Poincar\'e recurrences and Ulam method for the Chirikov standard map

TL;DR

This paper investigates the statistics of Poincaré recurrences in the Chirikov standard map and the separatrix map at critical parameters, using a generalized Ulam method and a new Monte Carlo approach to analyze long-time behavior.

Contribution

It introduces a generalized Ulam method and a novel survival Monte Carlo technique to study Poincaré recurrences over extensive timescales in complex dynamical maps.

Findings

Recurrences at long times are influenced by trajectory sticking near the critical golden curve.

Poincaré exponents are determined for the studied maps.

Eigenstates of the Ulam matrix exhibit localization properties related to recurrences.

Abstract

We study numerically the statistics of Poincar\'e recurrences for the Chirikov standard map and the separatrix map at parameters with a critical golden invariant curve. The properties of recurrences are analyzed with the help of a generalized Ulam method. This method allows to construct the corresponding Ulam matrix whose spectrum and eigenstates are analyzed by the powerful Arnoldi method. We also develop a new survival Monte Carlo method which allows us to study recurrences on times changing by ten orders of magnitude. We show that the recurrences at long times are determined by trajectory sticking in a vicinity of the critical golden curve and secondary resonance structures. The values of Poincar\'e exponents of recurrences are determined for the two maps studied. We also discuss the localization properties of eigenstates of the Ulam matrix and their relation with the Poincar\'e recurrences.