On the Detuned 2:4 Resonance

We consider families of Hamiltonian systems in two degrees of freedom with an equilibrium in 1 : 2 resonance. Under detuning, this “Fermi resonance” typically leads to normal modes losing their stability through period-doubling bifurcations. For cubic potentials, this concerns the short axial orbits, and in galactic dynamics, the resulting stable periodic orbits are called “banana” orbits. Galactic potentials are symmetric with respect to the coordinate planes whence the potential—and the normal form—both have no cubic terms. This Z2×Z2 symmetry turns the 1 : 2 resonance into a higher-order resonance, and one therefore also speaks of the 2 : 4 resonance. In this paper, we study the 2 : 4 resonance in its own right, not restricted to natural Hamiltonian systems where H=T+V would consist of kinetic and (positional) potential energy. The short axial orbit then turns out to be dynamically stable everywhere except at a simultaneous bifurcation of banana and “anti-banana” orbits, while it is now the long axial orbit that loses and regains stability through two successive period-doubling bifurcations.