Periodic solution of circular Sitnikov restricted four-body problem using multiple scales method

The main purpose of present work is to find approximated periodic solution in the compact form of the circular Sitnikov restricted four-body problem by getting liberate of the secular terms using the Multiple Scale Method. Also, we found numerical solutions of the prospective model and compared it with analytically obtained solutions for bounded small amplitude (\(z(0)\le 0.3\)). However, its solution goes away from the actual motion when \(z(0)>0.3\). We observed that the motion obtained by both methods is well ordered and periodical, whenever the infinitesimal body starts moving close to the primary’s common center of mass. Whereas, the motion starts a long way from the center of mass, the second-order approximation results always give a periodic solution and numerical result may not present periodic solution for long time. The initial conditions play most important part in numerical as well as logical system. The Multiple Scale Method is more realistic and accurate than the numerical because it always provide commitment that the motion will be periodic all time. We also discussed and compared an exact solution of this problem with MacMillan problem.