Jacobi stability and aperiodicity of Rikitake-Hide model based on KCC-theory

Abstract

The fluctuations of sunspots as an index of solar activity have an 11-year cycle and secular variation with grand minima. The basic features of fluctuations of sunspots are reproduced as fluctuations of magnetic energy using the Rikitake-Hide model, a coupled Faraday disks model combining the Rikitake and Hide models. This study discusses the Jacobi stability and aperiodicity of the Rikitake-Hide model using the second and third invariants in the KCC (Kosambi–Cartan–Chern) theory. The second and third KCC-invariants provide the Jacobi stability and aperiodicity (discrepancies) in the system and are expressed by the electric current and angle velocities in the model. From the calculations from the KCC-invariants, the Rikitake model that accounts for most of the magnetic energy is Jacobi unstable when the magnetic energy takes local minima. The aperiodicity of the electric–current trajectory in the Rikitake model does not accumulate over a long time, and the cycle behavior of magnetic energy basically persists because of the symmetric change in the electromagnetic field of the model. Meanwhile, the aperiodicity due to the Hide model intermittently reduces the amplitude of the magnetic energy and increases its cycle length. Especially, large aperiodicity due to the Hide model as a disturbance makes the trajectory in the Rikitake model take an unusual path at local minima that is Jacobi unstable, causing large-scale grand minima of magnetic energy without reversal of the magnetic field. Therefore, the Jacobi stability and aperiodicity are key factors in the behavior of the magnetic energy in the Rikitake-Hide model.