Dimensions and entropies of strange attractors from a fluctuating dynamics approach

It is shown that the fluctuations in the divergence of near-by trajectories on (strictly deterministic) strange attractors can be modelled by stochastic concepts. In particular, we propose Kramers-Moyal type equations for correlation functions between points on the attractor. The drift terms are the...

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Bibliographic Details
Published in:Physica. D 1984-01, Vol.13 (1), p.34-54
Main Authors: Grassberger, P., Procaccia, I.
Format: Article
Language:English
Subjects:
Exact sciences and technology
Geometry, differential geometry, and topology
Mathematical methods in physics
Physics
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Summary:It is shown that the fluctuations in the divergence of near-by trajectories on (strictly deterministic) strange attractors can be modelled by stochastic concepts. In particular, we propose Kramers-Moyal type equations for correlation functions between points on the attractor. The drift terms are the Lyapunov exponents, the diffusion terms depend on the above fluctuations. From this, we obtain bounds on generalized dimensions and entropies. Numerical results show that in nearly all studied cases (Hénon map, Zaslavskii map, Mackey-Glass eq.) the attractors are fractal measures in the sense of Farmer (information dimension ≠ Hausdorff dimension; metric entropy ≠ topological entropy).
ISSN:0167-2789
1872-8022
DOI:10.1016/0167-2789(84)90269-0