An Introduction To Chaotic Dynamical Systems

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.

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 * Sarlovskiis 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 Hnon 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

Robert Devaney