We investigate the onset of chaotic dynamics of the one-dimensional discrete nonlinear Schrödinger equation (DNLSE) with periodic boundary conditions in the presence of a single on-site defect. This model describes a ring of weakly-stochasticity in three different scenarios. We make use of a suitable Poincaré surface of section and continuation methods. The global stochasticity is characterized by chaotic symbolic synchronization between the population inversions of certain pairs of condensates. We were able to follow the motion along the stochastic layers of different resonances in the chaotic self-trapping regime. Moreover, the time series of the Poincaré cycles suggests that in the global stochasticity regime the dynamics is, to some extent, Markovian, in spite of the fact that the condensates are phase locked with almost the same phase. This phase locking induces a peculiar local interference in the matter waves of the condensates.