As　bistable　composite　laminated　plate　and　shell　structures　are　often　exposed　to　dynamic　environments　in　practical　applications,　the　global　and　local　dynamics　of　a　bistable　asymmetric　composite　laminated　shell　subjected　to　the　base　excitation　is　presented　in　this　paper.　Temperature　difference,　base　excitation　amplitude,　and　detuning　parameters　are　discussed.　With　the　change　of　temperature　difference,　the　super-critical　pitchfork　bifurcation　occurs.　Three　equilibrium　solutions　corresponding　to　three　equilibrium　configurations　(two　stable　configurations　and　one　unstable　configuration)　can　be　obtained.　With　the　increase　of　excitation　amplitude,　local　and　global　dynamics　play　a　leading　role　successively.　The　global　dynamics　between　the　two　stable　configurations　behave　as　the　periodic　vibration,　the　quasi-periodic　vibration,　the　chaotic　vibration　and　dynamic　snap-through　when　the　excitation　amplitude　is　large　enough.　The　local　dynamics　that　are　confined　to　a　single　stable　configuration　behave　as　1:2　internal　resonance,　saturation　and　permeation　when　the　excitation　amplitude　is　small.　Dynamic　snap-through　and　large-amplitude　vibrations　with　two　potential　wells　for　the　global　dynamics　will　lead　to　a　broad　application　prospect　of　the　bistable　asymmetric　composite　laminated　shell　in　energy　harvesting　devices.