Contracting singular cycles
A singular cycle of a vector-field is a finite set of hyperbolic singular and periodic orbits linked in a cyclic way by orbits in the intersections of the stable and unstable manifolds of the singularities and periodic orbits. Simple singular cycles are considered in the paper, i.e. singular cycles for vector-fields in \(\mathbb R^3\) with only one singular point and satisfying some additional properties. Such a cycle is called contracting provided the expanding eigenvalue of the singularity is smaller then the weakest contracting one. In the paper [\textit{M. J. Pacificio} and \textit{A. Rovella}, Ann. Sci. Ec. Norm. Super. (4) 26, 691-700 (1993; Zbl 0802.58036)], it was proved a result on the measure of the bifurcating set for generic one-parameter families through a simple contracting cycle with only one periodic orbit. The paper under review improves that result; in particular it treats the case of several periodic orbits.
scientific article; zbMATH DE number 1199912