{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,4]],"date-time":"2025-11-04T16:00:11Z","timestamp":1762272011946,"version":"build-2065373602"},"reference-count":64,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2016,2,27]],"date-time":"2016-02-27T00:00:00Z","timestamp":1456531200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Fonds National de la recherche scientifique (Belgium)"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>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.<\/jats:p>","DOI":"10.3390\/e18030064","type":"journal-article","created":{"date-parts":[[2016,2,29]],"date-time":"2016-02-29T10:55:59Z","timestamp":1456743359000},"page":"64","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":23,"title":["Self-Replicating Spots in the Brusselator Model and Extreme Events in the One-Dimensional Case with Delay"],"prefix":"10.3390","volume":"18","author":[{"given":"Mustapha","family":"Tlidi","sequence":"first","affiliation":[{"name":"Facult\u00e9 des Sciences, Universit\u00e9 Libre de Bruxelles, Campus Plaine, C.P. 231, Brussels B-1050, Belgium"}]},{"given":"Yerali","family":"Gandica","sequence":"additional","affiliation":[{"name":"Namur Center for Complex Systems (naXys), University of Namur, Rempart de la Vierge 8, Namur B-5000, Belgium"}]},{"given":"Giorgio","family":"Sonnino","sequence":"additional","affiliation":[{"name":"Facult\u00e9 des Sciences, Universit\u00e9 Libre de Bruxelles, Campus Plaine, C.P. 231, Brussels B-1050, Belgium"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5178-4077","authenticated-orcid":false,"given":"Etienne","family":"Averlant","sequence":"additional","affiliation":[{"name":"Facult\u00e9 des Sciences, Universit\u00e9 Libre de Bruxelles, Campus Plaine, C.P. 231, Brussels B-1050, Belgium"},{"name":"Department of Applied Physics and Photonics (IR-TONA), Vrije Universiteit Brussel, Pleinlaan 2, Brussels B-1050, Belgium"}]},{"given":"Krassimir","family":"Panajotov","sequence":"additional","affiliation":[{"name":"Department of Applied Physics and Photonics (IR-TONA), Vrije Universiteit Brussel, Pleinlaan 2, Brussels B-1050, Belgium"},{"name":"Institute of Solid State Physics, 72 Tzarigradsko Chaussee Blvd., Sofia 1784, Bulgaria"}]}],"member":"1968","published-online":{"date-parts":[[2016,2,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1695","DOI":"10.1063\/1.1668896","article-title":"Symmetry breaking instabilities in dissipative systems. 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