Details for: Holmgren R. A First Course in Discrete Dynamical Systems 1994

Holmgren R. A First Course in Discrete Dynamical Systems 1994

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Textbook in PDF format An increasing number of colleges and universities are offering undergraduate courses in discrete dynamical systems. This growth is due in part to the proliferation of inexpensive and powerful computers, which have provided access to the interesting and complex phenomena that are at the heart of dynamics. A second reason for introducing dynamics into the undergraduate curriculum is that it serves as a bridge from concrete, often algorithmic calculus courses, to the more abstract concepts of analysis and topology. Discrete dynamical systems are essentially iterated functions, and if there is one thing computers do well, it is iteration. It is now possible for anyone with access to a personal computer to generate beautiful images whose roots lie in discrete dynamical systems. The mathematics behind the pictures are beautiful in their own right and are the subject of this text. Every effort has been made to exploit this opportunity to illustrate the beauty and power of mathematics in an interesting and engaging way. This work is first and foremost a mathematics book. Individuals who read it and do the exercises will gain not only an understanding of dynamical systems, but an increased understanding of the related areas in analysis as well. Preface Introduction 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 Symbolic Dynamics and Chaos The Logistic Function, Part II: Topological Conjugacy The Logistic Function, Part III Newton's Method Numerical Solutions of Differential Equations The Dynamics of Complex Functions The Quadratic Family and the Mandelbrot Set Generating Julia and Mandelbrot Sets on a Computer Computer Algorithms Graphical Analysis References Dynamical Systems Topics in Mathematics General Interest Books on Dynamics Computer Programs and Algorithms

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