# A Matrix Valued Kuramoto Model

## TL;DR

This paper introduces a new matrix-valued Kuramoto model where symmetric matrices evolve to align their eigenframes, analyzing stability and dynamics, including the instability of twist states.

## Contribution

It proposes a novel matrix-valued Kuramoto model and provides stability analysis for phase-locked and twist states, extending synchronization theory to non-commuting matrices.

## Key findings

- Phase-locked state is stable when matrices align.
- Twist states are dynamically unstable.
- Eigenframe alignment leads to commuting matrices.

## Abstract

Beginning with the work of Lohe [14,15] there have been a number of papers [3,5,8,9,11] that have generalized the Kuramoto model for phase-locking to a non-commuting situation. Here we propose and analyze another such model. We consider a collection of symmetric 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 phase-locked state and show that it is stable. We also analyze a dynamic analog of the twist states arising in the standard Kuramoto model, and show that these twist states are dynamically unstable.