A first course in discrete dynamical systems分离动态系统第一课
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作者： Richard A. Holmgren著
出 版 社：
出版时间： 2000-8-1字数：版次： 1页数： 223印刷时间： 2000/08/01开本： 16开印次： 1纸张： 胶版纸I S B N ： 9780387947808包装： 平装内容简介
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.
目录
Preface
Rationale for the new edition
How to use this book.
Acknowledgments.
List of Symbols
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.Perlodic 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 Farallies of Functions and Bifurcations
Exercise Set 7
8.The Logistic Function Part l：Cantor Sets and Chaos
8.1.A First Look at the Logistic Function when r4
8.2.Cantor Sets.
8.3.Chaos and the Dynamics of the Logistic Function
8.4.A Few Additional Comments on Cantor Sets
Exercise Set 8
9.The Logistic Function Part Il：Topological Conjugacy
Exercise Set 9
10.The Logistic Function Part IIl：A Period-Doubling
C：ascade
Exercise Set 10.
11.The Logistic Function Part IV：Symbolic Dynamics
11.1.Symbolic Dynamics and Metric Spaces
1 1.2.Symbolic Dynamics and the Logistic Function
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 Diffrential 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
15.1.Generating Julia and Mandelbrot Sets on a Computer.
Exercise Set 15
Appendix.Mathematica Algorithms
A.1.Iterating FUnctions.
Finding the VaIue of a Point Under Iteration
Tables of Iterates
……
References
Index
