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

Abstract

We consider the operator $F(u)=u^{\prime }+f(t,u(t))$ acting on periodic real valued functions. Generically, critical points of $F$ are infinite dimensional Morin-like singularities and we provide operational characterizations of the singularities of different orders. A global Lyapunov-Schmidt decomposition of $F$ converts $F$ into adapted coordinates, $\Fbd(\tilde v, \overline u) = (\tilde v, \overline v)$, where $\tilde{v}$ is a function of average zero and both $\overline{u}$ and $\overline{v}$ are numbers. Thus, global geometric aspects of $F$ reduce to the study of a family of one-dimensional maps: we use this approach to obtain normal forms for several nonlinearities $f$. For example, we characterize autonomous nonlinearities giving rise to global folds and, in general, we show that $F$ is a global fold if all critical points are folds. Also, $f(t,x)=x^3-x$, or, more generally, the Cafagna-Donati nonlinearity, yield global cusps; for $F$ interpreted as a map between appropriate Hilbert spaces, the requested changes of variable to bring $F$ to normal form can be taken to be diffeomorphisms. A key ingredient in the argument is the contractibility of both the critical set and the set of non-folds for a generic autonomous nonlinearity. We also obtain a numerical example of a polynomial $f$ of degree 4 for which $F$ contains butterflies (Morin singularities of order 4)---% it then follows that $F(u)=v$ has six solutions for some $v$.

This is a corrected version of the paper published in 1997. 34 pages, 4 figures