Basins of convergence in the circular Sitnikov four-body problem with non-spherical primaries

TL;DR

This study explores how the shape of non-spherical primaries influences the convergence regions of the Newton-Raphson method in the Sitnikov four-body problem, revealing complex basin structures and their dependence on oblateness.

Contribution

It provides a detailed numerical analysis of the basins of convergence in the Sitnikov four-body problem with non-spherical primaries, highlighting the impact of oblateness on basin geometry and entropy.

Findings

Oblateness significantly alters basin structures.

Convergence regions depend on initial conditions and oblateness.

Basin entropy varies with the oblateness coefficient.

Abstract

The Newton-Raphson basins of convergence, related to the equilibrium points, in the Sitnikov four-body problem with non-spherical primaries are numerically investigated. We monitor the parametric evolution of the positions of the roots, as a function of the oblateness coefficient. The classical Newton-Raphson optimal method is used for revealing the basins of convergence, by classifying dense grids of initial conditions in several types of two-dimensional planes. We perform a systematic and thorough analysis in an attempt to understand how the oblateness coefficient affects the geometry as well as the basin entropy of the convergence regions. The convergence areas are related with the required number of iterations and also with the corresponding probability distributions.