Nice animation, great colors. Joan Birman has an interesting paper on arxiv "Lorenz Knots", which inspired me to try and get a better understanding of attractor topology. The Lorenz equations are a multiway system whose structure is essentially that of the tent map, as can be deduced by analysis of the knot template given by Birman. Existence of periodic limit cycles in the Lorenz flow is then seen roughly equivalent to existence of periodic cycles in the tent map. Anyone who has worked through Predrag Cvitanović's book on chaos knows what a difficulty it can be to find initial conditions, certainly not as easy as listing rational numbers. There seems to be much more to do in this field of chaotic oscillator analysis. Unfortunately, knot recognition is such a difficult problem, we can't easily generate topological data, which we would need to move forward. Happy Plotting!
