V. A. Avetisov 1, S. K. Nechaev 2, 3
Physical Review E: Statistical, Nonlinear, and Soft Matter Physics 81 (2010) 046211
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.
- 1. The Semenov Institute of Chemical Physics,
Russian Academy of Sciences - 2. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
CNRS : UMR8626 – Université Paris XI – Paris Sud - 3. P. N. Lebedev Physical Institute,
Russian Academy of Science