Stability and bifurcation analysis in a magnetic bearing system with time delays

A kind of magnetic bearing system with time delay is considered. Firstly, linear stability of the model is investigated by analyzing the distribution of the roots of the associated characteristic equation. According to the analysis results, the bifurcation diagram is drawn in the appropriate parameter plane. It is found that the Hopf bifurcation occurs when the delay passes through a sequence of critical values. Then the explicit algorithm for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived, using the theory of normal form and center manifold. Finally, some numerical simulations are carried out to illustrate the results found.