An Introduction To Chaotic Dynamical Systems

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