Multiple families of synodic resonant periodic orbits in the bicircular restricted four–body problem

Abstract

Periodic orbits in gravitational many–body problems have been fundamental special solutions for analyzing phase–space flows not only in autonomous systems but also in time–dependent non–autonomous systems. The growing interest in cislunar space may require deeper understandings of periodic orbits in the Earth–Moon system beyond the simplified dynamics in the circular restricted three–body problem. The present paper deals with two mechanisms of the generation of multiple families of synodic resonant periodic orbits in the bicircular restricted four–body problem through numerical examples adopting planar and three–dimensional retrograde periodic orbits around the Earth. The first part of the paper reveals complex interplays between a synodic resonant periodic orbit and multi–revolutional orbits originated from period–multiplying bifurcations. Extensions of continuation curves beyond the usual limit find another 12:11 synodic resonant planar retrograde periodic orbit around the Earth in addition to the one already found in the previous work. The second part shows that a doubly symmetric periodic orbit in the circular restricted three–body problem can generate four families of synodic resonant periodic orbits in the bicircular restricted four–body problem. This work identifies linearly stable, 1:1 synodic resonant, three–dimensional retrograde periodic orbits around the Earth and indicates their substantially large stability regions under solar gravitational perturbations.