What is the mechanism of power-law distributed Poincar\'e recurrences in higher-dimensional systems?

TL;DR

This paper investigates the mechanism behind power-law distributed Poincaré recurrences in higher-dimensional Hamiltonian systems, revealing that trapping occurs at the surface of regular regions and is dominated by resonance channels, unlike in 2D maps.

Contribution

It identifies the trapping mechanism in 4D symplectic maps as occurring outside the Arnold web and driven by resonance channels, providing new insights into higher-dimensional chaos.

Findings

Trapping occurs at the surface of regular regions, not due to phase space hierarchy.

Resonance channels extend into chaotic regions, influencing trapping.

Evidence of partial transport barriers and stochastic drift along resonance channels.

Abstract

The statistics of Poincar\'e recurrence times in Hamiltonian systems typically shows a power-law decay with chaotic trajectories sticking to some phase-space regions for long times. For higher-dimensional systems the mechanism of this power-law trapping is still unknown. We investigate trapped orbits of a generic 4D symplectic map in phase space and frequency space and find that, in contrast to 2D maps, the trapping is (i) not due to a hierarchy in phase space. Instead, it occurs at the surface of the regular region, (ii) outside of the Arnold web. The chaotic dynamics in this sticky region is (iii) dominated by resonance channels which reach far into the chaotic region: We observe (iii.a) clear signatures of some kind of partial transport barriers and conjecture (iii.b) a stochastic process with an effective drift along resonance channels. These two processes lay the basis for a future understanding of the mechanism of power-law trapping in higher-dimensional systems.