Ineffective perturbations in a planar elastica

The bifurcation diagram for the buckling of a planar elastica under a load λ is made up of a “trivial” branch of unbuckled configurations for all λ and a sequence of branches of buckled configurations that are connected to the trivial branch at pitchfork bifurcation points. We use several perturbation expansions to determine how this diagram perturbs with the addition of a small intrinsic shape in the elastica, focusing in particular on the effect near the bifurcation points. We find that for almost all intrinsic shapes f(s), the difference between the buckled solution and the trivial solution is O( 1/3), but for some “ineffective” f , this difference is O( ), and we find functions uj(s) so that f is ineffective at bifurcation point number j when 〈f, uj〉 = 0. These ineffective perturbations have important consequences in numerical simulations, in that the perturbed bifurcation diagram has sharper corners near the former bifurcation points, and there is a higher risk of a numerical simulation inadvertently hopping between branches near these corners.