A First Course in Chaotic Dynamical Systems: Theory and Experiment

by Devaney,Robert L.

• ISBN13: 9780201554069
• ISBN10: 0201554062
• Format: Hardcover
• Copyright: 2014-07-01
• Publisher: CRC Press

Summary

Written by one of the most respected mathematicians in the field, this book conveys the essential mathematical ideas in dynamical systems using a combination of theory and computer experimentation. This introductory look at dynamical systems includes investigating the rates of approach to attracting and indifferent fixed points to the discovery of Feigenbaum's constant; exploring the window structure in the orbit diagram; and understanding the periods of the bulbs in the Mandelbrot set.

Author Biography

Professor Robert L. Devaney received his A.B. from Holy Cross College and his Ph.D. from the University of California at Berkeley in 1973. He taught at Northwestern University, Tufts University, and the University of Maryland before coming to Boston University in 1980. He served there as chairman of the Department of Mathematics from 1983 to 1986. His main area of research is dynamical systems, including Hamiltonian systems, complex analytic dynamics, and computer experiments in dynamics. He is the author of An Introduction to Chaotic Dynamical Systems, and Chaos, Fractals, and Dynamics: Computer Experiments in Modern Mathematics, which aims to explain the beauty of chaotic dynamics to high school students and teachers.

Table of Contents

A Mathematical and Historical Tour	1	(8)
Images from Dynamical Systems	1	(4)
A Brief History of Dynamics	5	(4)
Examples of Dynamical Systems	9	(8)
An Example from Finance	9	(2)
An Example from Ecology	11	(1)
Finding Roots and Solving Equations	12	(3)
Differential Equations	15	(2)
Orbits	17	(12)
Iteration	17	(1)
Orbits	18	(1)
Types of Orbits	19	(3)
Other Orbits	22	(2)
The Doubling Function	24	(1)
Experiment: The Computer May Lie	25	(4)
Graphical Analysis	29	(7)
Graphical Analysis	29	(3)
Orbit Analysis	32	(1)
The Phase Portrait	33	(3)
Fixed and Periodic Points	36	(16)
A Fixed Point Theorem	36	(1)
Attraction and Repulsion	37	(1)
Calculus of Fixed Points	38	(4)
Why Is This True?	42	(4)
Periodic Points	46	(2)
Experiment: Rates of Convergence	48	(4)
Bifurcations	52	(17)
Dynamics of the Quadratic Map	52	(5)
The Saddle-Node Bifurcation	57	(4)
The Period-Doubling Bifurcation	61	(2)
Experiment: The Transition to Chaos	63	(6)
The Quadratic Family	69	(13)
The Case c = -2	69	(2)
The Case c < -2	71	(4)
The Cantor Middle-Thirds Set	75	(7)
Transition to Chaos	82	(15)
The Orbit Diagram	82	(7)
The Period-Doubling Route to Chaos	89	(3)
Experiment: Windows in the Orbit Diagram	92	(5)
Symbolic Dynamics	97	(17)
Itineraries	97	(1)
The Sequence Space	98	(5)
The Shift Map	103	(3)
Conjugacy	106	(8)
Chaos	114	(19)
Three Properties of a Chaotic System	114	(7)
Other Chaotic Systems	121	(5)
Manifestations of Chaos	126	(2)
Experiment: Feigenbaum's Constant	128	(5)
Sarkovskii's Theorem	133	(21)
Period 3 Implies Chaos	133	(4)
Sarkovskii's Theorem	137	(5)
The Period 3 Window	142	(4)
Subshifts of Finite Type	146	(8)
The Role of the Critical Orbit	154	(10)
The Schwarzian Derivative	154	(3)
The Critical Point and Basins of Attraction	157	(7)
Newton's Method	164	(12)
Basic Properties	164	(5)
Convergence and Nonconvergence	169	(7)
Fractals	176	(27)
The Chaos Game	176	(2)
The Cantor Set Revisited	178	(2)
The Sierpinski Triangle	180	(2)
The Koch Snowflake	182	(3)
Topological Dimension	185	(1)
Fractal Dimension	186	(4)
Iterated Function Systems	190	(7)
Experiment: Iterated Function Systems	197	(6)
Complex Functions	203	(18)
Complex Arthmetic	203	(4)
Complex Square Roots	207	(2)
Linear Complex Functions	209	(3)
Calculus of Complex Functions	212	(9)
The Julia Set	221	(25)
The Squaring Function	221	(5)
The Chaotic Quadratic Function	226	(1)
Cantor Sets Again	227	(6)
Computing the Filled Julia Set	233	(5)
Experiment: Filled Julia Sets and Critical Orbits	238	(1)
The Julia Set as a Repellor	239	(7)
The Mandelbrot Set	246	(17)
The Fundamental Dichotomy	246	(3)
The Mandelbrot Set	249	(4)
Experiment: Periods of Other Bulbs	253	(4)
Experiment: Periods of the Decorations	257	(1)
Experiment: Find the Julia Set	258	(1)
Experiment: Spokes and Antennae	259	(1)
Experiment: Similarity of the Mandelbrot and Julia Sets	260	(3)
Further Projects and Experiments	263	(16)
The Tricorn	263	(1)
Cubics	264	(3)
Exponential Functions	267	(3)
Trigonometric Functions	270	(3)
Complex Newton's Method	273	(6)
Appendix A. Mathematical Preliminaries	279	(8)
Appendix B. Algorithms	287	(8)
Appendix C. References	295	(4)
Index	299