Spatio-Temporal Patterns In Toroidally Coupled Oscillator Systems

Abstract

This paper presents a study of the spatio-temporal patterns of oscillations that are possible in systems of identical oscillators. These oscillators are symmetrically coupled with the symmetry of a discrete torus. The analysis deals with periodic motions of the entire array rather than individual cells. It exploits the symmetry of the array using results from equivariant bifurcation theory. This work presents a complete list of invariants, equivariants, normal forms, isotropy subgroups and fixed-points subspaces, for the cases with periodicity . It is carried out for the case of a rectangular array with toroidal symmetry. The analysis included all the generic equivariant Hopf bifurcations in this setting and determines the onset, stability and the generic behavior of spatio-temporal patterns for all primary branches. We also find all possible secondary patterns of oscillations using the Theorem.