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

TL;DR

This study investigates Poincaré recurrences in Hamiltonian systems with few degrees of freedom, revealing a universal power-law decay with an exponent around 1.3, independent of system specifics, through numerical simulations.

Contribution

It provides numerical evidence for a universal decay exponent in Poincaré recurrences across Hamiltonian systems with finite stability islands.

Findings

Decay of recurrences follows a power law with exponent ~1.3.

The exponent is consistent across different systems and parameters.

Mechanisms behind the decay remain to be fully understood.

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.