Bistable　structures　with　the　elastic　instability　caused　by　buckling　are　confirmed　to　perform　significantly　in　microelectromechanical　systems,　metamaterials,　energy　harvester,　vibration　isolation　and　morphing　structures.　The　target　of　this　paper　is　to　explore　the　dynamic　responses　of　cross-ply　bistable　composite　laminate,　focusing　on　the　nonlinear　effect　of　the　single-　and　double-well　vibration.　Both　theoretical　and　finite　element　(FE)　methods　are　employed　to　simulate　the　nonlinear　vibration　and　dynamic　snap-through　of　bistable　composite　laminate　under　the　foundation　excitation　at　the　center.　In　the　theoretical　model,　the　governing　equations　are　established　via　Lagrange＇s　equation　based　on　the　first-order　shear　deformation　theory,　von　Karman　nonlinear　straindisplacement　relation　and　Rayleigh-Ritz　method.　The　fourth-order　Runge-Kutta　method　is　adopted　to　solve　the　governing　equations,　and　the　numerical　results　are　validated　by　FE　model.　Subsequently,　the　details　of　the　dynamic　responses　are　analyzed　to　identify　the　nonlinear　effects　in　the　form　of　bifurcation　diagram,　phase　portrait,　time　history,　Poincare　maps　and　amplitude　spectrum.　The　dynamic　responses　are　examined　for　a　series　of　excitation　parameters　in　both　time　and　frequency　domain.　Through　fixed　frequency　and　frequency　sweep　tests,　the　nonlinear　phenomenon　of　the　single-　and　double-well　vibrations　are　analyzed　including　superharmonic　resonance,　stiffness　softening,　hysteresis　phenomenon,　various　periodic　and　chaotic　vibration.　The　diverse　responses　to　external　inputs　contribute　significantly　to　efficiently　predicting　mechanical　behaviors　in　real-world　conditions,　thereby　offering　indispensable　theoretical　support　for　structural　design　applications.