An Introduction To Chaotic Dynamical Systems

Description

There is an explosion of interest in dynamical systems in the mathematical community as well as in many areas of science. The results have been truly exciting: systems which once seemed completely intractable from an analytic point of view can now be understood in a geometric or qualitative sense rather easily. Scientists and engineers realize the power and the beauty of the geometric and qualitative techniques. These techniques apply to a number of important nonlinear problems ranging from physics and chemistry to ecology and economics. Computer graphics have allowed us to view the dynamical behavior geometrically. The appearance of incredibly beautiful and intricate objects such as the Mandelbrot set, the Julia set, and other fractals have really piqued interest in the field. This is text is aimed primarily at advanced undergraduate and beginning graduate students.  Throughout, the author emphasizes the mathematical aspects of the theory of discrete dynamical systems, not the many and diverse applications of this theory. The field of dynamical systems and especially the study of chaotic systems has been hailed as one of the important breakthroughs in science in the past century and its importance continues to expand. There is no question that the field is becoming more and more important in a variety of scientific disciplines. New to this edition: •Greatly expanded coverage complex dynamics now in Chapter 2 •The third chapter is now devoted to higher dimensional dynamical systems. •Chapters 2 and 3 are independent of one another. •New exercises have been added throughout. Contents Preface to the Third Edition Author Part I One Dimensional Dynamics 1 A Visual and Historical Tour 2 Examples of Dynamical Systems 3 Elementary Definitions 4 Hyperbolicity 5 An Example: The Logistic Family 6 Symbolic Dynamics 7 Topological Conjugacy 8 Chaos 9 Structural Stability 10 Sharkovsky’s Theorem 11 The Schwarzian Derivative 12 Bifurcations 13 Another View of Period Three 14 The Period-Doubling Route to Chaos 15 Homoclinic Points and Bifurcations 16 Maps of the Circle 17 Morse-Smale Diffeomorphisms Part II Complex Dynamics 18 Quadratic Maps Revisited 19 Normal Families and Exceptional Points 20 Periodic Points 21 Properties of the Julia Set 22 The Geometry of the Julia Sets 23 Neutral Periodic Points 24 The Mandelbrot Set 25 Rational Maps 26 The Exponential Family Part III 27 Dynamics of Linear Maps 28 The Smale Horseshoe Map 29 Hyperbolic Toral Automorphisms 30 Attractors 31 The Stable and Unstable Manifold Theorem 32 Global Results and Hyperbolic Maps 33 The Hopf Bifurcation 34 The Hénon Map A Mathematical Preliminaries Bibliography Index