Research On The Dynamic Behavior Of Several Typical Nonlinear Systems Excited By Random Noise

System is the ubiquitous model of the natural world.Studying the dynamic behavior of the system is helpful to understand the influence of the internal rules and external factors on the system.In recent years,some scholars paid more and more attention to the study of the dynamic behaviors of nonlinear systems driven by random forces.In this paper,we studied the instability probability density evolution of the bistable system driven by Gaussian colored noise and white noise,the stability probability density(SPD)evolution and stochastic resonance(SR)in a second-order and underdamped asymmetric bistable system driven by Lévy noise,and the effects of non-Gaussian noise and external periodic force on the time derivative of information entropy for a damped harmonic oscillator.Research methods and conclusions are as follows:We studied the instability probability density evolution of the bistable system driven by Gaussian colored noise and white noise.Firstly,the bistable system is linearized in the initial area by applying ? expansion theory of the Green function.Next,the non-stationary state solution p(x,t)of the Fokker-Planck equation is obtained by using eigenvalue and eigenvector theory.The influence of Gaussian white noise and color noise on p(x,t)are analyzed.Numerical computation results show that:p(x,t)appears a peak with the increasing of t.Increasing of white noise intensity D also make p(x,t)appear unimodal.The instability probability density evolution of the bistable system is influenced by the noise correlation time,which is different from the previous conclusions.The two-order under-damped asymmetric bistable system excited by Lévy noise is studied for the first time,new conclusions about the phase transition of the system is obtained.Lévy noise is generated by Janicki-Weron algorithm which is different from the usual Gaussian noise.The numerical solutions of system equation are obtained by the fourth-order stochastic Runge-Kutta algorithm.Then the stationary SPD are obtained by solving the equation of system.It can be known from the discussion:asymmetric parameter r,stability index ?,skewness parameters ? and noise intensity D can induce a transition from a bimodal to unimodal SPD.The influence of various parameters of the external periodic force and Lévy noise on SR phenomenon was discussed.On the basis of the previous research,the differential equation of the system was solved.Then take SNR as index,we found that the larger value of the stability index a of Lévy noise,signal amplitude A can give rise to the SR phenomenon.On the contrary,the larger values of skewness parameters? of Lévy noise and damping parameter y further weakens the occurrence of the SR phenomenon in the given system.For asymmetric parameter r,the more symmetric the system is,the more favorable it is for the system to generate SR.We also studied the effects of non-Gaussian noise and external periodic force on the time derivative of information entropy for a damped harmonic oscillator.The influence of parameter q of non-Gaussian noise and other parameters on the stability of the system is discussed.The FPK equation of the system is obtained by approximate treatment of non-Gaussian noise by path integral method,which is simplified by way of linear transformation.Finally,the upper bound of the time derivative of information entropy of this process is exactly obtained on the basis of the Schwartz inequality principle and the definition of Shannon's information entropy.Results show that the increase of q can slow down the system conversion from a non-equilibrium state relaxation to steady state.The stability of the system is promoted by y,D,and ?,?,A,?.impeded the stability of the system in different degrees.