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Abstract: The Kuramoto model captures various synchronization phenomena in biologicaland man-made systems of coupled oscillators. It is well-known that there existsa critical coupling strength among the oscillators at which a phase transitionfrom incoherency to synchronization occurs. This paper features fourcontributions. First, we characterize and distinguish the different notions ofsynchronization used throughout the literature and formally introduce theconcept of phase cohesiveness as an analysis tool and performance index forsynchronization. Second, we review the vast literature providing necessary,sufficient, implicit, and explicit estimates of the critical coupling strengthfor finite and infinite-dimensional, and for first and second-order Kuramotomodels. Third, we present the first explicit necessary and sufficient conditionon the critical coupling to achieve synchronization in the finite-dimensionalKuramoto model for an arbitrary distribution of the natural frequencies. Themultiplicative gap in the synchronization condition yields a practicalstability result determining the admissible initial and the guaranteed ultimatephase cohesiveness as well as the guaranteed asymptotic magnitude of the orderparameter. Fourth and finally, we extend our analysis to multi-rate Kuramotomodels consisting of second-order Kuramoto oscillators with inertia and viscousdamping together with first-order Kuramoto oscillators with multiple timeconstants. We prove that the multi-rate Kuramoto model is locally topologicallyconjugate to a first-order Kuramoto model with scaled natural frequencies, andwe present necessary and sufficient conditions for almost global phasesynchronization and local frequency synchronization. Interestingly, theseconditions do not depend on the inertiae which contradicts prior observationson the role of inertiae in synchronization of second-order Kuramoto models.



Author: Florian Dorfler, Francesco Bullo

Source: https://arxiv.org/







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