  It is known that a coherently oscillating axion field is a candidate of the
dark matter. In the presence of the oscillating axion, the photon can be
resonantly produced through the parametric amplification. In the universe,
there also exist cosmological magnetic fields which are coherent
electromagnetic fields. In the presence of magnetic fields, an axion can be
converted into a photon, and vice versa. Thus, it is interesting to investigate
what happens for the axion-photon system in the presence of both the axion dark
matter and the magnetic fields. This system can be regarded as a coupled system
of the axion and the photon whose equations contain the Mathieu type terms. We
find that the instability condition is changed in the presence of magnetic
fields in contrast to the conventional Mathieu equation. The positions of
bifurcation points between stable and unstable are shifted and new instability
bands appear. This is because the resonantly amplified axion can be converted
to photon, and vice versa.
