Analyzing the Effects of Fifth and Seventh Order Terms in a Generalized Henon-Heiles Potential

TL;DR

This work extends the classic Hénon–Heiles system by introducing a generalized potential with fifth- and seventh-order terms, controlled by parameters $\delta$ and $\alpha$, to study how higher-order contributions affect chaotic dynamics near the boundary between bounded and unbounded motion. The authors derive the generalized HHH Hamiltonian and analyze the system using equipotential-based energy thresholds, Poincaré sections, and maximal Lyapunov exponents to quantify chaos. Their key findings show that chaos strengthens as the energy approaches the critical threshold $E_{ ext{min}}$, while the fifth-order term ($\delta$) tends to regularize motion and the seventh-order term ($\alpha$) tends to enhance chaos; these trends are corroborated by Lyapunov statistics. The results illuminate how higher-order terms shape orbital structure in axisymmetric galactic potentials and provide a controlled framework for exploring chaos in generalized Henon–Heiles-type systems.

Abstract

In 1962, astronomers Michel Hénon and Carl Heiles studied orbits of stars around centers of galaxies to determine the third integral of motion in galactic dynamics. In order to do this, they reduced the system down to a 2-dimensional axisymmetric Hamiltonian system. Now this is known as the Hénon-Heiles (HH) System. Due to its apparent simplicity but extremely complicated dynamical behavior, this system is currently a paradigm in dynamical systems. In this paper, we perform a series expansion up to the seventh order of a potential with axial and reflection symmetries. After some transformations, this turns into the generalized Hénon-Heiles (GHH) system where we separate the fifth and seventh-order terms. We qualitatively analyze this system for energies near the threshold between bounded and unbounded motion with Poincaré sections and quantitatively analyze with Lyapunov Exponents. We find that particles far from the critical energy demonstrate less chaos. Additionally, the fifth-order term creates more regularity while the seventh-order term does the opposite.

Figures (12)

• Figure 1: HH-System Potential Landscape
• Figure 2: Contour plot at (a) $E=1/8$ and (b) $E=1/5$
• Figure 3: Poincaré sections of HH-System at $E=1/8$
• Figure 4: Contour Plot of HH Potential at $E=1/8$
• Figure 5: Transition from Bounded to Unbounded Motion