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

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 ∈ [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 ∈ ℕ there are a finite number of nontrivial symmetric 2Nπ-periodic solutions. All of them parabolic and unstable (in the Lyapunov sense) if we consider the corresponding autonomous equation as a 2π-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 2Nπ-periodic solutions in the circular case for nonnecessarily small values of the eccentricity e and in some cases for all values of e ∈ [0, 1[. However this approach does not provide information about the stability properties of these periodic solutions. We present a new method that quantifies the above-mentioned bifurcating families and their stability properties at least in first approximation. Our approach is based on two general results. The first one is provides an estimation of the growth of the canonical solutions for one-parametric differential equation of the form x¨ + a(t, λ)x = 0 with a ∈ C¹([0, T] × [0, Λ]). The second one gives stability criteria for the one-parametric Hill’s equation of the form x¨ + q(t, λ)x = 0, (*) where q(·, λ) is T periodic and q ∈ C³(R × [0, Λ]) such that for λ = 0 the equation (*) is parabolic. We apply these results to the case N = 1 and show that both branches are elliptic for low values of e.