Multicluster States in Adaptive Networks of Coupled Phase Oscillators

Abstract

In this chapter, we analyze multicluster states in a network of adaptively coupled phase oscillators. Multicluster states are composed of several one-clusters with distinct frequencies. Starting from random initial conditions, our numerical simulations show two different types of states. These are the splay and the antipodal type multicluster states. A third mixed type multicluster state is found by specially prepared initial conditions. For all these states the collective motion of oscillators, the shape of the coupling structure, and the interaction between the frequency clusters are presented in detail. It turns out that due to the adaptive dynamics of the coupling weights, the oscillators are able to form groups of strongly connected units. The interaction between the groups is weak compared to the interaction within the groups. We provide an analytical description for the existence of multicluster states by using methods from asymptotic analysis. Employing the fact of weak inter-cluster coupling, we find stability conditions for multicluster states.