Time-Inverted Kuramoto Dynamics for kappa-Clustered Circle Coverage

In this paper we analyse the equilibrium configurations for the time-inverted Kuramoto Model with homogeneous agents and a fixed ring topology, where time-inverted means that the coupling between the different states is via a negative factor. This model exhibits a dual behaviour with respect to the classic Kuramoto Model with a positive coupling. In the paper, we show the existence of two possible stable equilibrium configurations: the splay state formation (1-clustered coverage) and the deployment in clusters (kappa-clustered coverage). We provide sufficient conditions for the splay state formation and a stability analysis for the networked system. Moreover, we provide some initial results towards the controllability of the final equilibrium configurations. In particular, we lay the foundations to understand the conditions to switch between stable equilibria.