Kneading the Lorenz attractor

TL;DR

This paper demonstrates that certain regions of the Lorenz system's parameter space can be effectively modeled by one-dimensional Lorenz maps, capturing key dynamics and bifurcations of the attractor.

Contribution

It proves the existence of parameter regions where Lorenz attractor dynamics can be reduced to symmetric Lorenz maps with constant slope, linking one-dimensional models to complex bifurcations.

Findings

Lorenz maps encode essential features of the Lorenz attractor.

The dynamics of the Lorenz system can be reduced to these maps in certain parameter regions.

The maps govern bifurcations and explain observed phenomena in 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 $F_\beta$ with a constant slope $\beta\in(1,2]$. As we will show, not only the map $F_\beta$ 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.