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

The present 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 nonvanishing forcing strength, a semi-global normal form system is found by averaging and applying a van der Pol transformation. The bifurcation diagram is organised by a codimension 3 singularity of nilpotent-elliptic type. A fairly complete analysis of local bifurcations is given; moreover, all the nonlocal bifurcation curves predicted by Dumortier et al. (1991) are found numerically.