Chaotic Hamiltonian systems revisited: Survival probability

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