The study of Newton–Raphson basins of convergence in the three-dipole problem

We consider a system in which the charged particle orbits under the influence of the electromagnetic field of three dipoles located on a system of three celestial bodies. Using well-known bivariate iterative scheme, known as Newton–Raphson (NR) iterative scheme, we numerically evaluated the positions of the stationary points (SPs) or equilibrium points (EPs) or libration points (LPs) and the linked basins of convergence (BoCs), and we also evaluated their linear stability. Moreover, we unveiled how the parameters, entering the effective potential function, affect the convergence dynamics of the system. Moreover, we also unveiled how the involved parameters affect the geometry of the zero velocity curves (ZVCs). Further, the correlation with the required number of iterations and the regions of convergence as well as the probability distributions associated to the BoCs is illustrated. In order to quantify the degree of final-state uncertainty of the BoCs, the basin entropy (BE) and for the fractality of boundaries of BoCs, the boundary basin entropy (BBE) are computed.