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

To understand the mechanisms underlying the magnetic fields of planets, including the Earth and the Sun, previous studies have proposed disk dynamo models as a suitable reduction of the mean-field dynamo equations. One of these is the Rikitake-Hide model, which combines the Rikitake model (a two-disk model for geomagnetic reversal) and the Hide model (a disk model with a motor and mechanical friction). This model reproduces nearly equal intervals of magnetic field reversals with modulated cycles, resembling the fluctuations of sunspots. In this presentation, we discuss the Jacobi stability and aperiodicity of the Rikitake-Hide model using geometrical invariants in the KCC (Kosambi–Cartan–Chern) theory. In the KCC theory, the second and third KCC invariants are related to the Jacobi stability and aperiodicity of trajectories in the system, expressed through variables (electrical currents and angular velocities) and parameters. By calculating the Jacobi stability of the Rikitake-Hide model using the second KCC invariant, we find that the model is Jacobi unstable when fluctuations in magnetic energy (square of the electric current) reach local minima. The instability at local minima manifests as branches in the trajectories of electric currents in the model. Based on the third KCC invariant for the trajectories of electric currents in the Rikitake-Hide model, the aperiodicity of the model may arise from individual electric currents. Although the model is always accompanied by aperiodicity, nearly equal intervals of magnetic field reversals are preserved through the cancellation of aperiodicity by symmetric cyclical fluctuations of electric currents. On the other hand, asymmetric electric currents originating from the motor and mechanical friction in the model alter the periods of magnetic field intervals. Finally, we consider the correspondence between the fluctuations of magnetic energy in the Rikitake-Hide model and sunspots on the Sun and discuss the implications for solar activity.