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

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. This article is part of the theme issue ‘Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 2)’.

Stability and Hopf bifurcation in a state-dependent delayed Brusselator-type model

“…The Brusselator reaction-diffusion model has also been extensively studied (see [8,7,18]). Moreover, the diffusive Brusselator model with delayed feedback has been studied, including the Hopf bifurcation analysis, the stationary localized structures, the self-replicating spots (see [2,17,32]). Lv and Liu [21] investigated the Turing-Hopf bifurcation and normal form of a diffusive Brusselator model with gene expression time delay.…”

Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology, the legacy of Ilya Prigogine (part 2)

“…Numerical continuation techniques and weakly nonlinear theory are used to analyse in depth their global behaviour [8]. Bifurcation analysis of the localized spot and a self-replication instability leading to bound states of localized spots are analysed by using two-dimensional continuation techniques in [9].…”

Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology, the legacy of Ilya Prigogine (part 1)

“…Optical frequency combs are expected to have industrial applications such as the generation of ultra-stable lightwave and microwave signals for aerospace engineering, optical communication networks and microwave photonic systems. Next, the paper by Vladimir Zykov [9] presents a review of spiral wave formation in excitable media. Leon Brenig [10] reports on the reduction of nonlinear dynamical systems to canonical forms.…”