Recognition of resonance type in periodically forced oscillators

This paper deals with families of periodically forced oscillators undergoing a Hopf–Neĭmarck–Sacker bifurcation. The interest is in the corresponding resonance sets, regions in parameter space for which subharmonics occur. It is a classical result that the local geometry of these sets in the non-degenerate case is given by an Arnol’d resonance tongue. In a mildly degenerate situation a more complicated geometry given by a singular perturbation of a Whitney umbrella is encountered. Our main contribution is providing corresponding recognition conditions, that determine to which of these cases a given family of periodically forced oscillators corresponds. The conditions are constructed from known results for families of diffeomorphisms, which in the current context are given by Poincaré maps. Our approach also provides a skeleton for the local resonant Hopf–Neĭmarck–Sacker dynamics in the form of planar Poincaré–Takens vector fields. To illustrate our methods two case studies are included: A periodically forced generalized Duffing–Van der Pol oscillator and a parametrically forced generalized Volterra–Lotka system.