×

Symmetric periodic solutions in the Sitnikov problem. (English) Zbl 1354.34072

The author presents results for the Sitnikov problem – a simplified problem in celestial mechanics. In the paper, the author presents a number of theorems, lemmas, and propositions and thereby extends results of previous investigations. He shows that there exist odd periodic solutions with a number of zeros for the title problem.
Although very specialized, the paper is carefully written and should be of interest to theoreticians working in celestial mechanics and in nonlinear ordinary differential equations.

MSC:

34C25 Periodic solutions to ordinary differential equations
70F07 Three-body problems
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Bakker, L.; Simmons, S., A separating surface for Sitnikov-like n+1-body problems, J. Differential Equations, 258, 3063-3087, (2015) · Zbl 1395.70018
[2] Corbera, M.; Llibre, J., Periodic orbits of the Sitnikov problem via a Poincaré map, Celestial Mech. Dynam. Astronom., 77, 273-303, (2000) · Zbl 0986.70010
[3] Jiménez-Lara, L.; Escalona-Buendía, A., Symmetries and bifurcations in the Sitnikov problem, Celestial Mech. and Dynam. Astronom., 79, 97-117, (2001) · Zbl 0993.70010
[4] Llibre, J.; Ortega, R., On the families of periodic solutions of the Sitnikov problem, SIAM J. Appl. Dyn. Syst., 7, 561-576, (2008) · Zbl 1159.70010
[5] W. Magnus and S. Winkler, Hill’s equation, Corrected reprint of the 1966 edition, Dover Publications, Inc., New York, 1979. · Zbl 0993.70010
[6] J. Moser, Stable and random motions in dynamical systems, Princeton University Press, Princeton, N. J., 1973. · Zbl 0271.70009
[7] 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) · Zbl 1380.70028
[8] Pérez-Franco, L.; Gidea, M.; Levi, M.; Pérez-Chavela, E., Stability interchanges in a curved Sitnikov problem, Nonlinearity, 29, 1056-1079, (2016) · Zbl 1411.70014
[9] Rivera, A., Periodic solutions in the generalized Sitnikov (\(N\) + 1)-body problem, SIAM J. Appl. Dyn. Syst., 12, 1515-1540, (2013) · Zbl 1282.70017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.