Poincar\'e recurrences in Hamiltonian systems with a few degrees of freedom

Abstract

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.