# Equilibrium points and basins of convergence in the linear restricted four-body problem with angular velocity

## TL;DR

This study investigates how the parameters of mass distribution and angular velocity influence the equilibrium points and convergence basins in a linear restricted four-body problem, revealing their significant impact on the system's dynamics.

## Contribution

It provides a systematic numerical analysis of the effects of mass parameter and angular velocity on equilibrium points and their basins of attraction in the four-body problem.

## Key findings

- Parameters significantly affect the shape and fractality of convergence regions.
- The number of iterations correlates with the size of attraction basins.
- Both parameters are crucial in determining the system's dynamical behavior.

## Abstract

The planar linear restricted four-body problem is used in order to determine the Newton-Raphson basins of convergence associated with the equilibrium points. The parametric variation of the position as well as of the stability of the libration points is monitored when the values of the mass parameter $b$ as well as of the angular velocity $\omega$ vary in predefined intervals. The regions on the configuration $(x,y)$ plane occupied by the basins of attraction are revealed using the multivariate version of the Newton-Raphson iterative scheme. The correlations between the attracting domains of the equilibrium 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 parameters $b$ and $\omega$ influence the shape, the geometry and of course the fractality of the converging regions. Our numerical outcomes strongly indicate that these two parameters are indeed two of the most influential factors in this dynamical system.