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 Er of such functions with the Cr-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, 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.