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Wavelet analysis of nonlinear dynamical systems. (English) Zbl 1051.65124

Summary: A wavelet based technique for exploring chaos, randomness and periodicity in nonlinear dynamical systems is proposed. The approximate solution of a dynamical system, considered as a signal, is analyzed by studying suitable clusters of the wavelet coefficients. The wavelet coefficients (detail coefficients) are strictly related to the Newton’s finite differences discrete operators, thus giving information on the first and second order properties of the data. The energy and the Shannon information function are connected with Hurst exponent in order to show the autocorrelation of the details along the signal. This gives a wavelet based technique to connect the evolution on time of the detail coefficients with the chaotic evolution. As application, the Lorenz attractor is analyzed in a wavelet basis.

MSC:

65P20 Numerical chaos
65T60 Numerical methods for wavelets
37M25 Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy, etc.)
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37H10 Generation, random and stochastic difference and differential equations
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