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

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