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

TL;DR

This paper analyzes the global geometric structure of a class of differential operators with Morin singularities, providing normal forms, characterizations of folds and cusps, and demonstrating complex solution behaviors through examples.

Contribution

It introduces a global geometric framework for understanding Morin singularities in differential operators and derives normal forms and classifications for various nonlinearities.

Findings

Characterization of Morin singularities in differential operators

Normal forms for specific nonlinearities like cubic and Cafagna-Donati

Existence of polynomial examples with complex singularities such as butterflies

Abstract

We consider the operator $F(u) = u' + f(t,u(t))$ 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 $v$.