Tangency bifurcations of global Poincare maps

One tool to analyze the qualitative behavior of a periodic orbit of a vector field in R^n is to consider the Poincare return map to an (n-1)-dimensional section. The image under the Poincare map of points on this section that lie near the periodic orbit is obtained by following the orbits under the flow of the vector field until the next (local) intersection. It is well known that the Poincare map defined on a section transverse to a periodic orbit is a diffeomorphism locally near the periodic orbit. However, in practice, often an arbitrary global Poincare section is taken. Generically, there are points where the flow is tangent to the Poincare section, and these give rise to discontinuities of the Poincare map. In fact, the orbits of some points may not even return to the section, in which case the Poincare map is not defined at all. In this paper we study tangency bifurcations of invariant manifolds of Poincare maps on global sections of vector fields in R^2 and R^3. At such a bifurcation the manifold becomes tangent to the section, which results in a qualitative change of the Poincare map while the underlying flow itself does not undergo a bifurcation. Using tools from singularity theory, we present all normal forms of the codimension-one tangency bifurcations in the neighborhood of the respective tangency point. The study of these bifurcations is motivated by and illustrated with the examples of the (unforced) Van der Pol oscillator and a system modeling a semiconductor laser with optical injection. Finally, we present a framework for the generalization of our normal form results to arbitrary dimension and codimension.