On the phenomenon of mixed dynamics in Pikovsky-Topaj system of coupled rotators

TL;DR

This paper investigates mixed dynamics in a family of time-reversible coupled rotator systems, revealing persistent intersections of stable and unstable periodic orbits and analyzing bifurcations that indicate non-conservative behavior.

Contribution

It demonstrates the existence of mixed dynamics in the Pikovsky-Topaj system through indirect search for periodic orbits and bifurcation analysis, expanding understanding of non-conservative time-reversible systems.

Findings

Attractor and repeller intersect but do not coincide.

Mixed dynamics occurs over a large parameter range.

Bifurcation analysis helps detect mixed dynamics.

Abstract

A one-parameter family of time-reversible systems on $\mathbb{T}^3$ is considered. It is shown that the dynamics is not conservative, namely the attractor and repeller intersect but not coincide. We explain this as the manifestation of the so-called mixed dynamics phenomenon which corresponds to a persistent intersection of the closure of stable periodic orbits and the closure of the completely unstable periodic orbits. We search for the stable and unstable periodic orbits indirectly, by finding non-conservative saddle periodic orbits and heteroclinic connections between them. In this way, we are able to claim the existence of mixed dynamics for a large range of parameter values. We investigate local and global bifurcations that can be used for the detection of mixed dynamics.