Abstract
The association of the Sitnikov family with families of multiple three-dimensional periodic orbits is studied. In particular, the families consisting of three-dimensional periodic orbits bifurcating from self-resonant orbits of the Sitnikov family at double, triple and quadruple period of the bifurcation orbit are considered. The branch families close upon themselves and remain 3D up to their terminations having two common members with the Sitnikov family. By varying the mass parameter we also study the evolution of some of the computed families and find that they become isolas and disappear gradually in three-dimensions by shrinking to point size.
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Alfaro J.M., Chiralt C.: Invariant rotational curves in Sitnikov’s problem. Celest. Mech. Dyn. Astron. 55, 351–367 (1993)
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)
Dellwo D., Keller H.B., Matkowsky B.J., Reiss E.L.: On the birth of isolas. SIAM J. Appl. Math. 42, 956–963 (1981)
Douskos C., Kalantonis V., Markellos P., Perdios E.: On Sitnikov-like motions generating new kinds of 3D periodic orbits in the R3BP with prolate primaries. Astrophys. Space Sci. 337, 99–106 (2012)
Dvorak R.: Numerical results to the Sitnikov-problem. Celest. Mech. Dyn. Astron. 56, 71–80 (1993)
Faruque S.B.: Solution of the Sitnikov problem. Celest. Mech. Dyn. Astron. 87, 353–369 (2003)
Hagel J.: A new analytic approach to the Sitnikov problem. Celest. Mech. Dyn. Astron. 53, 267–292 (1992)
Hagel J., Lhotka C.: A high order perturbation analysis of the Sitnikov problem. Celest. Mech. Dyn. Astron. 93, 201–228 (2005)
Kalantonis V.S., Perdios E.A., Perdiou A.E.: The Sitnikov family and the associated families of 3D periodic orbits in the photogravitational RTBP with oblateness. Astrophys. Space Sci. 315, 323–334 (2008)
Kovács T., B.: Transient chaos in the Sitnikov problem. Celest. Mech. Dyn. Astron. 105, 289–304 (2009)
Lara L.J., Buendía A.E.: Symmetries and bifurcations in the Sitnikov problem. Celest. Mech. Dyn. Astron. 79, 97–117 (2001)
Llibre J., Ortega R.: On the families of periodic orbits of the Sitnikov problem. SIAM J. Appl. Dyn. Syst. 7, 561–576 (2008)
Mac Millan W.D.: An integrable case in the restricted problem of three bodies. Astron. J. 27, 11–13 (1913)
Pavanini, G.: Sopra una nuova categoria di solutioni periodiche nel problema dei tre corpi. Ann. Math. Serie III, Tomo XIII, 179–202 (1907)
Perdios E.A.: The manifold of families of 3D periodic orbits associated to Sitnikov motions in the restricted three-body problem. Celest. Mech. Dyn. Astr. 99, 85–104 (2007)
Perdios E.A., Kalantonis V.S., Douskos C.N.: Straight-line oscillations generating three-dimensional motions in the photogravitational restricted three-body problem. Astrophys. Space Sci. 314, 199–208 (2008)
Perdios E.A., Markellos V.V.: Stability and bifurcations of Sitnikov motion. Celest. Mech. 42, 187–200 (1988)
Robin I.A., Markellos V.V.: Numerical determination of three-dimensional periodic orbits generated from vertical self-resonant satellite orbits. Celest. Mech. 21, 395–434 (1980)
Robin I.A., Markellos V.V.: The mechanism of branching of three-dimensional periodic orbits from the plane. In: Markellos, V.V., Kozai, Y. (eds) Dynamical Trapping and Evolution in the Solar System, pp. 213–224. D.Reidel, Dordrecht (1983)
Ruzza S., Lhotka C.: High order normal form construction near the elliptic orbit of the Sitnikov problem. Celest. Mech. Dyn. Astron. 111, 449–464 (2011)
Sidorenko V.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)
Sitnikov K.: The existence of oscillatory motions in the three-body problem. Dokl. Akad. Nauk. 133, 303–306 (1960)
Szebehely V.: Theory of Orbits. Academic Press, Orlando (1967)
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Perdios, E.A., Kalantonis, V.S. Self-resonant bifurcations of the Sitnikov family and the appearance of 3D isolas in the restricted three-body problem. Celest Mech Dyn Astr 113, 377–386 (2012). https://doi.org/10.1007/s10569-012-9424-0
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DOI: https://doi.org/10.1007/s10569-012-9424-0