Classifying orbits in the classical Henon-Heiles Hamiltonian system

TL;DR

This study investigates how the total energy influences orbit types in the Henon-Heiles Hamiltonian system, revealing the transition from regular to chaotic motion and identifying various orbit families through numerical methods.

Contribution

It provides a comprehensive numerical analysis of orbit classification in the Henon-Heiles system across different energy levels, using SALI and spectral dynamics methods.

Findings

Low energy levels exhibit entirely regular motion with box orbits.

Increasing energy introduces chaos and multiple resonant orbit families.

Most resonant orbits are bifurcations from the 1:1 resonant family.

Abstract

The H\'{e}non-Heiles potential is undoubtedly one of the most simple, classical and characteristic Hamiltonian systems. The aim of this work is to reveal the influence of the value of the total orbital energy, which is the only parameter of the system, on the different families of orbits, by monitoring how the percentage of chaotic orbits, as well as the percentages of orbits composing the main regular families evolve when energy varies. In particular, we conduct a thorough numerical investigation distinguishing between ordered and chaotic orbits, considering only bounded motion for several energy levels. The smaller alignment index (SALI) was computed by numerically integrating the equations of motion as well as the variational equations to extensive samples of orbits in order to distinguish safely between ordered and chaotic motion. In addition, a method based on the concept of spectral dynamics that utilizes the Fourier transform of the time series of each coordinate is used to identify the various families of regular orbits and also to recognize the secondary resonances that bifurcate from them. Our exploration takes place both in the physical $(x,y)$ and the phase $(y,\dot{y})$ space for a better understanding of the orbital properties of the system. It was found, that for low energy levels the motion is entirely regular being the box orbits the most populated family, while as the value of the energy increases chaos and several resonant families appear. We also observed, that the vast majority of the resonant orbits belong in fact in bifurcated families of the main 1:1 resonant family. We have also compared our results with previous similar outcomes obtained using different chaos indicators.