Phase portrait of Eq. (5). The separatrix (black curve) splits the phase space in two different domains: inside the separatrix the region associated with the tadpole orbits (in red) and the horseshoe domain (blue orbits) outside. The phase portrait is symmetric with respect to ζ = 180°. The horizontal purple segment indicates the range of variation of ζ₀ while the vertical one shows the section used as initial condition to draw Fig. 5. See the text for more details.