Analytical Approximate Solutions of a Magnetic Spherical Pendulum: Stability Analysis

Abstract

Abstract Purpose Under certain conditions, the governing equation of motion of magnetic spherical pendulum results in a cubic-quintic Duffing equation. The current work aims to achieve an analytical bounded procedure of this equation. Methods This may be accomplished by grouping nonlinear expanded frequency, Homotopy perturbation method (HPM), and Laplace transforms. Therefore, this technique helps disregard the appearance of the source of secular terms. Results To validate the obtained explanation, based on the method of Runge–Kutta of the fourth order (RK4), the numerical calculation is performed. On the other hand, the linearized stability analysis is carried out to explore stability neighbouring the fixed points. Moreover, the time history of the attained solution and the corresponding phase plane plots are obtained to expose the influence of the affecting factors in the behavior of motion. Conclusions A comparison between both solutions gives a good matching between them, which explores the worthy accuracy of the approach in question. Several phase portraits are planned toward illustrating the different types of stability and instability near the equilibrium points, where the relation between the expanded and the cyclotron frequency (that are generated by the magnetic field) is characterized for diverse standards of the azimuthal angular velocity.