Chaos: From Theory to Applications
Springer Science & Business Media, 06.12.2012 - 274 Seiten
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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Estimating the Lyapunov Exponents from Time Series
EVIDENCE OF CHAOS IN CONTROLLEDAND UNCONTROLLED EXPERIMENTS
Nonlinear Electrical Circuits
Do LowDimensional Attractors Exist in Uncontrolled Physical Systems?
Delineating and Quantifying the Dynamics
Determining the Various Dimensions and Lyapunov Exponents for
An Obvious Question
Pitchfork Bifurcation and Period Doubling
Flip Bifurcation and Period Doubling
Universality and Routes to Chaos
A Synonym of Randomness and Beauty
PhaseSpace ReconstructionSingular System Approach
How Many Points Are Enough?
Distinguishing Chaotic Signals from Nonchaotic or from Random Fractal Sequences
Testing for Nonlinearity
Other Approaches to Estimate Dimensions
NONLINEAR TIME SERIES FORECASTING
Global and Local Approximations
Using Chaos in Weather Prediction
Chaos and Noise
Nonlinear Prediction as a Way of Distinguishing Chaotic Signals from Random Fractal Sequences
Statistical Noise Reduction
Some Additional Comments on the Effect of Truncation
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algorithm approach assume autocorrelation function becomes periodic behavior bifurcation called chaotic attractor Chapter conservative systems consider coordinates correlation coefficient correlation dimension corresponding defined DetA deterministic different initial conditions differential equations dimensional dissipative systems dynamical system eigenvalues Elsner embedding dimension equilibrium points ergodic estimate Euclidean example exhibit Figure courtesy finite fixed point fluctuations forecasting fractal dimension frequency Hamiltonian Hénon map integration interval iterations Koch curve limit cycle linear logistic equation logistic map Lorenz system low-dimensional Lyapunov exponents mathematical matrix neural network nonlinear prediction number of points observed obtain output pendulum period doubling phase portrait phase space Poincaré section power spectrum procedure quasi-periodic random reconstructed Reproduced by permission Rössler Rössler attractor scaling region self-similar sequence shown in Fig slope solution spectra spectral density stability statistical step structure topological torus Trace Tsonis unstable variable vector versus volume xn+1 zero