Systems: Bifurcation Techniques

Abstract

In many reaction-diffusion problems, it is the case that a uniform solution exists, but is stable only for certain ranges of the parameters present in the equation. Then it is typically also the case that when the parameters assume values near the “transition” zone between stability and instability of the uniform solution, other nonuniform, but small amplitude, solutions exist as well. Sometimes they are stable, and thus represent new solutions to which stability has been transferred from the uniform solution. This appearance of new solutions is considered a bifurcation phenomenon, because in parameter-amplitude diagrams, new solutions branch off the known uniform solution at critical values of the parameters.