Periodic orbit analysis at the onset of the unstable dimension variability and at the blowout bifurcation

TL;DR

This paper investigates the onset of unstable dimension variability (UDV) in chaotic systems, revealing a new mechanism involving infinite period orbits and analyzing the relation between UDV, blowout bifurcation, and on-off intermittency.

Contribution

It introduces a novel mechanism for UDV onset based on infinite period orbits and links it to blowout bifurcation and intermittency phenomena in low-dimensional chaotic systems.

Findings

UDV onset linked to infinite period orbits approaching stability loss

Blowout bifurcation causes the entire chaotic set to lose transversal stability

Chaotic trajectories near the invariant set show on-off intermittency scaling

Abstract

Many chaotic dynamical systems of physical interest present a strong form of nonhyperbolicity called unstable dimension variability (UDV), for which the chaotic invariant set contains periodic orbits possessing different numbers of unstable eigendirections. The onset of UDV is usually related to the loss of transversal stability of an unstable fixed point embedded in the chaotic set. In this paper, we present a new mechanism for the onset of UDV, whereby the period of the unstable orbits losing transversal stability tends to infinity as we approach the onset of UDV. This mechanism is unveiled by means of a periodic orbit analysis of the invariant chaotic attractor for two model dynamical systems with phase spaces of low dimensionality, and seems to depend heavily on the chaotic dynamics in the invariant set. We also described, for these systems, the blowout bifurcation (for which the chaotic set as a whole loses transversal stability) and its relation with the situation where the effects of UDV are the most intense. For the latter point, we found that chaotic trajectories off, but very close to, the invariant set exhibit the same scaling characteristic of the so-called on-off intermittency.