Difference resonances in a controlled van der Pol-Duffing oscillator involving time delay

A non-resonant interaction of two Hopf bifurcations may appear after the trivial solution of a controlled van der Pol-Duffing oscillator without external excitation loses its stability, when two critical time delays corresponding to two Hopf bifurcations have the same value. In the vicinity of the non-resonant Hopf bifurcations, the presence of a periodic excitation in the controlled oscillator can induce difference resonances in the forced response, when the forcing frequency and the frequencies of the two Hopf bifurcations satisfy certain relationships. It is found that the frequency response curves of the controlled system under difference resonances are an isolated closed curve. The difference resonance response may admit two stable motions on a three-dimensional torus consisting of three frequencies. Illustrative examples are given to show the quasi-periodic motions.