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Resonance tongues in the linear Sitnikov equation. (English) Zbl 1391.70032

Summary: In this paper, we deal with a Hill’s equation, depending on two parameters \(e\in [0,1)\) and \(\varLambda >0\), that has applications to some problems in Celestial Mechanics of the Sitnikov type. Due to the nonlinearity of the eccentricity parameter \(e\) and the coexistence problem, the stability diagram in the \((e,\varLambda )\)-plane presents unusual resonance tongues emerging from points \((0,(n/2)^2)\), \(n=1,2,\dots \) The tongues bounded by curves of eigenvalues corresponding to \(2\pi \)-periodic solutions collapse into a single curve of coexistence (for which there exist two independent \(2\pi \)-periodic eigenfunctions), whereas the remaining tongues have no pockets and are very thin. Unlike most of the literature related to resonance tongues and Sitnikov-type problems, the study of the tongues is made from a global point of view in the whole range of \(e\in [0,1)\). Indeed, an interesting behavior of the tongues is found: almost all of them concentrate in a small \(\varLambda \)-interval \([1,9/8]\) as \(e\rightarrow 1^-\). We apply the stability diagram of our equation to determine the regions for which the equilibrium of a Sitnikov \((N+1)\)-body problem is stable in the sense of Lyapunov and the regions having symmetric periodic solutions with a given number of zeros. We also study the Lyapunov stability of the equilibrium in the center of mass of a curved Sitnikov problem.

MSC:

70F10 \(n\)-body problems
70G60 Dynamical systems methods for problems in mechanics
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[1] Arnol’d, VI, Remarks on the perturbation theory for problems of Mathieu type, Rus. Math. Surv., 38, 215-233, (1983) · Zbl 0541.34035
[2] Bakker, L; Simmons, S, A separating surface for Sitnikov-like \(n+1\)-body problems, J. Differ. Equ., 258, 3063-3087, (2015) · Zbl 1395.70018
[3] Belbruno, E; Libre, J; Ollé, M, On the families of periodic orbits which bifurcate from the circular Sitnikov motions, Celest. Mech. Dyn. Astron., 60, 99-129, (1994) · Zbl 0818.70011
[4] Bountis, T; Papadakis, KE, The stability of the vertical motion in the \(N\)-body circular Sitnikov problem, Celest. Mech. Dyn. Astron., 104, 205-225, (2009) · Zbl 1165.70012
[5] Broer, HW; Simó, C, Hill’s equation with quasi-periodic forcing: resonance tongues, instability pockets and global phenomena, Bol. Soc. Brasil. Math., 29, 253-293, (1998) · Zbl 0917.34019
[6] Broer, HW; Levi, M, Geometrical aspects of stability theory of hill’s equations, Arch. Rat. Mech. Anal., 131, 225-240, (1995) · Zbl 0840.34047
[7] Brown, B.M., Eastham, M.S.P., Schmidt, K.M.: Periodic Differential Operators. Advances and Applications. Birkhäuser, Basel (2013) · Zbl 1267.34001
[8] Celletti, A, Analysis of resonances in the spin-orbit problem in celestial mechanics: the synchronous resonance (part I), J. Appl. Math. Phys., 41, 174-204, (1990) · Zbl 0699.70014
[9] Celletti, A; Chierchia, L, Measures of basins of attraction in spin-orbit dynamics, Celest. Mech. Dyn. Astron., 101, 159-170, (2008) · Zbl 1342.70035
[10] Coddington, E., Levinson, N.: Theory of Ordinary Differential Equations. Mc Graw Hill, New York (1955) · Zbl 0064.33002
[11] Dias L.B. and Cabral H.E.: Parametric stability in a Sitnikov-like restricted P-body problem. J. Dyn. Diff. Equ. (2016). https://doi.org/10.1007/s10884-016-9533-7 · Zbl 1390.70028
[12] Franco-Pérez, L; Gidea, M; Levi, M; Pérez-Chavela, E, Stability interchanges in a curved Sitnikov problem, Nonlinearity, 29, 1056-1079, (2016) · Zbl 1411.70014
[13] Gan, S; Zhang, M, Resonance pockets of hill’s equations with two-step potentials, SIAM J. Math. Anal., 32, 651-664, (2000) · Zbl 0973.34019
[14] Goldreich, P; Peale, S, Spin-orbit coupling in the solar system, Astron J., 71, 425-38, (1966)
[15] Havil, J.: Gamma. Exploring Euler’s Constant. Princeton University Press, Princeton (2003) · Zbl 1023.11001
[16] Kamke, E, A new proof of sturm’s comparison theorems, Amer. Math. Monthly, 46, 417-421, (1939) · Zbl 0061.17803
[17] Krantz, S.G., Parks, H.R.: The Implicit Function Theorem: History, Theory and Applications. Birkhäuser, Basel (2003) · Zbl 1012.58003
[18] Levy, DM; Keller, JB, Instability intervals of hill’s equation, Comm. Pure Appl. Math., 16, 469-479, (1963) · Zbl 0121.31202
[19] Llibre, J; Ortega, R, On the families of periodic orbits of the Sitnikov problem, SIAM J. Appl. Dyn. Syst., 7, 561-576, (2008) · Zbl 1159.70010
[20] Magnus, W., Winkler, S.: Hill’s equation. Dover, New York (1979) · Zbl 0158.09604
[21] Martínez Alfaro, J; Chiralt, C, Invariant rotational curves in sitnikov’s problem, Celest. Mech. Dyn. Astron., 55, 351-367, (1993) · Zbl 0773.70006
[22] Moser, J.: Stable and random motions in dynamical systems. Annals of Math Studies 77. Princeton University Press, New Jersey (1973) · Zbl 0271.70009
[23] Núñez, D; Ortega, R, Parabolic fixed points and stability criteria for non-linear hill’s equation, Zeitschrift für Angewandte Mathematik und Physik ZAMP, 51, 890-911, (2000) · Zbl 0973.34046
[24] Ortega, R, The stability of the equilibrium of a nonlinear hill’s equation, SIAM J. Math. Anal., 25, 1393-1401, (1994) · Zbl 0807.34065
[25] Ortega, R; Ferrera, J (ed.); López-Gómez, J (ed.); Ruiz del Portal, FR (ed.), The stability of the equilibrium: a search for the right approximation, 215-234, (2005), Amsterdam
[26] Ortega, R, Symmetric periodic solutions in the Sitnikov problem, Arch. Math., 107, 405-412, (2016) · Zbl 1354.34072
[27] 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
[28] Pustylnikov, LD, On certain final motions in the \(n\)-body problem, J. Appl. Math. Mech., 54, 272-274, (1990) · Zbl 0739.70007
[29] Rivera, A, Periodic solutions in the generalized Sitnikov \((N+1)\)-body problem, SIAM J. Appl. Dyn. Syst., 12, 1515-1540, (2013) · Zbl 1282.70017
[30] Sidorenko, VV, 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) · Zbl 1270.70033
[31] Suraj, MS; Hassan, MR, Sitnikov restricted four-body problem with radiation pressure, Astrophys. Space Sci., 349, 705-716, (2013)
[32] Pol, B; Strutt, MJO, On the stability of the solutions of mathieu’s equation, London Edinburgh Dublin Phil. Mag. J. Sci., 5, 18-38, (1928) · JFM 54.0469.02
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