Anomalous dynamics and the choice of Poincar\'e recurrence-set

TL;DR

This paper explores how the choice of recurrence-set affects Poincaré recurrence-times statistics in Hamiltonian maps, providing visualization methods and insights into long transient behaviors and regular motion detection.

Contribution

It introduces a novel visualization technique linking recurrence-set shape to return probability distributions in phase space, aiding in understanding and modifying recurrence properties.

Findings

Recurrence-set shape significantly influences return probability distributions.

The method detects tiny regions of regular motion effectively.

It explains the dependence of transient decay exponents on recurrence-set choice.

Abstract

We investigate the dependence of Poincar\'e recurrence-times statistics on the choice of recurrence-set, by sampling the dynamics of two- and four-dimensional Hamiltonian maps. We derive a method that allows us to visualize the direct relation between the shape of a recurrence-set and the values of its return probability distribution in arbitrary phase-space dimensions. Such procedure, which is shown to be quite effective in the detection of tiny regions of regular motion, allows to explain it why similar recurrence-sets have very different distributions and how to modify them in order to enhance their return probabilities. Applied on data, this permits to understand the co-existence of extremely long, transient power-like decays whose anomalous exponent depends on the chosen recurrence-set.