Stability and bifurcations of symmetrical 3D-periodic orbits in the Sitnikov five-body problem

Abstract

We have studied the variations in stability, bifurcation, critical velocity, Lagrange points, and critical periodic orbits in the context of the transition from the Sitnikov four-body system (as analyzed by Soulis et al. (Celest. Mech. Dyn. Astron. 100:251–266, 2008)) to the Sitnikov five-body system. The incorporation of one primary body results in a reduction in critical velocity and an increase in the number of stability intervals. We applied Floquet’s theory to study the stability/instability of the motion of negligible mass. For this, we assume \(z_{in}\) as family parameter and vary it in the interval \([0, 10]\). Upon slightly perturbing the negligible mass from the z-axis, we obtained 13 Lagrange points. We have determined three-dimensional families of periodic orbits which bifurcate from the critical points/bifurcation points. We observed that the bifurcation points lie within the interval \([1.0709360, 2.7944120]\). We have discussed stability/instability of periodic orbits bifurcating from the critical points.