### Abstract

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 language | English (US) |
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Pages (from-to) | 595-624 |

Number of pages | 30 |

Journal | Journal of Statistical Physics |

Volume | 178 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 2020 |

### Keywords

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

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

*Journal of Statistical Physics*,

*178*(2), 595-624. https://doi.org/10.1007/s10955-019-02442-w