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Efficient computational approaches to obtain periodic orbits in Hamiltonian systems: application to the motion of a lunar orbiter

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Abstract

In this paper, we study the problem of computing periodic orbits of Hamiltonian systems providing large families of such orbits. Periodic orbits constitute one of the most important invariants of a system, and this paper provides a comprehensive analysis of two efficient computational approaches for Hamiltonian systems. First, a new version of the grid search method, applied to problems with three degrees of freedom, has been considered to find, systematically, symmetric periodic orbits. To obtain non-symmetric periodic orbits, we use a modification of an optimization method based on an evolutionary strategy. Both methods require a great computational effort to find a big number of periodic orbits, and we apply parallelization tools to reduce the CPU time. Finally, we present a strategy to provide initial conditions of the periodic orbits with arbitrary precision. We apply all these algorithms to the problem of the motion of the lunar orbiter referred to the rotating reference frame of the Moon. The periodic orbits of this problem are very useful from the space engineering point of view because they provide low-cost orbits.

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  1. http://sourceforge.net/projects/tidesodes/

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Acknowledgments

A. Dena thanks Prof. G. Lord for his hospitality during her stay at the Heriot-Watt University, Edinburgh. The work of A. Dena was carried out under the HPC-EUROPA2 project (project number: 228398) with the support of the European Commission - Capacities Area - Research Infrastructures. Besides, this work made use of the facilities of HECToR, the UK’s national high-performance computing service, which is provided by UoE HPCx Ltd at the University of Edinburgh, Cray Inc and NAG Ltd, and funded by the Office of Science and Technology through EPSRC’s High-End Computing Programme. The authors A. Abad, R. Barrio and A. Dena were partially supported by the Spanish Research project MTM2012-31883.

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Authors and Affiliations

  1. Centro Universitario de la Defensa Zaragoza, Zaragoza, Spain

    Ángeles Dena

  2. Grupo de Mecánica Espacial, University of Zaragoza, Zaragoza, Spain

    Alberto Abad

  3. Departmento Matemática Aplicada, University of Zaragoza, Zaragoza, Spain

    Roberto Barrio

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  1. Ángeles Dena
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  2. Alberto Abad
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  3. Roberto Barrio
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Correspondence to Ángeles Dena.

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Dena, Á., Abad, A. & Barrio, R. Efficient computational approaches to obtain periodic orbits in Hamiltonian systems: application to the motion of a lunar orbiter. Celest Mech Dyn Astr 124, 51–71 (2016). https://doi.org/10.1007/s10569-015-9651-2

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