Design of Coupled Harmonic Oscillators for Synchronization and Coordination

Synchronization and coordination of coupled oscillators are fundamental behaviors in complex dynamical systems. This paper considers the design of coupled harmonic oscillators to generate an orbitally stable limit cycle of prescribed oscillation profile. Based on the Floquet theory and averaging techniques, necessary and sufficient conditions are obtained for nonlinear coupling functions to achieve local exponential convergence to a desired orbit. Unlike globally convergent methods based on contraction analysis, the result applies to oscillators without flow invariance properties. Insights into coordination mechanisms are gained through interpretation of the coupling structure as a directed graph. The theory is illustrated by simple tutorial examples.