Complex Singularities and the Lorenz Attractor

TL;DR

This paper explores the complex singularities of the Lorenz system by extending it into the complex plane, developing a formal analysis using psi series, and proving convergence to better understand its chaotic dynamics.

Contribution

It introduces a complete formal development of complex singularities of the Lorenz system using psi series and proves their convergence, advancing mathematical understanding of the system's complex behavior.

Findings

Psi series contain two undetermined constants

Singularities are characterized by a two-parameter family of solutions

Convergence of psi series is established using a novel technique

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 mathematical 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 singularities of the Lorenz system using {\it 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 questions, answers to which may bring us closer to understanding the connection of complex singularities to Lorenz dynamics.