Dynamical analysis of bounded and unbounded orbits in a generalized H\'enon-Heiles system

TL;DR

This paper introduces a generalized Hénon-Heiles system derived from a fifth-order potential expansion, analyzing its chaotic and regular behaviors in bounded and unbounded regimes using qualitative and quantitative methods.

Contribution

It presents a new generalized potential system and compares its dynamical behavior to the classic Hénon-Heiles system, revealing wider regular regions despite higher order terms.

Findings

Chaoticity decreases as energy moves away from critical energy.

The generalized system exhibits larger regular islands than the original Hénon-Heiles system.

Wider zones of regularity are observed despite higher order terms.

Abstract

The H\'enon-Heiles potential was first proposed as a simplified version of the gravitational potential experimented by a star in the presence of a galactic center. Currently, this system is considered a paradigm in dynamical systems because despite its simplicity exhibits a very complex dynamical behavior. In the present paper, we perform a series expansion up to the fifth-order of a potential with axial and reflection symmetries, which after some transformations, leads to a generalized H\'enon-Heiles potential. Such new system is analyzed qualitatively in both regimes of bounded and unbounded motion via the Poincar\'e sections method and plotting the exit basins. On the other hand, the quantitative analysis is performed through the Lyapunov exponents and the basin entropy, respectively. We find that in both regimes the chaoticity of the system decreases as long as the test particle energy gets far from the critical energy. Additionally, we may conclude that despite the inclusion of higher order terms in the series expansion, the new system shows wider zones of regularity (islands) than the ones present in the H\'enon-Heiles system.