Bifurcation and Hausdorff dimension in families of chaotically driven maps with multiplicative forcing

Gerhard Keller; Atsuya Otani

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arXiv · 1208.2888

Bifurcation and Hausdorff dimension in families of chaotically driven maps with multiplicative forcing

Gerhard Keller·Atsuya Otani

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Abstract

We study bifurcations of invariant graphs in skew product dynamical systems driven by hyperbolic surface maps T like Anosov surface diffeomorphisms or baker maps and with one-dimensional concave fibre maps under multiplicative forcing when the forcing is scaled by a parameter r=e^{-t}. For a range of parameters two invariant graphs (a trivial and a non-trivial one) coexist, and we use thermodynamic formalism to characterize the parameter dependence of the Hausdorff and packing dimension of the set of points where both graphs coincide. As a corollary we characterize the parameter dependence of the dimension of the global attractor A_t: Hausdorff and packing dimension have a common value dim(A_t), and there is a critical parameter t_c determined by the SRB measure of T^{-1} such that dim(A_t)=3 for t < t_c and t --> dim(A_t) is strictly decreasing for t_c < t < t_{max}.

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BibTeX RIS

Gerhard Keller, Atsuya Otani. 2012-08-14. Bifurcation and Hausdorff dimension in families of chaotically driven maps with multiplicative forcing. https://arxiv.org/abs/1208.2888

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