Superdiffusion in the Dissipative Standard Map

TL;DR

This paper investigates transport properties of the dissipative standard map, revealing mostly normal diffusion except near special parameters where anomalous superdiffusion occurs due to trajectory stickiness to invariant sets.

Contribution

It demonstrates the occurrence of superdiffusion caused by sticky invariant sets in the dissipative standard map, highlighting conditions for anomalous transport.

Findings

Diffusion is generally normal except near special parameters.

Anomalous superdiffusion occurs near crisises due to stickiness.

Distribution on sticky sets matches the SRB measure.

Abstract

We consider transport properties of the chaotic (strange) attractor along unfolded trajectories of the dissipative standard map. It is shown that the diffusion process is normal except of the cases when a control parameter is close to some special values that correspond to the ballistic mode dynamics. Diffusion near the related crisises is anomalous and non-uniform in time: there are large time intervals during which the transport is normal or ballistic, or even superballistic. The anomalous superdiffusion seems to be caused by stickiness of trajectories to a non-chaotic and nowhere dense invariant Cantor set that plays a similar role as cantori in Hamiltonian chaos. We provide a numerical example of such a sticky set. Distribution function on the sticky set almost coincides with the distribution function (SRB measure) of the chaotic attractor.