Chaotic Hamiltonian systems revisited: Survival probability

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.

7 pages, 3 figures: some points clarified, references added

References in corpus (6)

• Long-Time Correlations in the Stochastic Regime
• On universality of algebraic decays in Hamiltonian systems
• Universality of algebraic laws in Hamiltonian systems
• Calculation of Superdiffusion for the Chirikov-Taylor Model
• Instability statistics and mixing rates
• Random Hierarchical Matrices: Spectral Properties and Relation to Polymers on Disordered Trees

Cited by in corpus (5)

• Fluctuations in the relaxation dynamics of mixed chaotic systems
• Poincaré recurrences and Ulam method for the Chirikov standard map
• Native ultrametricity of sparse random ensembles
• Coarse-graining complex dynamics: Continuous Time Random Walks vs. Record Dynamics
• Quantitative universality for a class of weakly chaotic systems