Sang Woo Ryoo | Searcharxiv

arXiv2020

Constants of motion for the finite-dimensional Lohe type models with frustration and applications to emergent dynamics

We present constants of motion for the finite-dimensional Lohe type aggregation models with frustration and we apply them to analyze the emergence of collective behaviors. The Lohe type models have been proposed as possible non-abelian and higher-dimensional generalizations of the Kuramoto model, which is a prototype phase model for synchronization. The aim of this paper is to study the emergent collective dynamics of these models under the effect of (interaction) frustration, which generalizes phase-shift frustrations in the Kuramoto model. To this end, we present constants of motion, i.e., conserved quantities along the flow generated by the models under consideration, and, from the perspective of the low-dimensional dynamics thus so obtained, derive several results concerning the emergent asymptotic patterns of the Kuramoto and Lohe sphere models.

math-ph↗

arXiv2020

Asymptotic phase-locking dynamics and critical coupling strength for the Kuramoto model

We study the asymptotic clustering (phase-locking) dynamics for the Kuramoto model. For the analysis of emergent asymptotic patterns in the Kuramoto flow, we introduce the pathwise critical coupling strength which yields a sharp transition from partial phase-locking to complete phase-locking, and provide nontrivial upper bounds for the pathwise critical coupling strength. Numerical simulations suggest that multi- and mono-clusters can emerge asymptotically in the Kuramoto flow depending on the relative magnitude of the coupling strength compared to the sizes of natural frequencies. However, theoretical and rigorous analysis for such phase-locking dynamics of the Kuramoto flow still lacks a complete understanding, although there were some recent progress on the complete synchronization of the Kuramoto model in a sufficiently large coupling strength regime. In this paper, we present sufficient frameworks for partial phase-locking of a majority ensemble and the complete phase-locking in terms of the initial phase configuration, coupling strength and natural frequencies. As a by-product of our analysis, we obtain nontrivial upper bounds for the pathwise critical coupling strength in terms of the diameter of natural frequencies, initial Kuramoto order parameter and the system size $N$. We also show that phase-locked states whose order parameters are less than $N^{-\frac{1}{2}}$ are linearly unstable.

math.DS↗