Morin singularities and global geometry in a class of ordinary differential operators

(∗) u′(t) + f(t, u(t)) = g(t), where the unknown u is a real function on S1 and the nonlinearity f : S1×R → R can assume a number of forms. Our approach is to study the global geometry of the operator F : B1 → B0, u → u′ + f(t, u) where the domain is either C1(S1) (the Banach space of periodic functions with continuous derivatives) or the Hilbert space H1(S1) of periodic functions with square integrable derivative. Ideally, we search for global changes of variables in both domain and image taking the operator F to a simple normal form. This goal has been achieved in previous occasions, starting with the seminal work of A. A. Ambrosetti and G. Prodi ([AP]) and its geometric interpretation by M. S. Berger and P. T. Church ([BC]), who showed that the operator associated to a certain nonlinear Dirichlet problem gives rise to a global fold between infinite