Synchronization of first-order autonomous oscillators on Riemannian manifolds

Abstract

The present research work recalls a control-theoretic approach to the synchronization of a first-order master/slave oscillators pair on $\mathbb{R}^3$ and extends such technique to the case of curved Riemannian manifolds. As theoretical results, this paper proves the asymptotic convergence of the feedback controller and studies the entity of the 'control effort'. As a case study, the complete equations for the controller of a slave oscillator on the unit hypersphere $\mathbb{S}^{n-1}$ are laid out and are illustrated by numerical examples for $n = 3$ and $n = 10$, even in the hypothesis of noisy master-system state measurement.