Strange Attractors in Dissipative Nambu Mechanics : Classical and Quantum Aspects

TL;DR

This paper extends Nambu-Hamiltonian mechanics to dissipative systems, models classical strange attractors like Lorenz and R"ossler, and introduces a quantum framework with non-commutative coordinates, revealing quantum effects on attractor dynamics.

Contribution

It develops a dissipative extension of Nambu mechanics and provides a novel quantization of the Lorenz attractor using non-commutative phase space matrices.

Findings

Classical dissipative systems can be modeled within the extended Nambu framework.

Quantum Lorenz attractor exhibits an attracting ellipsoid in high-dimensional phase space.

Dissipation leads to violations of quantum commutation relations, indicating volume contraction.

Abstract

We extend the framework of Nambu-Hamiltonian Mechanics to include dissipation in $R^{3}$ phase space. We demonstrate that it accommodates the phase space dynamics of low dimensional dissipative systems such as the much studied Lorenz and R\"{o}ssler Strange attractors, as well as the more recent constructions of Chen and Leipnik-Newton. The rotational, volume preserving part of the flow preserves in time a family of two intersecting surfaces, the so called {\em Nambu Hamiltonians}. They foliate the entire phase space and are, in turn, deformed in time by Dissipation which represents their irrotational part of the flow. It is given by the gradient of a scalar function and is responsible for the emergence of the Strange Attractors. Based on our recent work on Quantum Nambu Mechanics, we provide an explicit quantization of the Lorenz attractor through the introduction of Non-commutative phase space coordinates as Hermitian $ N \times N $ matrices in $ R^{3}$. They satisfy the commutation relations induced by one of the two Nambu Hamiltonians, the second one generating a unique time evolution. Dissipation is incorporated quantum mechanically in a self-consistent way having the correct classical limit without the introduction of external degrees of freedom. Due to its volume phase space contraction it violates the quantum commutation relations. We demonstrate that the Heisenberg-Nambu evolution equations for the Quantum Lorenz system give rise to an attracting ellipsoid in the $3 N^{2}$ dimensional phase space.