Chaos: From Theory to Applications
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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algorithm approach assume autocorrelation function becomes periodic behavior bifurcation Brownian motion called chaotic attractor Chapter conservative systems consider constant coordinates correlation coefficient correlation dimension corresponding defined deterministic different initial conditions differential equations dimensional dissipative systems dynamical system eigenvalues embedding dimension equilibrium points ergodic estimate Euclidean example exhibit fBm's Figure courtesy finite first-order fixed point fluctuations forecasting fractal dimension frequency Hamiltonian Henon map integration interval iterations length limit cycle linear logistic map Lorenz system Lyapunov exponents mathematical matrix neural network noise reduction nonlinear prediction number of points observed obtain output pendulum period doubling periodic orbits phase portrait phase space power spectrum procedure quasi-periodic random reconstructed Reproduced by permission route to chaos scaling region self-similar sequence shown in Fig slope solution spectra spectral density stability statistical step strange attractor structure topological torus Trace Tsonis unstable variable vector volume zero