This paper studies bifurcations in the solution structure of an optimal control problem for mobile robotic formation control. In particular, this paper studies a group of mobile robots operating in a two-dimensional environment. Each robot has a predefined initial state and final state and we compute an optimal path between the two states for every robot. The path is optimized with respect to two factors, the control effort and the deviation from a desired “formation,” and a bifurcation parameter gives the relative weight given to each factor. Using an asymptotic analysis, we show that for small values of the bifurcation parameter (corresponding to heavily weighting the control effort) a single unique solution is expected, and that as the bifurcation parameter becomes large (corresponding to heavily weighting maintaining the formation) a large number of solutions is expected. Between the asymptotic extremes, a numerical investigation indicates a solution bifurcation structure with a cascade of increasing numbers of solutions, reminiscent, but not the same as, period-doubling bifurcations leading to chaos in dynamical systems. Furthermore, we show that if the system is symmetric, the bifurcation structure possesses symmetries, and also present a symmetry-breaking example of a non-holonomic system. Knowledge and understanding of the existence and structure of bifurcations in the solutions of this type of formation control problem are important for robotics engineers because common optimization approaches based on gradient-descent are only likely to converge to the single nearest solution, and a more global study provides a deeper and more comprehensive understanding of the nature of this important problem in robotics.