Aging generates regular motions in weakly chaotic systems

TL;DR

This paper explores how aging affects the statistical behavior of weakly chaotic systems with infinite invariant measures, revealing a coexistence of regular and weakly chaotic motions influenced by aging.

Contribution

It introduces a novel distributional limit theorem for time-averaged observables under aging conditions in weakly chaotic systems with infinite invariant measures.

Findings

A new distributional limit theorem for aging systems.

Regular motions become more probable with aging.

Coexistence of regular and weakly chaotic behaviors in aging regimes.

Abstract

Using intermittent maps with infinite invariant measures, we investigate the universality of time-averaged observables under aging conditions. According to Aaronson-Darling-Kac theorem, in non-aged dynamical systems with infinite invariant measures, the distribution of the normalized time averages of integrable functions converge to the Mittag-Leffler distribution. This well known theorem holds when the start of observations coincides with the start of the dynamical processes. Introducing a concept of the aging limit where the aging time $t_a$ and the total measurement time $t$ goes to infinity while the aging ratio $t_a/t$ is a constant, we obtain a novel distributional limit theorem of time-averaged observables integrable with respect to the infinite invariant density. Applying the theorem to the Lyapunov exponent in intermittent maps, we find that regular motions and a weakly chaotic behavior coexist in the aging limit. This mixed type of dynamics is controlled by the aging ratio and hence is very different from the usual scenario of regular and chaotic motions in Hamiltonian systems. The probability of finding regular motions in non-aged processes is zero, while in the aging regime it is finite and it increases when system ages.