Poincaré recurrences in Hamiltonian systems with a few degrees of freedom

Abstract

Hundred twenty years after the fundamental work of Poincaré, the statistics of Poincaré 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 β≈1.3. This value is smaller compared to the average exponent β≈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 Poincaré exponent has a universal average value β≈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. Poincaré recurrences in DNA are also discussed.