Stationary localized structures and the effect of the delayed feedback in the Brusselator model

TL;DR

This paper studies stationary localized structures in the Brusselator model and explores how delayed feedback can induce motion in these structures and patterns, combining bifurcation analysis with control mechanisms.

Contribution

It introduces a detailed bifurcation analysis of localized structures and demonstrates how delayed feedback can control their motion in the Brusselator model.

Findings

Localized spots emerge through bifurcations.

Transition from spots to extended square patterns.

Delayed feedback induces spontaneous motion of structures.

Abstract

The Brusselator reaction-diffusion model is a paradigm for the understanding of dissipative structures in systems out of equilibrium. In the first part of this paper, we investigate the formation of stationary localized structures in the Brusselator model. By using numerical continuation methods in two spatial dimensions, we establish a bifurcation diagram showing the emergence of localized spots. We characterize the transition from a single spot to an extended pattern in the form of squares. In the second part, we incorporate delayed feedback control and show that delayed feedback can induce a spontaneous motion of both localized and periodic dissipative structures. We characterize this motion by estimating the threshold and the velocity of the moving dissipative structures.