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We prove the existence of a multi-dimensional non-trivial invariant toroidalmanifold for the Kuramoto network with adaptive coupling. The constructedinvariant manifold corresponds to the multi-cluster behavior of the oscillatorsphases. Contrary to the static coupling, the adaptive coupling strengthsexhibit quasiperiodic oscillations preserving zero phase-difference withinclusters. The derived sufficient conditions for the existence of the invariantmanifold provide a trade-off between the natural frequencies of theoscillators, coupling plasticity parameters, and the interconnection structureof the network. Furthermore, we study the robustness of the invariant manifoldwith respect to the perturbations of the interconnection topology and establishstructural and quantitative constraints on the perturbation adjacency matrixpreserving the invariant manifold. Additionally, we demonstrate the applicationof the new results to the problem of interconnection topology design whichconsists in endowing the desired multi-cluster behavior to the network bycontrolling its interconnection structure.

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