Control of transversal instabilities in reaction-diffusion systems

Sonja Totz,Harald Engel,Jan Frederik Totz,Jakob Löber

doi:10.1088/1367-2630/aabce5

Abstract

In two-dimensional reaction-diffusion systems, local curvature perturbations on traveling waves are typically damped out and vanish. However, if the inhibitor diffuses much faster than the activator, transversal instabilities can arise, leading from flat to folded, spatio-temporally modulated waves and to spreading spiral turbulence. Here, we propose a scheme to induce or inhibit these instabilities via a spatio-temporal feedback loop. 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 stabilizing plane wave propagation. Conversely, in numerical simulations with the modified Oregonator model for the photosensitive Belousov–Zhabotinsky reaction, which does not exhibit transversal instabilities on its own, we demonstrate the feasibility of inducing transversal instabilities and study the emerging wave patterns in a well-controlled manner.

Highlights

A large variety of pattern forming processes can be understood in terms of the advancement of an interface between two or more spatial domains
We present a feedback loop to induce, control, and suppress transversal instabilities of reactiondiffusion waves
The control signal is calculated from the local curvature of the iso-concentration line of the wave

Summary

Introduction

A large variety of pattern forming processes can be understood in terms of the advancement of an interface between two or more spatial domains. For standard activator-inhibitor kinetics, these so-called transversal or lateral wave instabilities typically occur if the inhibitor diffuses much faster than the activator This result was analytically predicted first by Kuramoto for piecewise-linear reaction kinetics [13, 14]. The photosensitive BZ reaction (PBZR) is well-suited for experiments, due to the possibility of applying spatio-temporal external forcing or feedback-mediated control by exploiting the dependence of the local excitation threshold on the intensity of applied illumination [23]. Spiral wave drift in response to resonant external forcing and various feedback-mediated control loops have been extensively studied experimentally in PBZR systems, compare for example [27,28,29,30,31]. We destabilize a stable propagating planar reaction-diffusion wave by inducing transversal instabilities via feedback, and study the wave patterns emerging beyond the instability threshold. In addition we demonstrate the capability to actively select wave patterns by modifying feedback parameters accessible in a chemical experiment

The piecewise-linear FHN model

Curvature-dependent feedback control

Excitation of transversal instabilities

Conclusions