A Matrix-Valued Kuramoto Model

Jared C. Bronski, Thomas E. Carty, Sarah E. Simpson

Research output: Contribution to journalArticlepeer-review


The Kuramoto model is an important model for studying the onset of phase-locking in an ensemble of nonlinearly coupled phase oscillators. Each oscillator has a natural frequency (often taken to be random) and interacts with the other oscillators through the phase difference. It is known that, as the coupling strength is increased, there is a bifurcation in which the incoherent state becomes unstable and a stable phase-locked solution is born. Beginning with the work of Lohe there have been a number of paper that have generalized the Kuramoto model for phase-locking to a non-commuting setting. Here we propose and analyze another such model. We consider a collection of matrix-valued variables that evolve in such a way as to try to align their eigenvector frames. The phase-locked state is one where the eigenframes all align, and thus the matrices all commute. We analyze the stability of the equal frequency phase-locked state and show that it is stable. We also analyze (for the case of symmetric matrices) a dynamic analog of the twist states arising in the standard Kuramoto model, and show that these twist states are dynamically unstable.

Original languageEnglish (US)
Pages (from-to)595-624
Number of pages30
JournalJournal of Statistical Physics
Issue number2
StatePublished - Jan 1 2020


  • Kuramoto model
  • Lohe model
  • Phase-locking
  • Quantum synchronization
  • Synchronization

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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