Instability of Equilibrium Solutions of Hamiltonian Systems with n-Degrees of Freedom Under the Existence of Multiple Resonances and an Application to the Spatial Satellite Problem

  • Daniela Cárcamo-DíazEmail author
  • Claudio Vidal


In this paper, we prove the instability of one equilibrium point in a Hamiltonian system with n-degrees of freedom under two assumptions: the first is the existence of multiple resonance of odd order s (without resonance of lower order) but with the possible existence of resonance of higher order; and the second is the existence of an invariant ray solution of the truncated Hamiltonian system up to order s. It is shown that in the case of resonance without interaction, the necessary conditions for instability have important simplifications with respect to the general case. Examples in three, four and six degrees of freedom are given. An application of our main result to the spatial satellite problem is considered.


Hamiltonian system Equilibrium solution Stability Normal form Resonance Invariant ray solution Chetaev’s theorem 

Mathematics Subject Classification

37C75 34D20 34A25 



The authors would like to thank the referee for valuable comments, which improved an earlier version of this paper. Claudio Vidal was partially supported by project Fondecyt 1180288. This paper is part of the Daniela Cárcamo-Díaz Ph.D. thesis in the Program Doctorado en Matemática Aplicada, Universidad del Bío-Bío. Daniela Cárcamo-Díaz acknowledges funding from CONICYT PhD/2016-21161143.


  1. 1.
    Arnold, V.: Small denominators and the problems of stability of motion in classical and celestial mechanics. Usp. Matem. Nauka 18, 6 (1963)MathSciNetGoogle Scholar
  2. 2.
    Arnold, V.I., Kozlov, V.V., Neishtadt, A.I.: Mathematical Aspects of Classical and Celestial Mechanics. Encyclopaedia of Mathematical Sciences, 3rd edn. Springer, Berlin (2006)Google Scholar
  3. 3.
    Bardin, B., Markeev, A.: On the stability of planar oscillations and rotations of a satellite in a circular orbit. Celest. Mech. Dyn. Astron. 85, 51–66 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dos Santos, F., Vidal, C.: Stability of equilibrium solutions of autonomous and periodic Hamiltonian systems in the case of multiple resonances. J. Differ. Equ. 258, 3880–3901 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Goltser, Y.: On the stability of differential equations systems with the spectrum on the imaginary axis. Funct. Differ. Equ. 4(1–2), 47–63 (1997)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Goltser, Y.: Invariant rays and Lyapunov functions. Funct. Differ. Equ. 6(1–2), 91–110 (1999)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Khazin, L.: On the stability of Hamiltonian systems in the presence of resonances. J. Appl. Math. Mech. 35, 384–931 (1971)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Khazin, L.: Stability of the critical equilibrium state of Hamiltonian systems of differential equations (Russian). Akad. Nauk SSSR Inst. P.M.M. 133, 20 (1981)MathSciNetGoogle Scholar
  9. 9.
    Khazin, L.: Interaction of third-order resonances in problems of the stability of Hamiltonian systems. J. Appl. Math. Mech. 48(3), 356–360 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Khazin, L., Shnol, E.: Stability of the equilibrium states (Translated from the Russian). Nonlinear Science: Theory and Applications. Manchester University Press, Manchester (1991)Google Scholar
  11. 11.
    Kunitsyn, A., Medvedev, S.: On Stability in the presence of sereval resonances. J. Appl. Math. Mech. 41(3), 419–426 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kunitsyn, A., Perezhogin, A.A.: The stability of neutral system in the case of a multiple fourth-order resonance. Prikl. Mat. Mekh. 49, 72–77 (1985)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Kunitsyn, A., Tuyakbayev, A.: The stability of hamiltonian systems in the case of a mutiple fourth-order resonance. J. Appl. Math. Mech. 56(4), 572–576 (1992)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Markeev, A.P.: Librations Points in Celestial Mechanics and Space Dynamics (in Russian). Nauka, Moscow (1978)Google Scholar
  15. 15.
    Markeev, A.P., Sokolskii, A.G.: On the Stability of Relative Equilibrium of a Satellite in a Circular Orbit. Nauka, Moscow (1975)Google Scholar
  16. 16.
    Meyer, K., Hall, G.R.: Introduction to Hamiltonian Dynamical systems and the N-body Problem. Applied Mathematical Science, vol. 90. Springer, New York (1992)CrossRefGoogle Scholar
  17. 17.
    Molčanov, A.M.: Stability in the case of a neutral linear approximation (Russian). Dokl. Akad. Nauk SSSR 141, 24–27 (1961)MathSciNetGoogle Scholar
  18. 18.
    Moser, J.: New aspects in the theory of stability of Hamiltonian systems. Commun. Appl. Math. 11(1), 81–114 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Moser, J.: Lectures on Hamiltonian system. Mem. Am. Math. 81, 10–13 (1963)Google Scholar
  20. 20.
    Siegel, C., Moser, J.: Lectures on Celestial Mechanics. Springer, New York (1971)CrossRefzbMATHGoogle Scholar
  21. 21.
    Zhavnerchik, V.E.: On instability on the presence of several resonances (Russian). Prikl. Mat. Mekh. 43(6), 970–974 (1979)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Zhavnerchik, V.E.: On the stability of autonomous systems in the presence of several resonances (Russian). Prikl. Mat. Mekh. 43(2), 229–234 (1979)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Zhavnerchik, V.E.: On stability in the presence of multiple resonance of odd order (Russian). Prikl. Mat. Mekh. 44(6), 971–976 (1980)MathSciNetGoogle Scholar

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Authors and Affiliations

  1. 1.Departamento de Matemática, Facultad de CienciasUniversidad del Bío-BíoConcepción, VIII-RegiónChile
  2. 2.Grupo de Investigación en Sistemas Dinámicos y Aplicaciones-GISDA, Departamento de Matemática, Facultad de CienciasUniversidad del Bío-BíoConcepción, VIII-RegiónChile

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