<p>This paper discusses the existence and uniqueness of random periodic paths of stochastic periodic semi-flows. Random periodic attractors are introduced and synchronization for stochastic periodic semi-flows is proved under some conditions to find the unique random periodic path. The multiplicative ergodic theorem of stochastic periodic semi-flows is proved to characterize Lyapunov exponents. The Benzi–Parisi–Sutera–Vulpiani climate model is an example to verify the results by estimating the negative Lyapunov exponent constructed by the density function from the Fokker–Planck equation. Numerical approximations are performed with great agreement. A case of gradient systems is considered to be another example of a negative Lyapunov exponent.</p>