On the persistence of quasi-periodic invariant tori for double Hopf bifurcation of vector fields

We analyze the persistence of quasi-periodic invariant 2- and 3-tori for the double Hopf (Hopf–Hopf) bifurcation by using the KAM method. We prove that in a sufficiently small neighborhood of the bifurcation point, the full system has quasi-periodic 2-tori for most of the parameter sets where its truncated normal form possesses 2-tori. Under appropriate conditions we obtain that the full system also has quasi-periodic 3-tori for most parameters near the Hopf bifurcation curve of its truncated normal form and along the direction of the bifurcation, and these 3-tori bifurcate from invariant 2-tori. We also give concrete formulas on the existence of quasi-periodic invariant 2- and 3-tori, which are based on coefficients of the truncated normal form.