Preprint No. A-01-04

Arnd Scheel

Radially symmetric patterns of reaction-diffusion systems

Abstract: We study pattern formation from a spatially homogeneous equilibrium in reaction diffusion systems posed on $\R^n$. When a parameter is varied such that the equilibrium looses stability, we show that various types of radially symmetric patterns may arise, depending on the nature of the instability. Among others, we find stationary focus patterns and oscillatory target patterns. The patterns are found as heteroclinic orbits for the elliptic and parabolic equations, rewritten as a dynamical system in the radial variable $r$. A systematic bifurcation theory is developped for these systems, including a center-manifold reduction, a normal form theory for the far-field, and a matching procedure between core region and far-field.

Keywords: reaction-diffusion systems, defects, target patterns, center manifolds, normal forms, Turing instability, radial symmetry

Mathematics Subject Classification (MSC2000): 35B32, 58J50, 35B40, 37G05, 37G10, 34C37

Language: ENG

Available: Pr-A-01-04.ps, Pr-A-01-04.ps.gz

Contact: Arnd Scheel, Freie Universität Berlin, Fachbereich Mathematik und Informatik, Arnimallee 2-6, D-14195 Berlin, Germany (scheel@math.fu-berlin.de)

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