Onset of chaotic symbolic synchronization between population inversions in an array of weakly coupled Bose-Einstein condensates.
We investigate the onset of chaotic dynamics of the one-dimensional discrete nonlinear Schrödinger equation with periodic boundary conditions in the presence of a single on-site defect. This model describes a ring of weakly coupled Bose-Einstein condensates with attractive interactions. We focus on the transition to global stochasticity in three different scenarios as the defect is changed. We make use of a suitable Poincaré section and study different families of stationary solutions, where certain bifurcations lead to global stochasticity. The global stochasticity is characterized by chaotic symbolic synchronization between the population inversions of certain pairs of condensates. We have seen that the Poincaré cycles are useful to gain insight in the dynamics of this Hamiltonian system. Indeed, the return maps of the Poincaré cycles have been used successfully to follow the orbit 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 of the matter waves of the condensates.