Equivalence of Synchronization States in the Hybrid Kuramoto Flow

Abstract

We establish a unified synchronization framework for the all-to-all hybrid Kuramoto model that couples first- and second-order oscillators within a single dynamical system. Although the Kuramoto model has become one of the most widely used paradigms for describing synchronization phenomena-appearing in more than 100,000 scientific studies-the fundamental relationships among distinct synchronization states remain unresolved. In this work, we rigorously prove that full phase-locking, phase-locking, frequency synchronization, and order-parameter synchronization are equivalent for arbitrary hybrid ensembles. The proof combines dissipative energy methods, LaSalle-type compactness arguments, the Poincar{\'e}-Bendixson theorem, and Thieme's asymptotically autonomous theory to demonstrate that synchronization equivalence is topological, determined solely by the finite equilibrium structure of the all-to-all network. This result provides a complete mathematical characterization of synchronization in finite oscillator systems and clarifies its geometric invariance across first-, second-, and hybrid-order models.