Nonresonant Hopf bifurcations of a controlled van der Pol–Duffing oscillator

The trivial equilibrium of a van der Pol–Duffing oscillator with a nonlinear feedback control may lose its stability via Hopf bifurcations, when the time delay involved in the feedback control reaches certain values. Nonresonant Hopf–Hopf interactions may occur in the controlled van der Pol–Duffing oscillator when the corresponding characteristic equation has two pairs of purely imaginary roots. With the aid of normal form theory and centre manifold theorem as well as a perturbation method, the dynamic behaviour of the nonresonant co-dimension two bifurcation is investigated by studying the possible solutions and their stability of the four-dimensional ordinary differential equations on the centre manifold. In the vicinity of the nonresonant Hopf bifurcation, the oscillator may exhibit the initial equilibrium solution, two periodic solutions as well as a quasi-periodic solution on a two-dimensional torus, depending on the dummy unfolding parameters and nonlinear terms. The analytical predictions are found to be in good agreement with the results of numerical integration of the original delay differential equation.