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 $\beta \approx 1.3$. 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 Poncaré 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.

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

References in corpus (4)

• Long-Time Correlations in the Stochastic Regime
• On universality of algebraic decays in Hamiltonian systems
• Hypothesis of strong chaos and anomalous diffusion in coupled symplectic maps
• Instability statistics and mixing rates

Cited by in corpus (6)

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• What is the mechanism of power-law distributed Poincaré recurrences in higher-dimensional systems?
• Intermittent stickiness synchronization
• Geometry of complex instability and escape in four-dimensional symplectic maps
• Anomalous dynamics and the choice of Poincaré recurrence-set
• Time periodicity and dynamical stability in two-boson systems