Clustering in globally coupled oscillators

Abstract

In this paper we describe approaches to clustering in systems of globally coupled identical oscillators. The first of these approaches is based entirely on the symmetry of such systems, and gives information about the behaviour of the systems near degenerate bifurcation points. We summarize existing results from such analysis, and indicate why further techniques are required to augment the symmetry-based methods. This leads to a second approach based on constructing certain reduced models. This modelling approach relies indirectly on symmetry, using the fact that the systems in question have many invariant subspaces as a result of their symmetry. It is shown how knowledge of behaviour on certain subspaces can be used to predict behaviour on others subspaces, even when their dimensions are different. In applications, this approach can be used to predict the stable clustering behaviour that cannot be predicted by other approaches and may be hard to find numerically. All results are illustrated with examples.