VARIETY AND GENERALITY OF CLUSTERING IN GLOBALLY COUPLED OSCILLATORS

Clustering behavior in globally coupled identical oscillators is studied by using the phase model. Any pair of oscillators is assumed to interact through their phase difference. Without specifying the coupling function, the stability of symmetric cluster states is analyzed, where 'symmetric' means that the clusters contain the same number of oscillators. It is found that higher harmonics in the coupling function are essential to clustering. Examples show that various cluster states appear depending on the coupling form, including those with oscillating intervals, anti-phase states and chaotic long transient. Apart from the phase model, we also study a general model for globally coupled limit-cycle oscillators in the vicinity of the Hopf bifurcation, and find that such a system does not exhibit clustering within the lowest-order approximation.