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Analytical and numerical manifolds in a symplectic 4-D map

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Abstract

We study analytically the orbits along the asymptotic manifolds from a complex unstable periodic orbit in a symplectic 4-D Froeschlé map. The orbits are given as convergent series. We compare the analytic results by truncating the series at various orders with the corresponding numerical results and we find agreement along a more extended length, as the order of truncation increases. The agreement is improved when the parameters approach those of the stability domain. Along the manifolds no terms with small divisors appear in the series. The same result is found if we use a parametrization method along the asymptotic curves. In the case of orbits starting close to the manifolds small divisors appear, but the orbits remain close to the manifolds for an extended period of time. If the parameters of the map are close to the stable domain the orbits recede and approach the origin several times and remain confined in a certain volume around the origin for a long time before escaping to large distances. For special sets of parameters we see resonance phenomena and the orbits take particular forms near every resonance.

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Notes

  1. The original map (Froeschlé and Gonczi 1980) has \(L_1=L_2\).

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Acknowledgments

This research is partially supported by the Research Committee of the Academy of Athens through the Project 200/815.

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  1. Research Center for Astronomy and Applied Mathematics of the Academy of Athens, Soranou Efessiou 4, 11527, Athens, Greece

    N. Delis & G. Contopoulos

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  1. N. Delis
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  2. G. Contopoulos
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Correspondence to N. Delis.

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Delis, N., Contopoulos, G. Analytical and numerical manifolds in a symplectic 4-D map. Celest Mech Dyn Astr 126, 313–337 (2016). https://doi.org/10.1007/s10569-016-9697-9

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