Semi-global analysis of periodic and quasi-periodic normal-internal k : 1 and k : 2 resonances

This paper investigates a family of nonlinear oscillators at Hopf bifurcation, driven by a small quasi-periodic forcing. In particular, we are interested in the situation that at bifurcation and for vanishing forcing strength, the driving frequency and the normal frequency are in k : 1 or k : 2 resonance. For small but non-vanishing forcing strength, a semi-global normal form system is found by averaging and applying a van der Pol transformation. The bifurcation diagram is organized by a codimension 3 singularity of nilpotent-elliptic type. A fairly complete analysis of local bifurcations is given; moreover, all the non-local bifurcation curves predicted by Dumortier et al (1991 Bifurcations of Planar Vector Fields (Lecture Notes in Mathematics vol 1480) (Berlin: Springer)), excepting boundary bifurcations, are found numerically.