Nonlinear waves on circle networks with excitable nodes

TL;DR

This paper introduces a simplified Kuramoto-type model for nonlinear wave dynamics on circle networks with excitable nodes, enabling analytical classification of traveling wave solutions and revealing universal features.

Contribution

The authors develop a new, analytically tractable model for nonlinear waves on circle networks, connecting it to the Frenkel-Kontorova model and classifying wave solutions.

Findings

Model simplifies analysis of nonlinear waves on circle networks

Traveling wave solutions are classified and shown to be universal

Perturbation analysis and numerical methods confirm findings

Abstract

Nonlinear wave formation and propagation on a complex network with excitable node dynamics is of fundamental interest in diverse fields in science and engineering. Here, we propose a new model of the Kuramoto type to study nonlinear wave generation and propagation on circular subgraphs of a complex network. On circle networks, in the continuum limit, this model is equivalent to the over-damped Frenkel-Kontorova model. The new model is shown to keep the essential features of those well-known models such as the diffusively coupled B\"ar-Eiswirth model but with much simplified expression such that analytic analysis becomes possible. We classify traveling wave solutions on circle networks and show the universality of its features with perturbation analysis and numerical computation.