Watanabe–Strogatz invariants in the Liouvillian dynamics of coupled phase oscillators via the Koopman framework

Abstract In dynamical systems, invariants, i.e. constants of motion conserved along the trajectory, play important roles in characterizing the system's dynamical behaviour. Recent applications of the Koopman operator framework to nonlinear dynamical systems have provided new insights into the invariants. For a certain class of globally coupled phase oscillators, which serve as models for various synchronization phenomena, Watanabe and Strogatz proved the existence of N−3 invariants in N oscillator systems. In this study, we derive these invariants from an operator-theoretic perspective by exploiting the relation between Liouvillian (Perron–Frobenius) and Koopman descriptions of the dynamics. Exploiting a simple multiplicative property of functions under the action of the Liouvillian and Koopman operators, we explicitly construct a family of functions whose ratios yield the invariants of the underlying dynamics. Our analysis successfully reproduces the full set of N−3 invariants known in Watanabe–Strogatz theory, and offers an alternative spectral perspective. We demonstrate this approach for a well-studied class of phase models, including the Ermentrout–Kopell, pairwise Kuramoto, and higher-order Kuramoto models.