Frequency clusters in adaptive networks

We investigate the mechanisms of cluster formation in adaptive networks of oscillatory systems. The adaptation of coupling weights is an important self-control mechanism in various systems such as power grid networks, social networks as well as neuronal networks. In neuronal systems, for instance, the adaptation is called neuronal plasticity and it regulates the changes of the synaptic weights depending on the neuronal activity. This paper is devoted to paradigmatic model of adaptively coupled phase oscillators. We describe the splitting of the network into clusters of oscillators with different frequencies. We prove under which conditions one-cluster and multi-cluster states exist. We show that the clusters can be of different nature: in- or anti-phase synchronized, rotating wave type and others. Interestingly, the existence of multi-cluster states is shown, where different clusters exhibit different patterns. We also provide conditions for the stability of cluster states, which reveal a high level of multistability of such states.