Determining the Newton-Raphson basins of attraction in the electromagnetic Copenhagen problem

Abstract

The Copenhagen problem where the primaries of equal masses are magnetic dipoles is used in order to determine the Newton-Raphson basins of attraction associated with the equilibrium points. The parametric variation of the position as well as of the stability of the Lagrange points are monitored when the value of the ratio λ of the magnetic moments varies in predefined intervals. The regions on the configuration (x,y) plane occupied by the basins of convergence are revealed using the multivariate version of the Newton-Raphson iterative scheme. The correlations between the basins of attraction of the libration points and the corresponding number of iterations needed for obtaining the desired accuracy are also illustrated. We perform a thorough and systematic numerical investigation by demonstrating how the dynamical quantity λ influences the shape, the geometry and also the degree of fractality of the attracting domains. Our numerical results strongly indicate that the ratio λ is indeed a very influential parameter in the electromagnetic binary system.