On the Equivalence of Synchronization Definitions in the Kuramoto Flow: A Unified Approach

We present a unified dynamical framework that rigorously establishes the equivalence between various synchronization notions in generalized Kuramoto models. Our formulation encompasses both first- and second-order models with heterogeneous inertia, damping, natural frequencies, and general symmetric coupling coefficients, including attractive, repulsive, and mixed interactions. We demonstrate that, under minimal assumptions, full phase-locking, phase-locking, frequency synchronization, order parameter synchronization, and acceleration synchronization coincide. This resolves long-standing ambiguities in the nonlinear theory of synchronization and provides a systematic classification of asymptotic behaviors in finite-size oscillator networks. The framework further provides sharpened necessary conditions for synchronization across both first- and second-order classical Kuramoto flows, by establishing upper and lower bounds on the long-time behavior of the order parameter. Beyond its mathematical depth, our theory offers a broadly applicable framework that may inform future studies in nonlinear optics, quantum synchronization, and collective behavior in open systems. In particular, by rigorously linking microscopic dynamics to macroscopic observables such as the Kuramoto order parameter, it has the potential to clarify the dynamical mechanisms underlying coherence, dissipation, and emergent order in mesoscopic and many-body settings.