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Discrete dynamical systems are essentially iterated functions. Given the ease with which computers can do iteration, it is now possible for anyone with access to a personal computer to generate beautiful images whose roots lie in discrete dynamical systems. Images of Mandelbrot and Julia sets abound in publications both mathematical and not. The mathematics behind the pictures are beautiful in their own right and are the subject of this text. The level of the presentation is suitable for advanced undergraduates with a year of calculus behind them. Students in the author's courses using this material have come from numerous disciplines; many have been majors in other disciplines who are taking mathematics courses out of general interest. Concepts from calculus are reviewed as necessary. Mathematica programs that illustrate the dynamics and that will aid the student in doing the exercises are included in an appendix.
INDICE: Preface.- List of Symbols.- A Quick Look at Functions.- The Topology of the Real Numbers.- Periodic Points and Stable Sets.- Sarkovskii's Theorem.- Differentiability and Its Implications.- Parametrized Families of Functions and Bifurcations.- The Logistic Function Part I: Cantor Sets and Chaos.- The Logistic Function Part II: Topological Conjugacy.- The Logistic Function Part III: A Period-Doubling.- The Logistic Function Part IV: Symbolic Dynamics.- Newton's Method.- Numerical Solutions of Differential Equations.- The Dynamics of Complex Functions.- The Quadratic Family and the Mandelbrot Set.- Appendix. Mathematica Algorithms.- References.- Index.
Students of mathematics, students of physics, engineering and biology
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