DOUBLE-HOPF BIFURCATION IN AN OSCILLATOR WITH EXTERNAL FORCING AND TIME-DELAYED FEEDBACK CONTROL

摘要

In this paper an oscillator with time delayed velocity feedback controls is studied in detail. The particular attention is focused on internal double-Hopf bifurcation with an external exciting force. Linear analysis is used to find the critical conditions under which a double-Hopf bifurcation occurs. Then center manifold theory is applied to obtain an ODE system described on a four-dimensional center manifold. Further, the technique of multiple-time scales is employed to find the approximate solutions of periodic and quasi-periodic motions. Finally, numerical simulation results are presented to verify the analytical predictions. Also, for some certain parameter values, numerical results show chaotic attractors.