Stochastic Stability And Bifurcation Analyze For The Stoch Astic Dynamic Systems With Random Parameters

Stochastic factors exist in the course of the practice objectively, which will lead to the conclusion of studying the dynamic system's behavior by deterministic method larger errors. Therefore, the stochastic factors must be taken into account in the description of the system. In this study, we studied the dynamic system with random parameters by Gegenbauer, Chebyshev orthogonal polynomial approximation method and the Hamilton theory, and base on the theory analysis and method exploration of the available structural system with stochastic parameters, we further explored the stochastic stability and stochastic bifurcation of nonlinear stochastic dynamic system.The main contents of this paper are as follows:1.In this article, the present status and development of the probabilistic method about stochastic structures with uncertain parameters, stochastic stability and the stochastic bifurcation will be reviewed. We also elaborated the concept of Stochastic Hopf bifurcation, Lyapunov coefficient, stochastic stability, stochastic bifurcation and Lyapunov index method.2.The probability density function with respect of a parameter l( ?-PDF for short) was mentioned. Pointed out that ?-PDF not only can be used to imitate some promblems that solved by asymmetric probability density function in the engineering, but also can be used to approximate the truncated normal distribution of probability density function that often appears in the engineering. The dynamical behavior of the stochastic Hopf bifurcation in the stochastic fianacial system was researched by the Gegenbauer orthogonal polynomial approximation method, which reduces the stochastic fianacial system into its equal deterministic fianacial system. The stochastic stability and Hopf bifurcation behavior of the financial system with random parameters was also be studied. Finally, the results was verified by numerical analysis.3.when ?-1, ?-PDF becomes the arch probability density function, the dynamical behavior of stochastic Duffing oscillator system could be researched by the Chebyshev orthogonal polynomial approximation method. According to the Routh-Hurwitz criterion, the stability of the system could be analysised and the condition of the stability of the system would be obtained. Next, the response was carried on the numerical analysis.4. On the basis of Oseledec multiplicative ergodic theorem and the maximum Lyapunov index, the probability to 1 asymptotic stability was discussed by using the theory of Hamilton. In order to calculate the maximum Lyapunov index, the first is making sure the form of invariant measure solution to analyze the diffusion process of singularity and singular boundary. We also studied the stochastic stability of the financial system which includes white noise excitation and stochastic bifurcation in detail. Further more, using Lyapunov function to determine the stability, and distinguish D-bifurcation by the change of plus or minus of the Lyapunov index. The stochastic bifurcation was simulated finally.