Fractal dimension of transport coefficients in a deterministic dynamical system

DOI10.1088/0305-4470/37/45/009zbMATH Open1064.82034arXivnlin/0405006OpenAlexW3099630555MaRDI QIDQ4652964

Zbigniew Koza

Publication date: 28 February 2005

Published in: Journal of Physics A: Mathematical and General (Search for Journal in Brave)

Abstract: In many low-dimensional dynamical systems transport coefficients are very irregular, perhaps even fractal functions of control parameters. To analyse this phenomenon we study a dynamical system defined by a piece-wise linear map and investigate the dependence of transport coefficients on the slope of the map. We present analytical arguments, supported by numerical calculations, showing that both the Minkowski-Bouligand and Hausdorff fractal dimension of the graphs of these functions is 1 with a logarithmic correction, and find that the exponent $gamma$ controlling this correction is bounded from above by 1 or 2, depending on some detailed properties of the system. Using numerical techniques we show local self-similarity of the graphs. The local self-similarity scaling transformations turn out to depend (irregularly) on the values of the system control parameters.

Full work available at URL: https://arxiv.org/abs/nlin/0405006

Mathematics Subject Classification ID

Transport processes in time-dependent statistical mechanics (82C70) Low-dimensional dynamical systems (37E99)

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