Basins of convergence of equilibrium points in the pseudo-Newtonian planar circular restricted three-body problem

TL;DR

This study explores how the basins of convergence for equilibrium points in a pseudo-Newtonian three-body system evolve with a transition parameter, revealing complex geometries and iteration behaviors through extensive numerical analysis.

Contribution

It provides a detailed numerical investigation of the basins of attraction in a pseudo-Newtonian three-body problem, highlighting the influence of the transition parameter on their geometry and stability.

Findings

Basins of attraction exhibit complex, evolving geometries as the transition parameter varies.

The number of iterations required for convergence correlates with basin structures.

The evolution of basins is highly sensitive and dynamically intricate.

Abstract

The Newton-Raphson basins of attraction, associated with the libration points (attractors), are revealed in the pseudo-Newtonian planar circular restricted three-body problem, where the primaries have equal masses. The parametric variation of the position as well as of the stability of the equilibrium points is determined, when the value of the transition parameter $\epsilon$ varies in the interval $[0,1]$. The multivariate Newton-Raphson iterative scheme is used to determine the attracting domains on several types of two-dimensional planes. A systematic and thorough numerical investigation is performed in order to demonstrate the influence of the transition parameter on the geometry of the basins of convergence. The correlations between the basins of attraction and the corresponding required number of iterations are also illustrated and discussed. Our numerical analysis strongly indicates that the evolution of the attracting regions in this dynamical system is an extremely complicated yet very interesting issue.