  We analyze the chaotic dynamics of a one-dimensional discrete nonlinear
Schr\"odinger equation. This nonintegrable model, ubiquitous in several fields
of physics, describes the behavior of an array of coupled complex oscillators
with a local nonlinear potential. We explore the Lyapunov spectrum for
different values of the energy density, finding that the maximal value of the
Kolmogorov-Sinai entropy is attained at infinite temperatures. Moreover, we
revisit the dynamical freezing of relaxation to equilibrium, occurring when
large localized states (discrete breathers) are superposed to a generic
finite-temperature background. We show that the localized excitations induce a
number of very small, yet not vanishing, Lyapunov exponents, which signal the
presence of extremely long characteristic time-scales. We widen our analysis by
computing the related Lyapunov covariant vectors, to investigate the
interaction of a single breather with the various degrees of freedom.
