Connection between three-body configuration and four-body configuration of the Sitnikov problem when one of the masses approaches zero: circular case

In this manuscript we have established averaged equation of motion of the Sitnikov restricted three- body and four-body problem when all the primaries are point masses, by applying the Van der Pol transformation and averaging technique of J. Guckenheimer and P. Holmes (in Nonlinear Oscillations, Dynamical System Bifurcations of Vector Fields, Springer, Berlin, 1983). In addition to the resonance criterion at the 3/2 commensurability we have chosen ω=2n/3,n=4, ω is the angular velocity of the coordinate system. Further we established the Series solution of the three-body and four-body problem in the sense of Sitnikov. Lastly the periodicities of the solutions have been examined by the Poincare section and four-body and three-body problem have been compared by different comparative graphs and surfaces.