Bifurcations of Regular Limit Periodic Sets

Abstract

In this chapter, (X _λ) will be a smooth or analytic (in Section 3) family of vector fields on a phase space S, with parameter λ ∈ P, as in Chapter 1. Periodic orbits and elliptic singular points which are limits of sequences of limit cycles are called regular limit periodic sets. The reason for this terminology is that for such a limit periodic set r one can define local return maps on transversal segments, which are as smooth as the family itself. The limit cycles near r will be given by a smooth equation and the theory of bifurcations of limit cycles from r will reduce to the theory of unfoldings of differentiable functions. In fact, we will just need the Preparation Theorem and not the whole Catastrophe Theory to treat finite codimension unfoldings.