Abstract

The well-known Lorenz system has become a paradigm example of a seemingly simple, low-dimensional system displaying extreme sensitivity and chaos. Derived by meteorologist Edward Lorenz in the 1960 as a much simplified model of scrolls in the atmosphere, the three differential equations now bearing his name have been found to describe the chaotic dynamics of systems as diverse as class-C lasers and leaky water wheels. The chaotic dynamic has been represented mathematically by the much publicised Lorenz attractor.

This talk will discuss how the dynamics of the Lorenz system is organised globally throughout phase space, not just on the Lorenz attractor itself. To this end, we consider, compute and visualise a surface known as the Lorenz manifold, consisting of all trajectories that end up at the origin rather than move over the Lorenz attractor forever. The Lorenz manifold changes in an amazing way during the transition from simple to chaotic dynamics. It is an intriguing and beautiful geometric object — and has even been turned into a piece of art.