On Regularity of Certain Linear Expansions of Dynamical Systems on a Torus

Abstract

We investigate the problem of the existence of the Green–Samoilenko function for linear expansions of dynamical systems on a torus of the form

$$\frac{{d\phi }}{{dt}} = a(\phi ),{\text{ }}C(\phi )\frac{{d\phi }}{{dt}} + \frac{1}{2}\dot C(\phi )x = A(\phi )x,$$

where C(ϕ) ∈ C′(T _m; a) is a nondegenerate symmetric matrix.