Imperfect bifurcation in the presence of symmetry

Consider the familiar principle that typically (or generically) a system of m scalar equations in n variables where m>n has no solutions. This principle can be reformulated geometrically as follows. If S is a submanifold of a manifold X with codimension m (i.e. m = άimX — dimS) and iϊf:R-*X is a smooth mapping where m>n, then usually or generically Image /nS is empty. One of the basic tenets in the application of singularity theory is that this principle holds in a general way in function spaces. In the next few paragraphs we shall try to explain this more general situation as well as to explain its relevance to bifurcation problems. First we describe an example through which these ideas may be understood. Consider the buckling of an Euler column. Let λ denote the applied load and x denote the maximum deflection of the column. After an application of the Lyapunov-Schmidt procedure the potential energy function Ffor this system may be written as a function of x and λ alone and hence the steady-state configurations of the column may be found by solving