It is mathematically and thoroughly proved in this paper that the nonlinear stochastic ocean-atmosphere oscillator model possesses a stable limit cycle; then the model equations are transformed into the FokkerPlanck equation (FPE), and the evolution of El Nino-Southern Oscillation (ENSO) from unstable state to stable state is studied from the point of view of nonequilibrium system dynamics. The study results reveal that although the complex nonlinear ocean-atmosphere oscillator model possesses multiequilibrium states,the real climatic system possesses only a quasi-normal state and a strong ENSO cycle stable state. The first passage time between states is also given in this paper, and the theoretical computational results agree with observational data.