An efficient method for studying weak resonant double Hopf bifurcation in nonlinear systems with delayed feedbacks

An efficient method, called the perturbation-incremental scheme (PIS), is proposed to study, both qualitatively and quantitatively, the delay-induced weak or high-order resonant double Hopf bifurcation and the dynamics arising from the bifurcation of nonlinear systems with delayed feedback. The scheme is described in two steps, namely, the perturbation and the incremental steps, when the time delay and the feedback gain are taken as the bifurcation parameters. As for applications, the method is employed to investigate the delay-induced weak resonant double Hopf bifurcation and dynamics in the van der Pol-Duffing and the Stuart-Landau systems with delayed feedback. For bifurcation parameters close to a double Hopf point, all solutions arising from the resonant bifurcation are classified qualitatively and expressed approximately in a closed form by the perturbation step of the PIS. Although the analytical expression may not be accurate enough for bifurcation parameters away from the double Hopf point, it is used as an initial guess for the incremental step which updates the approximate expression iteratively and performs parametric continuation. The analytical predictions on the two systems show that the delayed feedback can, on the one hand, drive a periodic solution into an amplitude death island where the motion vanishes and, on the other hand, create complex dynamics such as quasi-periodic and coexisting motions. The approximate expression of periodic solutions with parameter varying far away from the double Hopf point can be calculated to any desired accuracy by the incremental step. The validity of the results is shown by their consistency with numerical simulations. We show that as an analytical tool the PIS is simple but efficient.