Dynamics of a Duffing-van der Pol oscillator with time delayed position feedback

摘要

The two main objectives of this article are (1) to determine the bifurcating periodic responses accurately and to study the effects of time delay and feedback gain on the steady state response in autonomous oscillators by means of the residue harmonic balance method; and (2) to study the dynamics of both the autonomous and non-autonomous Duffing-van der Pol oscillators with delayed position feedback as examples. The results of the steady state response to frequency and amplitude excitation versus the time delay, positive and negative feedback gains are studied. Both saddle-node and subcritical Hopf bifurcations are found and discussed. The obtained second-order solutions are in excellent agreement with numerical integration results. By combining both harmonic balance and polynomial homotopy continuation techniques, the superharmonic resonance responses of the non-autonomous oscillator are investigated in order to obtain all possible steady state solutions. The effects of the feedback gains and time delay on the steady state response are found analytically. It is shown that the fundamental and superharmonic amplitudes decrease with increasing feedback gain. Furthermore, the multiple symmetric solutions of the response amplitudes versus those of external excitation are detected when time delay becomes smaller.