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

TL;DR

This paper investigates recurrence-time statistics in three-dimensional non-Hamiltonian volume-preserving systems, revealing how extra dimensions influence trapping times, trajectory penetration, and decay behaviors, with implications for understanding complex dynamical regimes.

Contribution

It introduces a detailed analysis of RTS in 3D non-Hamiltonian VPS, highlighting the role of extra dimensions in trapping and penetration dynamics, and compares these with dissipative and noisy standard maps.

Findings

Plateaus in RTS relate to long trapping inside tubes.

Trajectories penetrate islands diffusively after long trapping.

Exponential RTS decay linked to quasi-regular motion.

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 before impenetrable regular islands from the uncoupled case. Essentially the plateaus found in the RTS are related to trajectories that stay long times inside trapping tubes, not allowing recurrences, and then penetrates 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 (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 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.