Kneading the Lorenz attractor

Abstract

A Lorenz map $f:[0,1]\to[0,1]$ is a piecewise continuous map, modeled after an idealized version of the Lorenz attractor. In this paper we settle the following question - how much of the dynamics of the Lorenz attractor can be modeled by such one-dimensional model? In this paper we will prove there exist open regions in the parameter space of the Lorenz system where one can canonically reduce the dynamics of the Lorenz attractor into those of a symmetric Lorenz map $F_β$ with a constant slope $β\in(1,2]$. As we will show, not only the map $F_β$ encodes many of the essential features of the Lorenz attractor, it also governs many of its bifurcations. As such, our results correlate closely with the results of numerical studies, and possibly explain the bifurcation phenomena observed in the Lorenz attractor.