Riddling Bifurcation in Chaotic Dynamical Systems

Abstract

When a chaotic attractor lies in an invariant subspace, as in systems with symmetry, riddling can occur. Riddling refers to the situation where the basin of a chaotic attractor is riddled with holes that belong to the basin of another attractor. We establish properties of the riddling bifurcation that occurs when an unstable periodic orbit embedded in the chaotic attractor, usually of low period, becomes transversely unstable. An immediate physical consequence of the riddling bifurcation is that an extraordinarily low fraction of the trajectories in the invariant subspace diverge when there is a symmetry breaking. [S0031-9007(96)00503-0]

Intermittency in a Locally Forced Plane Couette Flow

“…In some sense, the structure is spatially frozen and spatiotemporal intermittency is converted into temporal intermittency, which suggests to interpret phenomena within the theory of lowdimensional dynamical systems. In this framework, several routes to chaos have been recognized [1]; however features characteristic of known scenarios (e.g., [13][14][15]) have not been clearly identified up to now. Some enlightening is expected from the detailed study of the streak breakdown and relaminarization steps that is currently underway [16].…”

An introduction to the synchronization of chaotic systems: coupled skew tent maps

“…Usually, basins of attraction are open sets with a simple geometric structure. In the context of synchronization of chaos, riddled basins have been observed [7][8][9][10][11][12], i.e. basins that nowhere contain any open balls, even not very tiny ones, but nevertheless have positive Lebesgue measure.…”

“…[5,6]). Recently, various papers in mathematics and physics have been published, which lead to a better understanding of the phenomenon of synchronization [7][8][9][10][11][12]. The purpose of the present paper is to explain some of their content on the simple example of two coupled skew tent maps, keeping engineering applications in mind.…”

Role of the Absorbing Area in Chaotic Synchronization

“…The riddling bifurcation (also referred to as the bubbling transition [5]) in which the first orbit embedded in the chaotic attractor loses its transverse stability has recently been investigated in detail by Lai et al [8]. They suggest that the bifurcation takes place as two repellers located symmetrically on either side of the invariant subspace approach the chaotic attractor and collide with a saddle embedded in this attractor.…”