  Magnetoelectric phase diagrams of the perovskite manganites, Eu1-xYxMnO3 and
Gd1-xTbxMnO3, are theoretically studied. We first construct a microscopic
model, and then analyze the model using the Monte-Carlo method. We reproduce
the diagrams, which contain two different multiferroic states, i.e., the
ab-plane spin cycloid with electric polarization P//a and the bc-plane spin
cycloid with P//c. We reveal that their competition originates from a conflict
between the single-ion anisotropy and the Dzyaloshinsky-Moriya interaction,
which is controlled by the second-neighbor spin exchanges enhanced by the
GdFeO3-type distortion. This leads to a P flop from a to c with increasing x in
agreement with the experiments.
