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| The Sitnikov problem is a restricted three body problem where two point masses $m_{1}=m_{2}$ (primaries) moves around the center of mass in elliptic orbits lying on the plane $x,y$ as solutions of the $2$ body problem and a infinitesimal mass $m_{3}$ moves on the $z$ axis. In appropiate units the equation of motion of the infinitesimal mass is given by
\begin{equation}\label{sp}
\ddot{z}+\frac{z}{(z^{2}+r(t,e)^{2})^{3/2}}=0,
\end{equation}
where $e\in [0,1[$ is the eccentricity of the ellipses described by the primaries and $r(t,e)$ is the distance from the primaries to the center of mass.
In this talk, we consider the Sitnikov problem with $N$ primaries bodies with equal mass moving around the origin in the plane $x,y$ as generalized solutions of the Lagrange problem. We call this problem the Generalized Sitnikov Problem. This special restricted N+1-body problem can be reduced to the Sitnikov problem with an appropriate positive parameter $\lambda $. More precisely we study the differential equation
\begin{equation}\label{sp2}
\ddot{z}+\frac{\lambda z}{(z^{2}+r(t,e)^{2})^{3/2}}=0,
\end{equation}
where the parameter $\lambda$ is given by
\begin{equation*}
\lambda=\frac{N}{2\sum_{k=1}^{N-1} \csc (\frac{k\pi}{N})}.
\end{equation*}
According to the number of bodies we proved the existence (or non-existence) of a finite number (or infinite number) of symmetric families of periodic solutions that bifurcate from the equilibrium $z=0$ (center of mass of the system) at certain values of the eccentricity. Worth mentioning that the type of bifurcation from the equilibrium that one expects is the pitchfork bifurcation, but the standard transversality conditions are not easy to check, therefore we present a slight variant of the theorem on pitchfork bifurcation adapted to our problem. See [5,6]
Keywords: N+1-body problem, Sitnikov problem,
periodic orbits, Sturm-Liouville theory, global continuation, bifurcations.
[1] E. Belbruno, J. Llibre, M. Oll\'e. \textit{On the families of the periodic orbits which bifurcate from the circular Sitnikov motions}, Celestial Mech. Dynam. Astronm., 60 (1994), 99-129.
[2] L. Jim\'enez-Lara, A. Escalona-Buend\'ia. \textit{Symmetries and bifurcations in the Sitnikov problem},
Celestial Mech. Dynam. Astronom., 79 (2001) 97-117.
[3] J. Llibre, R. Ortega. \textit{On the families of periodic orbits of the Sitnikov problem}, Siam J. Applied Dynamical Systems Vol. 7, (2008) 561{-}576.
[4] J. Moser, \textit{Stable and Random Motions in Dynamical Systems}, Annals of Math. Studies 77, Princeton
University Press, Princeton, NJ, 1973.
[5] R. Ortega, Andr\'es Rivera. \textit{Global bifurcations from the center of mass in the Sitnikov problem}, Discrete and Continuos Dynamical Systems series B 14 (2010), 719-732.
[6] Andr\'es Rivera, \textit{Periodic solutions in the Generalized Sitnikov (N+1)-body problem}, Siam J. Applied Dynamical Systems Vol. 12, No. (2013) 1515-1540. |
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