Knotted Strange Attractors and Matrix Lorenz Systems

TL;DR

This paper introduces a generalized Lorenz system with variables in a Lie algebra, revealing knotted attractors and quantum fluctuation dynamics, with potential implications for understanding chaotic quantum systems.

Contribution

It proposes a novel Lie algebra-based generalization of Lorenz equations, analyzing conditions for non-linear chaos and quantum fluctuations within this framework.

Findings

Knotted strange attractors identified in the system.

Bimodal distribution of the largest Lyapunov exponent.

Dynamics confined within the Cartan subalgebra, modeling quantum fluctuations.

Abstract

A generalization of the Lorenz equations is proposed where the variables take values in a Lie algebra. The finite dimensionality of the representation encodes the quantum fluctuations, while the non-linear nature of the equations can describe chaotic fluctuations. We identify a criterion, for the appearance of such non-linear terms. This depends on whether an invariant, symmetric tensor of the algebra can vanish or not. This proposal is studied in detail for the fundamental representation of $\mathfrak{u}(2)$. We find a knotted structure for the attractor, a bimodal distribution for the largest Lyapunov exponent and that the dynamics takes place within the Cartan subalgebra, that does not contain only the identity matrix, thereby can describe the quantum fluctuations.