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In this paper, an oscillator with time delayed velocity feedback controls is studied in detail. Particular attention is given to internal double-Hopf bifurcation with an external exciting force. Linear analysis is used to find the critical conditions under which 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 not only validate the analytical predictions, but also show chaotic motions for some certain parameter values.
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Chen et al. (2006) studied this question.
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