Orthogonal Polynomial Approximation Of Stochastic Bifurcation Analysis

The study of nonlinear stochastic dynamic system is quickly developed through new computational tools and new methods of mathematic analysis, but in this field there are more unknown facets we have not recognized. This thesis is devoted to explore bifurcation problems in nonlinear stochastic dynamics system with random parameter via orthogonal polynomial approximation method. The main works as follows:Firstly, stochastic van der Pol system with random parameter subject to an arch-like probability density function is reduced into its equivalent deterministic one using Chebyshev polynomial approximation method. The numerical results show that similar to their counterpart in deterministic system the symmetry-breaking bifurcation and period-doubling bifurcation occur in the stochastic van der Pol system. But for random factor the point of period-doubling bifurcation can move in the stochastic system as increasing the frequency or amplitude of harmonic excitation.Secondly, the A -probability density function is more commonly applied in engineering and physics is introduced. Then we reduce stochastic Duffing-van der Pol system into its equivalent deterministic system by Gegenbauer polynomial approximation method. The stochastic bifurcations of stochastic system are analyzed by numerical method. Comparing with deterministic system, the point of bifurcation in stochastic Duffing-van der Pol system is moved as increasing the intensity of random parameter. These results are available to practical application.Thirdly, we generalize the orthogonal polynomial method to nonlinear system with two mutual independent random parameters. Taking stochastic Duffing system as an example, the stochastic bifurcations are explored by numerical method. Under effect of the intensity of random parameters, the period-doubling bifurcations in stochastic Duffing system are more complex. From these results we can prove that this method can solve the bifurcation problems of the nonlinear system with several random parameters.