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With the help of discontinuous mapping and tail mapping, the maximum Lyapunov exponent of the system with the above three structures was constructed using Khasminskii transform. For the shape memory alloy beam model with unilateral constraints, the corresponding Lyapunov exponent sequences were analyzed and the bifurcation and chaotic motions of the system were discussed. The results show that the validity of the maximum Lyapunov exponent algorithm is verified by combining the time history diagram and phase diagram.