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Title of Journal: Nonlinear Dyn

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Abbravation: Nonlinear Dynamics

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Springer Netherlands

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DOI

10.1007/bf02759469

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1573-269X

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Classifying orbits in the classical Hénon–Heiles H

Authors: Euaggelos E Zotos
Publish Date: 2014/10/31
Volume: 79, Issue: 3, Pages: 1665-1677
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Abstract

The Hénon–Heiles potential is undoubtedly one of the most simple classical and characteristic Hamiltonian systems The aim of this work was 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 xy and the phase ydoty 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 11 resonant family We have also compared our results with previous similar outcomes obtained using different chaos indicators

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