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

Solution 1:

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!