It is shown that, for pertinent values of the fabrication and control parameters, an attractive finite coupling fixed point emerges in the phase diagram of a Y-junction of superconducting Josephson chains.  The new fixed point arises only when the dimensionless flux f  piercing the central loop of the network equals \pi.  A detailed account of the Josephson current-phase relationship along the arms of the network, near each one of the allowed fixed points, is provided.  The results evidence remarkable similarities between the phase diagram accessible to a Y-junction of superconducting Josephson chains and the one found in the analysis.