Bifurcation of singular solutions in reversible systems and applications to reaction - Diffusion equations

Dynamical systems that are reversible in the sense of Moser are investigated and bifurcation of trajectories connecting saddle points from stationary solutions is studied. As an application, reaction-diffusion models in one space dimension are considered. These equations are studied in the neighborhood of a point, where the set of spatially homogeneous solutions displays a Hopf bifurcation. It is shown that from such a point branches of solutions bifurcate, which can be described as waves travelling to or from a center. These waves may be exponentially damped at infinity or not. They can be regarded as one-dimensional analogues of “target patterns” or “spiral waves.