Geometry and dynamics of mildly degenerate Hopf–Neĭmarck–Sacker families near resonance

Abstract

This paper deals with families of planar diffeomorphisms undergoing a Hopf–Neĭmarck–Sacker bifurcation and focuses on bifurcation diagrams of the periodic dynamics. In our universal study the corresponding geometry is classified using Lyapunov–Schmidt reduction and contact-equivalence singularity theory, equivariant under an appropriate cyclic group. This approach recovers the non-degenerate standard Arnol'd resonance tongues. Our main concern is a mildly degenerate case, for which we analyse a 4-parameter universal model in detail. The corresponding resonance set has a Whitney stratification, which we explain by giving its incidence diagram and by giving both two- and three-dimensional cross-sections of parameter space. We investigate the further complexity of the dynamics of the universal model by performing a bifurcation study of an approximating family of Takens normal form vector fields. In particular, this study demonstrates how the bifurcation set extends an approximation of the resonance set of the universal model.