# Revealing the basins of convergence in the planar equilateral restricted four-body problem

## TL;DR

This study investigates how the mass parameter affects the shape and fractality of convergence basins in the planar equilateral restricted four-body problem, revealing the influence of system parameters on dynamical behavior.

## Contribution

It provides a systematic numerical analysis of the Newton-Raphson basins of convergence and their dependence on the mass parameter in this specific four-body problem.

## Key findings

- Mass parameter significantly influences basin shape and fractality.
- Convergence regions vary with changes in $m_3$.
- Attracting domains correlate with iteration counts.

## Abstract

The planar equilateral restricted four-body problem where two of the primaries have equal masses is used in order to determine the Newton-Raphson basins of convergence associated with the equilibrium points. The parametric variation of the position of the libration points is monitored when the value of the mass parameter $m_3$ varies 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 dynamical parameter $m_3$ influences the shape, the geometry and the degree of fractality of the converging regions. Our numerical outcomes strongly indicate that the mass parameter is indeed one of the most influential factors in this dynamical system.