as well [18] . As one can see, the system is characterized by two bare transition temperatures, T$_{f}$ and T$_{M}$ representing ferroelectric and ferromagnetic transitions, respectively. The form of the magnetoelectic coupling term ∝ γ ( PM ) 2 can be explained by using general symmetry arguments. Namely, the onset of ferroelectric order requires the breaking of spatial inversion symmetry, whereas the appearance of a spontaneous magnetization is connected with the breakdown of time reversal symmetry. The coupling term in (2) obeys those two conditions and allows for ferroic order with simultaneous nonzero P and M . In further calculations we use a dimensionless form of the free energy expansion (B5) which is obtained from (2) by dividing both sides of (2) by the constant value a 2 $_{0}$T 2 M b . In effect, we obtain: where: Δ F = b a 2 $_{0}$T 2 M [ F ( P,M,T ) − F$_{0}$ ( T )], h = b 1 2 ( a$_{0}$T$_{M}$ ) 3 2 H$_{a}$ ≡ s$_{h}$H$_{a}$ , l = α 2 $_{0}$b a 2 $_{$_{0}$β}$, e = b a 2 $_{0}$T 2 M √ α$_{0}$T$_{f}$ β E$_{a}$ , ˜ M = M $_{m$_{0}$}$, m$_{0}$ = √ a$_{0}$T$_{M}$ b , ˜ P = P $_{p$_{0}$}$, p$_{0}$ = √ α$_{0}$T$_{f}$ β , γ$_{m}$ = γα$_{0}$ $_{βa$_{0}$}$. Within this formulation T$_{f}$ $_{T$_{M}$}$, l , and γ$_{m}$ are the material parameters. In this case, the bare magnetic transition temperature T$_{M}$ is assumed as always nonzero, whereas the ferroelectric temperature is T$_{f}$ ≥ 0. Taking derivatives of (B5) with respect to ˜ M and ˜ P , respectively, we obtain the following system of nonlinear equations: In the limit of zero electric field, we obtain the following relation between ˜ P and ˜ M from (4): 4 ˜ P = ±$^{√}$ 1 − T T$_{f}$ − γ$_{m}$ l T$_{M}$ T$_{f}$ ˜ M $^{2}$. (5) l$^{(}$ T$_{f}$ T$_{M}$$^{)}$$^{2}$ ( T/T$_{f}$ − 1) ˜ P + l$^{(}$ T$_{f}$ T$_{M}$$^{)}$$^{2}$ ˜ P 3 + γ$_{m}$ T$_{f}$ T$_{M}$ ˜ M 2 ˜ P − e = 0 . (4) ( T/T$_{M}$ − 1) ˜ M + ˜ M 3 + γ$_{m}$ T$_{f}$ T$_{M}$ ˜ M ˜ P 2 − h = 0 , (3) Δ F = 1 2$^{(}$ T T$_{M}$ − 1$^{)}$ ˜ M 2 + 1 4 ˜ M 4 (2) + 1 2 l$^{(}$ T$_{f}$ T$_{M}$$^{)}$$^{2}$$^{(}$ T T$_{f}$ − 1$^{)}$ ˜ P 2 + 1 4 l$^{(}$ T$_{f}$ T$_{M}$$^{)}$$^{2}$ ˜ P 4 + 1 2 γ$_{m}$ T$_{f}$ T$_{M}$ ˜ P 2 ˜ M 2 − ˜ Pe − ˜ Mh,