Bifurcations in a Hamiltonian system with two degrees of freedom associated with the reversible hyperbolic umbilic

We deal with Hamiltonian bifurcations associated with the reversible umbilic in two degrees of freedom systems defined by 0:1 resonance, i.e. the unperturbed equilibrium has two purely imaginary eigenvalues and a semisimple double-zero one. The Hamiltonian is written as the sum of integrable Hamiltonian N (1/2 (x(2) + y(2)), q, p; lambda) and a small perturbation P(x, y, q, p; lambda) by the normalization procedure. The phase portrait of N on each level of the integral I-1 = 1/2 (x(2) + y(2)) is then studied in detail, obtaining an unfolding related to the reversible hyperbolic umbilic catastrophe. The persistence of 2-tori (i.e. two-dimensional tori) for the full system is analysed via KAM theory pointed out just as in Broer et al. (Z Angew Math Phys 44: 389-432, 1993), (in: Langford, Nagata (eds) Normal forms and homoclinic chaos, Waterloo, (1992), Fields Institute Communications 4 (1995)). In a sense, our results can be seen as a four-dimensional extension of a planar problem studied by Hanssmann (Phys D 112: 81-94, 1998).