Hopf bifurcation analysis of man-machine system involving time delays

To discuss the effects' of change of time delay on the dynamics of a nonlinear system, the Hopf bifurcation analyses on the role played by the delay in the control force of a structurally unstable man-machine system were investigated. By analyzing the associated characteristic equation, it is found that the Hopf bifurcation occurs when the delay passes through a critical value. The explicit algorithm for determining the direction of the Hopf bifurcation and the stability of bifurcated periodic solutions were derived using the theory of normal form and center manifold. Results of numerical simulations agree well with the theoretical analysis results, which means that the system control can be realized by adjusting the magnitude of time delay.