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Elucidating the escape dynamics of the four hill potential

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Abstract

The escape mechanism of the four hill potential is explored. A thorough numerical investigation takes place in several types of two-dimensional planes and also in a three-dimensional subspace of the entire four-dimensional phase space in order to distinguish between bounded (ordered and chaotic) and escaping orbits. The determination of the location of the basins of escape toward the different escape channels and their correlations with the corresponding escape time of the orbits is undoubtedly an issue of paramount importance. It was found that in all examined cases all initial conditions correspond to escaping orbits, while there is no numerical indication of stable bounded motion, apart from some isolated unstable periodic orbits. Furthermore, we monitor how the fractality evolves when the total orbital energy varies. The larger escape periods have been measured for orbits with initial conditions in the fractal basin boundaries, while the lowest escape rates belong to orbits with initial conditions inside the basins of escape. We hope that our numerical analysis will be useful for a further understanding of the escape dynamics of orbits in open Hamiltonian systems with two degrees of freedom.

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Notes

  1. A sticky orbit is a special type of orbit which behaves as a regular one for long time intervals before it exhibits its true chaotic nature.

  2. The phenomenon of almost zero fractality, at the basin boundaries, at relatively high energy levels seems to be common in several types of Hamiltonian systems. For example, the same behavior is also observed in the Hénon-Heiles system but only for high enough values of the total orbital energy \((E > 2)\).

  3. The dimension of the complete phase space of a Hamiltonian system with N degrees of freedom is 2N. This phase space is foliated into \(2N-1\)-dimensional invariant leaves corresponding to the numerical values of the Hamiltonian. However, the dimension of the entire phase space always remains 2N, irrelevantly of any possible invariant foliation.

  4. The same phenomenon of the loss of the symmetry of a dynamical system due to the particular choice of the initial conditions of the orbits also applies for other types of Hamiltonian systems. For example, in the Hénon–Heiles system the \(2\pi /3\) symmetry is observable only in the configuration (x, y) space and only by using polar coordinates. On the other hand, for initial conditions of orbits in the \((y,\dot{y})\) phase space the three escape channels are no longer equiprobable (see, e.g., the series of papers on the Hénon-Heiles system by Sanjuán and collaborators).

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Acknowledgements

I would like to gratefully and sincerely thank Dr. Christof Jung for all the illuminating discussions during this research. My warmest thanks also go to the two anonymous referees for the careful reading of the manuscript as well as for all the apt suggestions and comments which allowed us to improve both the quality and the clarity of the paper.

Funding The author states that he has not received any research grants.

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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. Elucidating the escape dynamics of the four hill potential. Nonlinear Dyn 89, 135–151 (2017). https://doi.org/10.1007/s11071-017-3441-1

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