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We analyze the recurrence-time statistics (RTS) in three-dimensionalnon-Hamiltonian volume preserving systems (VPS): an extended standard map, anda fluid model. The extended map is a standard map weakly coupled to anextra-dimension which contains a deterministic regular, mixed (regular andchaotic) or chaotic motion. The extra-dimension strongly enhances the trappingtimes inducing plateaus and distinct algebraic and exponential decays in theRTS plots. The combined analysis of the RTS with the classification of orderedand chaotic regimes and scaling properties, allows us to describe the intricateway trajectories penetrate the before impenetrable regular islands from theuncoupled case. Essentially the plateaus found in the RTS are related totrajectories that stay long times inside trapping tubes, not allowingrecurrences, and then penetrates diffusively the islands (from the uncoupledcase) by a diffusive motion along such tubes in the extra-dimension. Allasymptotic exponential decays for the RTS are related to an ordered regime(quasi-regular motion) and a mixing dynamics is conjectured for the model.These results are compared to the RTS of the standard map with dissipation ornoise, showing the peculiarities obtained by using three-dimensional VPS. Wealso analyze the RTS for a fluid model and show remarkable similarities to theRTS in the extended standard map problem.

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