The Complex Dynamics Of A Relative Rotation System And Its Control By Delay Velocity Feedback

In this article, the complex dynamics in a typical relative rotation system such as chaos and erosion of safe basin induced by the variation of system parameters is considered. We plan to employ delayed velocity feedback to suppress the irregular complex dynamical behaviors in the system for keeping the reliability of the oscillation.Firstly, the dynamic model of two-mass relative rotation system is established. On the basis of the establishment, the dynamic model of a relative rotation system with the delayed velocity feedback control is established; according to the two models, global bifurcation is obtained, and the critical excitation for chaos and bifurcation of safe basin of the relative rotation system will be obtained by the Melnikov method. Then the critical excitation for chaos and bifurcation of safe basin of the delayed velocity feedback controlled relative rotation system will be obtained by the Melnikov method. By discussing the critical condition for the losing of stability of equilibrium causing by delay velocity feedback in the linear system, we aim to obtain the ranges of the delay and feedback gain for controlling the complex dynamics.Second, fourth order Runge-Kutta method is used to draw the Poincare map, bifurcation diagram numerically for stimulating complex dynamical behaviors of the systems under different system parameters at the same time. The validity of the theoretical analysis can be verified by the consistency of the numerical simulation and analytical results. The results we have obtained show that the delayed velocity feedback has bi-directional "switch" controlling effect on the complex dynamical behavior of the relative rotation system. Under a positive gain, the delayed velocity feedback can suppress the chaotic motion and erosion of safe basins; the safe basin area of relative rotation system with chaotic motion increases evidently; the safe basin area of relative rotation system without chaotic motion also increases. While under a negative gain, the delayed velocity feedback makes the simple motion of the relative rotation system becomes complicated. The periodic motion of the relative rotation system becomes gradually into period-doubling bifurcation to chaos with negative feedback gains becomes smaller.Finally, we compare the delay velocity feedback control of the relative rotation system with the delay displacement feedback control of the relative rotation system, and analyze the similarities and differences of two kinds of feedback control. The stability of equilibrium of delayed velocity feedback control system does not switch, so the delay is positive. While stability of equilibrium of the delay delayed displacement feedback will switch with the time delay.