Upper, down, two-sided Lorenz attractor, collisions, merging, and switching

Abstract We present a modified version of the well-known geometric Lorenz attractor. It consists of a C¹ open set O of vector fields in R³ having an attracting region U satisfying three properties. Namely, a unique singularity ; a unique attractor including the singular point and the maximal invariant in U has at most two chain recurrence classes, which are and (at most) one hyperbolic horseshoe. The horseshoe and the singular attractor have a collision along with the union of 2 codimension 1 submanifolds which split O into three regions. By crossing this collision locus, the attractor and the horseshoe may merge into a two-sided Lorenz attractor, or they may exchange their nature: the Lorenz attractor expels the singular point and becomes a horseshoe, and the horseshoe absorbs becoming a Lorenz attractor.