Bifurcations in Integrable Systems with Three Degrees of Freedom. I

We study the local structure of a real analytic integrable Hamiltonian system with three degrees of freedom in the neighborhoods of compact singular orbits. In such systems, one-dimensional compact orbits of the related Hamiltonian action usually form one-parameter families, and two-dimensional orbits form two-parameter families. Therefore, changes in the local orbit structure may occur along the families. In this paper, we study neighborhoods of compact one-dimensional orbits (i.e., semilocal rank $1$ corank $2$ singularities of the energy–momentum map). Using the results of Zung and Kudryavtseva on the existence of a local Hamiltonian action of the $2$-torus, we analyze bifurcations of the semilocal orbit structure near degenerate orbits corresponding to resonances of various types. We show that these bifurcations are structurally stable with respect to analytic integrable perturbations of the system. In all cases, we construct standard polynomial Hamiltonians, which, together with quadratic and linear first integrals, provide a $C^\omega$ left–right classification of the energy–momentum maps in the neighborhoods of degenerate compact orbits. We also present phase portraits and bifurcation diagrams of some standard systems with the corresponding bifurcations.