Normal forms of non-resonance and weak resonance double Hopf bifurcation in the retarded functional differential equations and applications

•The norm forms of the double Hopf bifurcation for a general DDE is established.•The van der Pol–Duffing oscillator with delayed position and velocity feedback is investigated.•The dynamical classification near the double Hopf bifurcation point is studied.•The stable periodic-doubling solutions and unstable three-dimensional torus are found.

In this paper, we firstly present the general framework of calculation of normal forms of non-resonance and weak resonance double Hopf bifurcation for the general retarded functional differential equations by using the normal form theory of delay differential equations due to Faria and Magalha˜es. Then, the dynamical behavior of van der Pol–Duffing oscillator with delayed position and velocity feedback is considered. Specifically, the dynamical classification near the double Hopf bifurcation point is investigated by analyzing the obtained normal form. Finally, the numerical simulations support the theoretical results and present some interesting phenomena.