$Bi_{0.5+x}(Na_{0.78}K_{0.22})_{0.5-3x}TiO_3$ ceramics with an excess $Bi^{3+}$ and a deficiency of $Na^+$ and $K^+$ were synthesized by a conventional solid state reaction method. The structure and morphology of $Bi_{0.5+x}(Na_{0.78}K_{0.22})_{0.5-3x}TiO_3$ ceramics were characterized by X-ray diffraction and field emission scanning electron microscopy. The electric polarization and mechanical strain induced by external electric field, and the temperature dependence of dielectric constant were investigated. These results demonstrated that an ergodic relaxor phase can be induced by controls of the mole ratio of $Bi^{3+}$, $Na^+$ and $K^+$. A phase boundary between non-ergodic and ergodic relaxor phases can be observed at ambient temperature. The ergodic relaxor phase can be transferred to the ferroelectric phase by application of the electric field. The stability of the induced ferroelectric phases strongly depends on the mole ratio of $Bi^{3+}$, $Na^+$ and $K^+$. The maximum strain of 0.31% was observed in $Bi_{0.51}(Na_{0.78}K_{0.22})_{0.47}TiO_3$ ceramics sintered at $1,150^{circ}C$ for 2 h.