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

Xiaoxue Zhao; Zhuchun Li; Xiaoping Xue; Xiaoxue Zhao; Zhuchun Li; Xiaoping Xue

doi:10.3934/nhm.2018014

Networks and Heterogeneous Media

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

Xiaoxue Zhao ^{1
,},
Zhuchun Li ^{2
,
,},
Xiaoping Xue ^{2
,}

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

Received: 01 September 2017
34C15, 34D06, 92D25

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.
- Kuramoto oscillators coupled in a ring,
- phase-locked state,
- stability,
- basin
Citation: Xiaoxue Zhao, Zhuchun Li, Xiaoping Xue. Formation, stability and basin of phase-locking for Kuramoto oscillators bidirectionally coupled in a ring[J]. Networks and Heterogeneous Media, 2018, 13(2): 323-337. doi: 10.3934/nhm.2018014

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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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