Paper Info
Title
STABILITY AND BIFURCATION OF MIXING IN THE KURAMOTO MODEL WITH INERTIA
Abstract
The Kuramoto model of coupled second order damped oscillators on convergent sequences of graphs is analyzed in this work. The oscillators in this model have random intrinsic frequencies and interact with each other via nonlinear coupling. The connectivity of the coupled system is assigned by a graph which may be random as well. In the thermodynamic limit the behavior of the system is captured by the Vlasov equation, a hyperbolic partial differential equation for the probability distribution of the oscillators in the phase space. We study stability of mixing, a steady state solution of the Vlasov equation, corresponding to the uniform distribution of phases. Specifically, we identify a critical value of the strength of coupling, at which the system undergoes a pitchfork bifurcation. It corresponds to the loss of stability of mixing and marks the onset of synchronization. As for the classical Kuramoto model, the presence of the continuous spectrum on the imaginary axis poses the main difficulty for the stability analysis. To overcome this problem, we use the methods from the generalized spectral theory developed for the original Kuramoto model. The analytical results are illustrated with numerical bifurcation diagrams computed for the Kuramoto model on ErdoH \s--Re'\nyi and small-world graphs. Applications of the second order Kuramoto model include power networks, coupled pendula, and various biological networks. The analysis in this paper provides a mathematical description of the onset of synchronization in these systems.
Year
DOI
Venue
2022
10.1137/21M1427000
SIAM JOURNAL ON MATHEMATICAL ANALYSIS
Keywords
DocType
Volume
&nbsp, Vlasov equation, Kuramoto model with inertia, mixing, stability, bifurcation, synchronization
Journal
54
Issue
ISSN
Citations 
2
0036-1410
0
PageRank 
References 
Authors
0.34
0
2
Name
Order
Citations
PageRank
Hayato Chiba100.34
Georgi S. Medvedev200.34