  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.
