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

## TL;DR

This paper investigates how the magnetic moment ratio affects the Newton-Raphson basins of attraction in a binary system with magnetic primaries, revealing complex and fractal basin structures.

## Contribution

It introduces a systematic numerical analysis of the influence of magnetic moment ratio on attraction basins in the electromagnetic Copenhagen problem.

## Key findings

- The magnetic moment ratio significantly alters basin shapes and fractality.
- The number of iterations correlates with basin geometry.
- The ratio λ strongly influences the stability and geometry of equilibrium points.

## 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 $\lambda$ 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 $\lambda$ influences the shape, the geometry and also the degree of fractality of the attracting domains. Our numerical results strongly indicate that the ratio $\lambda$ is indeed a very influential parameter in the electromagnetic binary system.