Codimension-one tangency bifurcations of global Poincaré maps of four-dimensional vector fields

When one considers a Poincaré return map on a general unbounded (n-1)-dimensional section in a vector field in R^n there are typically points where the flow is tangent to the section. The only notable exception is when the system is (equivalent to) a periodically forced system. The tangencies cause bifurcations of the Poincaré return map when the section is moved when there are no bifurcations in the underlying vector field. The interaction of invariant manifolds and the tangency loci on the surface gives rise to discontinuities of the Poincaré map and there can be open regions where the map is not defined. We study the case of the four-dimensional phase space R^4. Specifically, we make use of tools from singularity theory and flowbox theory to present normal forms of the codimension-one tangency bifurcations in the neighbourhood of a tangency point.