Recurrence-time statistics in non-Hamiltonian volume-preserving maps and flows.

Abstract

We analyze the recurrence-time statistics (RTS) in three-dimensional non-Hamiltonian volume-preserving systems (VPS): an extended standard map and a fluid model. The extended map is a standard map weakly coupled to an extra dimension which contains a deterministic regular, mixed (regular and chaotic), or chaotic motion. The extra dimension strongly enhances the trapping times inducing plateaus and distinct algebraic and exponential decays in the RTS plots. The combined analysis of the RTS with the classification of ordered and chaotic regimes and scaling properties allows us to describe the intricate way trajectories penetrate the previously impenetrable regular islands from the uncoupled case. Essentially the plateaus found in the RTS are related to trajectories that stay for long times inside trapping tubes, not allowing recurrences, and then penetrate diffusively the islands (from the uncoupled case) by a diffusive motion along such tubes in the extra dimension. All asymptotic exponential decays for the RTS are related to an ordered regime (quasiregular motion), and a mixing dynamics is conjectured for the model. These results are compared to the RTS of the standard map with dissipation or noise, showing the peculiarities obtained by using three-dimensional VPS. We also analyze the RTS for a fluid model and show remarkable similarities to the RTS in the extended standard map problem.