Steady-state Probability Density Of A Three-stable System Under Different Noise Excitations

The stationary probability distribution function is important physical quantities of the nonlinear dynamical system. Noise-induced transition is a crucial phenomenon in stochastic dynamics. The research of theory and the potential applications have already caused wide public concern. Based on the theory and technique of random process and nonlinear dynamical systems. Through the Langevin equation of system, the Fokker-Planck equation is obtained by using some approximative approaches. By the combination of numerical analysis and numerical simulation, the effect of the stationary probability distribution in Gaussian white noise% Gaussian color noise and non-Gaussian noise are presented.The main contents and conclusions are as follows:1.The effect of the stationary properties of a triple-well potential system with white noises are investigated. By virtue of the Liouville and Novikov theorem, the Fokker-Planck equation is obtained. The explicit expressions of the stationary prob-ability distribution function is presented. Based on the results, effects of following parameters, including the intensity of cross-correlation ?, the intensity of additive noise Q and the intensity of multiplicative noise P. Using the numerical compu-tation, the results show that the intensity of cross-correlation and the intensity of multiplicative noise can induce phase transition as the parameter ? and P are in-creased; However, the intensity of additive noise can not induce phase transition with parameter Q increases. To verify the validity of the stationary probability dis-tribution function. Numerical simulation by Euler algorithm, the time course of x(t) and stationary probability distribution function ?st(x)indicate the following and are in good agreement with the theoretical results.2.The effect of the stationary properties of a triple-well potential system with colored noises are presented. Making use of the unified colored noise approxima-tion and stochastic equivalence theory, the expressions of the steady steady state probability distribution function is obtained. After numerical calculation, results show that the multiplicative noise D and the self-correlated time ?1 can induce the transition, the additive noise Q and the self-correlated time ?2 can not induce the transition. What's more, the Box-Mueller algorithm is used to generate Gaussian white noise. By using numerical simulation and analyzing the time course of x(t) and stationary probability distribution function ?(x), a good agreement is found by comparing analysis and numerical simulation.3.The effect of the stationary properties of a triple-well potential system with non-Gaussian noise is showed. By virtue of the path-integral and the unified colored noise approximation, the expressions of the stationary probability distribution func-tion is investigated. The results show that the intensity of multiplicative noise D, the intensity of additive noise Q, the derivation parameter q and the self-correlated time ? can change structures of the stationary probability distribution function. Nu-merical simulation the time course of x(t) and the stationary probability distribution ?st(x). It is shown that the theoretical results are consistent with the simulation results.