Long time evolution of the H\'enon-Heiles system for small energy

TL;DR

This paper investigates the long-term behavior of the Hénon-Heiles system at small energies, deriving asymptotic invariants that persist over extended timescales and enhance understanding of its complex dynamics.

Contribution

It introduces a novel analytical framework to derive long-lived invariants for the Hénon-Heiles system at low energies, surpassing traditional perturbation methods.

Findings

Derived asymptotic constants of motion valid over long timescales

Excellent agreement between formulas and numerical simulations

Enhanced understanding of the system's long-term dynamical behavior

Abstract

The H\'enon-Heiles system, initially introduced as a simplified model of galactic dynamics, has become a paradigmatic example in the study of nonlinear systems. Despite its simplicity, it exhibits remarkably rich dynamical behavior, including the interplay between regular and chaotic orbital dynamics, resonances, and stochastic regions in phase space, which have inspired extensive research in nonlinear dynamics. In this work, we investigate the system's solutions at small energy levels, deriving asymptotic constants of motion that remain valid over remarkably long timescales -- far exceeding the range of validity of conventional perturbation techniques. Our approach leverages the system's inherent two-scale dynamics, employing a novel analytical framework to uncover these long-lived invariants. The derived formulas exhibit excellent agreement with numerical simulations, providing a deeper understanding of the system's long-term behavior.