Based On The Chebyshev Orthogonal Polynomial Approximation With Bounded Random Parameter Bifurcation And Chaotic Phenomena

This paper studies bifurcation and chaos phenomena of systems with random parameter under harmonic excitations or both harmonic and impulsive excitations. The main contents of this paper are as follows:Chapter one introduces the development history and the current developments of stochastic dynamic systems. Some numerical methods in the study of random parameter structure are also introduced mainly on the development history and their defaults. The main numerical methods which are used in this paper are given in detail. At the end of this part, the main contents of this paper are presented.In chapter two, we study bifurcation and chaos of a double-well Duffing-van der Pol system with bounded random parameters and subject to harmonic excitations. With the aid of Chebyshev polynomials, the stochastic Duffing-van der Pol system is transformed at first into its equivalent deterministic one, and then through which bifurcation phenomena of the original stochastic system can be explored by deterministic numerical methods. Numerical results show that similar to the deterministic mean-parameter system various forms of symmetry-breaking bifurcation, period-doubling bifurcation, and Neimark bifurcation may occur in the stochastic Duffing-van der Pol system, and illuminate that period-doubling bifurcation of the random parameter system has its own characteristics. Numerical results for the original stochastic Duffing-van der Pol system obtained separately by the Mento-Carlo method and the Chebyshev polynomial approximation are compared to show the effectiveness and efficiency of the latter method in solving the nonlinear dynamical problems of system with bounded random parameters.In the following two chapter, the Chebyshev polynomials approximation are applied to investigate bifurcation behaviors of symmetry-breaking and period-doubling for a coupled double-well Duffing system with bounded random parameters and subject to harmonic excitations(chapter three) and a Duffing system with bounded random parameters, but under both harmonic excitations and pulsed excitations(chapter four). Both the numerical results of these two chapters show that the Chebyshev polynomial approximation is an effective method in the study of system with random parameter.The last chapter concludes the work and points out some aspects to be further studied on dynamic behaviors of system with random parameters by applying the orthogonal polynomial approximation.