The time-dependent intramolecular vibrational dynamics of acetylene is investigated with the vibrogram method for the effective Hamiltonian obtained by Abbouti Temsamani and Herman [J. Chem. Phys. 102, 6371 (1995)]. The quantum recurrences evidenced by the vibrogram are shown to be in correspondence with emerging periodic orbits, especially, of the bending type. In this model, we find different kinds of bifurcations and, in particular, a birth of bending local modes as well as further bifurcations causing the destabilization of bending periodic orbits. This destabilization leads to a faster damping of the corresponding recurrences and to an increase in the intramolecular vibrational relaxation. A comparison with monodeuterated acetylene is also carried out.