Hopf Bifurcation in Spatially Extended Reaction—Diffusion Systems

Abstract

We consider weakly unstable reaction—diffusion systems defined on domains 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. This instability leads to an almost spatially homogeneous Hopf-bifurcation in time. A standard example is the Brusselator in certain parameter 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 based on normal form methods. As a consequence of the results, the global existence in time of all small bifurcating solutions and the upper-semicontinuity of the original system attractor towards the associated Ginzburg-Landau attractor follows.