Pacemakers in large arrays of oscillators with nonlocal coupling

TL;DR

This paper models how localized heterogeneities in large one-dimensional oscillator arrays with nonlocal coupling can act as pacemakers, generating wave sources that influence the entire system, unlike in higher-dimensional systems.

Contribution

It demonstrates the existence of steady wave-source solutions caused by localized heterogeneities in nonlocal oscillator arrays, using a novel analytical approach involving isomorphisms and Kondratiev spaces.

Findings

Localized heterogeneities act as pacemakers in 1D oscillator arrays.

Steady wave-source solutions exist due to heterogeneities with positive average.

The approach relates the nonlocal problem to the viscous eikonal equation.

Abstract

We model pacemaker effects of an algebraically localized heterogeneity in a 1 dimensional array of oscillators with nonlocal coupling. We assume the oscillators obey simple phase dynamics and that the array is large enough so that it can be approximated by a continuous nonlocal evolution equation. We concentrate on the case of heterogeneities with positive average and show that steady solutions to the nonlocal problem exist. In particular, we show that these heterogeneities act as a wave source, sending out waves in the far field. This effect is not possible in 3 dimensional systems, such as the complex Ginzburg-Landau equation, where the wavenumber of weak sources decays at infinity. To obtain our results we use a series of isomorphisms to relate the nonlocal problem to the viscous eikonal equation. We then use Fredholm properties of the Laplace operator in Kondratiev spaces to obtain solutions to the eikonal equation, and by extension to the nonlocal problem.