Symmetric periodic orbits near binary collision in a restricted four-body problem for the figure-eight choreography

We study symmetric periodic orbits near collision in a nonautonomous restricted planar four-body problem. The restricted problem consists of a massless particle moving under the gravitational influence due to three bodies with the same positive mass (the primaries), following the figure-eight choreography. We use regularized coordinates, in order to deal numerically with motions near collision between the massless particle and one of the primaries. By means of reversing symmetries, we characterize the symmetric periodic orbits near collision. The initial conditions for these orbits were computed by solving numerically some boundary value problems. We explain theoretically, and confirm numerically, how different parts of the diagram of initial conditions are related due to the symmetry of the figure-eight choreography.