Control And Application Of Hopf Bifurcation For Nonlinear Systems

Hopf bifurcation as an important dynamic bifurcation is closely related to some self-excited vibration phenomenon and has a high theoretical value in dynamic bifurcation and limited cycle research. The research of Hopf bifurcation control can avoid the adverse consequence and has guiding meaning for improving the stability and reliability of systems.The control effects of Hopf bifurcation controller in nonlinear systems are discussed through stability analysis and Normal Form direct method. The method that using analytical expressions express the Normal Form coefficient can be used in Hopf bifurcation control of high dimensional nonlinear systems. The main work includes the following three parts:Part I: The washout filter is adopted to control Hopf bifurcation of R?ssler system. The effect of its parameters on Hopf bifurcation is discussed through stability analysis and calculating the Normal Form coefficient using Normal Form direct method. Its linear parameters can control the position of bifurcation point, while the nonlinear parameters control the bifurcation type and amplitude of limit cycles.Part II: The robustness of washout filter is discussed. The stability region of system parameters can be obtained through stability analysis firstly, and then the robust region where controller is also effective is found out. By using Normal Form direct method, the analytical expression of Normal Form coefficient is solved and the effect of nonlinear parameters on Hopf bifurcation type is discussed in detail.Part III: The control effects of shape memory alloy fibers in panel are researched. Firstly, the longitudinal motion equation of panel according to lager deformation theory is established, then, the ordinary differential equations are obtained through Galerkin discrete and reducing orders. The effect of SMA fibers on the stability of panel is discussed by solving the algebraic condition of panel flutter, and the effect of temperature is also obtained.All the theoretical analysis results obtained are verified by numerical simulation.