Bifurcations in integrable systems with three degrees of freedom — I

The local structure of a real-analytic integrable Hamiltonian system with three degrees of freedom in neighborhoods of its compact singular orbits is studied. In such systems, one-dimensional compact orbits of the related Hamiltonian action are usually met in one-parameter families, and two-dimensional orbits form two-parameter families. Therefore, changes in the local orbit structure are possible along the families. The paper studies neighborhoods of compact one-dimensional orbits (i.e., semi-local singularities of rank 1 and corank 2 of the energy-momentum mapping). On the basis of results by N. T. Zung and E. A. Kudryavtseva on the existence of a local Hamiltonian action of 2-torus, bifurcations of the semi-local orbit structure near degenerate orbits corresponding to resonances of various types are investigated. It is shown that these bifurcations are structurally stable with respect to analytic integrable perturbations of the system. In all cases, standard polynomial Hamiltonians are constructed, which, together with quadratic and linear first integrals, provide $C^\infty$-left-right classification for the energy-momentum mappings in neighborhhods of degenerate compact orbits. Phase portraits and bifurcation diagrams of some reduced systems with corresponding bifurcations are also presented.