The gluing bifurcation: I. Symbolic dynamics of the closed curves

We study the periodic orbits which can occur in a neighbourhood of a codimension-two gluing bifurcation involving two trajectories bi-asymptotic to the same stationary point. Provided some simple conditions are satisfied we prove that there are either zero, one or two closed curves and that these hove a specific symbolic form which, in particular, allows us to associate a rotation number with each of them. Furthermore, pairs of orbits which can coexist are identified: the two rotation numbers must be Farey neighbours.