Ulam method for the Chirikov standard map

TL;DR

This paper introduces a generalized Ulam method for symplectic maps with divided phase space, demonstrating convergence of the Perron-Frobenius operator approximation and revealing power law decay in relaxation modes linked to Poincaré recurrences.

Contribution

The paper develops a new generalized Ulam method applicable to symplectic maps with divided phase space and shows its convergence and spectral properties through extensive numerical analysis.

Findings

Ulam approximant converges to a continuous limit in chaotic regions.

Relaxation mode spectrum exhibits power law decay.

Decay exponent matches Poincaré recurrence exponent.

Abstract

We introduce a generalized Ulam method and apply it to symplectic dynamical maps with a divided phase space. Our extensive numerical studies based on the Arnoldi method show that the Ulam approximant of the Perron-Frobenius operator on a chaotic component converges to a continuous limit. Typically, in this regime the spectrum of relaxation modes is characterized by a power law decay for small relaxation rates. Our numerical data show that the exponent of this decay is approximately equal to the exponent of Poincar\'e recurrences in such systems. The eigenmodes show links with trajectories sticking around stability islands.