Fractal Basins of Convergence of a Seventh-Order Generalized Hénon–Heiles Potential

Abstract

This article aims to investigate the points of equilibrium and the associated convergence basins in a seventh-order generalized Hénon–Heiles potential. Using the well-known Newton–Raphson iterator, we numerically locate the positions of the points of equilibrium, while we also obtain their linear stability. Furthermore, we demonstrate how the two variable parameters, entering the generalized Hénon–Heiles potential, affect the convergence dynamics of the system as well as the fractal degree of the basin diagrams. The fractal degree is derived by computing the (boundary) basin entropy as well as the uncertainty dimension.