2025/03/25 by Hsiao, Ting-Yang, Lo, Yun-Feng, Zhu, Chengbin · 2 citations
#34C15 #34D06 #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2503.19781
We present a rigorous mathematical framework establishing the equivalence of four classical notions of synchronization full phase-locking, phase-locking, frequency synchronization, and order parameter synchronization in generalized Kuramoto models, via a non-perturbative, finite-dimensional analysis. Our approach avoids linearization, mean-field limits, and restrictions on initial conditions, relying instead on global phase-space geometry, periodic vector field structure, and compactness arguments based on contradiction. These results clarify the foundational role of the order parameter and provide a unified understanding of synchronization across a broad class of heterogeneous oscillator networks.