Chaos and Fractals
豆瓣
An Elementary Introduction
Feldman, David P.
简介
This book provides the reader with an elementary introduction to chaos and fractals, suitable for students with a background in elementary algebra, without assuming prior coursework in calculus or physics. It introduces the key phenomena of chaos - aperiodicity, sensitive dependence on initial conditions, bifurcations - via simple iterated functions. Fractals are introduced as self-similar geometric objects and analyzed with the self-similarity and box-counting dimensions. After a brief discussion of power laws, subsequent chapters explore Julia Sets and the Mandelbrot Set. The last part of the book examines two-dimensional dynamical systems, strange attractors, cellular automata, and chaotic differential equations. The book is richly illustrated and includes over 200 end-of-chapter exercises. A flexible format and a clear and succinct writing style make it a good choice for introductory courses in chaos and fractals.
contents
I Introducing Discrete Dynamical Systems
0 Opening Remarks
1 Functions
2 Iterating Functions
3 Qualitative Dynamics: The Fate of the Orbit
4 Time Series Plots
5 Graphical Iteration
6 Iterating Linear Functions
7 Population Models
8 Newton, Laplace, and Determinism
II Chaos
9 Chaos and the Logistic Equation
10 The Butterfly Effect
11 The Bifurcation Diagram
12 Universality
13 Statistical Stability of Chaos
14 Determinism, Randomness, and Nonlinearity
III Fractals
15 Introducing Fractals
16 Dimensions
17 Random Fractals
18 The Box-Counting Dimension
19 When do Averages Exist?
20 Power Laws and Long Tails
21 Infinities, Big and Small
IV Julia Sets and the Mandelbrot Set
22 Introducing Julia Sets
23 Complex Numbers
24 Julia Sets for the Quadratic Family
25 The Mandelbrot Set
V Higher-Dimensional Systems
26 Two-Dimensional Discrete Dynamical Systems
27 Cellular Automata
28 Introduction to Differential Equations
29 One-Dimensional Differential Equations
30 Two-Dimensional Differential Equations
31 Chaotic Differential Equations and Strange Attractors
VI Conclusion
32 Conclusion
VII Appendices
A Review of Selected Topics from Algebra
B Histograms and Distributions
C Suggestions for Further Reading