# Revealing the Newton-Raphson basins of convergence in the circular pseudo-Newtonian Sitnikov problem

## TL;DR

This study investigates how the Newton-Raphson method's convergence behavior and basin structures vary in the pseudo-Newtonian Sitnikov problem with three and four primaries, highlighting the influence of a transition parameter.

## Contribution

It provides a systematic analysis of convergence basins, fractal boundaries, and basin entropy variations in the pseudo-Newtonian Sitnikov problem, which was not previously explored.

## Key findings

- Convergence basins are significantly affected by the transition parameter.
- Fractal basin boundaries are identified and characterized.
- Basin entropy varies systematically with the transition parameter.

## Abstract

In this paper we numerically explore the convergence properties of the pseudo-Newtonian circular restricted problem of three and four primary bodies. The classical Newton-Raphson iterative scheme is used for revealing the basins of convergence and their respective fractal basin boundaries on the complex plane. A thorough and systematic analysis is conducted in an attempt to determine the influence of the transition parameter on the convergence properties of the system. Additionally, the roots (numerical attractors) of the system and the basin entropy of the convergence diagrams are monitored as a function of the transition parameter, thus allowing us to extract useful conclusions. The probability distributions, as well as the distributions of the required number of iterations are also correlated with the corresponding basins of convergence.