Synchronization analysis of coupled planar oscillators by averaging

Sufficient conditions for synchronization of coupled Lienard-type oscillators are investigated via averaging technique. Coupling considered here is pairwise, unidirectional, and described by a nonlinear function (whose graph resides in the first and third quadrants) of some projection of the relative distance (between the states of the pair being coupled) vector. Under the assumption that the interconnection topology defines a connected graph, it is shown that the solutions of oscillators can be made converge arbitrarily close to each other, while let initially be arbitrarily far apart, provided that the frequency of oscillations is large enough and the initial phases of oscillators all lie in an open semicircle. It is also shown that (almost) synchronized oscillations always take place at some fixed magnitude independent of the initial conditions. Similar results are generated for nonlinearly-coupled harmonic oscillators.