Chaotic Hamiltonian systems revisited: Survival probability

TL;DR

This paper analyzes the survival probability in the standard mapping, showing it follows a power-law decay and introducing a semi-phenomenological approach that maps the system near chaos to ultrametric diffusion, estimating the decay exponent.

Contribution

The paper introduces a semi-phenomenological method linking the dynamics near chaos to ultrametric diffusion and estimates the survival probability decay exponent.

Findings

Survival probability follows a power-law decay with exponent approximately 1.44.

Mapping near chaos border to ultrametric diffusion provides new insights into the system's dynamics.

Estimated decay exponent matches theoretical predictions based on hierarchical transition rates.

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.