Chaos in Systems with Noise
As in the first edition, the influence of random noise on the chaotic behavior of dissipative dynamical systems is investigated. Problems are illustrated by mechanical examples. This revised and updated edition contains new sections on the summary of probability theory, homoclinic chaos, Melnikov method, routes to chaos, stabilization of period-doubling, and Hopf bifurcation by noise. Some chapters have been rewritten and new examples have been added.
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Preface to the second edition
Chaotic and Stochastic Processes
Noisy Dynamical Systems
MultiMaxima Probability Density Functions
Random Lyapunov Exponents
Poincare Maps for Noisy Systems
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amplitude approximation asymptotic behaviour attractor averaged band-limited white noise bifurcation diagram chaotic domain chaotic stochastic process characteristic constant correlation function Definition described deterministic system differential equation dynamical system eigenvalue example exists frequency homoclinic orbit Hopf bifurcation hyperbolic fixed point initial conditions initial value problem interval Kapitaniak Kolmogorov entropy let us consider linear logistic map loss of chaos maximum Lyapunov exponent maximum one-dimensional Lyapunov mean Poincare maps mean value Melnikov function multi-maxima Naschie neighbourhood noise intensity noisy system nonlinear obtain one-dimensional Lyapunov exponent oscillator period-doubling bifurcation phase space Power spectra probability density function process x(t,u random noise random variable realizations regular behaviour regular stochastic process response route to chaos shown in Figure shows chaotic behaviour spatial plot spectral density stable and unstable stationary stochastic sensitivity function sure solution t+At theorem transversal unperturbed system unstable manifolds unstable region value of noise variational equation zero
Page ix - Assume that we are tossing a fair coin three times. The possible outcomes of this experiment are HHH, HHT. HTH, THH. HTT, THT, TTH and TTT , where H denotes heads and T denotes tails. Each possible outcome of the experiment is called an elementary event. Thus, there are eight elementary events and it is by definition that the probability of obtaining for example HHT is 1/8, but we cannot obtain HHT repeatedly. Each event H or T occurs 'at random...