Symmetry and secondary bifurcation of elastic structures

We consider non-linear bifurcation problems for elastic structures modeled by the operator equation F[w;α]=0 where F:X×Rk→Y,X,Y are Banach spaces and X⊂Y. We focus attention on problems whose bifurcation equations are of the formfi(α1,α2;λ,μ)=(aiμ+biλ)αi+piαi3+qiαi∑j=1,j≠ikαj+12+αihi(λ,μ;α1,α2,…αk)i=1,2,…kwhich emanates from bifurcation problems for which the linearization of F is Fredholm operators of index 0. Under the assumption of F being odd we prove an important theorem of existence of secondary bifurcation. Under this same assumption we prove a symmetry condition for the reduced equations and consequently we got an existence result for secondary bifurcation. We also include a stability analysis of the bifurcating solutions.