  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 co-ordinate
planes whence the potential -- and the normal form -- both have no cubic terms.
This $\mathbb{Z}_2 \times \mathbb{Z}_2$-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.
