Kubichka A. Studies of degenerate cases cases in the perturbation theory of coisotropic invariant tori of Hamiltonian systems

The goal of this dissertation is to study degenerate cases in the perturbation theory of coisotropic invariant tori (c.i.t) of Hamiltonian systems. We establish new conditions for the existence of ergodic quasiperiodic motions in perturbation problem for а Hamiltonian system whose c.i.t. fill a manifold of nonzero co-dimension, and show that c.i.t. of the perturbed system form a Whitney-smooth family. In the case when not only the system's Hamiltonian but also the symplectic structure is deformed, it is shown that near a manifold of elliptic quasistationary points there appears a Whitney-smooth family of coisotropic invariant tori of the perturbed system. In the perturbation problem for systems with low-dimensional toric non-poissonian symmetries, a Whitney-smooth family of c.i.t. is constructed near a relative equilibria manifold of the averaged system of first approximation. For nonautonomous Hamiltonian system with rapidly oscillating time-quasiperiodic Hamiltonian we establish the metric stabilityof elliptic quasistationary point. By means of this result we give a mathematical treatment for effects of vibrational stabilization of rigid body motions about a suspension point performing fast quasiperiodic oscillations of small amplitude along a non-vertical axis.