Hölder regularity and chaotic attractors
نویسنده
چکیده
We demonstrate how the Hölder regularity of a given signal is a lower bound for the Grassberger-Procaccia correlation dimension of strange attractors PACS numbers : 05.45D,47.52 It is known from the celebrated work by Lorenz [1] that even low-dimensional deterministic dynamical systems may exhibit chaotic behavior. In the context of turbulence in fluid dynamics, Ruelle and Takens [2] have shown that the usual attractors of the asymptotic flow in the phase-space (fixed points, periodic and quasiperiodic motion) cannot explain the sensitive dependence of the solutions on the initial conditions . Attractors which display chaotic features have been called by Ruelle and Takens “strange attactors”. Very often a strange attractor is a fractal object with different topological and metric (Hausdorff) dimensions. Grassberger and Procaccia [3] introduced the correlation dimension ν of strange attractors as a new measure related to the fractal dimension D by the relation ν < D. The Grassberger and Procaccia method [3] is a practical algorithm to extract dimensional information from experimental data. Given a signal ~ Xi=1,N = ~ X(t+ ıτ)i=1,N , the correlation integral is defined as the standard correlation function of the time series on the attractor:
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تاریخ انتشار 2001