Power-law trapping in the volume-preserving Arnold-Beltrami-Childress map

TL;DR

This paper investigates the intermittent, power-law decay of Poincaré recurrence times in a three-dimensional volume-preserving chaotic system, revealing the role of resonances and partial barriers in long-term orbit trapping.

Contribution

It introduces a resonance-based classification of long-trapped orbits in the 3D Arnold-Beltrami-Childress map, explaining power-law recurrence decay and oscillations.

Findings

Power-law decay of recurrence times with oscillations observed.

Resonances and partial barriers identified as key factors.

Classification of long-trapped orbits explains decay behavior.

Abstract

Understanding stickiness and power-law behavior of Poincar\'e recurrence statistics is an open problem for higher-dimensional systems, in contrast to the well-understood case of systems with two degrees-of-freedom. We study such intermittent behavior of chaotic orbits in three-dimensional volume-preserving systems using the example of the Arnold-Beltrami-Childress map. The map has a mixed phase space with a cylindrical regular region surrounded by a chaotic sea for the considered parameters. We observe a characteristic overall power-law decay of the cumulative Poincar\'e recurrence statistics with significant oscillations superimposed. This slow decay is caused by orbits which spend long times close to the surface of the regular region. Representing such long-trapped orbits in frequency space shows clear signatures of partial barriers and reveals that coupled resonances play an essential role. Using a small number of the most relevant resonances allows for classifying long-trapped orbits. From this the Poincar\'e recurrence statistics can be divided into different exponentially decaying contributions which very accurately explains the overall power-law behavior including the oscillations.