The topology of critical sets of some ordinary differential operators

We survey recent work of Burghelea, Malta and both authors on the topology of critical sets of nonlinear ordinary differential operators. For a generic nonlinearity f , the critical set of the first order nonlinear operator F1(u)(t) = u (t) + f(u(t)) acting on the Sobolev space H1 p of periodic functions is either empty or ambient diffeomorphic to a hyperplane. For the second order operator F2(u)(t) = −u(t) + f(u(t)) on H2 D (Dirichlet boundary conditions), the critical set is ambient diffeomorphic to a union of isolated parallel hyperplanes. For second order operators on H2 p , the critical set is not a Hilbert manifold but is still contractible and admits a normal form. The third order case is topologically far more complicated.