Poincare recurrences and transient chaos in systems with leaks

TL;DR

This paper develops a theoretical framework for understanding escape rates and Poincare recurrences in chaotic systems with leaks, linking transient chaos to observable decay behaviors and system properties.

Contribution

It introduces an improved expression for escape rates in systems with leaks, connecting transient chaos theory with Poincare recurrence analysis.

Findings

Derived an expression for escape rate considering leak position and size.

Established the equivalence between leaks in open systems and Poincare recurrences in closed systems.

Justified the division of invariant saddle into hyperbolic and nonhyperbolic parts for decay analysis.

Abstract

In order to simulate observational and experimental situations, we consider a leak in the phase space of a chaotic dynamical system. We obtain an expression for the escape rate of the survival probability applying the theory of transient chaos. This expression improves previous estimates based on the properties of the closed system and explains dependencies on the position and size of the leak and on the initial ensemble. With a subtle choice of the initial ensemble, we obtain an equivalence to the classical problem of Poincare recurrences in closed systems, which is treated in the same framework. Finally, we show how our results apply to weakly chaotic systems and justify a split of the invariant saddle in hyperbolic and nonhyperbolic components, related, respectively, to the intermediate exponential and asymptotic power-law decays of the survival probability.