Persistent singular attractors arising from singular cycle under symmetric conditions

Abstract

We prove that generic symmetric $C^{r}$-vector field families on$\mathbb{R}^{3}$ unfolding a contracting singular cycle, exhibitssingular attractors for a positive lebesgue measure set of parametervalues. Essentially the cycle is formed by a real contractingsingularity, like those in the geometric contracting Lorenzattractor, whose unstable branches go to periodic orbits in thecycle. We obtain a lower estimate for the density of this set at thefirst bifurcation value. Furthermore, for parameter values in thisset the corresponding vector field admits a unique SRB measure,whose support coincides with the attractor.