Low order normal resonances in quasi-periodically forced systems
by
Florian Wagener
Center for Nonlinear Dynamics in Economics and Finance (CeNDEF), Universiteit van Amsterdam

A family of planar vector fields at Hopf bifurcation is investigated, with a small quasi-periodic perturbation added. The driving frequency and the normal frequency are assumed to be at, or close to k:1 or k:2 resonance at bifurcation for vanishing perturbation strength. This parameter region is in the heart of the so-called Chenciner bubbles.

For small but nonvanishing perturbation strength, integrable semi-local normal form systems are found by applying averaging and van der Pol transformation techniques (depending on the resonance k), and by truncation of quasi-periodic terms of high order in the perturbation strength. Local bifurcation diagrams of the integrable system persist if the quasi-periodic terms are taken into account again, excepting strongly resonant Hopf bifurcations. For these, the analysis can be reapplied, yielding a (finite) cascade of resonances within resonances.

Date received: March 30, 2001