The Birkhoff/Gustavson normalizations provide a good account of phase portrait, especially of the surface of sections in sufficiently small vicinities of a stable equilibrium point of Hamiltonian systems. The B/G normalizations suit to the cases that the frequency ratio of the linearized oscillation around the equilibrium point is irrational/ rational. Even if the ratio may range over a very small interval, the switching of normalizations may give rise to the question as to whether any non-negligible difference takes place or not between the B/G normalizations. In this paper, a classical mechanical model for the Josephson junction prism qubit (JJPQ) is taken as a testing model to the question above: no non-negligible difference is observed in the nested structures of the invariant tori between the Birkhoff and the Gustavson normal-form approximations for the JJPQ.

Birkhoff normalization, Gustavson normalization, Josephson junction qubit, periodic orbits.