Take a look at a standard picture of the Lorenz attractor (with the classical parameter values):

lorenz attractor

The Lorenz attractor is a complicated fractal subset of $\mathbb{R}^3$ of Hausdorff dimension > 2. Does this fractal include the two critical points at the centers of the spirals and/or the one at the origin?

According to what formal definition of attractor?


Let $\varphi$ be the flow of the Lorenz system with the classical parameter values. There exists a forward invariant open set $U$ (a double torus) containing the origin but not the other two critical points. The Lorenz attractor is $$\mathcal{A} = \bigcap_{t\geqslant 0} \varphi(U,t).$$ Since the origin is an equilibrium point contained in $U$, it is also contained in $\mathcal{A}$. Since $U$ is forward invariant and does not contain the other two critical points, $\mathcal{A}$ also does not contain the other two critical points.

(Adopted from Warwick Tucker's thesis The Lorenz attractor exists p. 9, bottom.)


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