On the families of periodic orbits which bifurcate from the circular Sitnikov motions. (English) Zbl 0818.70011

Summary: We deal with the circular Sitnikov problem as a subsystem of the three- dimensional circular restricted three-body problem. The paper has a first analytical part, where by using elliptic functions we give the analytical expressions for the solutions of the circular Sitnikov problem and for the period function of its family of periodic orbits. We also analyze the qualitative and quantitative behavior of the period function. In the second numerical part, we study the linear stability of the family of periodic orbits of the Sitnikov problem, and of the families of periodic orbits of the three-dimensional circular restricted three-body problem which bifurcate from them; we follow these bifurcated families until they end in families of periodic orbits of the planar circular restricted three-body problem. Finally, the characteristic curves of some bifurcated families obtained for the mass parameter close to \(1/2\) are also described.


70F07 Three-body problems
37G99 Local and nonlocal bifurcation theory for dynamical systems
Full Text: DOI


[1] Alekseev, V.M.: 1968a, ?Quasirandom dynamical systems I?,Math. USSR Sbornik,5, 73-128 · Zbl 0198.56903
[2] Alekseev, V.M.: 1968b, ?Quasirandom dynamical systems II?,Math. USSR Sbornik,6, 505-560 · Zbl 0198.57001
[3] Alekseev, V.M.: 1969, ?Quasirandom dynamical systems III?,Math. USSR Sbornik,7, 1-43 · Zbl 0198.57002
[4] Broucke, R.: 1969, ?Stability of Periodic Orbits in the Elliptic Restricted Three-Body Problem?,AIAA J.,7, 1003-1009 · Zbl 0179.53301
[5] Byrd, P.F. and Friedman, M.D.: 1954,Handbook of elliptic integrals for engineers and physicists, Springer-Verlag, Berlin. · Zbl 0055.11905
[6] Dvorak, R.: 1993, ?Numerical results to the Sitnikov problem?,Celest. Mech. 56, 71-80
[7] Gómez, G., Llibre, J., Martínez, R. and Simó, C.: 1985, ?Station keeping of libration point orbits?, ESOC Contract no. 5648/83/D/JS (SC).
[8] Gradshteyn, I.S. and Ryzhik, I.M.: 1981,Table of integrals, series and products, Academic Press, New York. · Zbl 0918.65002
[9] Hagel, J.: 1992, ?A new analytical approach to the Sitnikov Problem?,Celest. Mech.,53, 267-292 · Zbl 0757.70006
[10] Hagel, J. and Trenkler T.: 1993, ?A computer aided analysis of the Sitnikov problem?,Celest. Mech.,56, 81-98 · Zbl 0780.70005
[11] Hénon, M.: 1965, ?Exploration numérique du problème restreint II, Masses égales?,Ann. d’Astrophysique,28, 992-1007
[12] Hénon, M.: 1973, ?Vertical stability of periodic orbits in the Restricted Problem I, Equal masses?,Astronomy and Astrophys.,28, 415-426 · Zbl 0272.70023
[13] Jie Liu and Yi-Sui Sun: 1990, ?On the Stinikov Problem?,Celest. Mech.,49, 285-302 · Zbl 0718.70005
[14] Katsiaris, G.: 1971, ?Two families of simply symmetric orbits inE 3?,Astrophys. and Space Sci.,10, 71-86
[15] Llibre, J. and Simó, C.: 1980, ?Estudio cualitativo del problema de Sitnikov?,Pub. Mat. U.A.B., 49-71
[16] MacMillan, W.D.: 1913, ?An Integrable Case in the Restricted Problem of Three Bodies?,Astron. J.,27, 11 · JFM 44.0841.12
[17] Martinez-Alfaro, J. and Chiralt, C.: 1993, ?Invariant rotational curves in Sitnikov’s problem?,Celest. Mech.,55, 351-367 · Zbl 0773.70006
[18] Moser, J.: 1973,Stable and random motions in dynamical systems, Annals of Math. Studies77, Princeton Univ. Press. · Zbl 0271.70009
[19] Perdios, E. and Markellos, V.V.: 1988, ?Stability and bifurcations of Sitnikov motions?,Celest. Mech.,42, 187-200
[20] Siegel, C.L. and Moser, J.K.: 1971,Lectures on Celestial Mechanics, Springer-Verlag, Berlin. · Zbl 0312.70017
[21] Sitnikov, K.: 1960, ?Existence of oscillating motions for the three-body problem?,Dokl. Akad. Nauk., USSR,133, 303-306 · Zbl 0108.18603
[22] Stiefel, E.L. and Scheifele, G.: 1971,Linear and regular Celestial Mechanics, Springer-Verlag, Berlin. · Zbl 0226.70005
[23] Stumpff, K.: 1965,Himmelsmeckanik, Band II, VEB, Berlin, 73-79
[24] Szebehely, V.: 1967,Theory of orbits, Academic Press, New York. · Zbl 0158.43206
[25] Wintner, A.: 1941,The analytical foundations of Celestial Mechanics, Princeton Univ. Press. · Zbl 0026.02302
[26] Wodnar, K.: 1993, ?The original Sitnikov article-New insights?,Celest. Mech.,56, 99-101 · Zbl 0789.70009
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