Some problems of the linear theory

Abstract

We shall be considering a system of equations of the form

$$\frac{{d\phi }}{{dt}} = a\left( \phi \right),\frac{{dx}}{{dt}} = P\left( \phi \right)x + f\left( \phi \right)$$

((1.1))

, where a $a,P,f \in {{C}^{r}}\left( {{{\mathcal{T}}_{m}}} \right),\phi = \left( {{{\phi }_{1}}, \ldots ,{{\phi }_{m}}} \right),x = \left( {{{x}_{1}}, \ldots ,{{x}_{n}}} \right)$. Underlining its linearity with respect to the variable x we shall call it a linear non-homogeneous system of equations defined on the direct product of the m-dimensional torus T _m and the Euclidean space Eⁿ. An invariant manifold of system of equations (1.1) of the form

$$ x = u\left( \phi \right), \phi \in {T_m}, $$

where u ∈C^s(T _m ), will be called an m-dimensional s times continuously differentiable invariant torus of system (1.1). For s = 0 an invariant manifold of system (1.1) of the form (1.2) will be called an invariant torus of this system.