Chaos: From Theory to Applications

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Springer Science & Business Media, 06.12.2012 - 274 Seiten
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Based on chaos theory two very important points are clear: (I) random looking aperiodic behavior may be the product of determinism, and (2) nonlinear problems should be treated as nonlinear problems and not as simplified linear problems. The theoretical aspects ofchaos have been presented in great detail in several excellent books published in the last five years or so. However, while the problems associated with applications of the theory-such as dimension and Lyapunov exponentsestimation, chaosand nonlinear pre diction, and noise reduction-have been discussed in workshops and ar ticles, they have not been presented in book form. This book has been prepared to fill this gap between theory and ap plicationsand to assist studentsand scientists wishingto apply ideas from the theory ofnonlinear dynamical systems to problems from their areas of interest. The book is intended to be used as a text for an upper-level undergraduate or graduate-level course, as well as a reference source for researchers. My philosophy behind writing this book was to keep it simple and informative without compromising accuracy. I have made an effort to presentthe conceptsby usingsimplesystemsand step-by-stepderivations. Anyone with an understanding ofbasic differential equations and matrix theory should follow the text without difficulty. The book was designed to be self-contained. When applicable, examples accompany the theory. The reader will notice, however, that in the later chapters specific examples become less frequent. This is purposely done in the hope that individuals will draw on their own ideas and research projects for examples.
 

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Inhalt

CHAPTER
5
CHAPTER 1
7
Stability Analysis
17
CHAPTER 3
33
Integrable and Nonintegrable Dynamical Systems
42
CHAPTER 4
49
Fractality in Time Series
55
ATTRACTORS
67
Estimating the Lyapunov Exponents from Time Series
179
CHAPTER 9
182
EVIDENCE OF CHAOS IN CONTROLLEDAND UNCONTROLLED EXPERIMENTS
189
Nonlinear Electrical Circuits
193
CouetteTaylor System
194
RayleighBénard Convection
196
Other Experiments
200
Do LowDimensional Attractors Exist in Uncontrolled Physical Systems?
202

Strange Attractors
73
Delineating and Quantifying the Dynamics
83
Determining the Various Dimensions and Lyapunov Exponents for
94
An Obvious Question
100
Pitchfork Bifurcation and Period Doubling
107
Flip Bifurcation and Period Doubling
115
Universality and Routes to Chaos
126
A Synonym of Randomness and Beauty
132
Quantum Chaos
141
CHAPTER 8
149
PhaseSpace ReconstructionSingular System Approach
156
Estimating Dimensions
158
How Many Points Are Enough?
162
Distinguishing Chaotic Signals from Nonchaotic or from Random Fractal Sequences
173
Testing for Nonlinearity
177
Other Approaches to Estimate Dimensions
178
CHAPTER 10
211
NONLINEAR TIME SERIES FORECASTING
213
Global and Local Approximations
215
Examples
222
InputOutput Systems
224
Neural Networks
226
Examples
231
Using Chaos in Weather Prediction
235
Chaos and Noise
238
Nonlinear Prediction as a Way of Distinguishing Chaotic Signals from Random Fractal Sequences
241
CHAPTER 11
245
Noise Reduction
247
Statistical Noise Reduction
255
Some Additional Comments on the Effect of Truncation
256
INDEX 271
267
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