Quantitative Stability of Certain Families of Periodic Solutions in the Sitnikov Problem

Abstract

The Sitnikov problem is a special case of the restricted three-body problem where the primaries move in elliptic orbits of the two-body problem with eccentricity $ e\in [0,1[$ and the massless body moves on a straight line perpendicular to the plane of motion of the primaries through their barycenter. It is well known that for the circular case ($e=0$) and a given $N\in \mathbb{N}$ there are a finite number of nontrivial symmetric $2N\pi$-periodic solutions. All of them parabolic and unstable (in the Lyapunov sense) if we consider the corresponding autonomous equation as a $2\pi$-periodic equation. The authors in [J. Llibre and R. Ortega, SIAM J. Appl. Dyn. Syst., 7 (2008), pp. 561--576] proved that these families of periodic solutions can be continued from the known $2N\pi$-periodic solutions in the circular case for nonnecessarily small values of the eccentricity $e$ and in some cases for all values of $e\in [0,1[$. However this approach does not provide information about the stability properties of the...