Kuramoto oscillators with inertia: A fast-slow dynamical systems approach

Choi, Young-Pil; Ha, Seung-Yeal; Jung, Sungeun; Slemrod, Marshall

doi:10.1090/qam/1380

Authors: Young-Pil Choi, Seung-Yeal Ha, Sungeun Jung and Marshall Slemrod
Journal: Quart. Appl. Math. 73 (2015), 467-482
MSC (2010): Primary 91G80, 97M30
DOI: https://doi.org/10.1090/qam/1380
Published electronically: June 18, 2015
MathSciNet review: 3400753
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Abstract: We present a fast-slow dynamical systems theory for a Kuramoto type model with inertia. The fast part of the system consists of $N$-decoupled pendulum equations with constant friction and torque as the phase of individual oscillators, whereas the slow part governs the evolution of order parameters that represent the amplitude and phase of the centroid of the oscillators. In our new formulation, order parameters serve as orthogonal observables in the framework of Artstein-Kevrekidis-Slemrod-Titi’s unified theory of singular perturbation. We show that Kuramoto’s order parameters become stationary regardless of the coupling strength. This generalizes an earlier result (Ha and Slemrod (2011)) for Kuramoto oscillators without inertia.

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