Semi-Analytical Closed-Form Solutions for the Rikitake-Type System through the Optimal Homotopy Perturbation Method

The goal of this work is to build semi-analytical solutions of the Rikitake-type system by means of the optimal homotopy perturbation method (OHPM) using only two iterations. The chaotic behaviors are excepted. By taking into consideration the geometrical properties of the Rikitake-type system, the closed-form solutions can be established. The obtained solutions have a periodical behavior. These geometrical properties allow reducing the initial system to a second-order nonlinear differential equation. The latter equation is solved analytically using the OHPM procedure. The validation of the OHPM method is presented for three cases of the physical parameters. The advantages of the OHPM technique, such as the small number of iterations (the efficiency), the convergence control (in the sense that the semi-analytical solutions are approaching the exact solution), and the writing of the solutions in an effective form, are shown graphically and with tables. The accuracy of the results provides good agreement between the analytical and corresponding numerical results. Other dynamic systems with similar geometrical properties could be successfully solved using the same procedure.