Kuramoto variables as eigenvalues of unitary matrices

TL;DR

This paper extends the Kuramoto model by representing variables as eigenvalues of unitary matrices, exploring their dynamics, synchronization, and connections to quantum transport, revealing new behaviors with matrix coupling.

Contribution

It introduces a novel interpretation of the Kuramoto model using eigenvalues of unitary matrices and analyzes their dynamics, including synchronization and quantum transport connections.

Findings

Eigenvalues of unitary matrices can model Kuramoto oscillators.

Synchronization occurs when the matrix evolves into a multiple of the identity.

Matrix coupling leads to unexpected dynamical behaviors.

Abstract

We generalize the Kuramoto model by interpreting the $N$ variables on the unit circle as eigenvalues of a $N$-dimensional unitary matrix $U$, in three versions: general unitary, symmetric unitary and special orthogonal. The time evolution is generated by $N^2$ coupled differential equations for the matrix elements of $U$, and synchronization happens when $U$ evolves into a multiple of the identity. The Ott-Antonsen ansatz is related to the Poisson kernels that are so useful in quantum transport, and we prove it in the case of identical natural frequencies. When the coupling constant is a matrix, we find some surprising new dynamical behaviors.