The dynamics of perturbations of the contracting Lorenz attractor

Abstract

We show here that by modifying the eigenvalues λ₂ < λ₃ < 0 < λ₁ of the geometric Lorenz attractor, replacing the usualexpanding condition λ₃+λ₁ > 0 by acontracting condition λ₃+λ₁ < 0, we can obtain vector fields exhibiting transitive non-hyperbolic attractors which are persistent in the following measure theoretical sense: They correspond to a positive Lebesgue measure set in a twoparameter space. Actually, there is a codimension-two submanifold in the space of all vector fields, whose elements are full density points for the set of vector fields that exhibit a contracting Lorenz-like attractor in generic two parameter families through them. On the other hand, for an open and dense set of perturbations, the attractor breaks into one or at most two attracting periodic orbits, the singularity, a hyperbolic set and a set of wandering orbits linking these objects.