Basins of convergence of equilibrium points in the generalized Henon-Heiles system

TL;DR

This study investigates how the basins of convergence for equilibrium points in the generalized Hénon-Heiles system evolve with perturbations, revealing their geometry, stability, and convergence properties using numerical methods.

Contribution

It provides a systematic numerical analysis of the Newton-Raphson basins of convergence and their dependence on the perturbation parameter in the GHH system.

Findings

Attracting regions are mapped on the (x,y) plane.

The geometry and basin entropy of convergence regions depend on the perturbation parameter.

Number of iterations required varies with the basin regions.

Abstract

We numerically explore the Newton-Raphson basins of convergence, related to the libration points (which act as attractors of the convergence process), in the generalized H\'{e}non-Heiles system (GHH). The evolution of the position as well as of the linear stability of the equilibrium points is determined, as a function of the value of the perturbation parameter. The attracting regions, on the configuration $(x,y)$ plane, are revealed by using the multivariate version of the classical Newton-Raphson iterative algorithm. We perform a systematic investigation in an attempt to understand how the perturbation parameter affects the geometry as well as of the basin entropy of the attracting domains. The convergence regions are also related with the required number of iterations.