Poincare recurrences from the perspective of transient chaos

TL;DR

This paper links Poincaré recurrences in chaotic systems to transient chaos by showing their distribution equivalence through a leaking system approach, revealing decay rate similarities despite differences in average times.

Contribution

It introduces a novel perspective connecting recurrence times to escape times via transient chaos theory, applicable to Hamiltonian systems with mixed phase space.

Findings

Decay rates of recurrence and escape distributions are always the same.

Average recurrence and escape times differ for general initial ensembles.

Results validate the division of the chaotic saddle into hyperbolic and non-hyperbolic parts.

Abstract

We obtain a description of the Poincar\'e recurrences of chaotic systems in terms of the ergodic theory of transient chaos. It is based on the equivalence between the recurrence time distribution and an escape time distribution obtained by leaking the system and taking a special initial ensemble. This ensemble is atypical in terms of the natural measure of the leaked system, the conditionally invariant measure. Accordingly, for general initial ensembles, the average recurrence and escape times are different. However, we show that the decay rate of these distributions is always the same. Our results remain valid for Hamiltonian systems with mixed phase space and validate a split of the chaotic saddle in hyperbolic and non-hyperbolic components.