# Lorenz Attractor, its Geometric Model, and 14th Smale's problem.

I've found a post with a beautifully animated video that states the following:

In 2001 mathematician Warwick Tucker proved that the paper model accurately describes the motion on the Lorenz attractor. For every trajectory on the attractor, there is a trajectory on the paper model that behaves exactly the same way (illustration below: paper model on the left and trajectory on Lorenz Attractor on the right).

As a try to understand this fact, I've found Dr. Warwick's Ph.D. dissertation and his 2002 paper (which turned out to be a solution to 14th Smale's problem). However, with my humble knowledge in this area, I am struggling to understand what part of these works implies the statement above?

If you are familiar with this work, please explain it in simpler and intuitive terms on how one can imply the statement above. Or, please give me a hint on how to approach these works to get the above implication. Any help is appreciated.

UPDATE: There is the following part of the 2002 paper that might lead to the answer of my question, although I don't know how:

Problem Number 14 reads as follows: Is the dynamics of the ordinary differential equations of Lorenz that of the geometric Lorenz attractor of Williams, Guckenheimer, and Yorke?

As an affirmative answer to Smale’s question, we are now ready to state the sole theorem of this paper:
Main Theorem. For the classical parameter values, the Lorenz equations support a robust strange attractor $A$. Furthermore, the flow admits a unique SRB measure $\mu_\phi$ with $\text{supp}(\mu_\phi) = A$.

How is this theorem answers Smale's question and implies the statement above about trajectories?

• Thank you, the post and the video are beautiful. I think that the key words are "support a strange attractor". This proves that all trajectories eventually converge on some set. The precise description of that set, and the dynamics on it (which corresponds to the movement of the little car in the video), are probably buried in the proofs. At least, these are my 2 cents May 27 '18 at 23:00
• Note that (at least formally) the first picture from the blog post corresponds to Birman-Guckenheimer-Williams model of Lorenz attractor, while Tucker's work is more about the so called geometric model, studied by Guckenheimer-Williams and Afraimovich-Bykov-Shilnikov. May 28 '18 at 7:45

The original paper by Tucker (from 1999) proving that the Lorenz attractor exists can be found following this link:

https://ac.els-cdn.com/S076444429980439X/1-s2.0-S076444429980439X-main.pdf?_tid=9bb4fb3b-49c9-4ed3-9bb4-d2aaab811f47&acdnat=1528699700_c2d66242bd506d5d84cee6aaf65517ee

A Nature resume of the paper is the following:

https://www.nature.com/articles/35023206

As far as I understand the problem, Tucker's idea is to subdivide a certain portion of the phase space into small boxes, hence showing that the flow of the dynamical system stays "trapped" inside these boxes. Somehow, the fact the flow is actually trapped is given by some simulation with ODE solver. In the Nature paper, the use of numerical tools for a theoretical proof is celebrated:

"At last, the embarrassing gap between what we think we know about a nonlinear dynamical system from numerical simulations, and what we actually know in full logical rigour, is starting to close."

I hope this helps!