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.

中文翻译：

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七阶广义 Hénon-Heiles 势收敛的分形盆地

本文旨在研究七阶广义 Hénon-Heiles 势中的平衡点和相关的收敛盆地。使用众所周知的 Newton-Raphson 迭代器，我们在数值上定位了平衡点的位置，同时我们还获得了它们的线性稳定性。此外，我们演示了进入广义 Hénon-Heiles 势的两个变量参数如何影响系统的收敛动力学以及盆地图的分形度。分形度是通过计算（边界）盆地熵以及不确定性维数得出的。