Kuramoto Oscillators With Asymmetric Coupling

TL;DR

This paper investigates asymmetric coupling in Kuramoto oscillators with phase delays, revealing unique phase-locked states and synchronization phenomena inspired by neuroscience models.

Contribution

It introduces a partly solvable model of asymmetric Kuramoto oscillators with phase delays, highlighting novel synchronization behaviors not seen in symmetric cases.

Findings

Asymmetric coupling enables phase-locked states with phase differences.

Symmetric coupling with phase delay π/2 shows no synchronization.

The model exhibits unusual synchronization phenomena inspired by neuroscience.

Abstract

We study a system of coupled oscillators of the Sakaguchi-Kuramoto type with interactions including a phase delay. We consider the case of a coupling matrix such that oscillators with large natural frequencies drive all slower ones but not the reverse. This scheme is inspired by Hebbian learning in neuroscience. We propose a simple model which is partly solvable analytically and shows many unusual features. For example, for the case of phase delay $\pi/2$ (a symmetric phase response curve) with symmetric coupling there is no synchronous behavior. But in the asymmetric coupling case, there are phase-locked states, i.e., states where many oscillators have the same frequency, but with substantial phase differences.