The Lorenz Renormalization Conjecture

TL;DR

This paper investigates whether the hyperbolic horseshoe paradigm for low-dimensional dynamical systems extends to higher-dimensional Lorenz systems, proposing a conjecture on Lorenz renormalization supported by numerical evidence.

Contribution

It introduces a conjecture classifying Lorenz renormalization dynamics and provides numerical support, advancing understanding of higher-dimensional chaotic systems.

Findings

Proposes a conjecture on Lorenz renormalization dynamics.

Provides numerical evidence supporting the conjecture.

Addresses the extension of hyperbolic paradigms to higher dimensions.

Abstract

The renormalization paradigm for low-dimensional dynamical systems is that of hyperbolic horseshoe dynamics. Does this paradigm survive a transition to more physically relevant systems in higher dimensions? This article addresses this question in the context of Lorenz dynamics which originates in homoclinic bifurcations of flows in three dimensions and higher. A conjecture classifying the dynamics of the Lorenz renormalization operator is stated and supported with numerical evidence.