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A Turing-Hopf Bifurcation Scenario for Pattern Formation on Growing Domains
Jorge A Castillo1, Faustino Sánchez-Garduño2, Pablo Padilla3
1Universidad Autónoma de Guerrero, C. Carlos E. Adame #54, Col. Garita, C.P. 39650, Acapulco, Gro., Mexico.
Abstract:
In this paper, we study the emergence of different patterns that are formed on both static and growing domains and their bifurcation structure. One of these is the so-called Turing-Hopf morphogenetic mechanism. The reactive part we consider is of FitzHugh-Nagumo type. The analysis was carried out on a flat square by considering both fixed and growing domain. In both scenarios, sufficient conditions on the parameter values are given for the formation of specific space-time structures or patterns. A series of numerical solutions of the corresponding initial and boundary value problems are obtained, and a comparison between the resulting patterns on the fixed domain and those arising when the domain grows is established. We emphasize the role of growth of the domain in the selection of patterns. The paper ends by listing some open problems in this area.
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