Periodic solutions for a dumbbell satellite equation

  • Zaitao LiangEmail author
  • Fangfang Liao
Original Paper


In this paper, we study the existence of at least two geometrically distinct periodic solutions for a differential equation which models the planar oscillations of a dumbbell satellite under the influence of the gravity field generated by an oblate body, considering the effect of the zonal harmonic parameter \(J_{2}\). And at least one of such two periodic solutions is unstable. The proof is based on the version of the Poincaré–Birkhoff theorem due to Franks. Moreover, we also study the existence and multiplicity of periodic solutions and subharmonic solutions with winding number.


Dumbbell satellite Geometrically distinct periodic solutions Unstable Poincaré–Birkhoff theorem 

Mathematics Subject Classification

34C25 37C25 



We would like to express our great thanks to the referees for their valuable suggestions. We also would like to show our thanks to Professor Jifeng Chu (Shanghai Normal University) for his constant supervision and support. Zaitao Liang was jointly supported by the Key Program of Scientific Research Fund for Young Teachers of Anhui University of Science and Technology (QN2018109). Fangfang Liao was supported by the National Natural Science Foundation of China (Grant No. 11701375) and QingLan project of Jiangsu Province.


  1. 1.
    Abouelmagd, E.I., Guirao, J.L.G., Hobiny, A., Alzahrani, F.: Stability of equilibria points for a dumbbell satellite when the central body is oblate spheroid. Discrete Contin. Dyn. Syst. Ser. S 8(6), 1047–1054 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Abouelmagd, E.I., Guirao, J.L.G., Vera, J.A.: Dynamics of a dumbbell satellite under the zonal harmonic effect of an oblate body. Commun. Nonlinear Sci. Numer. Simul. 20(3), 1057–1069 (2015)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bardin, B.S., Chekina, E.A., Chekin, A.M.: On the stability of a planar resonant rotation of a satellite in an elliptic orbit. Regul. Chaotic Dyn. 20(1), 63–73 (2015)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Burov, A.A., Kosenko, I.I., Troger, H.: On periodic motions of an orbital dumbbell-shaped body with a cabin-elevator. Mech. Solids 47(3), 269–284 (2012)CrossRefGoogle Scholar
  5. 5.
    Belestky, V.V.: Motion of an artificial satellite about a center of mass. Israel Program for Scientific Translations, Jerusalem (1966)Google Scholar
  6. 6.
    Birkhoff, G.D.: An extension of Poincaré’s last geometric theorem. Acta Math. 47(4), 297–311 (1926)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Brereton, R.C., Modi, V.J.: On the stability of planar librations of a dumb-bell satellite in an elliptic orbit. Aeronaut. J. 70, 1098–1102 (1966)CrossRefGoogle Scholar
  8. 8.
    Celletti, A., Sidorenko, V.: Some properties of the dumbbell satellite attitude. Celest. Mech. Dyn. Astron. 101(1–2), 105–126 (2008)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chu, J., Liang, Z., Torres, P.J., Zhou, Z.: Existence and stability of periodic oscillations of a rigid dumbbell satellite around its center of mass. Discrete Contin. Dyn. Syst. Ser. B 22(7), 2669–2685 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Elipe, A., Palacios, M., Pretka-Ziomek, H.: Equilibria of the two-body problem with rigid dumb-bell satellite. Chaos Solitons Fractals 35, 830–842 (2008)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Fernández-Martínez, M., López, M.A., Vera, J.A.: On the dynamics of planar oscillations for a dumbbell satellite in \(J_{2}\) problem. Nonlinear Dyn. 84(1), 143–151 (2016)CrossRefGoogle Scholar
  12. 12.
