Stability and bifurcations of even periodic orbits in the Sitnikov problem

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


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.


Periodic orbits Sitnikov problem Numerical continuation 

Mathematics Subject Classification

70F15 34B15 37G15 37N05 



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.


  1. Belbruno, E., Llibre, J., Ollé, M.: On the families of periodic orbits which bifurcate from the circular Sitnikov motions. Celest. Mech. Dyn. Astron. 60, 99–129 (1994)ADSMathSciNetCrossRefGoogle Scholar
  2. Corbera, M., Llibre, J.: Periodic orbits of the Sitnikov problem via a Poincaré map. Celest. Mech. Dyn. Astron. 77, 273–303 (2000)ADSCrossRefGoogle Scholar
  3. Corbera, M., Llibre, J.: On symmetric periodic orbits of the elliptic Sitnikov problem via the analytic continuation method. In: Chenciner, A., Cushman, R., Robinson, C., Jeff Xia, Z. (eds.) Celestial Mechanics. Contemporary Mathematics, vol. 292, pp. 91–127. AMS, Providence (2002)Google Scholar
  4. Doedel, E.J., Champneys, A.R., Fairgrieve, T.F., Kuznetsov, Y.A., Sandstede, B., Wang, X.J.: AUTO97: Continuation and Bifurcation Software for Ordinary Differential Equations (1997).
  5. Dvorak, R.: Numerical results to the Sitnikov problem. Celest. Mech. Dyn. Astron. 87, 71–80 (1993)ADSMathSciNetCrossRefGoogle Scholar
  6. Dormand, J.R., Prince, P.J.: A family of embedded Runge–Kutta formulae. J. Comput. Appl. Math. 6, 19–26 (1980)MathSciNetCrossRefGoogle Scholar
  7. Galán, J., Núñez, D., Rivera, A.: Quantitative stability of certain families of periodic solutions in the Sitnikov problem. SIAM J. App. Dyn. Syst. 17, 52–77 (2018)MathSciNetCrossRefGoogle Scholar
  8. Galán-Vioque, J., Muñoz-Almaraz, F.J., Freire, E., Freire, E.: Continuation of periodic orbits in symmetric hamiltonian and conservative sytems. Eur. Phys. J. Top. 223(13), 2705–2722 (2014)CrossRefGoogle Scholar
  9. Jury, E.I.: Inners and Stability of Dynamic Systems. Wiley, Hoboken (1975)CrossRefGoogle Scholar
  10. Jiménez-Lara, L., Escalona-Buendía, A.: Symmetries and bifurcations in the Sitnikov problem. Celest. Mech. Dyn. Astron. 79, 97–117 (2001)ADSMathSciNetCrossRefGoogle Scholar
  11. Libre, J., Simó, C.: Estudio cualitativo del problema de Sitnikov. Publicacions Mathemàtiques U.A.B 18, 49–71 (1980)CrossRefGoogle Scholar
  12. Llibre, J., Ortega, R.: On the families of periodic orbits of the Sitnikov problem. SIAM J. Appl. Dyn. Syst. 7, 561–576 (2008)ADSMathSciNetCrossRefGoogle Scholar
  13. MacMillian, W.D.: An integrable case in the restricted problem of three bodies. Astron. J. 27, 11–13 (1913)ADSCrossRefGoogle Scholar
  14. Magnus, W., Winkler, S.: Hill’s Equation. Dover, New York (1979)zbMATHGoogle Scholar
  15. Martínez-Alfaro, J., Chiralt, C.: Invariant rotational curves in Sitnikov’s problem. Celest. Mech. Dyn. Astron. 55, 351–367 (1993)ADSMathSciNetCrossRefGoogle Scholar
  16. Muñoz-Almaraz, F.J., Freire, E., Galán-Vioque, J., Vanderbauwhede, A.: Continuation of normal doubly symmetric orbits in conservative reversible systems. Celest. Mech. Dyn. Astron. 97(1), 17–47 (2007)ADSMathSciNetCrossRefGoogle Scholar
  17. Muñoz-Almaraz, F.J., Freire, E., Galán-Vioque, J., Doedel, E.J., Vanderbauwhede, A.: Continuation of periodic orbits in conservative Hamiltonian systems. Physica D 181, 1–38 (2003)ADSMathSciNetCrossRefGoogle Scholar
  18. Ortega, R., Rivera, A.: Global bifurcations from the center of mass in the Sitnikov problem. Discrete Contin. Dyn. Syst. Ser. B 14, 719–732 (2010)MathSciNetCrossRefGoogle Scholar
  19. Ortega, R.: The twist coefficient of periodic solutions of a time-dependent Newton’s equation. J. Dyn. Differ. Equ. 4, 651–665 (1992)MathSciNetCrossRefGoogle Scholar
  20. Ortega, R.: Periodic solutions of a newtonian equation: stability by the third approximation. J. Differ. Equ. 128, 491–518 (1996)ADSMathSciNetCrossRefGoogle Scholar
  21. Pavanini, G.: Sopra una nuova categoria di soluzioni periodiche nel problema di tre corpi. Annali di Mathematica, Serie III, Tomo XIII, pp. 179–202 (1907)Google Scholar
  22. Perdios, E., Markellos, V.V.: Stability and bifurcations of Sitnikov motions. Celest. Mech. Dyn. Astron. 42, 187–200 (1988)MathSciNetCrossRefGoogle Scholar
  23. Rivera, A.: Bifurcación de soluciones periódicas en el problema de Sitnikov. Ph.D. Thesis, Universidad de Granada (2012)Google Scholar
  24. Robinson, C.: Uniform sub-harmonic orbit for Sitinikov problem. Discrete Contin. Dyn. Syst. Ser. S 1, 647–652 (2008)MathSciNetCrossRefGoogle Scholar
  25. Sidorenko, V.: On the circular Sitnikov problem: the alternation of stability and instability in the family of vertical motions. Celest. Mech. Dyn. Astron. 109, 367–384 (2011)ADSMathSciNetCrossRefGoogle Scholar
  26. Soulis, P., Bountis, T., Dvorak, R.: Stability of motion in the Sitnikov 3-body problem. Celest. Mech. Dyn. Astron. 99, 129–148 (2007)ADSMathSciNetCrossRefGoogle Scholar
  27. Sitnikov, K.A.: Existence of oscillating motion for the three-body problem. Dokl. Akad. Nauk 133, 303–306 (1960)MathSciNetGoogle Scholar
  28. Tkhai, V.N.: Periodic motions of a reversible second-order mechanical system. Application to the Sitnikov problem. J. Appl. Math. Mech. 70, 734–753 (2006)MathSciNetCrossRefGoogle Scholar
  29. Zhang, M., Cen, X., Cheng, X.: Linearized stability and instability of nonconstant periodic solutions of Lagrangian equations. Math. Methods Appl. Sci. 41, 4853–4866 (2018)ADSMathSciNetCrossRefGoogle Scholar

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© 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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