KCC-theory and geometry of the Rikitake system
KCC-theory and geometry of the Rikitake system
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DOI：
10.1088/1751-8113/40/11/011
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发表时间：
2007-02
期刊：
影响因子：
--
通讯作者：
T. Yajima;H. Nagahama
中科院分区：
文献类型：
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作者：
T. Yajima;H. Nagahama
地球磁场经历非周期性反转。这些可以通过简单的两盘发电机系统（Rikitake 系统）来解释。在本文中，基于微分几何（Kosambi-Cartan-Chern 理论）研究了 Rikitake 系统。运动的电气和机械方程源自法拉第定律以及磁流体动力学方程。从几何理论来看，Rikitake系统的解可以看作是切丛上的一条轨迹。因此，Rikitake 系统中存在五个几何不变量。作为扭转张量的第三个不变量可以用互感来表示，互感是引起非周期性磁反转的电气和机械相互作用的结果。这种非周期性行为对应于拓扑不变量（例如表达环形流和极向流之间相互作用的陈-西蒙斯数）的磁流体动力湍流运动。该 Rikitake 系统与其他非线性动力系统等效。因此，各种非线性动力系统的混沌行为可以通过五个几何不变量和拓扑不变量（陈-西蒙斯数）来统一研究。
The Earth's magnetic field undergoes aperiodical reversals. These can be explained by a simple two-disc dynamo system (Rikitake system). In this paper, the Rikitake system is studied based on a differential geometry (theory of Kosambi–Cartan–Chern). The electrical and mechanical equations of motion are derived from Faraday's law as well as from magnetohydrodynamic equations. From the geometric theory, the solution of the Rikitake system can be regarded as a trajectory on the tangent bundle. Accordingly, there exist five geometrical invariants in the Rikitake system. The third invariant as a torsion tensor can be expressed by mutual-inductances as a result of electrical and mechanical interactions which cause the aperiodic magnetic reversal. This aperiodic behaviour corresponds to a magnetohydrodynamic turbulent motion by a topological invariant such as Chern–Simons number which expresses the interaction between the toroidal and poloidal currents. This Rikitake system is equivalent to other nonlinear dynamical systems. Thus, chaotic behaviours of various nonlinear dynamical systems can be uniformly investigated by the five geometrical invariants and the topological invariant (the Chern–Simons number).