The Research Of Stochastic Hopf Bifurcation Based On Chebyshev Orthogonal Polynomial Approximation Theory

Stochastic bifurcation is a hot topic in the area of mechanics in the past few decades. Based on the theory of stochastic structures and stochastic dynamical system, the response, bifurcation and chaos in nonlinear stochastic dynamical system are explored by the method of Chebyshev polynomial approximation. The Hopf bifurcations of stochastic nonlinear dynamical system with random parameters are investigated in this paper. Main content of this paper are as follows:First, we proposed a new two-dimensional chaotic system with random parameters, a motor system with random parameters, a financial system with random parameters. The stability and the existence of the equilibrium of the system are analyzed, and the conditions for generating Hopf bifurcation is investigated by choosing the appropriate bifurcation parameter. More precisely,we respectively choose parameter a, b as the bifurcation parameter, and the Hopf bifurcation occurs when bifurcation parameter a, b passes through the critical value0 a,0b. In order to study the dynamical behavior of the kind of system mentioned above, Chebyshev polynomial approximation is applied to transform them into their equivalent deterministic system. The parameter condition to ensure the appearance of Hopf bifurcation in that system are obtained by the first Lyapunov coefficient method. with aid of the Maple program. In the process of numerical calculation,we can get some important conclusions of occurring Hopf bifurcation in high-dimensional deterministic system by Maple and Matlab. By analyzing we see that the systems occur supercritical Hopf bifurcation or subcritical Hopf bifurcation. And it can be changed from an unstable state to a steady state if the system occurs supercritical Hopf bifurcation and satisfies some certain conditions. We can avoid sharp fluctuation,explain and predict some practical problems by changing the system parameter appropriately.Second, by the study of stochastic system with the deterministic system theory, we find its have some characteristics similar to that of the deterministic system, and also show some special characteristics of stochastic system. Different from the deterministic system, the critical value of stochastic Hopf bifurcation is determined not only by stochastic parameters in stochastic system, but also by the intensity of random parameter. As the intensity of random parameter is changed, the critical value of stochastic Hopf bifurcation is also changed.At last, numerical simulations results show the effectiveness of the method and the correctness of the theoretical results in the paper. Apparently, there are more interesting problems about the kind of system in terms of complexity, control, and synchronization, which deserve further investigation.