Global Phase Synchronization Decoupled from Amplitude Dynamics

Abstract: The Kuramoto model is a canonical framework for analyzing phase synchronization, yet its utility is restricted to the vicinity of the oscillator's unperturbed limit cycle. Here, we present a method to construct coupled-oscillator models that globally preserve Kuramoto-type phase dynamics by using phase-amplitude coordinates defined via Koopman operator theory. We introduce a solvable model, termed Kuramoto-Stuart-Landau model, which exhibits nontrivial synchronized dynamics far from the limit cycle. We also construct three phase synchronized oscillators whose amplitudes exhibit Lorenz-type chaos. Our method is applicable to general limit-cycle systems, achieving phase synchronization globally while preserving arbitrarily complex amplitude dynamics.