Planar and screw-shaped solutions for a system of two reaction-diffusion equations on the circle

Matthias Büger

doi:10.3934/dcds.2006.16.745

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2006, Volume 16, Issue 4: 745-756. Doi: 10.3934/dcds.2006.16.745

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Planar and screw-shaped solutions for a system of two reaction-diffusion equations on the circle

Matthias Büger^1,

1.
Mathematisches Institut, Justus-Liebig-Universität Gießen, Beskidenstr. 9, D-35398 Gießen

Received: October 2004

Revised: May 2006

Published: December 2006

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Abstract

We describe the dynamics of a system of two reaction-diffusion equations on the circle. We show that the elements of the $\omega$-limit sets of every solution can be classified by the number of times which they wind around the circle line --- they look either flat or screw-shaped. We prove a Poincaré-Bendixson result. Furthermore, we give a criterion under which screw-shaped stationary or periodic solutions are unstable.

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Mathematics Subject Classification: Primary: 35K57.

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Planar and screw-shaped solutions for a system of two reaction-diffusion equations on the circle

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