Lorenz equations and the figure eight knot

Abstract

Lorenz equations were first presented in 1963 by Edward Lorenz, they depend on three real positive parameters. For some of these parameters which are called T-points, there are two heteroclinic orbits connecting the three singular points in the equations. The heteroclinic connections can be extended into an invariant curve passing through infinity. We consider the system at the second T-point parameter, and develop a geometric model for the flow that simulates the Lorenz dynamics there. We show that the model contains infinitely many periodic orbits, and that as knots they are all positive, prime and fibered.