Bifurcation of Euler buckling problem, revisited (Q1982562)

The authors develop a long and detailed exposition of the treatment by Golubitsky and Schaeffer of the bifurcation theory for the classical bifurcation for the Euler buckling problem. The main tool used here is the singularity theory due to the same authors. The problem is treated in terms of the minimization of the associated energies in some Sobolev space. The problem is reduced mainly to the study of the change of zeroes of some function \(\Phi\) and then use a Lyapunov-Schmidt reduction. After some preliminaries it is proved that \(\Phi\) is smooth and its Taylor coefficients are considered. A bifurcation equation \(F=0\) is studied and the Taylor coefficients of \(F\) as well. The notion of \(p\)-\(K\)-versal unfolding (or universal unfolding) is introduced in Section 7 and the bifurcation and hysteresis sets are studied in Section 8. Finally, several approximation figures for these quantities are provided at the end of the paper.