Motion of the two co-orbital bodies (red and blue) and their barycentre (purple) in a co-rotating frame with frequency n. Tadpole (left) and horseshoe (right). δ = 0.6. Here μ = 2 × 10^-4 and the planets are located at 1 AU from the star. By eliminating the influence of n, one can see the long-term motion of the barycentre of the planets. P_ν is the period of the periodic trajectories represented by the coloured lines. See the text for more details.