Basins of convergence of equilibrium points in the generalized Hill problem

TL;DR

This paper investigates how the basins of attraction for equilibrium points in the generalized Hill problem change with parameters, revealing complex geometries and stability properties using numerical methods.

Contribution

It provides a detailed numerical analysis of the basins of convergence and their dependence on the perturbation parameter in the generalized Hill problem.

Findings

Basins of attraction vary significantly with perturbation parameter.

The geometry of convergence domains is highly complex.

Basin entropy and iteration counts are correlated.

Abstract

The Newton-Raphson basins of attraction, associated with the libration points (attractors), are revealed in the generalized Hill problem. The parametric variation of the position and the linear stability of the equilibrium points is determined, when the value of the perturbation parameter $\epsilon$ varies. 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 perturbation parameter on the geometry as well as of the basin entropy 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.