Strong selection or rejection of spatially periodic patterns in degenerate bifurcations

Abstract When studying bifurcating space-periodic solutions one often has a situation, common in many problems of fluid dynamics, in which in the R-k plane (R a control parameter such as Reynolds number, k wave number) the passage from linear stability to instability is characterized by a parabola-like curve, having a minimum at Rc, kc. We analyse classes of problems in which the coefficients of nonlinear terms in the amplitude equation go through zero near kc. Various examples of such degenerate problems can be found in the literature, but have not been studied yet. We give an extensive classification of bifurcation pictures, which display rather unusual behaviour. We further study the stability of these periodic solutions subject to quite general perturbation. We find as a general result that all bifurcating solutions are unstable except for a small neighbourhood of a curve Γ in the R-k plane. There is hence a strong selection mechanism of (periodic) patterns, which fixes, for each value of R, with a very small uncertainty, the wave number of the stable bifurcating periodic solution. For one class of problems the curve Γ continues to exist for all values of R for which our theory is consistent. For another class the curve stops at a value R ∗ > R c . For R > R ∗ all bifurcating periodic solutions are unstable. In this case instability provides a mechanism of rejection of all periodic patterns. In the last section we analyse the particular case of the Blasius boundary-layer flow.