Independent dimensional phase transition on a two-dimensional Kuramoto model with matrix coupling

TL;DR

This paper explores a novel matrix coupling mechanism in a two-dimensional Kuramoto model, revealing independent phase transitions in each dimension and diverse synchronization behaviors, advancing understanding of high-dimensional oscillator dynamics.

Contribution

It introduces a matrix coupling approach for 2D Kuramoto oscillators, enabling independent transitions and revealing new synchronization modes not seen in scalar-coupled models.

Findings

Matrix coupling allows independent synchronization transitions in each dimension.

Four distinct system modes emerge from different matrix configurations.

Transitions between modes can be continuous or explosive.

Abstract

The high-dimensional generalization of the one-dimensional Kuramoto paradigm has been an essential step in bringing about a more faithful depiction of the dynamics of real-world systems. Despite the multi-dimensional nature of the oscillators in these generalized models, the interacting schemes so far have been dominated by a scalar factor unanimously between any pair of oscillators that leads eventually to synchronization on all dimensions. As a natural extension of the scalar coupling befitting for the one-dimensional case, we take a tentative step in studying numerically and theoretically the coupling mechanism of $2\times2$ real matrices on two-dimensional Kuramoto oscillators. One of the features stemmed from this new mechanism is that the matrix coupling enables the two dimensions of the oscillators to separate their transitions to either synchronization or desynchronization which has not been seen in other high-dimensional generalizations. Under various matrix configurations, the synchronization and desynchronization of the two dimensions combine into four qualitatively distinct modes of position and motion of the system. We demonstrate that as one matrix is morphed into another in a specific manner, the system mode also switches correspondingly either through continuous or explosive transitions of the order parameters, thus mimicking a range of behaviors in information science and biology.