Control of transversal instabilities in reaction-diffusion systems

TL;DR

This paper investigates how to control and induce transversal instabilities in reaction-diffusion systems, particularly in the photosensitive Belousov-Zhabotinsky reaction, using spatio-temporal feedback mechanisms to manipulate wave patterns.

Contribution

It introduces a novel control scheme to artificially induce or suppress transversal wave instabilities in reaction-diffusion systems based on wave shape feedback.

Findings

Feasibility demonstrated in numerical simulations with the Oregonator model.

Control scheme can induce instabilities in PBZR with realistic parameters.

Control can suppress instabilities and stabilize flat wave propagation.

Abstract

In two-dimensional reaction-diffusion systems, local curvature perturbations in the shape of traveling waves are typically damped out and disappear in the course of time. If, however, the inhibitor diffuses much faster than the activator, transversal instabilities can arise, leading from flat to folded, spatio-temporally modulated wave shapes and to spreading spiral turbulence. For experimentally relevant parameter values, the photosensitive Belousov-Zhabotinsky reaction (PBZR) does not exhibit transversal wave instabilities. Here, we propose a mechanism to artificially induce these instabilities via a wave shape dependent spatio-temporal feedback loop, and study the emerging wave patterns. In numerical simulations with the modified Oregonator model for the PBZR using experimentally realistic parameter values we demonstrate the feasibility of this control scheme. Conversely, in a piecewise-linear version of the FitzHugh-Nagumo model transversal instabilities and spiral turbulence in the uncontrolled system are shown to be suppressed in the presence of control, thereby stabilising flat wave propagation.