Local characterization of transient chaos on finite times in open systems

TL;DR

This paper introduces local finite-time measures, such as escape rate and fractal dimensions, to characterize transient chaos in open systems, demonstrating their spatial variation and relation to asymptotic laws.

Contribution

It develops new local quantifiers for transient chaos in open systems and shows their consistency with known asymptotic relationships in a non-asymptotic regime.

Findings

Quantifiers vary significantly across the domain.

Spatial variation aligns with known asymptotic relationships.

Deviations are smaller than differences between locations.

Abstract

To characterize local finite-time properties associated with transient chaos in open dynamical systems, we introduce an escape rate and fractal dimensions suitable for this purpose in a coarse-grained description. We numerically illustrate that these quantifiers have a considerable spread across the domain of the dynamics, but their spatial variation, especially on long but non-asymptotic integration times, is approximately consistent with the relationship that was recognized by Kantz and Grassberger for temporally asymptotic quantifiers. In particular, deviations from this relationship are smaller than differences between various locations, which confirms the existence of such a dynamical law and the suitability of our quantifiers to represent underlying dynamical properties in the non-asymptotic regime.