Alexandr Karabanov

We address the dynamics of near-integrable Hamiltonian systems with 3 degrees of freedom in extended vicinities of unperturbed resonant invariant Liouville tori. The main attention is paid to the case where the unperturbed torus satisfies two independent resonance conditions. In this case the average dynamics is 4-dimensional, reduced to a generalised motion under a conservative force on the 2-torus and is generically non-integrable. Methods of differential topology are applied to full description of equilibrium states and phase foliations of the average system. The results are illustrated by a simple model combining the non-degeneracy and non-integrability of the isoenergetically reduced system.

The problem of coupled oscillations is considered in the case when a stable equilibrium of a globally averaged system passes through a resonance curve. Questions of persistence of invariant tori and transition to a self-synchronization are particularly discussed.

The problem of qualitative behaviour of four-dimensional quasi-Hamiltonian system in a neighbourhood of a fixed resonance is considered. The general analytical grounds of the problem are touched upon. We turn to the global analysis of special three-dimensional system averaged near the resonance. Furthermore, a checking the theory against numerical simulations is made. Two physical examples, revealing an irregular resonant dynamics, are studied.