An increasing number of colleges and universities are offering undergradu ate 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 undergradu ate 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.
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Illustrationen
ISBN-13
978-1-4684-0222-3 (9781468402223)
DOI
10.1007/978-1-4684-0222-3
Schweitzer Klassifikation
1 Introduction.- 1.1 Phase Portraits.- Exercise Set 1.- 2 A Quick Look at Functions.- Exercise Set 2.- 3 The Topology of the Real Numbers.- Exercise Set 3.- 4 Periodic Points and Stable Sets.- 4.1 Graphical Analysis.- Exercise Set 4.- 5 Sarkovskii's Theorem.- Exercise Set 5.- 6 Differentiability and Its Implications.- Exercise Set 6.- 7 Parametrized Families of Functions and Bifurcations.- Exercise Set 7.- 8 The Logistic Function, Part I.- Exercise Set 8.- 9 Symbolic Dynamics and Chaos.- Exercise Set 9.- 10 The Logistic Function, Part II: Topological Conjugacy.- Exercise Set 10.- 11 The Logistic Function, Part III.- Exercise Set 11.- 12 Newton's Method.- 12.1 Newton's Method for Quadratic Functions.- 12.2 Newton's Method for Cubic Functions.- 12.3 Intervals and Rates of Convergence.- Exercise Set 12.- 13 Numerical Solutions of Differential Equations.- Exercise Set 13.- 14 The Dynamics of Complex Functions.- 14.1 The Complex Numbers.- 14.2 Complex Functions.- 14.3 The Dynamics of Complex Functions.- 14.4 The Riemann Sphere.- 14.5 Newton's Method in the Complex Plane.- Exercise Set 14.- 15 The Quadratic Family and the Mandelbrot Set.- Generating Julia and Mandelbrot Sets on a Computer.- Exercise Set 15.- Appendix A. Computer Algorithms.- A.1 Iterating Functions.- Finding the Value of a Point Under Iteration.- Tables of Iterates.- Controlling the Precision of the Computations.- Graphing Iterated Functions.- A.2 Graphical Analysis.- A.3 Bifurcation Diagrams.- A.4 Julia Sets.- A.5 The Mandelbrot Set.- A.6 Stable Sets of Newton's Method.- References.- Dynamical Systems.- Topics in Mathematics.- General Interest Books on Dynamics.- Computer Programs and Algorithms.- List of Symbols.
