Kuramoto Oscillators With Asymmetric Coupling

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 $\pi \mathrm{/}2$ (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.