The response of a Duffing–van der Pol oscillator under delayed feedback control

Abstract The nonlinear dynamics of a Duffing–van der Pol oscillator under linear-plus-nonlinear state feedback control with a time delay are investigated. By means of the averaging method and Taylor expansion, two slow-flow equations for the amplitude and phase of the primary resonance response are derived, from which the relations between the amplitude and phase of the primary resonance response and all other parameters are obtained, respectively. The singularity analysis of the equation governing the amplitude of the primary resonance response shows that the bifurcation modes are perturbations of the pitchfork bifurcation. Conditions preventing multiple solutions, corresponding to two different kinds of bifurcation modes, are given, since cases for which multiple solutions are available should be avoided. The stable condition for steady-state response is also given by the Routh–Hurwitz criterion. It is also shown that coupled nonlinear state feedback control can be replaced by uncoupled nonlinear state feedback control.