A first course in chaotic dynamical systems: theory and experiment

Table of Content

Ch. 1. A Mathematical and Historical Tour.
1.1. Images from Dynamical Systems.
1.2. A Brief History of Dynamics
Ch. 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
Ch. 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
Ch. 4. Graphical Analysis.
4.1. Graphical Analysis.
4.2. Orbit Analysis.
4.3. The Phase Portrait
Ch. 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
Ch. 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
Ch. 7. The Quadratic Family.
7.1. The Case c =​ -2.
7.2. The Case c [actual symbol not reproducible] -2.
7.3. The Cantor Middle-Thirds Set
Ch. 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
Ch. 9. Symbolic Dynamics.
9.1. Itineraries.
9.2. The Sequence Space.
9.3. The Shift Map.
9.4. Conjugacy
Ch. 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
Ch. 11. Sarkovskii's Theorem.
11.1. Period 3 Implies Chaos.
11.2. Sarkovskii's Theorem.
11.3. The Period 3 Window.
11.4. Subshifts of Finite Type
Ch. 12. The Role of the Critical Orbit.
12.1. The Schwarzian Derivative.
12.2. The Critical Point and Basins of Attraction
Ch. 13. Newton's Method.
13.1. Basic Properties.
13.2. Convergence and Nonconvergence
Ch. 14. Fractals.
14.1. The Chaos Game.
14.2. The Cantor Set Revisited.
14.3. The Sierpinski Triangle.
14.4. The Koch Snowflake.
14.5. Topological Dimension.
14.6. Fractal Dimension.
14.7. Iterated Function Systems.
14.8. Experiment: Iterated Function Systems
Ch. 15. Complex Functions.
15.1. Complex Arithmetic.
15.2. Complex Square Roots.
15.3. Linear Complex Functions.
15.4. Calculus of Complex Functions
Ch. 16. The Julia Set.
16.1. The Squaring Function.
16.2. The 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 Repellor
Ch. 17. The Mandelbrot Set.
17.1. The Fundamental Dichotomy.
17.2. The Mandelbrot Set.
17.3. Experiment: Periods of Other Bulbs.
17.4. Experiment: Periods of the Decorations.
17.5. Experiment: Find the Julia Set.
17.6. Experiment: Spokes and Antennae.
17.7. Experiment: Similarity of the Mandelbrot and Julia Sets
Ch. 18. Further Projects and Experiments.
18.1. The Tricorn.
18.2. Cubics.
18.3. Exponential Functions.
18.4. Trigonometric Functions.
18.5. Complex Newton's Method
Appendix A. Mathematical Preliminaries
Appendix B. Algorithms
Appendix C. References.

A First Course in Chaotic Dynamical Systems: Theory and Experiment is the first book to introduce modern topics in dynamical systems at the undergraduate level. Accessible to readers with only a background in calculus, the book integrates both theory and computer experiments into its coverage of contemporary ideas in dynamics. It is designed as a gradual introduction to the basic mathematical ideas behind such topics as chaos, fractals, Newton's method, symbolic dynamics, the Julia set, and the Mandelbrot set and includes biographies of some of the leading researchers in the field of dynamical systems. Mathematical and computer experiments are integrated throughout the text to help illustrate the meaning of the theorems presented. Chaotic Dynamical Systems Software, Labs 1?6 is a supplementary laboratory software package, available separately, that allows a more intuitive understanding of the mathematics behind dynamical systems theory. Combined with A First Course in Chaotic Dynamical Systems, it leads to a rich understanding of this emerging field.

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