Quasi-periodic vibrations in a delayed van der Pol oscillator with time-periodic delay amplitude

Quasi-periodic vibrations in a delayed van der Pol oscillator with time-periodic delay amplitude is investigated in this paper. The case where the delay amplitude in the position is modulated with a certain amplitude and a resonant frequency is considered. Application of the double perturbation method enables approximation of the amplitudes of quasi-periodic vibrations of the oscillator near a parametric resonance for which the frequency of the delay amplitude modulation is near twice the natural frequency of the oscillator. Analytical approximations supported by numerical simulations provide the regions in parameter space where quasi-periodic vibrations exist. Results show that the modulation of the delay amplitude in the position not only gives birth to quasi-periodic vibrations in a large range of parameters, but also with broadband large-amplitude covering a wide range of excitation frequency.