The geometry of the critical set of nonlinear periodic Sturm–Liouville operators

We study the critical set C of the nonlinear differential operator F(u)=−u″+f(u) defined on a Sobolev space of periodic functions Hp(S1), p⩾1. Let be the plane z=0 and, for n>0, let be the cone x2+y2=tan2z, |z−2πn|<π/2; also set . For a generic smooth nonlinearity f:R→R with surjective derivative, we show that there is a diffeomorphism between the pairs (Hp(S1),C) and (R3,Σ)×H where H is a real separable infinite-dimensional Hilbert space.