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.

中文翻译：

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上、下、两侧洛伦兹吸引子、碰撞、合并和切换

我们提出了著名的几何洛伦兹吸引子的修改版本。它由一个 $C^1$ 开集 ${\mathcalO}$ 中的向量场 ${\mathbb R}^3$ 有一个有吸引力的区域 ${\mathcal U}$ 满足三个性质。也就是说，独特的奇点 $\西格玛$ ; 独特的吸引子 $\Lambda$ 包括奇异点和最大不变量 ${\mathcal U}$ 至多有两个链递归类，分别是 $\Lambda$ 和（至多）一个双曲线马蹄铁。马蹄形和奇异吸引子随着并集发生碰撞 $2$ 余维数 $1$ 分裂的子流形 ${\mathcalO}$ 分为三个区域。通过穿过这个碰撞轨迹，吸引子和马蹄形可能会合并成两侧洛伦兹吸引子，或者它们可能会交换性质：洛伦兹吸引子驱逐奇点 $\西格玛$ 变成马蹄铁，马蹄铁吸收 $\西格玛$ 成为洛伦兹吸引子。