Families of asymmetric periodic solutions in the restricted four-body problem

Abstract

Very recently, we presented five of the basic families of the network of periodic orbits of the restricted four-body problem which are simple, i.e. one intersection with the horizontal $x$ -axis at the half period, symmetric with respect to the same axis and asymmetric with respect to the vertical $y$ -axis. In the present work, using these families, we found series of asymmetric critical orbits for various values of the primaries $m_{2}$ and $m_{3}$ . From these critical orbits we calculate and present five new families of simple periodic orbits which are asymmetric with respect to both the $x$ - and $y$ -axis. Additionally, we describe a grid method in the $(x_{0}, \dot{x}_{0})$ plane and we obtain initial conditions for new asymmetric double-periodic orbits. We determine ten families of asymmetric double-periodic orbits from the bifurcations of the previous five asymmetric families using the special generating horizontally critical periodic orbits. The stability of each calculated asymmetric periodic orbit is also studied. Characteristic curves as well as stability diagrams of these families are illustrated. In the last section we present the evolution of the five basic families of simple asymmetric periodic orbits when the primaries are the Sun the Jupiter and the 2797 Teucer Asteroid.