Comparing the basins of attraction for several methods in the circular Sitnikov problem with spheroid primaries

TL;DR

This study compares various numerical methods in analyzing the basins of attraction in the circular Sitnikov problem with spheroid primaries, revealing differences in convergence structures and efficiency across methods.

Contribution

It provides a systematic comparison of multiple iterative schemes in the context of the Sitnikov problem with spheroid primaries, highlighting their convergence behaviors and fractal basin boundaries.

Findings

Most methods show similar convergence structures.

Some methods have highly fractal basin boundaries.

Efficiency varies significantly between methods.

Abstract

The circular Sitnikov problem, where the two primary bodies are prolate or oblate spheroids, is numerically investigated. In particular, the basins of convergence on the complex plane are revealed by using a large collection of numerical methods of several order. We consider four cases, regarding the value of the oblateness coefficient which determines the nature of the roots (attractors) of the system. For all cases we use the iterative schemes for performing a thorough and systematic classification of the nodes on the complex plane. The distribution of the iterations as well as the probability and their correlations with the corresponding basins of convergence are also discussed. Our numerical computations indicate that most of the iterative schemes provide relatively similar convergence structures on the complex plane. However, there are some numerical methods for which the corresponding basins of attraction are extremely complicated with highly fractal basin boundaries. Moreover, it is proved that the efficiency strongly varies between the numerical methods.