Classification of periodic differential equations by degrees of non-roughniss

A differential equation of the form $x' = f(t, x)$ with the right part $f(t, x)$ having continuous derivatives up to $r$-th order inclusive, $1$-periodic in $t$, we identify with the function $f$ and consider as an element of the Banach space $E^{r}$ of such functions with the $C^{r}$-norm. The equation $f$ defines a dynamical system on a cylindrical phase space. An equation $f$ is called rough if any equation close enough to it is topologically equivalent to $f$, that is, it has the same topological structure of the phase portrait. An equation $f$ has the $k$-th degree of non-roughness if any non-rough equation sufficiently close to it either has a degree of non-roughness less than $k$, or is topologically equivalent to $f$. The paper describes the set of equations of the $k$-th degree of non-roughness ($k < r$), shows that it form an embedded submanifold of codimension $k$ in $E^{r}$, are open and everywhere dense in the set of all non-rough equations that do not have a degree of non-roughness less than $k$.