On the stability of symmetric periodic orbits of the elliptic Sitnikov problem. (English) Zbl 1476.70015
In this paper the authors study symmetric periodic solutions coming from nonconstant periodic solutions of autonomous equations. They first provide a criterion for linearized stability and instability of odd periodic solutions, the proof techniques using the theory of Hill’s equations. Then they prove that the odd (2,1)-periodic solutions of elliptic Sitnikov problem are hyperbolic and unstable for small eccentricity, while the corresponding even (2,1)-periodic solutions are elliptic and linearized stable.
Reviewer: Cristian Vladimirescu (Craiova)
MSC:
| 70F07 | Three-body problems |
| 34C25 | Periodic solutions to ordinary differential equations |
| 34C23 | Bifurcation theory for ordinary differential equations |
| 34D20 | Stability of solutions to ordinary differential equations |
References:
| [1] | E. Belbruno, J. Llibre, and M. Ollé, On the families of periodic orbits which bifurcate from the circular Sitnikov motions, Celestial Mech. Dynam. Astronom., 60 (1994), pp. 99-129. · Zbl 0818.70011 |
| [2] | K.-C. Chen, Variational constructions for some satellite orbits in periodic gravitational force fields, Amer. J. Math., 132 (2010), pp. 681-709. · Zbl 1250.70012 |
| [3] | A. Chenciner, Four lectures on the \(N\)-body problem, in Hamiltonian Dynamical Systems and Applications, NATO Sci. Peace Secur. Ser. B Phys. Biophys., Springer, Dordrecht, 2008, pp. 21-52. · Zbl 1163.70005 |
| [4] | S. B. Faruque, Solution of the Sitnikov problem, Celest. Mech. Dyn. Astr., 87 (2003), pp. 353-369. · Zbl 1106.70323 |
| [5] | J. Galán, D. Nún͂ez, and A. Rivera, Quantitative stability of certain families of periodic solutions in the Sitnikov problem, SIAM J. Appl. Dyn. Syst., 17 (2018), pp. 52-77, https://doi.org/10.1137/17M1113990. · Zbl 1390.70017 |
| [6] | J. Galán-Vioque, D. Nun͂ez, A. Rivera, and C. Riccio, Stability and bifurcations of even periodic orbits in the Sitnikov problem, Celestial Mech. Dynam. Astronom., 130 (2018), 82. · Zbl 1448.70037 |
| [7] | X. Hu, Y. Long, and S. Sun, Linear stability of elliptic Lagrangian solutions of the planar three-body problem via index theory, Arch. Ration. Mech. Anal., 213 (2014), pp. 993-1045. · Zbl 1301.70007 |
| [8] | J. Lei, X. Li, P. Yan, and M. Zhang, Twist character of the least amplitude periodic solution of the forced pendulum, SIAM J. Math. Anal., 35 (2003), pp. 844-867, https://doi.org/10.1137/S003614100241037X. · Zbl 1189.37064 |
| [9] | M. Levi, Quasiperiodic motions in superquadratic time-periodic potentials, Comm. Math. Phys., 143 (1991), pp. 43-83. · Zbl 0744.34043 |
| [10] | J. Liu and Y.-S. Sun, On the Sitnikov problem, Celestial Mech. Dynam. Astronom., 49 (1990), pp. 285-302. · Zbl 0718.70005 |
| [11] | J. Llibre and R. Ortega, On the families of periodic orbits of the Sitnikov problem, SIAM J. Appl. Dyn. Syst., 7 (2008), pp. 561-576, https://doi.org/10.1137/070695253. · Zbl 1159.70010 |
| [12] | J. Llibre and C. Simó, Estudio cualitativo del problema de Sitnikov, Publ. Mat. U.A.B., 18 (1980), pp. 49-71. · Zbl 1535.37077 |
| [13] | W. Magnus and S. Winkler, Hill’s Equations, John Wiley, New York, 1966. · Zbl 0158.09604 |
| [14] | M. Misquero, Resonance tongues in the linear Sitnikov equation, Celestial Mech. Dynam. Astronom., 130 (2018), 30. · Zbl 1391.70032 |
| [15] | R. Ortega, Periodic solutions of a Newtonian equation: Stability by the third approximation, J. Differential Equations, 128 (1996), pp. 491-518. · Zbl 0855.34058 |
| [16] | R. Ortega, Symmetric periodic solutions in the Sitnikov problem, Arch. Math. (Basel), 107 (2016), pp. 405-412. · Zbl 1354.34072 |
| [17] | R. Ortega, Stability of periodic solutions of Hamiltonian systems with low dimension, Rend. Semin. Mat. Univ. Politec. Torino, 75 (2017), pp. 53-78. · Zbl 1440.37059 |
| [18] | R. Ortega and A. Rivera, Global bifurcations from the center of mass in the Sitnikov problem, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), pp. 719-732. · Zbl 1380.70028 |
| [19] | C. L. Siegel and J. K. Moser, Lectures on Celestial Mechanics, Springer-Verlag, Berlin, 1971. · Zbl 0312.70017 |
| [20] | K. A. Sitnikov, Existence of oscillating motion for the three-body problem, Dokl. Akad. Nauk SSSR, 133 (1960), pp. 303-306. · Zbl 0108.18603 |
| [21] | M. Zhang, X. Cen, and X. Cheng, Linearized stability and instability of nonconstant periodic solutions of Lagrangian equations, Math. Methods Appl. Sci., 41 (2018), pp. 4853-4866. · Zbl 1404.34069 |
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