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

Abstract

A one-parameter family of time-reversible systems on 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.

On three types of dynamics and the notion of attractor

“…In several papers [1,37,26,27,2,19,28,8] a new, third type of chaotic dynamics was identified -the so-called "mixed dynamics" characterized by the attractor-repeller merger. Below we propose a scheme which could formalize this idea.…”

“…On the other hand, there are various examples of reversible systems having symmetric generic elliptic points, see e.g. [38,8,6,7]. In particular, in the Suslov model [7] and in the Pikovsky-Topaj model [38], the involution g of the corresponding two-dimensional Poincaré map is x → x, y → −y and, thus, the attractor and repeller are always symmetric with respect to the axes y = 0, see Fig.…”

“…In this way, based on the analysis of stability of equilibrium states of (8), it was shown in [19] that stable periodic orbits can be born at bifurcations of elliptic periodic points in reversible maps. Here, we need a more detailed investigation of the dynamics of equation (8).…”

“…The attractor-repeller collision means that this function must have very degenerate critical points, so the intuition based on the Conley theorem might suggest that this phenomenon should be very exotic. However, in reality, the attractor-repeller merger appears quite robustly and persists for significant regions of parameter values in the models where it was detected [5,8]. As we will argue below, it is a generic phenomenon for the non-conservative time-reversible systems and, more generally, for systems belonging to the so-called absolute Newhouse domain [10,11].…”

“…Attractor (a) and repeller (b) for the the Suslov model (the pictures are taken from[7]). Attractor (c) and repeller (d) for the the Pikovsky-Topaj model (the pictures are taken from[8]). …”