Phase balancing in globally connected networks of Liénard oscillators

Abstract

We synthesize a feedback for a fully connected network of identical Liénard-type oscillators such that the phase-balanced equilibrium - the mode where the centroid of the coupled oscillators in polar coordinates is at the origin - is asymptotically stable, and the phase-synchronized equilibrium is unstable. Our approach hinges on a coordinate transformation of the oscillator dynamics to polar coordinates, and periodic averaging theory to simplify the examination of multiple time-scale behavior. Using Lyapunov- and linearization-based arguments, we demonstrate that the oscillator dynamics have the same radii and balanced phases in steady state for a large set of initial conditions. Numerical simulation results are presented to validate the analyses.