Synchronization and multi-cluster capabilities of oscillatory networks with adaptive coupling

We prove the existence of a multi-dimensional non-trivial invariant toroidal manifold for the Kuramoto network with adaptive coupling. The constructed invariant manifold corresponds to the multi-cluster behavior of the oscillators phases. Contrary to the static coupling, the adaptive coupling strengths exhibit quasiperiodic oscillations preserving zero phase-difference within clusters. The derived sufficient conditions for the existence of the invariant manifold provide a trade-off between the natural frequencies of the oscillators, coupling plasticity parameters, and the interconnection structure of the network. Furthermore, we study the robustness of the invariant manifold with respect to the perturbations of the interconnection topology and establish structural and quantitative constraints on the perturbation adjacency matrix preserving the invariant manifold. Additionally, we demonstrate the application of the new results to the problem of interconnection topology design which consists in endowing the desired multi-cluster behavior to the network by controlling its interconnection structure.