Abstract
In this paper, we study the problem of determining whether a global family of even periodic solutions of a generalized Sitnikov problem, which emerges from a circular generalized Sitnikov problem, continues for all values of eccentricity in [0, 1) or ends in the equilibrium.
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References
Alekseev, V.M.: Quasirandom dynamical systems. II. One-dimensional nonlinear oscillations in a field with periodic perturbation. Math. USSR-Sbornik 6(4), 505–560 (1968)
Alfaro, J.M., Chiralt, C.: Invariant rotational curves in Sitnikov’s problem. Celest. Mech. Dyn. Astron. 55(4), 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(1), 99–129 (1994)
Beltritti, G., Mazzone, F., Oviedo, M.: The Sitnikov problem for several primary bodies configurations. Celest. Mech. Dyn. Astron. 130(7), 45 (2018)
Bountis, T., Papadakis, K.: The stability of vertical motion in the n-body circular Sitnikov problem. Celest. Mech. Dyn. Astron. 104(1), 205–225 (2009)
Corbera, M., Llibre, J.: Periodic orbits of the Sitnikov problem via a Poincaré map. Celest. Mech. Dyn. Astron. 77(4), 273–303 (2000)
Corbera, M., Llibre, J.: On symmetric periodic orbits of the elliptic Sitnikov problem via the analytic continuation method. Contemp. Math. 292, 91–128 (2002)
Dvorak, R.: Numerical results to the Sitnikov-problem. Celest. Mech. Dyn. Astron. 56(1–2), 71–80 (1993)
Hartman, P.: Ordinary differential equations, Classics in Applied Mathematics, vol. 38. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2002). Corrected reprint of the second (1982) edition
Jiménez-Lara, L., Escalona-Buendía, A.: Symmetries and bifurcations in the Sitnikov problem. Celest. Mech. Dyn. Astron. 79(2), 97–117 (2001)
Llibre, J., Ortega, R.: On the families of periodic orbits of the Sitnikov problem. SIAM J. Appl. Dyn. Syst. 7(2), 561–576 (2008)
Llibre, J., Simó, C.: Estudio cualitativo del problema de Sitnikov. Publicacions de la Secció de Matemàtiques, pp. 49–71(1980)
Llibre, J., Moeckel, R., Simó, C.: Central configurations, periodic orbits, and Hamiltonian systems. Advanced Courses in Mathematics - CRM Barcelona. Birkhauser (2015)
Magnus, W., Winkler, S.: Hill’s Equation. Courier Corporation, Chelmsford (2013)
Marchesin, M., Vidal, C.: Spatial restricted rhomboidal five-body problem and horizontal stability of its periodic solutions. Celest. Mech. Dyn. Astron. 115(3), 261–279 (2013)
Moser, J.: Stable and Random Motions in Dynamical Systems: With Special Emphasis on Celestial Mechanics, Annals Mathematics Studies, vol. 1. Princeton University Press, Princeton (1973)
Murray, C.D., Dermott, S.F.: Solar System Dynamics. Cambridge University Press, Cambridge (1999)
Ortega, R.: Symmetric periodic solutions in the Sitnikov problem. Archiv der Mathematik 107(4), 405–412 (2016)
Ortega, R., Rivera, A.: Global bifurcations from the center of mass in the Sitnikov problem. Discrete Contin. Dyn. Syst. B 14(2), 719–732 (2010)
Pandey, L.P., Ahmad, I.: Periodic orbits and bifurcations in the Sitnikov four-body problem when all primaries are oblate. Astrophys. Space Sci. 345(1), 73–83 (2013)
Perdios, E., Markellos, V.V.: Stability and bifurcations of Sitnikov motions. Celest. Mech. Dyn. Astron. 42(1–4), 187–200 (1987)
Rabinowitz, P.H.: Some global results for nonlinear eigenvalue problems. J. Funct. Anal. 7(3), 487–513 (1971)
Rivera, A.: Periodic solutions in the generalized Sitnikov \((n+1)\)-body problem. SIAM J. Appl. Dyn. Syst. 12(3), 1515–1540 (2013)
Rivera, A., Andrés, M.: Bifurcación de soluciones periódicas en el problema de Sitnikov. Universidad de Granada, Granada (2012)
Sitnikov, K.: The existence of oscillatory motions in the three-body problem. Dokl. Akad. Nauk SSSR 133, 303–306 (1960)
Soulis, P.S., Papadakis, K.E., Bountis, T.: Periodic orbits and bifurcations in the Sitnikov four-body problem. Celest. Mech. Dyn. Astron. 100(4), 251–266 (2008)
Zanini, C.: Rotation numbers, eigenvalues, and the Poincaré–Birkhoff theorem. J. Math. Anal. Appl. 279(1), 290–307 (2003)
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Appendix
Appendix
In this section, we enunciate and demonstrate some technical lemmas, which deal with Taylor series of \(\sqrt{q_\lambda (u,e)}\) and \(\int _{0}^{\pi }\sqrt{q_\lambda (u,e)}\mathrm{d}u\) and are used to demonstrate other results of the article.
Before enunciating the first lemma, it should be noted the definition of combinatory number, \(\begin{pmatrix} \gamma \\ n \end{pmatrix}:=\frac{\prod _{i=1}^{n}(\gamma -i+1)}{n!}\), where \(\gamma \in {\mathbb {R}}\) and \(n\in {\mathbb {N}}\).
Lemma 6
Let \(q_\lambda (u,e)\) be the function defined in (16). Then, for \(u \in (0,\pi )\), we have that
where
with \(a_\lambda (u)=\frac{\cos ^2(u)-3}{32\lambda }\), \(b_\lambda (u)=-\frac{32\lambda -2}{32\lambda }\cos (u)\).
