# American Institute of Mathematical Sciences

December  2006, 16(4): 745-756. doi: 10.3934/dcds.2006.16.745

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

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

Received  October 2004 Revised  May 2006 Published  September 2006

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.
Citation: Matthias Büger. Planar and screw-shaped solutions for a system of two reaction-diffusion equations on the circle. Discrete & Continuous Dynamical Systems, 2006, 16 (4) : 745-756. doi: 10.3934/dcds.2006.16.745
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