Disconnected large bifurcation supports and Cartesian products of bifurcations

Abstract

A bifurcation that occurs in a multiparameter family is a Cartesian product if it splits into two factors in the sense that one bifurcation takes place in one part of the phase portrait, another one -- in another part, and they are in a sense independent, do not interact with each other. To understand how a family bifurcates, it is sufficient to study it in a neighborhood of the so-called large bifurcation support. Given a family of vector fields on $S^2$ that unfolds a field $v_0$, the respective large bifurcation support is a closed $v_0$-invariant subset of the sphere indicating parts of the phase portrait of $v_0$ affected by bifurcations. One should consider disconnected large bifurcation supports in order to obtain Cartesian products for sure. We prove that, if the large bifurcation support is disconnected and the restriction of the original family to some neighborhood of each connected component is structurally stable (plus some mild extra conditions), then the original family is a Cartesian product of the bifurcations that occur near the components of the large bifurcation support. We also show that the structural stability requirement cannot be omitted.