Conservation and bifurcation of an invariant torus of a vector field (Q1274001)

The author deals with the existence of invariant tori for small perturbed systems of a smooth vector field in \(\mathbb{R}^{n+m}\) possessing an invariant torus \(T^m\). It is assumed that the flow on \(T^m\) is quasiperiodic with m basic frequencies satisfying Diophantine conditions and the matrix \(\Omega\) of the variational equation with respect to the invariant torus is constant. When \(\Omega\) is a nonsingular matrix that can have purely imaginary eigenvalues, the invariant tori of different dimensions of perturbed systems are shown under certain conditions. If \(\Omega\) has no purely imaginary eigenvalues, the existence of invariant tori of perturbed systems is discussed in [\textit{N. N. Bogolyubov, Yu. A. Mitropolskii} and \textit{A. M. Samoilenko}, Methods of accelerated convergence in nonlinear mechanics. Berlin-Heidelberg-New York: Springer-Verlag. VIII (1976; Zbl 0331.34049)].