Equivalence of Synchronization States in the Hybrid Kuramoto Flow

TL;DR

This paper proves that various synchronization states in hybrid Kuramoto oscillator networks are mathematically equivalent, providing a comprehensive framework for understanding synchronization phenomena across different oscillator types.

Contribution

It establishes the first rigorous proof of the equivalence of multiple synchronization states in hybrid Kuramoto models, unifying their mathematical characterization.

Findings

Synchronization states are topologically equivalent in hybrid ensembles.

The equivalence depends solely on the finite equilibrium structure.

The proof uses advanced dynamical systems methods.

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.