On the Emergence and Orbital Stability of Phase-Locked States for the Lohe Model

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• 作者：Seung-Yeal Ha ; Sang Woo Ryoo
• 关键词：Emergence ; Kuramoto model ; Lohe model ; Quantum network ; Quantum synchronization
• 刊名：Journal of Statistical Physics
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：163
• 期：2
• 页码：411-439
• 全文大小：623 KB
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• 作者单位：Seung-Yeal Ha (1) (2)
  Sang Woo Ryoo (3)
  
  1. Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul, 151-747, Korea
  2. Korea Institute for Advanced Study, Hoegiro 87, Seoul, 130-722, Korea
  3. Department of Mathematical Sciences, Seoul National University, Seoul, 151-747, Korea
  
• 刊物类别：Physics and Astronomy
• 刊物主题：Physics
  Statistical Physics
  Mathematical and Computational Physics
  Physical Chemistry
  Quantum Physics
  
• 出版者：Springer Netherlands
• ISSN：1572-9613

文摘

We study the emergence and orbital stability of phase-locked states of the Lohe model, which was proposed as a non-abelian generalization of the Kuramoto phase model for synchronization. Lohe introduced a first-order system of matrix-valued ordinary differential equations for quantum synchronization and numerically observed the asymptotic formation and orbital stability of phase-locked states of the Lohe model. In this paper, we provide an analytical framework to confirm Lohe’s observations of emergent phase-locked states. This extends earlier special results on lower dimensions to any finite dimension. For the construction and orbital stability of phase-locked states, we introduce Lyapunov functions to measure the ensemble diameter and dissimilarity between two Lohe flows, and using the time-evolution estimates of these Lyapunov functions, we present an admissible set of initial states, and show that an admissible initial state leads to a unique phase-locked asymptotic state.