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arXiv · 0909.4513

Chaotic Hamiltonian systems revisited: Survival probability

V. A. Avetisov·S. K. Nechaev
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Abstract

We consider the dynamical system described by the area--preserving standard mapping. It is known for this system that $P(t)$, the normalized number of recurrences staying in some given domain of the phase space at time $t$ (so-clled "survival probability") has the power--law asymptotics, $P(t)\sim t^{-\nu}$. We present new semi--phenomenological arguments which enable us to map the dynamical system near the chaos border onto the effective "ultrametric diffusion" on the boundary of a tree--like space with hierarchically organized transition rates. In the frameworks of our approach we have estimated the exponent $\nu$ as $\nu=\ln 2/\ln (1+r_g)\approx 1.44$, where $r_g=(\sqrt{5}-1)/2$ is the critical rotation number.

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V. A. Avetisov, S. K. Nechaev. 2009-09-24. Chaotic Hamiltonian systems revisited: Survival probability. https://doi.org/10.1103/physreve.81.046211

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