Geodesic
Classification of periodic differential equations by degrees of non-roughniss
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 14 (2022) no. 3, pp. 52-59 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $k$.
Keywords: periodic differential equation, cylindrical phase space, structural stability, degree of structural instability, bifurcation manifold. 
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V. Sh. Roitenberg. Classification of periodic differential equations by degrees of non-roughniss. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 14 (2022) no. 3, pp. 52-59. http://geodesic.mathdoc.fr/item/VYURM_2022_14_3_a5/

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