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

Abstract

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.

Stability of cluster formations in adaptive Kuramoto networks

“…In the present paper, new sufficient conditions for the local asymptotic stability of multi-cluster formations of adaptive Kuramoto networks are proposed, which are formulated in terms of the interconnection topology of the networks and plasticity parameters of adaptive coupling. By this, the paper extends some results of Menara et al (2019a) to the case of adaptive networks and complements the results of Feketa et al (2019b) with sufficient conditions on the stability of multicluster formations. These new sufficient conditions provide qualitative relations between the intra-and inter-cluster network connectivity and the plasticity parameters of the corresponding adaptive couplings.…”

“…The network exhibits cluster synchronization when the oscillators can be partitioned into subsets so that the phases of the oscillators in each subset evolve identically. This type of behavior corresponds to the existence of an invariant toroidal manifold of system (1), see Feketa et al (2019b).…”

“…Together with the phases of oscillators which are defined on a circle, this leads to a dynamical system defined in the product of a multidimensional torus and Euclidean space. The emergence of synchrony in such networks corresponds to the existence of invariant manifolds of a lower dimension and of a special topological structure (Feketa et al, 2019b).…”

“…In Scardovi (2010), a class of phase models with state-dependent coupling that are gradients of suitable potential functions is identified. In Feketa et al (2019b), sufficient conditions are derived for the existence of multi-dimensional invariant toroidal manifolds which correspond to the multi-cluster behavior in Kuramoto networks with adaptive coupling. In the present paper, new sufficient conditions for the local asymptotic stability of multi-cluster formations of adaptive Kuramoto networks are proposed, which are formulated in terms of the interconnection topology of the networks and plasticity parameters of adaptive coupling.…”

What adaptive neuronal networks teach us about power grids

“…Adaptive networks have been reported for chemical [5], epidemic [6], biological, and social systems [7]. A paradigmatic example of adaptively coupled phase oscillators has recently attracted much attention [8][9][10][11][12][13][14][15][16][17] and it appears to be useful for predicting and describing phenomena in more realistic and detailed models [18][19][20][21].…”