# On the Stability of Symmetric Periodic Orbits of the Elliptic Sitnikov Problem

## TL;DR

This paper analytically investigates the stability of symmetric periodic orbits in the elliptic Sitnikov problem, revealing that odd solutions are unstable and even solutions are stable for small eccentricities, filling a gap in existing research.

## Contribution

It introduces a new stability criterion for odd symmetric periodic solutions and applies it to establish the stability properties of solutions in the elliptic Sitnikov problem.

## Key findings

- Odd $(2p,p)$-periodic solutions are hyperbolic and Lyapunov unstable for small eccentricity.
- Even $(2p,p)$-periodic solutions are elliptic and linearly stable.
- First analytical results on the stability of nonconstant periodic orbits in this problem.

## Abstract

Motivated by the recent works on the stability of symmetric periodic orbits of the elliptic Sitnikov problem, for time-periodic Newtonian equations with symmetries, we will study symmetric periodic solutions which are emanated from nonconstant periodic solutions of autonomous equations. By using the theory of Hill's equations, we will first deduce in this paper a criterion for the linearized stability and instability of periodic solutions which are odd in time. Such a criterion is complementary to that for periodic solutions which are even in time, obtained recently by the present authors. Applying these criteria to the elliptic Sitnikov problem, we will prove in an analytical way that the odd $(2p,p)$-periodic solutions of the elliptic Sitnikov problem are hyperbolic and therefore are Lyapunov unstable when the eccentricity is small, while the corresponding even $(2p,p)$-periodic solutions are elliptic and linearized stable. These are the first analytical results on the stability of nonconstant periodic orbits of the elliptic Sitnikov problem.