# Unveiling the basins of convergence in the pseudo-Newtonian planar circular restricted four-body problem

## TL;DR

This paper investigates how the transition parameter affects the convergence basins and stability of libration points in the pseudo-Newtonian four-body problem with equal masses, revealing complex dynamical behaviors.

## Contribution

It introduces a multivariate Newton-Raphson scheme to analyze basins of attraction and systematically studies the influence of the transition parameter on convergence topology.

## Key findings

- Transition parameter significantly alters libration point stability.
- Basins of attraction exhibit complex, evolving structures.
- Number of iterations correlates with basin topology.

## Abstract

The dynamics of the pseudo-Newtonian restricted four-body problem has been studied in the present paper, where the primaries have equal masses. The parametric variation of the existence as well as the position of the libration points are determined, when the value of the transition parameter $\epsilon \in [0, 1]$. The stability of these libration points has also been discussed. Our study reveals that the Jacobi constant as well as transition parameter $\epsilon$ have substantial effect on the regions of possible motion, where the fourth body is free to move. The multivariate version of Newton-Raphson iterative scheme is introduced for determining the basins of attraction in the configuration $(x,y)$ plane. A systematic numerical investigation is executed to reveal the influence of the transition parameter on the topology of the basins of convergence. In parallel, the required number of iterations is also noted to show its correlations to the corresponding basins of convergence. It is unveiled that the evolution of the attracting regions in the pseudo-Newtonian restricted four-body problem is a highly complicated yet worth studying problem.