Stability And Bifurcation Analysis Of Delayed Magnetically Levitated System

In magnetically levitated system, the electromagnetic operating force is a nonlinear function for the displacement of the object and the operating currents, the effects of the nonlinear forces will make the object produce considerable vibrations when the parameters belong to certain domain. So analyzing nonlinear vibrant properties and the stability of the system is an important subject in the study of the magnetically levitated system. Besides, it is inevitable that time delay will exist in the control system of the magnetically levitated system. And it makes the phase space of the dynamical system change from finite dimensions to infinite dimensions. Time delay may also affect the performance of the dynamical system, such as stability of bifurcation. Thus investigating the magnetically levitated system with time delay is of great theoretical and practical significance.In this thesis, at first, we introduce one kind of magnetically levitated system with time delay. Second, we analyze the stability of the equilibrium by using linearizing stability method. When the eigenvalues of the linear part have a pair of pure imaginary numbers, we obtain the corresponding delay value. The stability of the steady state is changed when the delay passes through the critical value, as well as there will be a family of periodic solutions bifurcation from the steady states, and we find that the stability switch occurs when delay varies. Then, we derive the formulae for determining the direction of the Hopf bifurcation and the stability of these periodic solutions bifurcating from the steady states, by using the normal form method and center manifold theorem. Finally some numerical simulations are carried out by using MATLAB, and the results are consistent with our analysis results.