    Franks, J.: Generalization of Poincaré–Birkhoff theorem. Ann. Math. 128(1), 139–151 (1988)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Fonda, A., Sabatini, M., Zanolin, F.: Periodic solutions of perturbed Hamiltonian systems in the plane by the use of the Poincaré–Birkhoff theorem. Topol. Methods Nonlinear Anal. 40(1), 29–52 (2012)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Krupa, M., Steindl, A., Troger, H.: Stability of relative equilibria. Part II: dumbbell satellites. Meccanica 35, 353–371 (2001)CrossRefGoogle Scholar
  15. 15.
    Guirao, J.L.G., Vera, J.A., Wade, B.A.: On the periodic solutions of a rigid dumbbell satellite in a circular orbit. Astrophys. Space Sci. 346(2), 437–442 (2013)CrossRefGoogle Scholar
  16. 16.
    Guirao, J.L.G., Llibre, J., Vera, J.A.: On the dynamics of the rigid body with a fixed point: periodic orbits and integrability. Nonlinear Dyn. 74(1–2), 327–333 (2013)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Marò, S.: Periodic solutions of a forced relativistic pendulum via twist dynamics. Topol. Methods Nonlinear Anal. 42(1), 51–75 (2013)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Nakanishi, K., Kojima, H., Watanabe, T.: Trajectories of in-plane periodic solutions of tethered satellite system projected on van der Pol planes. Acta Astronaut. 68(7–8), 1024–1030 (2011)CrossRefGoogle Scholar
  19. 19.
    Nuñez, D., Torres, P.J.: Stable odd solutions of some periodic equations modeling satellite motion. J. Math. Anal. Appl. 279(2), 700–709 (2003)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Poincaré, H.: Sur un théorème de géométrie. Rend. Circ. Mat. Palermo 33, 375–407 (1912)CrossRefGoogle Scholar
  21. 21.
    Petryshyn, W.V., Yu, Z.S.: On the solvability of an equation describing the periodic motions of a satellite in its elliptic orbit. Nonlinear Anal. 9(9), 969–975 (1985)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Rodnikov, A.V.: Equilibrium positions of a weight on a cable fixed to a dumbbell-shaped space station moving along a circular geocentric orbit. Cosmic Res. 44(1), 58–68 (2006)CrossRefGoogle Scholar
  23. 23.
    Schutte, A.D., Udwadia, F.E., Lam, T.: Nonlinear dynamics and control of a dumbbell spacecraft system. In: Proceedings of the 11th Aerospace Division International Conference on Engineering, Science, Construction, and Operations in Challenging Environments, American Society of Civil Engineers, Long Beach (2008)Google Scholar
  24. 24.
    Schutte, A.D., Udwadia, F.E.: New approach to the modeling of complex multibody dynamical systems. J. Appl. Mech. 78(2), 1–11 (2010)Google Scholar
  25. 25.
    Sanyal, A.K., Shen, J., McClamroch, N.H., Bloch, A.M.: Stability and stabilization of relative equilibria of dumbbell bodies in central gravity. J. Guid. Control Dyn. 28(5), 833–842 (2005)CrossRefGoogle Scholar
  26. 26.
    Sanyal, A.K., Shen, J., McClamroch, N.H.: Dynamics and control of an elastic dumbbell spacecraft in a central gravitational field. In: Proceedings of 42nd conference on decision and control, pp 2798–2803 (2003)Google Scholar
  27. 27.
    Vera, J.A.: On the periodic solutions of a rigid dumbbell satellite placed at L4 of the restricted three body problem. Int. J. Non-Linear Mech. 51, 152–156 (2013)CrossRefGoogle Scholar
  28. 28.
    Zevin, A.A.: On oscillations of a satellite in the plane of elliptic orbit. Kosmich. Issled XIX, 674–679 (1981)Google Scholar
  29. 29.
    Zevin, A.A., Pinsky, M.A.: Qualitative analysis of periodic oscillations of an earth satellite with magnetic attitude stabilization. Discrete Contin. Dyn. Syst. 6(2), 193–297 (2000)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Zlatoustov, V.A., Markeev, A.P.: Stability of planar oscillations of a satellite in an elliptic orbit. Celest. Mech. 7, 31–45 (1973)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.School of Mathematics and Big DataAnhui University of Science and TechnologyHuainanChina
  2. 2.Department of MathematicsSoutheast UniversityNanjingChina

Personalised recommendations