Nonlinear stability of stationary solutions to the Kuramoto-Sakaguchi equation with frustration

Seung-Yeal Ha; Hansol Park; Yinglong Zhang; Seung-Yeal Ha; Hansol Park; Yinglong Zhang

doi:10.3934/nhm.2020026

Networks and Heterogeneous Media

2020, Volume 15, Issue 3: 427-461. doi: 10.3934/nhm.2020026

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Nonlinear stability of stationary solutions to the Kuramoto-Sakaguchi equation with frustration

1.
Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Republic of Korea
2.
Korea Institute for Advanced Study, Hoegiro 85, Seoul, 02455, Republic of Korea
3.
Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea
4.
Stochastic Analysis and Application Research Center, Korea Advanced Institute of Science and Technology, Daejeon 34141, Republic of Korea

Received: 01 August 2019 Revised: 01 May 2020 Published: 09 September 2020
Primary: 92B25, 35Q92; Secondary: 35B40

We study measurable stationary solutions for the kinetic Kuramoto-Sakaguchi (in short K-S) equation with frustration and their stability analysis. In the presence of frustration, the total phase is not a conserved quantity anymore, but it is time-varying. Thus, we can not expect the genuinely stationary solutions for the K-S equation. To overcome this lack of conserved quantity, we introduce new variables whose total phase is conserved. In the transformed K-S equation in new variables, we derive all measurable stationary solution representing the incoherent state, complete and partial phase-locked states. We also provide several frameworks in which the complete phase-locked state is stable, whereas partial phase-locked state is semi-stable in the space of Radon measures. In particular, we show that the incoherent state is nonlinearly stable in a large frustration regime, whereas it can exhibit stable behavior or concentration phenomenon in a small frustration regime.
- Kuramoto-Sakaguchi equation,
- frustration,
- stationary solution,
- stability
Citation: Seung-Yeal Ha, Hansol Park, Yinglong Zhang. Nonlinear stability of stationary solutions to the Kuramoto-Sakaguchi equation with frustration[J]. Networks and Heterogeneous Media, 2020, 15(3): 427-461. doi: 10.3934/nhm.2020026

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Abstract

We study measurable stationary solutions for the kinetic Kuramoto-Sakaguchi (in short K-S) equation with frustration and their stability analysis. In the presence of frustration, the total phase is not a conserved quantity anymore, but it is time-varying. Thus, we can not expect the genuinely stationary solutions for the K-S equation. To overcome this lack of conserved quantity, we introduce new variables whose total phase is conserved. In the transformed K-S equation in new variables, we derive all measurable stationary solution representing the incoherent state, complete and partial phase-locked states. We also provide several frameworks in which the complete phase-locked state is stable, whereas partial phase-locked state is semi-stable in the space of Radon measures. In particular, we show that the incoherent state is nonlinearly stable in a large frustration regime, whereas it can exhibit stable behavior or concentration phenomenon in a small frustration regime.

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