Lyapunov spectra instability of chaotic dynamo Ricci flows in twisted magnetic flux tubes

TL;DR

This paper applies Ricci flow and Riemannian geometry to analyze the Lyapunov spectra of chaotic dynamo flows in twisted magnetic flux tubes, revealing conditions for chaos and potential for laboratory dynamo modeling.

Contribution

It introduces a geometric framework using Ricci flow to study Lyapunov spectra in twisted magnetic flux tubes, linking curvature to chaos onset.

Findings

Positive Lyapunov exponent proportional to radial flow indicates chaos.

Constant sectional Ricci curvature enables geodesic deviation analysis.

Chaotic behavior is guaranteed by the positive Lyapunov exponent.

Abstract

Previously Casetti, Clementi and Pettini [\textbf{Phys.Rev.E \textbf{54},6,(1996)}] have investigated the Lyapunov spectrum of Hamiltonian flows for several Hamiltonian systems by making use of the Riemannian geometry. Basically the Lyapunov stability analysis was substituted by the Ricci sectional curvature analysis. In this report we apply Pettini's geometrical framework to determine the potential energy of a twisted magnetic flux tube, from its curved Riemannian geometry. Actually the Lyapunov exponents, are connected to a Riemann metric tensor, of the twisted magnetic flux tubes (MFTs). The Hamiltonian flow inside the tube is actually given by Perelman Ricci flows constraints in twisted magnetic flux tubes, where the Lyapunov eigenvalue spectra for the Ricci tensor associated with the Ricci flow equation in MFTs leads to a finite-time Lyapunov exponential stretching along the toroidal direction of the tube and a contraction along the radial direction of the tube. The Jacobi equation for the MFTs is shown to have a constant sectional Ricci curvature which allows us to compute the Jacobi-Levi-Civita (JLC) geodesic deviation for the spread of lines on the tube manifold and chaotic action through the greatest of its Lyapunov exponents. By analyzing the spectra of the twisted MFT, it is shown that the greater exponent is positive and proportional to the random radial flow of the tube, which allows the onset of chaos is guaranted. The randomness in the twisted flow reminds a discussed here is similar of a recent work by Shukurov, Stepanov, and Sokoloff on dynamo action on Moebius flow [\textbf{Phys Rev E 78 (2008)}]. The dynamo action in twisted flux tubes discussed here may also serve as model for dynamo experiments in laboratory.