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Stability and bifurcations of even periodic orbits in the Sitnikov problem

  • Jorge Galán-VioqueEmail author
  • Daniel Nuñez
  • Andrés Rivera
  • Camila Riccio
Original Article
  • 88 Downloads
Part of the following topical collections:
  1. Recent advances in the study of the dynamics of N-body problem

Abstract

We study different families of even periodic solutions in the classical Sitnikov problem that emanate from the circular case as the eccentricity is increased. The families can be classified by the number N of full revolutions of the primaries and labelled by the number of zeroes p of the vertical coordinate of the massless body in half a period. We give a linear stability criterion of these branches depending on even N, based on the sign for the initial slope of the discriminant function for the associated Hill’s equation, in a computable interval of eccentricities. All families for \(N=2\) are linearly stable for small and computable e. The results show a fundamental symmetry-driven difference between the even and odd N cases.

Keywords

Periodic orbits Sitnikov problem Numerical continuation 

Mathematics Subject Classification

70F15 34B15 37G15 37N05 

Notes

Acknowledgements

The authors acknowledge fruitful discussions with Rafael Ortega on related topics. JGV’s research has been financially supported by the Spanish Ministry of Economy through Grant MTM2015-65608-P and Junta de Andalucía Excellence Grant P12-FQM-1658. The authors D. Nuñez and A. Rivera have been financially supported by the Capital Semilla (2015–2016) project 020100480.

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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Applied Mathematics, Escuela Técnica Superior de IngenieríaUniversidad de SevillaSevilleSpain
  2. 2.Pontificia Universidad Javeriana-CaliCaliColombia

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