An introduction to chaotic dynamical systems

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 2The 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.

Previous edition: Boulder, Colo.: Westview, 2003.

<P>I One Dimensional Dynamics</P><P>1.A Visual and Historical Tour<BR>2.Examples of Dynamical Systems<BR>3.Elementary Definitions<BR>4.Hyperbolicity<BR>5.An Example: The Logistic Family<BR>6.Symbolic Dynamics<BR>7.Topological Conjugacy<BR>8.Chaos<BR>9.Structural Stability<BR>10.Sharkovsky's Theorem<BR>11.The Schwarzian Derivative <BR>12.Bifurcations<BR>13.Another View of Period Three<BR>14.Period-Doubling Route to Chaos<BR>15.Homoclinic Points and Bifurcations<BR>16.Maps of the Circle<BR>17.Morse-Smale Diffeomorphisms</P><P>II Complex Dynamics</P><P>18.Quadratic Maps Revisited<BR>19.Normal Families and Exceptional Points <BR>20.Periodic Points<BR>21.Properties of the Julia Set<BR>22.The Geometry of the Julia Sets<BR>23.Neutral Periodic Points<BR>24.The Mandelbrot Set<BR>25.Rational Maps<BR>26.The Exponential Family</P><P>III Higher Dimensional Dynamics</P><P>27.Dynamics of Linear Maps<BR>28.The Smale Horseshoe Map<BR>29.Hyperbolic Toral Automorphisms<BR>30.Attractors<BR>31.The Stable and Unstable Manifold Theorem<BR>32.Global Results and Hyperbolic Maps<BR>33.The Hopf Bifurcation<BR>34.The Herron Map</P><P>Appendix: Mathematical Preliminaries</P>