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Upper, down, two-sided Lorenz attractor, collisions, merging, and switching
Published online by Cambridge University Press: 21 February 2024
Abstract
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.
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- © The Author(s), 2024. Published by Cambridge University Press