Phase Space Geometry and Chaotic Attractors in Dissipative Nambu Mechanics

TL;DR

This paper explores the geometric structure of the Lorenz system within Nambu mechanics, classifying manifolds and linking the Lorenz attractor to physical systems like a charged rigid body in a magnetic field, leading to new strange attractors.

Contribution

It introduces a geometric classification of manifolds in Nambu mechanics and relates the Lorenz system to physical models, extending to new strange attractors.

Findings

Lorenz system generated by intersection of quadratic surfaces

Classification of manifolds into four distinct classes

Identification of Lorenz system as a charged rigid body in magnetic field

Abstract

Following the Nambu mechanics framework we demonstrate that the non-dissipative part of the Lorenz system can be generated by the intersection of two quadratic surfaces that form a doublet under the group SL(2,R). All manifolds are classified into four dinstict classes; parabolic, elliptical, cylindrical and hyperbolic. The Lorenz attractor is localized by a specific infinite set of one parameter family of these surfaces. The different classes correspond to different physical systems. The Lorenz system is identified as a charged rigid body in a uniform magnetic field with external torque and this system is generalized to give new strange attractors.