HOPF-BIFURCATION IN SPATIALLY EXTENDED REACTION-DIFFUSION SYSTEMS

We consider weakly unstable reaction-diffusion systems defined on doma ins with one or more unbounded space-directions. In the systems which we have in mind, at criticality, the most unstable eigenvalue belongs to the wave vector zero and possesses a nonvanishing imaginary part. T his instability leads to an almost spatially homogeneous Hopf-bifurcat ion in time. A standard example is the Brusselator in certain paramete r ranges. Using multiple scaling analysis we derive a Ginzburg-Landau equation and show that all small solutions develop in such a way that they can be approximated after a certain time by the solutions of the Ginzburg-Landau equation. The proof differs essentially from the case when the bifurcating pattern is oscillatory in space. Our proof is bas ed on normal form methods. As a consequence of the results, the global existence in time of all small bifurcating solutions and the upper-se micontinuity of the original system attractor towards the associated G inzburg-Landau attractor follows.