Moduli of continuity of the derivatives of invariant tori for linear extensions of dynamical systems (Q5926584)

The system of differential equations of the form \[ \frac{d \varphi}{dt} = a(\varphi), \qquad \frac{dx}{dt} = A(\varphi)x + f(\varphi), \tag{1} \] with \(x\in{\mathbb R}^n\), \(\varphi \in T_m\), \(T_m\) is the \(m\)-dimensional torus, \(a(\varphi)\), \(A(\varphi)\), \(f(\varphi) \in C^0 (T_m),\) and \(C^0 (T_m)\) is the space of functions jointly continuous in \(\varphi\) and \(2\pi\)-periodic in each \(\varphi_j\), \(j=1,\dots,m\), is considered. It is supposed that system (1) has an invariant torus \(x=u(\varphi)\) or the Green function \(G_0(\tau, \varphi)\). A behavior of the moduli of continuity for higher-order derivatives of the Green function and the invariant torus of system (1) is studied. Some estimates and convergence conditions on the derivatives are obtained.