SADDLE-FOCUS SINGULAR CYCLES AND PREVALENCE OF HYPERBOLICITY

The objective of this work is to study the 2-parameter unfolding of an intricate bifurcating dynamical structure in dimension 3, namely a sa ddle-focus singular cycle. It is showed that hyperbolicity is a preval ent phenomena: the initial system is a total density point of hyperbol ic dynamics. This dynamic exhibit either simple critical elements or n ontrivial basic set, It seems to be the first time that non-trivial hy perbolicity (as well as its prevalence) is proved for the maximal inva riant set for a perturbation of a vector field with a cycle containing a saddle-focus singularity. (C) Elsevier, Paris.