Proof
Let us first observe that
where \(a_\lambda (u)=\frac{\cos ^2(u)-3}{32\lambda }\) and \(b_\lambda (u)=-\frac{32\lambda -2}{32\lambda }\cos (u)\). As \(\sqrt{1+x}=\sum _{i=0}^{\infty }\begin{pmatrix} \frac{1}{2}\\ i \end{pmatrix}x^i\) for all \(-1<x< 1\), we have that
where \(f_\lambda (u,e):=\sum _{n=0}^{\infty }\begin{pmatrix} \frac{1}{2}\\ n \end{pmatrix}e^n(b_\lambda (u)+ea_\lambda (u))^n\).
Since \(\sqrt{1+x}\) is analytic in \((-1,1)\) and \(b_\lambda (u)e+a_\lambda (u)e^2\) is analytic, as a function of e in \({\mathbb {R}}\), we have that, for every \(u\in (0,\pi )\), \(\sqrt{1+b_\lambda (u)e+a_\lambda (u)e^2}=f_\lambda (u,e)\) is analytic, as a function of e, in \((-1,1)\). Then, for \(e\in [0,1)\)
Let us calculate \(\frac{\partial ^nf_\lambda (u,0)}{\partial e^n}\).
Now,
where
Considering \(j=n-m\), and \(k=n-2j\), we have that
\(\square \)
Lemma 7
Let \(q_\lambda (u,e)\) be the function defined in (16). Then
Proof
Note first that \(\begin{pmatrix} \frac{1}{2}\\ k \end{pmatrix}(-1)^k<0\) if \(k\ge 1\), and \(\begin{pmatrix} \frac{1}{2}-j\\ k \end{pmatrix}(-1)^k>0\) if \(j\ge 1\) and \(k\ge 0\). Since the function \(-1<\frac{a_\lambda (u)}{1+b_\lambda (u)} < 0\) for \(u\in (0,\pi )\), and \(\sum _{k=0}^{\infty }\begin{pmatrix} \gamma \\ k \end{pmatrix}\beta ^k=(1+\beta )^{\gamma }\) for \(\gamma \in {\mathbb {R}}\) and \(|\beta |<1\), if \(u\in (0,\pi )\), we have that
Thus, by the dominated convergence theorem we have that
for every \(0\le e<1\). Taking into account the expression of \(c_{\lambda ,l}(u)\), if l is an odd number then \(\int _{0}^{\pi }c_{\lambda ,l}(u)\mathrm{d}u=0\). Therefore,
\(\square \)
Remark 5
From (33) in Lemma 6, if we define \(h_\tau (u,\omega ):=\cos ^{2\tau -2\omega }(u)(3-\cos ^2(u))^\omega \), we have that
Now, taking into account that \(\int _{0}^{\pi }\cos ^i(x)\mathrm{d}x=\frac{i-1}{i}\int _{0}^{\pi }\cos ^{i-2}(x)\mathrm{d}x\) for \(i\ge 2\), if \(\tau \ge n-m\ge 0\) then
Hence,
Note that \(\int _{0}^{\pi }c_{\lambda ,2\tau }(u)\mathrm{d}u\) is always a product between a number \(g_\lambda (\tau )\) and \(\pi \), so we can calculate with a symbolic computation software the expression of \(g_\lambda (\tau )\) without error. Thus, for example, we can verify that
In general it seems to be that \(\int _{0}^{\pi }c_{\lambda ,2\tau }(u)\mathrm{d}u>0\) for all \(\lambda \ge \frac{1}{8}\) and \(\tau \in {\mathbb {N}}_0\).
The code used in the Python library “Sympy” with which we calculate \(\int _{0}^{\pi }c_{\lambda ,2\tau }(u)\mathrm{d}u\) is described below
The function \(\text {gt(tau,lam)}\) calculates \(g_\lambda (\tau )\) with error zero.
Lemma 8
Let \(c_{1,l}(u)\) be the coefficient defined in (32). Then
for all \(\tau \in {\mathbb {N}}_0\).
Proof
For \(u\in (0,\frac{\pi }{2})\), we have that
Since the function \(\frac{a_1(u)}{(1+b_1(u))^{\frac{1}{2}}}\) is negative and increasing in \((0,\frac{\pi }{2})\) (because its derivative is equal to \( \frac{ \left( 45 \cos ^{2}{\left( u \right) } - 64 \cos {\left( u \right) } + 45\right) \sin {\left( u \right) }}{16 \left( 16 -15 \cos {\left( u \right) }\right) ^{\frac{3}{2}}} \), which is positive for all \(u\in (0,\frac{\pi }{2})\)), and the function \(\frac{-a_1(u)}{1+b_1(u)}\) is positive and decreasing in \((0,\frac{\pi }{2})\) (since its derivative \(- \frac{\left( 15 \cos ^{2}{\left( u \right) } - 32 \cos {\left( u \right) } + 45\right) \sin {\left( u \right) }}{2 \left( 15 \cos {\left( u \right) } - 16\right) ^{2}}\) is negative for \(u\in (0,\frac{\pi }{2})\)), we have that the last expression in the chain of inequalities mentioned above is increasing. Then, it reaches its minimum in \(\left[ 0,\frac{\pi }{2}\right] \) at \(u=0\). Hence \(c_{1,2\tau }(u)\ge c_{1,2\tau }(0)= - \frac{1}{4}\) for every \(u\in (0,\frac{\pi }{2})\). Therefore
\(\square \)
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Beltritti, G. Periodic solutions of a generalized Sitnikov problem. Celest Mech Dyn Astr 133, 6 (2021). https://doi.org/10.1007/s10569-021-10005-z
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DOI: https://doi.org/10.1007/s10569-021-10005-z