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Classifying orbits in the classical Hénon–Heiles Hamiltonian system

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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 \((x,y)\) 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.

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Notes

  1. If \(S\) is the \(2N\) dimensional phase space (four dimensional, in our case) where the orbits of a dynamical system evolve on, then a deviation vector \(\mathbf{w}\), which describes a small perturbation of a specific orbit \(\mathbf{x}\), evolves on a \(2N\) dimensional space \(T_{x}S\) tangent to \(S\).

  2. Generally, dynamical methods are broadly split into two types: (i) those based on the evolution of sets of deviation vectors in order to characterize an orbit and (ii) those based on the frequencies of the orbits which extract information about the nature of motion only through the basic orbital elements without the use of deviation vectors.

  3. For every orbital family there is a parent (or mother) periodic orbit, that is, an orbit that describes a closed figure. Perturbing the initial conditions which define the exact position of a periodic orbit we generate quasi-periodic orbits that belong to the same orbital family and librate around their closed parent periodic orbit.

  4. It is known that in dynamical systems with a finite energy of escape where escape is possible (open Hamiltonian systems), chaos appears and dominates at values of energy very close to the critical escape energy which determines the transition from bounded to unbounded motion.

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Acknowledgments

The author would like to express his warmest thanks to all the anonymous referees for the careful reading of the manuscript and for all the apt suggestions and comments which allowed us to improve both the quality and the clarity of the paper.

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  1. Department of Physics, School of Science, Aristotle University of Thessaloniki, 541 24, Thessaloniki, Greece

    Euaggelos E. Zotos

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Zotos, E.E. Classifying orbits in the classical Hénon–Heiles Hamiltonian system. Nonlinear Dyn 79, 1665–1677 (2015). https://doi.org/10.1007/s11071-014-1766-6

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