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 moves 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 like a $2\pi$-periodic equation. Using the method of global continuation of Leray-Schauder, J.Llibre and R.Ortega (J.Llibre $\&$ R. Ortega, 2008) 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 say anything about the stability properties of this periodic solutions. In this document we present a new method that quantifies the mentioned bifurcating families and them stabilities properties at least in first approximation. Our approach proposes two general methods: The first one is to estimate the growing of the canonical solutions for one-parametric differential equation of the form \[ \ddot{x}+a(t,\lambda)x=0, \] with $a\in C^{1}([0,T] \times [0,\Lambda])$. The second one gives stability criteria for one-parametric Hill's equation of the form \[ \ddot{x}+q(t,\lambda)x=0, \quad (\ast) \] where $q(\cdot,\lambda)$ is $T$-periodic and $q\in C^{3}(\mathbb{R}\times [0,\Lambda])$, such that for $\lambda=0$ the equation $(*)$ is parabolic.