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2018, Volume 13, Issue 2: 323-337. Doi: 10.3934/nhm.2018014

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Formation, stability and basin of phase-locking for Kuramoto oscillators bidirectionally coupled in a ring

1.
Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
2.
Department of Mathematics and Institute for Advanced Study in Mathematics, Harbin Institute of Technology, Harbin 150001, China

¹Corresponding Author: Zhuchun Li
¹Corresponding Author: Zhuchun Li

Received: September 2017

Published: June 2018

This work was supported by National Natural Science Foundation of China Grants 11401135, 11671109 and 11731010. Z. Li was also supported by the Fundamental Research Funds for the Central Universities (HIT.BRETIII.201501 and HIT.PIRS.201610).

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Abstract

We consider the dynamics of bidirectionally coupled identical Kuramoto oscillators in a ring, where each oscillator is influenced sinusoidally by two neighboring oscillator. Our purpose is to understand its dynamics in the following aspects: 1. identify all the phase-locked states (or equilibria) with stability or instability; 2. estimate the basins for stable phase-locked states; 3. identify the convergence rate towards phase-locked states. The crucial tool in this work is the celebrated theory of Łojasiewicz inequality.

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Mathematics Subject Classification: 34C15, 34D06, 92D25.

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References

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