotic dynamics to high school students and teachers.
Table of Contents
Part One: One-Dimensional Dynamics
Examples of Dynamical Systems
Preliminaries from Calculus
Elementary Definitions
Hyperbolicity
An example: the quadratic family
An Example: the Quadratic Family
Symbolic Dynamics
Topological Conjugacy
Chaos
Structural Stability
Sarlovskii's Theorem
The Schwarzian Derivative
Bifurcation Theory
Another View of Period Three
Maps of the Circle
Morse-Smale Diffeomorphisms
Homoclinic Points and Bifurcations
The Period-Doubling Route to Chaos
The Kneeding Theory
Geneaology of Periodic Units
Part Two: Higher Dimensional Dynamics
Preliminaries from Linear Algebra and Advanced Calculus
The Dynamics of Linear Maps: Two and Three Dimensions
The Horseshoe Map
Hyperbolic Toral Automorphisms
Hyperbolicm Toral Automorphisms
Attractors
The Stable and Unstable Manifold Theorem
Global Results and Hyperbolic Sets
The Hopf Bifurcation
The Hénon Map
Part Three: Complex Analytic Dynamics
Preliminaries from Complex Analysis
Quadratic Maps Revisited
Normal Families and Exceptional Points
Periodic Points
The Julia Set
The Geometry of Julia Sets
Neutral Periodic Points
The Mandelbrot Set
An Example: the Exponential Function