Complex Singularities and the Lorenz Attractor

Abstract

The Lorenz attractor is one of the best-known examples of applied mathematics. However, much of what is known about it is a result of numerical calculations and not of mathemat- ical analysis. As a step toward mathematical analysis, we allow the time variable in the three-dimensional Lorenz system to be complex, hoping that solutions that have resisted analysis on the real line will give up their secrets in the complex plane. Knowledge of singularities being fundamental to any investigation in the complex plane, we build upon earlier work and give a complete and consistent formal development of complex singular- ities of the Lorenz system using the psi series. The psi series contain two undetermined constants. In addition, the location of the singularity is undetermined as a consequence of the autonomous nature of the Lorenz system. We prove that the psi series converge, using a technique that is simpler and more powerful than that of Hille, thus implying a two-parameter family of singular solutions of the Lorenz system. We pose three ques- tions, answers to which may bring us closer to understanding the connection of complex singularities to Lorenz dynamics.