Analytical study of The Lorenz system: existence of infinitely many periodic orbits and their topological characterization

TL;DR

This paper analytically links the Lorenz equations to a geometric model for certain parameters, revealing the existence of infinitely many periodic orbits and using topological methods for 3D flows.

Contribution

It introduces a new analytical approach to connect Lorenz equations with geometric models for specific parameters, expanding understanding of their complex dynamics.

Findings

Existence of infinitely many periodic orbits in the Lorenz system.

Analytical relation established between Lorenz equations and geometric model.

Application of topological tools to three-dimensional flows.

Abstract

We consider the Lorenz equations, a system of three dimensional ordinary differential equations modeling atmospheric convection. These equations are chaotic and hard to study even numerically, and so a simpler "geometric model" has been introduced in the seventies. One of the classical problems in dynamical systems is to relate the original equations to the geometric model. This has been achieved numerically by Tucker for the classical parameter values, and remains open for general values. In this paper we establish analytically a relation to the geometric model, for a different set of parameter values that we prove must exist. This is facilitated by finding a novel way to apply topological tools developed for the study of surface dynamics to the more intricate case of three dimensional flows.