On the quasi-periodic response in the delayed forced Duffing oscillator

We explore the quasi-periodic (QP) vibrations in a delayed Duffing equation submitted to periodic forcing. The second-step perturbation method is applied on the slow flow of the oscillator to derive the slow–slow flow near the primary resonance. The QP solution corresponding to the nontrivial equilibrium of the slow–slow flow as well as its modulation envelope is predicted analytically. The influence of different system parameters on the QP response is reported and discussed. The analytical results show that for weak nonlinearity and small damping large-amplitude QP vibration induced by destabilization of limit cycle via Neimark–Sacker bifurcation occurs in a broadband of the excitation frequency and in large range of delay parameters.