Eur. Phys. J. B 76, 57-68 (2010)
https://doi.org/10.1140/epjb/e2010-00190-6

Ulam method for the Chirikov standard map

Laboratoire de Physique Théorique du CNRS, IRSAMC, Université de Toulouse, UPS, 31062 Toulouse, France

Corresponding author: ^a dima@irsamc.ups-tlse.fr

Received: 8 April 2010
Revised: 21 April 2010
Published online: 24 June 2010

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 Poincaré recurrences in such systems. The eigenmodes show links with trajectories sticking around stability islands.