Quantitative stability of certain families of periodic solutions in the Sitnikov problem

TL;DR

This paper introduces new methods to analyze the stability and bifurcation of periodic solutions in the Sitnikov problem, extending previous continuation results to include stability properties for solutions with varying eccentricity.

Contribution

It develops two general approaches to estimate solution growth and stability in parametric differential equations, applied to the Sitnikov problem's periodic solutions.

Findings

Quantifies bifurcating families of solutions.

Provides stability criteria for Hill's equations.

Extends stability analysis beyond known continuation results.

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.