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- Title
On the Finiteness of Collisions and Phase-Locked States for the Kuramoto Model.
- Authors
Ha, Seung-Yeal; Kim, Hwa; Ryoo, Sang
- Abstract
Synchronization phenomenon is ubiquitous in our complex systems, and many phenomenological models have been proposed and studied analytically and numerically. Among them, the Kuramoto model serves as a prototype model for the phase synchronization of weakly coupled oscillators. In this paper, we study the finiteness of collisions (crossings) among Kuramoto oscillators in the relaxation process toward the phase-locked states and the total number of phase-locked states with positive (Kuramoto) order parameters. For identical oscillators, it is well known that collisions between distinct oscillators cannot occur in finite-time, and we show that there are only a finite number of phase-locked states with positive order parameters. However, for non-identical oscillators, oscillators with different natural frequencies can cross each other in their relaxation process, and estimating the total number of phase-locked states is a nontrivial matter. We show that, for the non-identical case, asymptotic phase-locking is equivalent to the finiteness of collisions, and the total number of phase-locked states with positive order parameters is bounded above by $$2^N$$ , where N is the number of oscillators.
- Subjects
SYNCHRONIZATION; PHENOMENOLOGICAL equations; MATHEMATICAL models; PARAMETRONS; DYNAMICAL systems
- Publication
Journal of Statistical Physics, 2016, Vol 163, Issue 6, p1394
- ISSN
0022-4715
- Publication type
Article
- DOI
10.1007/s10955-016-1528-6