Families of periodic solutions of the 4- and 6-body problem using a gradient-free continuation method

In this paper, we describe a gradient-free method to solve a system of equations, and we use it to construct two families of pseudo-periodic planar solutions of the 4- and 6-body problem. The method is a stochastic black-box procedure that uses only function evaluations. For the 4-body problem, bodies 1 and 2 have mass 1 and move opposite to each other, and bodies 3 and 4 have mass $m_2$ and also move opposite to each other. For the 6-body problem, bodies 1, 2, and 3 have mass 1 and move on the vertices of an equilateral triangle centered at the origin, and bodies 4, 5, and 6 have mass $m_2$ and also move on the vertices of an equilateral triangle. In both cases, we compute families of periodic solutions by imposing return conditions up to rotation and relabeling.