扭曲磁通量管中混沌Ricci发电机流的李雅普诺夫谱不稳定性
Lyapunov spectra instability of chaotic dynamo Ricci flows in twisted magnetic flux tubes
摘要：
以前Casetti，Clementi和Pettini [ \textbf{Phys.Rev.E \textbf{54},6,(1996)} ]利用黎曼几何研究了几个Hamilton系统的Hamilton流的李雅普诺夫谱。基本上，李雅普诺夫稳定性分析被取代的里奇截面曲率分析。在这份报告中，我们应用Pettini的几何框架，以确定一个扭曲的磁通量管的势能，从其弯曲的黎曼几何。实际上，扭曲磁通量管的李雅普诺夫指数与黎曼度量张量有关。管内的哈密顿流实际上是由扭曲的磁通量管中的Perelman Ricci流约束给出的，其中与MFTs中的Ricci流方程相关联的Ricci张量的Lyapunov本征值谱导致沿着管的环形方向的有限时间Lyapunov指数拉伸和沿着管的径向方向的收缩。的MFTs的Jacobi方程示出有一个恒定的截面Ricci曲率，使我们能够计算的Jacobi-Levi-Civita（JLC）测地偏差的传播线管歧管和混沌行动，通过其最大的李雅普诺夫指数。通过对扭曲MFT的频谱分析，发现其大指数为正，且与管内的随机径向流成正比，从而抑制了混沌的发生。扭曲流的随机性使人想起这里讨论的类似于Shukurov、Stepanov和Sokoloff最近关于Moebius流发电机作用的工作[ \textbf{Phys Rev E 78 (2008)} ]。本文所讨论的扭曲磁通管中的发电机作用也可作为实验室发电机实验的模型。
英文摘要：
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.
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