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

• Published in: Journal of Differential Equations Vol. 260; no. 10; pp. 7320 - 7357
• Main Author: Li, Xuemei
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.05.2016
• Subjects: Double Hopf bifurcation; KAM technique; Persistence; Quasi-periodic invariant torus; Persistence; KAM technique; Double Hopf bifurcation; Quasi-periodic invariant torus
• ISSN: 0022-0396; 1090-2732
• DOI: 10.1016/j.jde.2016.01.025

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.