A theoretical model for statics and dynamics of bistable asymmetric cross-ply composite laminated square plates is established. Three equilibrium states are determined by curing analysis in statics. Meanwhile the super-critical pitchfork bifurcation with temperature difference as the control parameter is explicated in the process of curing. Two stable states and one unstable state are demonstrated by stability analysis. The potential energy curve with two potential wells is depicted，which can contribute to studying dynamic snap-through. Moreover，the dynamic bifurcation for the equilibrium points is induced by introducing damping in dynamics. The influence of the base excitation frequency on the dynamics is discussed by numerical simulation. When the excitation frequency is located in a certain range，the large-amplitude dynamic snap-through and nonlinear vibrations with two potential wells can occur. The dynamics behaviors of the bistable system are overwhelmingly dominated by periodic vibration，quasi-periodic vibration and chaotic vibration. The certain frequency range，where the large-amplitude dynamic snap-through and nonlinear vibrations with two potential wells can occur，can be defined as a certain frequency broadband which proofs bistable asymmetric cross-ply composite laminated square plates to be applicable to bistable energy harvesters.