Branch switching at a corank-4 bifurcation point of semi-linear elliptic problems with symmetry
Eugene L. Allgower, Klaus Böhmer, Zhen Mei · IMA Journal of Numerical Analysis · 1994
Branch switching of the problem {Δu+λf(u)=0 in Ω:=[0,1]×[0,1]u=0 on ∂Ω at a corank-4 bifurcation point is investigated by exploiting symmetry and other properties of the operator. Here f :R→R is a smooth odd function. The singularity of the corank-4 bifurcation point is decomposed into various subspaces via symmetries such that all solution branches can be followed by continuation methods and their modifications. At the same time, a modified Lyapunov-Schmidt method is used to determine solution branches having little symmetry on a slightly enlarged system in appropriate subspaces. We find altogether 40 different solution branches. Among them we have 13 solutions, such that none of these solutions can be generated by conjugation or hidden symmetries from any other solutions.