Computer assisted proof of the existence of the Lorenz attractor in the Shimizu-Morioka system

TL;DR

This paper rigorously proves the existence of Lorenz attractors in the Shimizu-Morioka system and extends the result to a class of three-dimensional polynomial diffeomorphisms, using numerical methods.

Contribution

It introduces a computer-assisted proof technique to establish Lorenz attractors in specific dynamical systems, expanding the understanding of chaotic attractors.

Findings

Lorenz attractor exists in the Shimizu-Morioka system for certain parameters

Discrete Lorenz attractors are shown to exist in polynomial diffeomorphisms

Rigorous numerics are effective in proving complex dynamical phenomena

Abstract

We prove, by employing rigorous numerics, that Shimizu-Morioka system has a Lorenz attractor for an open set of parameter values. Using this result, we prove the existence of a discrete version of the Lorenz attractor for a class of three-dimensional polynomial diffeomorphisms.