Universality of Pattern Description near Threshold

Abstract

In this chapter we unearth the universal equation describing every bifurcation from which spatial order can emerge. Any system which can lead to this spatial order can be described by this universal equation (by definition, an equation independent of the specific system under consideration). We will first derive this equation from a concrete example, and then use symmetry to argue for its universal application. We will see that even for fixed control parameters, there exist an infinite number of possible stable solutions, distinguishable by the resulting pattern’s wavelength and amplitude. We will study what is called a secondary instability – an instability which the periodic pattern, itself resulting from an instability (primary instability) of a homogeneous state, experiences. Known as the “Eckhaus instability”, it leads to a change in the pattern’s wavelength by the creation or destruction of a few repetitions of the pattern motif. Though the Eckhaus instability reduces the range of possible wavelengths for the periodic solutions, we will still be left with a band of possible wavelengths, all of which can be a priori realized within a given system depending on initial conditions. This will lead us naturally to the notion of wavelength selection, which will be addressed later in the book.