Correlation sum and recurrence determinism for interval maps

TL;DR

This paper investigates the asymptotic behavior of correlation sum and recurrence determinism in interval maps, providing conditions for their existence, formulas for computation, and examples illustrating their properties in zero-entropy systems.

Contribution

It offers new insights into the asymptotic properties of recurrence quantification measures for interval maps, including explicit formulas and counterexamples.

Findings

Asymptotic correlation sum exists in certain cases.

Provided formulas for computing correlation sum based on omega-limit set.

Recurrence determinism can be less than one in non-chaotic interval maps.

Abstract

Recurrence quantification analysis is a method for measuring the complexity of dynamical systems. Recurrence determinism is a fundamental characteristic of it, closely related to correlation sum. In this paper, we study asymptotic behavior of these quantities for interval maps. We show for which cases the asymptotic correlation sum exists. An example of an interval map with zero entropy and a point with the finite $\omega$-limit set for which the asymptotic correlation sum does not exist is given. Moreover, we present formulas for computation of the asymptotic correlation sum with respect to the cardinality of the $\omega$-limit set or to the configuration of the intervals forming it, respectively. We also show that for a not Li-Yorke chaotic (and hence zero entropy) interval map, the limit of recurrence determinism as distance threshold converges to zero can be strictly smaller than one.