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Entropy 2016, 18(3), 64; doi:10.3390/e18030064

Self-Replicating Spots in the Brusselator Model and Extreme Events in the One-Dimensional Case with Delay

1
Faculté des Sciences, Université Libre de Bruxelles, Campus Plaine, C.P. 231, Brussels B-1050, Belgium
2
Namur Center for Complex Systems (naXys), University of Namur, Rempart de la Vierge 8, Namur B-5000, Belgium
3
Department of Applied Physics and Photonics (IR-TONA), Vrije Universiteit Brussel, Pleinlaan 2, Brussels B-1050, Belgium
4
Institute of Solid State Physics, 72 Tzarigradsko Chaussee Blvd., Sofia 1784, Bulgaria
These authors contributed equally to this work.
*
Author to whom correspondence should be addressed.
Academic Editor: Raúl Alcaraz Martínez
Received: 29 November 2015 / Revised: 7 February 2016 / Accepted: 18 February 2016 / Published: 27 February 2016
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Abstract

We consider the paradigmatic Brusselator model for the study of dissipative structures in far from equilibrium systems. In two dimensions, we show the occurrence of a self-replication phenomenon leading to the fragmentation of a single localized spot into four daughter spots. This instability affects the new spots and leads to splitting behavior until the system reaches a hexagonal stationary pattern. This phenomenon occurs in the absence of delay feedback. In addition, we incorporate a time-delayed feedback loop in the Brusselator model. In one dimension, we show that the delay feedback induces extreme events in a chemical reaction diffusion system. We characterize their formation by computing the probability distribution of the pulse height. The long-tailed statistical distribution, which is often considered as a signature of the presence of rogue waves, appears for sufficiently strong feedback intensity. The generality of our analysis suggests that the feedback-induced instability leading to the spontaneous formation of rogue waves in a controllable way is a universal phenomenon. View Full-Text
Keywords: localized structures; spot self-replication; extreme events; rogue waves localized structures; spot self-replication; extreme events; rogue waves
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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MDPI and ACS Style

Tlidi, M.; Gandica, Y.; Sonnino, G.; Averlant, E.; Panajotov, K. Self-Replicating Spots in the Brusselator Model and Extreme Events in the One-Dimensional Case with Delay. Entropy 2016, 18, 64.

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