BIFURCATION OF CONTRACTING SINGULAR CYCLES

The aim of this work is to continue the analysis of a new mechanism, t he singular cycles, through which a vector field, depending on paramet er, may evolve when the parameter Varies from a vector field exhibitin g simple dynamics into one having non-trivial dynamics. Specifically; if we start with a Morse - Smale vector field and move through a gener ic one - parameter family of vector fields to a contracting singular c ycle and beyond, we reach a region filled up mostly with hyperbolic fl ows. In fact, the Lebesgue measure of parameter values corresponding t o non Axiom A flows is zero. Moreover we provide a complete descriptio n of the bifurcation set that appear in these families.