ISSN:
1572-9036
Keywords:
34C
;
34D
;
65N
;
65P
;
92
;
numerical methods
;
bifurcation theory
;
enzyme kinetics
;
morphogenesis
;
Galerkin method
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract This paper deals with the use of the Galerkin approximation for calculating branches of steady-state solutions. It is motivated by the analysis of a reaction-diffusion system modeled by a pair of nonlinear partial differential equations on a two-dimensional domain. The goal is to check the possibility of closed loops emerging from a ‘trivial’ branch. This issue is of importance in recent theories on morphogenesis in embryos (Kauffman et al. [3]). Numerical methods for continuing Galerkin approximations of the steady states give arcs of stable or unstable solutions. The numerical results are in agreement with the predictions of Brezzi et al. [6–8]. In particular, bifurcations from the trivial steady-state or symmetry-breaking bifurcations remain bifurcations for the approximate problem. The whole connected set of solutions thus obtained gives new insight into the behavior of solutions to reaction-diffusion equations and strongly advocates Kauffman's theory.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00046602
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