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Upper, down, two-sided Lorenz attractor, collisions, merging, and switching
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 44 / Issue 10 / October 2024
- Published online by Cambridge University Press:
- 21 February 2024, pp. 2737-2781
- Print publication:
- October 2024
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- Article
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We present a modified version of the well-known geometric Lorenz attractor. It consists of a
$C^1$ open set 
${\mathcal O}$ of vector fields in 
${\mathbb R}^3$ having an attracting region 
${\mathcal U}$ satisfying three properties. Namely, a unique singularity 
$\sigma $; a unique attractor 
$\Lambda $ including the singular point and the maximal invariant in 
${\mathcal U}$ has at most two chain recurrence classes, which are 
$\Lambda $ 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 
${\mathcal O}$ 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 
$\sigma $ and becomes a horseshoe, and the horseshoe absorbs 
$\sigma $ becoming a Lorenz attractor.
