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On a Numerical Liapunov-Schmidt Spectral Method and Its Application to Biological Pattern Formation

K. Böhmer, C. Geiger and J. D. Rodriguez
SIAM Journal on Numerical Analysis
Vol. 40, No. 2 (2003), pp. 683-701
Stable URL: http://www.jstor.org/stable/4100973
Page Count: 19
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On a Numerical Liapunov-Schmidt Spectral Method and Its Application to Biological Pattern Formation
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Abstract

Spectral expansions are used to provide a basis which preserves continuous symmetries. We show that spectral methods satisfy the conditions for convergence of numerical Liapunov-Schmidt methods. An explicit algorithm for the calculation of stationary bifurcation scenarios near primary instabilities in general continuous symmetric equations is given. The above convergence is extended to Γ-equivalent discrete and original bifurcation scenarios. The method is applied to a biologically motivated reaction-diffusion system with spherical symmetry forming patterns. A specific singularity of a generic steady state bifurcation is investigated in detail.

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