Poincaré recurrences in Hamiltonian systems with a few degrees of freedom

Details Poincaré recurrences in Hamiltonian systems with a few degrees of freedom Poincaré recurrences in Hamiltonian systems with a few degrees of freedom

2010-08-09

arxiv.org/abs/1008.1465v2

Phys. Rev. E v.82, p.055202(R) (2010)

Nonlinear Sciences

Chaotic Dynamics

revtex 4 pages, 4 figs; Refs. and discussion added

Scientific paper

10.1103/PhysRevE.82.055202

Hundred twenty years after the fundamental work of Poincar\'e, the statistics of Poincar\'e recurrences in Hamiltonian systems with a few degrees of freedom is studied by numerical simulations. The obtained results show that in a regime, where the measure of stability islands is significant, the decay of recurrences is characterized by a power law at asymptotically large times. The exponent of this decay is found to be $\beta \approx 1.3$. This value is smaller compared to the average exponent $\beta \approx 1.5$ found previously for two-dimensional symplectic maps with divided phase space. On the basis of previous and present results a conjecture is put forward that, in a generic case with a finite measure of stability islands, the Poncar\'e exponent has a universal average value $\beta \approx 1.3$ being independent of number of degrees of freedom and chaos parameter. The detailed mechanisms of this slow algebraic decay are still to be determined.

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