Morin singularities and global geometry in a class of ordinary differential operators
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 Morinlike singularities and we provide operational characterizations of the singularities of different orders. A global LyapunovSchmidt 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 onedimensional 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 CafagnaDonati 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 nonfolds 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 $v$.
 Publication:

arXiv eprints
 Pub Date:
 October 2007
 arXiv:
 arXiv:0710.1774
 Bibcode:
 2007arXiv0710.1774M
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 34L30;
 34B15;
 46N20
 EPrint:
 This is a corrected version of the paper published in 1997. 34 pages, 4 figures