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Imperfect Bifurcation Near a Double Eigenvalue: Transitions Between Nonsymmetric and Symmetric Patterns

Erneux, T. and Cohen, D. S. (1983) Imperfect Bifurcation Near a Double Eigenvalue: Transitions Between Nonsymmetric and Symmetric Patterns. SIAM Journal on Applied Mathematics, 43 (5). pp. 1042-1060. ISSN 0036-1399. doi:10.1137/0143068.

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We examine the existence of nonsymmetric and symmetric steady state solutions of a general class of reaction-diffusion equations. Our study consists of two parts: (i) By analyzing the bifurcation from a uniform reference state to nonuniform regimes, we demonstrate the existence of a unique symmetric solution (basic wave number two) which becomes linearly stable when it surpasses a critical amplitude. (We assume that the first bifurcation point corresponds to the emergence of the simplest nonsymmetric steady state solutions.) (ii) This result is not affected when a parameter is nonuniformly distributed in the system. However, one of the two possible branches of nonsymmetric solutions may disappear from the bifurcation diagram. Our analysis is motivated by the fact that experimental observations of pattern transitions during morphogenesis are interpreted in terms of the dynamics of stable concentration gradients. We have shown that in addition to the values of the physico-chemical parameters, these structures can be selected by two different mechanisms: (i) the linear stability of the nonuniform patterns, (ii) the effects of a small and nonuniform variation of a parameter in the spatial domain.

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Additional Information:© 1983 Society for Industrial and Applied Mathematics. Received by the editors January 7, 1981, and in revised form October 20, 1982. We thank Professor E. L. Reiss for many suggestions and for the critical reading of the manuscript. T.E. is "Chargé de Recherches" of F.N.R.S. (Belgium).
Issue or Number:5
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ID Code:32388
Deposited On:12 Jul 2012 20:36
Last Modified:09 Nov 2021 21:27

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