Multiclusters in networks of adaptively coupled phase oscillators

Dynamical systems on networks with adaptive couplings appear naturally in real-world systems such as power grid networks, social networks, and neuronal networks. We investigate a paradigmatic system of adaptively coupled phase oscillators inspired by neuronal networks with synaptic plasticity. One important behavior of such systems reveals splitting of the network into clusters of oscillators with the same frequencies, where different clusters correspond to different frequencies. Starting from one-cluster solutions we provide existence criteria for multicluster solutions and present their explicit form. The phases of the oscillators within one cluster can be organized in different patterns: antipodal, double antipodal, and splay type. Interestingly, multiclusters are shown to exist where different clusters exhibit different patterns. For instance, an antipodal cluster can coexist with a splay cluster. We also provide stability conditions for one- and multicluster solutions. These conditions, in particular, reveal a high level of multistability.