Kuramoto meets Koopman: Constants of motion, symmetries, and network motifs

Partial integrability in phase-oscillator dynamics is typically examined for identically connected oscillators or groups thereof. Yet, the precise connectivity conditions that ensure conserved quantities on general networks remain unclear. Using Koopman theory, we rigorously derive conditions for the existence of distinct conserved quantities in the Kuramoto model with heterogeneous phase lags on any weighted, directed, and signed graph. In the process, we establish a criterion for Lie symmetries in general dynamical systems in terms of the Koopman generator, allowing us to identify symmetries that generate new conserved quantities. These results reveal a broad class of network motifs that support conserved quantities and we detect these motifs in hundreds of complex empirical networks. Our results provide a rigorous theoretical application of Koopman’s framework to nonlinear dynamics on complex networks and reveal new possibilities for preserving phase relationships over time in complex systems through a careful design of their structure.