Stability Analysis Of The Elliptic Sitnikov Problem Using The Monodromy Matrix And Floquet Multipliers

Abstract

This study investigates the stability and instability of solutions of the Sitnikov problem using the monodromy matrix and Fluquet multiplier approach. The variational equations associated with the periodic
solutions were analysed, and the corresponding monodromy matrix was constructed to determine the Floquet multipliers and assess the stability of the motion. Numerical simulations were carried out in
Mathcad for different values of the orbital eccentricity. The results show that the periodic solution is stable in the circular case, where all Floquet multipliers lie on the unit circle. However, as the eccentricity
increases, the Floquet multipliers gradually move away from the unit circle, indicating a transition from stable to unstable motion. The magnitude of the dominant eigenvalues increases with eccentricity, demonstrating that higher eccentricities lead to stronger instability of the periodic solutions. Phase portraits further illustrate the change in the qualitative behaviour of the solutions as the eccentricity varies. These findings confirm that orbital eccentricity plays a significant role in determining the stability characteristics of the Sitnikov problem and demonstrate the effectiveness of the monodromy matrix and Floquet theory as tools for stability analysis..