A First Course In Chaotic Dynamical Systems: Theory And Experiment

Cover
Half Title
Series Page
Title Page
Copyright Page
Contents
Preface to the Second Edition
1. A Visual and Historical Tour
	1.1 Images from Dynamical Systems
	1.2 A Brief History of Dynamics
2. Examples of Dynamical Systems
	2.1 An Example from Finance
	2.2 An Example from Ecology
	2.3 Finding Roots and Solving Equations
	2.4 Differential Equations
3. Orbits
	3.1 Iteration
	3.2 Orbits
	3.3 Types of Orbits
	3.4 Other Orbits
	3.5 The Doubling Function
	3.6 Experiment: The Computer May Lie
4. Graphical Analysis
	4.1 Graphical Analysis
	4.2 Orbit Analysis
	4.3 The Phase Portrait
5. Fixed and Periodic Points
	5.1 A Fixed Point Theorem
	5.2 Attraction and Repulsion
	5.3 Calculus of Fixed Points
	5.4 Why Is This True?
	5.5 Periodic Points
	5.6 Experiment: Rates of Convergence
6. Bifurcations
	6.1 Dynamics of the Quadratic Map
	6.2 The Saddle-Node Bifurcation
	6.3 The Period-Doubling Bifurcation
	6.4 Experiment: The Transition to Chaos
7. The Quadratic Family
	7.1 The Case c = −2
	7.2 The Case c < −2
	7.3 The Cantor Middle-Thirds Set
8. Transition to Chaos
	8.1 The Orbit Diagram
	8.2 The Period-Doubling Route to Chaos
	8.3 Experiment: Windows in the Orbit Diagram
9. Symbolic Dynamics
	9.1 Itineraries
	9.2 The Sequence Space
	9.3 The Shift Map
	9.4 Conjugacy
10. Chaos
	10.1 Three Properties of a Chaotic System
	10.2 Other Chaotic Systems
	10.3 Manifestations of Chaos
	10.4 Experiment: Feigenbaum’s Constant
11. Sharkovsky’s Theorem
	11.1 Period 3 Implies Chaos
	11.2 Sharkovsky’s Theorem
	11.3 The Period-3 Window
	11.4 Subshifts of Finite Type
12. Role of the Critical Point
	12.1 The Schwarzian Derivative
	12.2 Critical Points and Basins of Attraction
13. Newton’s Method
	13.1 Basic Properties
	13.2 Convergence and Nonconvergence
14. Fractals
	14.1 The Chaos Game
	14.2 The Cantor Set Revisited
	14.3 The Sierpinski Triangle
	14.4 The Sierpinski Carpet
	14.5 The Koch Snowflake
	14.6 Topological Dimension
	14.7 Fractal Dimension
	14.8 Iterated Function Systems
	14.9 Experiment: Find the Iterated Function Systems
	14.10 Experiment: A “Real” Chaos Game
15. Complex Functions
	15.1 Complex Arithmetic
	15.2 Complex Square Roots
	15.3 Linear Complex Functions
	15.4 Calculus of Complex Functions
16. The Julia Set
	16.1 The Squaring Function
	16.2 Another Chaotic Quadratic Function
	16.3 Cantor Sets Again
	16.4 Computing the Filled Julia Set
	16.5 Experiment: Filled Julia Sets and Critical Orbits
	16.6 The Julia Set as a Repeller
17. The Mandelbrot Set
	17.1 The Fundamental Dichotomy
	17.2 The Mandelbrot Set
	17.3 Complex Bifurcations
	17.4 Experiment: Periods of the Bulbs
	17.5 Experiment: Periods of the Other Bulbs
	17.6 Experiment: How to Add
	17.7 Experiment: Find the Julia Set
	17.8 Experiment: Similarity of the Mandelbrot Set and Julia Sets
18. Other Complex Dynamical Systems
	18.1 Cubic Polynomials
	18.2 Rational Maps
	18.3 Exponential Functions
	18.4 Trigonometric Functions
	18.5 Complex Newton’s Method
Appendix A: Mathematical Preliminaries
	A.1 Functions
	A.2 Some Ideas from Calculus
	A.3 Open and Closed Sets
	A.4 Other Topological Concepts
Bibliography
Index