Lorenz attractor as a 2-dimensional set in $\mathbb{R}^3$

Question

Why is it correct to say that Lorenz attractor is a two-dimensional set in $\mathbb{R}^3$?

Thank you!

Answer 1

It depends on your definition of two-dimensional, more precisely your notion of dimension:

• Topological dimensions are 1. (There may be some notion of the topology of the attractor according to which it is two-dimensional, but not more than this.)

• Fractal dimensions are strictly larger than 2 (also see the other answer).

• If you look for the lowest-dimensional manifold that contains the attractor, this needs to be larger than two since you cannot have a continuous-time chaotic dynamics on a two-dimensional manifold: You need at least one dimension for temporal evolution, one for stretching, and one for folding.

Note that the last two points apply to all chaotic attractors of continuous-time systems.

Answer 2

No.

Wikipedia says that Hausdorff dimension of the Lorenz attractor is estimated to be $2.06 \pm 0.01$, for the classical parameters $\rho = 28$, $\sigma = 10$, $\beta = 8/3$.