Bifurcation into wave patterns and turbulence in reaction-diffusion equations

Abstract

A variety of wave patterns bifurcating at a highly degenerate eigenvalue are constructed and sorted out by stability analysis. Depending on the character of nonlinear interaction between excited modes, the stable postcritical state may be a single propagating or standing wave, an ordered pattern of standing waves, or an incoherent pattern comprising a large number of randomly phased modes. In addition, oscillatory destabilization of wave patterns and runaway to large-amplitude solutions are possible. Both ordered and chaotic solutions are generated by one and the same mechanism, and turbulent states which may be numerous and markedly dissimilar, always retain some elements of order.