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

Abstract: We present a slightly modified version of the well known "geometric Lorenz attractor". It consists in a C1 open set O of vector fields in R3 having an attracting region U containing: (1) a unique singular saddle point sigma; (2) a unique attractor Lambda containing the singular point; (3) the maximal invariant in U contains at most 2 chain recurrence classes, which are Lambda and (at most) one hyperbolic horseshoe. The horseshoe and the singular attractor have a collision along the union of 2 co-dimension 1 sub-manifolds which divide O in 3 regions. By crossing this collision locus, the attractor and the horseshoe may merge in a two-sided Lorenz attractor, or they may exchange their nature: the Lorenz attractor expel the singular point sigma and becomes a horseshoe and the horseshoe absorbs sigma becoming a Lorenz attractor. By crossing this collision locus, the attractor and the horseshoe may merge in a two-sided Lorenz attractor, or they may exchange their nature: the Lorenz attractor expel the singular point sigma and becomes a horseshoe and the horseshoe absorbs sigma becoming a Lorenz attractor.