Arnold cat map, Ulam method and time reversal

Details Arnold cat map, Ulam method and time reversal Arnold cat map, Ulam method and time reversal

2011-07-03

arxiv.org/abs/1107.0437v1

Physica D 241, 514-518 (2012)

Nonlinear Sciences

Chaotic Dynamics

7 pages, 8 figures. Research done at Quantware http://www.quantware.ups-tlse.fr/

Scientific paper

10.1016/j.physd.2011.11.012

We study the properties of the Arnold cap map on a torus with a several periodic sections using the Ulam method. This approach generates a Markov chain with the Ulam matrix approximant. We study numerically the spectrum and eigenstates of this matrix showing their relation with the Fokker-Plank relaxation and the Kolmogorov-Sinai entropy. We show that, in the frame of the Ulam method, the time reversal property of the map is preserved only on a short Ulam time which grows only logarithmically with the matrix size. Parallels with the evolution in a regime of quantum chaos are also discussed.

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