Control of spatiotemporal patterns in the Gray-Scott model

TL;DR

This paper explores how time-delayed feedback control influences the formation and stability of complex spatiotemporal patterns in the Gray-Scott reaction-diffusion model, revealing various dynamical regimes and control limitations.

Contribution

It investigates different control schemes and their effectiveness in managing chaos and pattern formation in the Gray-Scott model, highlighting stability boundaries and control failures.

Findings

Control can stabilize uniform steady states or induce bistability.

Diagonal control fails to manage spatiotemporal chaos.

Control stability depends on strength and delay parameters.

Abstract

This paper studies the effects of a time-delayed feedback control on the appearance and development of spatiotemporal patterns in a reaction-diffusion system. Different types of control schemes are investigated, including single-species, diagonal, and mixed control. This approach helps to unveil different dynamical regimes, which arise from chaotic state or from traveling waves. In the case of spatiotemporal chaos, the control can either stabilize uniform steady states or lead to bistability between a trivial steady state and a propagating traveling wave. Furthermore, when the basic state is a stable traveling pulse, the control is able to advance stationary Turing patterns or yield the above-mentioned bistability regime. In each case, the stability boundary is found in the parameter space of the control strength and the time delay, and numerical simulations suggest that diagonal control fails to control the spatiotemporal chaos.