Ulam method for the Chirikov standard map

Details Ulam method for the Chirikov standard map Ulam method for the Chirikov standard map

2010-04-08

arxiv.org/abs/1004.1349v2

Eur. Phys. J. B 76, 57-68 (2010)

Nonlinear Sciences

Chaotic Dynamics

13 pages, 13 figures, high resolution figures available at: http://www.quantware.ups-tlse.fr/QWLIB/ulammethod/ minor correctio

Scientific paper

10.1140/epjb/e2010-00190-6

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.

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