Chaotic Maps: Dynamics, Fractals, and Rapid Fluctuations

✍ Scribed by Goong Chen, Yu Huang

• Publisher: Morgan & Claypool Publishers
• Year: 2011
• Tongue: English
• Leaves: 243
• Series: Synthesis Lectures on Mathematics and Statistics
• Edition: 1
• Category: Library

✦ Synopsis

This book consists of lecture notes for a semester-long introductory graduate course on dynamical systems and chaos taught by the authors at Texas A&M University and Zhongshan University, China. There are ten chapters in the main body of the book, covering an elementary theory of chaotic maps in finite-dimensional spaces. The topics include one-dimensional dynamical systems (interval maps), bifurcations, general topological, symbolic dynamical systems, fractals and a class of infinite-dimensional dynamical systems which are induced by interval maps, plus rapid fluctuations of chaotic maps as a new viewpoint developed by the authors in recent years. Two appendices are also provided in order to ease the transitions for the readership from discrete-time dynamical systems to continuous-time dynamical systems, governed by ordinary and partial differential equations. Table of Contents: Simple Interval Maps and Their Iterations / Total Variations of Iterates of Maps / Ordering among Periods: The Sharkovski Theorem / Bifurcation Theorems for Maps / Homoclinicity. Lyapunoff Exponents / Symbolic Dynamics, Conjugacy and Shift Invariant Sets / The Smale Horseshoe / Fractals / Rapid Fluctuations of Chaotic Maps on RN / Infinite-dimensional Systems Induced by Continuous-Time Difference Equations

✦ Table of Contents

Preface......Page 13
Introduction......Page 17
The Inverse and Implicit Function Theorems......Page 24
Visualizing from the Graphics of Iterations of the Quadratic Map......Page 27
Notes for Chapter 1......Page 36
The Use of Total Variations as a Measure of Chaos......Page 37
Notes for Chapter 2......Page 44
Ordering among Periods: The Sharkovski Theorem......Page 45
Notes for Chapter 3......Page 48
The Period-Doubling Bifurcation Theorem......Page 51
Saddle-Node Bifurcations......Page 56
The Pitchfork Bifurcation......Page 60
Hopf Bifurcation......Page 63
Notes for Chapter 4......Page 71
Homoclinic Orbits......Page 73
Lyapunoff Exponents......Page 77
Notes for Chapter 5......Page 84
The Itinerary of an Orbit......Page 85
Properties of the shift map......Page 87
Symbolic Dynamical Systems k and +k......Page 94
The Dynamics of (k+,+) and Chaos......Page 97
Topological Conjugacy and Semiconjugacy......Page 109
Construction of Shift Invariant Sets......Page 113
Snap-back Repeller as a Shift Invariant Set......Page 122
Notes for Chapter 6......Page 125
The Standard Smale Horseshoe......Page 127
The General Horseshoe......Page 132
Notes for Chapter 7......Page 140
Examples of Fractals......Page 141
Hausdorff Dimension and the Hausdorff Measure......Page 142
Iterated Function Systems (IFS)......Page 146
Notes for Chapter 8......Page 155
Total Variation for Vector-Value Maps......Page 157
Rapid Fluctuations of Maps on RN......Page 161
Rapid Fluctuations of Systems with Quasi-shift Invariant Sets......Page 163
Rapid Fluctuations of Systems Containing Topological Horseshoes......Page 165
Examples of Applications of Rapid Fluctuations......Page 168
Notes for Chapter 9......Page 176
I3DS......Page 177
Rates of Growth of Total Variations of Iterates......Page 178
Properties of the Set B(f)......Page 180
Properties of the Set U(f)......Page 182
Properties of the Set E(f)......Page 190
Notes for Chapter 10......Page 193
The Local Behavior of 2-Dimensional Nonlinear Systems......Page 195
Index for Two-Dimensional Systems......Page 209
The Poincaré Map for a Periodic Orbit in RN......Page 211
The Mathematical Model and Motivations......Page 221
Chaotic Vibration of the Wave Equation......Page 225
Bibliography......Page 233
Authors' Biographies......Page 239
Index......Page 241