A New Perspective to Synchronization in Networks of Coupled Oscillators: Reverse Engineering and Convex Relaxation

We take a new approach to investigate synchronization in networks of coupled oscillators. We show that the coupled oscillator system when restricted to a proper region is a distributed partial primal-dual gradient algorithm for solving a well-defined convex optimization problem and its dual. We characterize conditions for synchronization solution of the KKT system of the optimization problem, based on which we derive conditions for synchronization equilibrium of the coupled oscillator network. This new approach reduces the hard problem of synchronization of coupled oscillators to a simple problem of verifying synchronization solution of a system of linear equations, and leads to a complete characterization of synchronization condition for the coupled oscillator network in an interesting and practically important region. Our synchronization condition is stated elegantly as the existence of solution for a system of linear equations, of which one best existing synchronization condition is a special sufficient condition case. In addition, we formulate a non-convex optimization problem with the force balance constraint for which the afore convex optimization problem is relaxation, and show that the coupled oscillator system is also a distributed algorithm for solving this non-convex problem. This has interesting implication on exact convex relaxation, and confirms the insight that a physical system usually solves a convex problem even though it may have a non-convex representation.