Nonlocal coupling among oscillators mediated by a slowly diffusing substance

TL;DR

This paper develops a comprehensive theory for nonlinear oscillator coupling via a diffusing chemical, extending previous models by allowing arbitrary diffusion times and solving the diffusion equation exactly in multiple dimensions.

Contribution

It generalizes Kuramoto's model by removing the fast diffusion assumption, providing exact solutions for diffusion with arbitrary times in various spatial dimensions.

Findings

Exact solutions for diffusion in 1D, 2D, and 3D.

Extension of oscillator coupling theory to arbitrary diffusion times.

Applicability to different boundary conditions.

Abstract

A general theory is presented for the coupling among nonlinear oscillators mediated by a diffusing chemical substance. We extend a model originally developed by Kuramoto, who supposed that the diffusion characteristic time is much shorter than the oscillator main period, such that diffusion occurs very fast. We eliminate this constraint and consider diffusion to have an arbitrary characteristic time, by solving exactly the diffusion equation using suitable Green functions. We present results in one, two and three dimension, with and without boundary conditions.