Synchronization Relations in the Hybrid Kuramoto Flow: Equivalence and a High-Coherence Criterion
We study four synchronization notions for the all-to-all hybrid Kuramoto model containing both first- and second-order oscillators with heterogeneous inertias and damping coefficients. We prove that full phase locking, bounded phase locking, and frequency synchronization are equivalent for arbitrary hybrid ensembles, and that each of these properties implies convergence of the complex order parameter. Conversely, if an order-parameter synchronized trajectory has limiting order parameter $Z^*$ satisfying $|Z^*|>\max\left\{\frac{ω_M}λ,\,1-\frac{2}{N}\right\}$, then the trajectory is frequency synchronized and hence fully phase locked. The converse proof is entirely real-dynamical. The omega-limit set is internally chain transitive and, because the order parameter is constant on that set, the dynamics reduce there to a product of frozen scalar equations. A tilted-energy argument shows that every nonstationary scalar factor of an internally chain transitive frozen set must cover the whole phase circle. Evaluating the constant mean field at an antipodal phase then forces $|Z^*|\le 1-2/N$, a contradiction. For zero natural frequencies, convergence follows from the analytic periodic gradient structure with degenerate inertia.