Accelerator modes and anomalous diffusion in 3D volume-preserving maps

TL;DR

This paper constructs a 3D volume-preserving map with accelerator modes and demonstrates that near bifurcations, the system exhibits trapping and anomalous diffusion, similar to 2D area-preserving maps.

Contribution

It introduces a new 3D volume-preserving map with accelerator modes created at Hopf bifurcations and analyzes their impact on diffusion and trapping times.

Findings

Presence of invariant tori bubbles near bifurcations

Algebraic decay of trapping times around these bubbles

Emergence of anomalous diffusion in the action variable

Abstract

Angle-action maps that are periodic in the action direction can have accelerator modes: orbits that are periodic when projected onto the torus, but that lift to unbounded orbits in an action variable. In this paper we construct a volume-preserving family of maps, with two angles and one action, that have accelerator modes created at Hopf-one (or saddle-center-Hopf) bifurcations. Near such a bifurcation we show that there is often a bubble of invariant tori. Computations of chaotic orbits near such a bubble show that the trapping times have an algebraic decay similar to that seen around stability islands in area-preserving maps. As in the 2D case, this gives rise to anomalous diffusive properties of the action in our 3D map.