Phase-coupled Oscillators with Plastic Coupling: Synchronization and Stability

TL;DR

This paper analyzes synchronization and stability in systems of homogeneous phase-coupled oscillators with plastic coupling, providing new stability criteria and conditions for phase-locking and unique stable equilibria.

Contribution

It introduces a gradient system framework for oscillators with plastic coupling and derives novel stability and instability conditions based on graph theory.

Findings

Systems always achieve frequency synchronization.

Provided necessary and sufficient stability conditions for tree topologies.

Identified conditions for almost global stability and phase-locking.

Abstract

In this article we study synchronization of systems of homogeneous phase-coupled oscillators with plastic coupling strengths and arbitrary underlying topology. The dynamics of the coupling strength between two oscillators is governed by the phase difference between these oscillators. We show that, under mild assumptions, such systems are gradient systems, and always achieve frequency synchronization. Furthermore, we provide sufficient stability and instability conditions that are based on results from algebraic graph theory. For a special case when underlying topology is a tree, we formulate a criterion (necessary and sufficient condition) of stability of equilibria. For both, tree and arbitrary topologies, we provide sufficient conditions for phase-locking, i.e. convergence to a stable equilibrium almost surely. We additionally find conditions when the system possesses a unique stable equilibrium, and thus, almost global stability follows. Several examples are used to demonstrate variety of equilibria the system has, their dependence on system's parameters, and to illustrate differences in behavior of systems with constant and plastic coupling strengths.