Tangency Bifurcations of Global Poincaré Maps

Abstract

One tool to analyze the qualitative behavior of a periodic orbit of a vector field in $\mathbb{R}^n$ is to consider the Poincaré return map to an $(n-1)$-dimensional section. The image under the Poincaré map of a point on this section that lies near the periodic orbit is obtained by following the flow of the vector field until the next (local) intersection. It is well known that the Poincaré map defined on a section transverse to a periodic orbit is a diffeomorphism locally near the periodic orbit. However, in practice one often considers the Poincaré map not only locally but also on a much larger global and typically unbounded section. Generically, there are then points where the flow is tangent to the section, and these give rise to discontinuities of the Poincaré map. In fact, the orbits of some points may not even return to the section, in which case the Poincaré map is not defined at all. In this paper we study tangency bifurcations of invariant manifolds of Poincaré maps on global sections of vector fields in $\mathbb{R}^2$ and $\mathbb{R}^3$. At such a bifurcation the manifold becomes tangent to the section, which results in a qualitative change of the Poincaré map while the underlying flow itself does not undergo a bifurcation. Using tools from singularity theory, we present 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.