Loveykin Y. Studies of multifrequency oscillations of near-integrable locally Hamiltonian systems

The dissertation is devoted to studing of multifrequency oscillations of near-integrable locally Hamiltonian systems. The theorem on perturbation of coisotropic invariant tori of locally Hamiltonain systems close to broadly integrable is proved. The invariant tori of perturbated system form a Whitney-smooth family. An existence of three dimensional coisotropic invariant tori in four dimensional phase space of Lagrange system that describes electron motions on two dimesional torus under the influence of electromagnetic field is established. We also consider locally Hamiltonian perturbations of Liouville integrable systems under simultaneous deformation of symplectic structure, and show that for sufficiently small values of small parameter a Cantor set of coisotropic invariant tori branches from the manifold of unperturbed lagrangian tori corresponding to family of the so-called nondegenerate quasistationary points of elliptic type. A theorem on existence of quasiperiodic motions on invariant tori of locally Hamiltonian systems close to conditionally integrable is proved. The bifurcation of invariant tori under locally Hamiltonian perturbations of completely integrable systems in a neighborhood of manifold of general type non-degenerate quasistationary points is studied. The problem on perturbation of irreducible invariant tori that foliate a central manifold of conditionally integrable locally Hamiltonian system is solved.