  We demonstrate the existence of localized oscillatory breathers for
quasi-one-dimensional Bose-Einstein condensates confined in periodic
potentials. The breathing behavior corresponds to position-oscillations of
individual condensates about the minima of the potential lattice. We deduce the
structural stability of the localized oscillations from the construction. The
stability is confirmed numerically for perturbations to the initial state of
the condensate, to the potential trap, as well as for external noise. We also
construct periodic and chaotic extended oscillations for the chain of
condensates. All our findings are verified by direct numerical integration of
the Gross-Pitaevskii equation in one dimension.
