Analytic proof of the existence of the Lorenz attractor in the extended Lorenz model

TL;DR

This paper provides an analytic proof for the existence of the Lorenz attractor in the classical Lorenz model and extends the proof to a family of Henon-like diffeomorphisms, avoiding computational methods.

Contribution

It offers the first computer-free analytic proof of the Lorenz attractor's existence and applies this to demonstrate a robust strange attractor in a related dynamical system.

Findings

Analytic proof of Lorenz attractor in Yudovich-Morioka-Shimizu model.

Verification of Shilnikov criteria for attractor birth.

Existence of a pseudohyperbolic strange attractor in Henon-like maps.

Abstract

We give an analytic (free of computer assistance) proof of the existence of a classical Lorenz attractor for an open set of parameter values of the Lorenz model in the form of Yudovich-Morioka-Shimizu. The proof is based on detection of a homoclinic butterfly with a zero saddle value and rigorous verification of one of the Shilnikov criteria for the birth of the Lorenz attractor; we also supply a proof for this criterion. The results are applied in order to give an analytic proof of the existence of a robust, pseudohyperbolic strange attractor (the so-called discrete Lorenz attractor) for an open set of parameter values in a 4-parameter family of three-dimensional Henon-like diffeomorphisms.