Lyapunov exponent is an important physical quantity indicating a system is in regular motion or in chaotic motion. In view of this, in this paper, we use the Lyapunov exponent to study the chaotic properties of the system, and the mutations in climate. The results show that the Lyapunov exponent is a reliable mutation detection method. For the ideal sequence or actual atmosphere sequence the method can be used to accurately find out the mutation position; the predictability of nonlinear local Lyapunov exponent is used to evidence the reliability of the early warning signals of abrupt climate change based on the phenomenon of critical slowing down, through calculating the largest Lyapunov exponent of each time the system intrinsic properties can be reflected and the chaos characteristics can be studied, thereby providing a theoretical basis for the wide applications of the present method in real observation data.