TY - JOUR
T1 - Stationary localized structures and the effect of the delayed feedback in the brusselator model
AU - Kostet, B.
AU - Tlidi, M.
AU - Tabbert, F.
AU - Frohoff-Hülsmann, T.
AU - Gurevich, S. V.
AU - Averlant, E.
AU - Rojas, R.
AU - Sonnino, G.
AU - Panajotov, K.
N1 - Funding Information:
Data accessibility. This article has no additional data. Competing interests. We declare we have no competing interests. Funding. F.T. received funds from the German Scholarship Foundation and the Center for Nonlinear Science Münster. M.T. received support from the Fonds National de la Recherche Scientifique (Belgium). K.P. acknowledges the Methusalem Foundation. Acknowledgements. Stimulating discussions with Thomas Erneux are gratefully acknowledged.
Publisher Copyright:
© 2018 The Author(s) Published by the Royal Society. All rights reserved.
PY - 2018/12/28
Y1 - 2018/12/28
N2 - 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)’.
AB - 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)’.
KW - Bifurcations
KW - Delayed feedback
KW - Drift instability
KW - Localized structures
KW - Pattern formation
KW - Reaction–diffusion systems
UR - http://www.scopus.com/inward/record.url?scp=85056343272&partnerID=8YFLogxK
U2 - 10.1098/rsta.2017.0385
DO - 10.1098/rsta.2017.0385
M3 - Article
C2 - 30420547
AN - SCOPUS:85056343272
VL - 376
JO - Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
JF - Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
SN - 1364-503X
IS - 2135
M1 - 20170385
ER -