KCC-theory and geometry of the Rikitake system

Abstract

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).