Synchronisation of Weakly Coupled Oscillators

Funding Information: This work was partially supported by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the project UIDB/00297/2020 (Centro de Matemática e Aplicações). Publisher Copyright: © 2021, Springer Nature Switzerland AG. The synchronization phenomenon was reported for the first time by Christiaan Huygens, when he noticed the strange tendency of a couple of clocks to synchronise their movements. More recently this phenomena was shown to be ubiquitous in nature and it is broadly studied by its applications, for example in biological cycles. We consider the problem of synchronization of a general network of linearly coupled oscillators, not necessarily identical. In this case the existence of a linear synchronization space is not expected, so we present an approach based on the proof of the existence of a synchronization manifold, the so-called generalised synchronization. Based on some results developed by R. Smith and on Wazewski’s principle, a general theory on the existence of invariant manifolds that attract the solutions of the system that are bounded in the future, is presented. Applications and estimates on parameters for the existence of synchronization are presented for several examples: systems of coupled pendulum type equations, coupled Lorenz systems of equations, and oscillators coupled through a medium, among many others. authorsversion published