  We study the effect of atomic interaction on the localization and the
associated dynamics of Bose-Einstein condensates in a one-dimensional
quasiperiodic optical lattice and random Gaussian disordered potentials. When
the interactions are absent, the condensates exhibit localization, which
weakens as we increase the interaction strength beyond a threshold value for
both potential types. We inspect the localized and delocalized states by
perturbing the system via quenching the interaction strength instantaneously to
zero and studying the dynamics of the condensate, which we further corroborate
using the out-of-time-order correlator. The temporal behaviour of the time
correlator displays regular dynamics for the localized state, while it shows
temporal chaos for the delocalized state. We confirm this dynamical behaviour
by analyzing the power spectral density of the time correlator. We further
identify that the condensate admits a quasiperiodic route to chaotic dynamics
for both potentials. Finally, we present the variation of the maximal Lyapunov
exponents for different nonlinearity and disorder strengths that have a
positive value in the regime where the time correlator function shows chaotic
behaviour. Through this, we establish the strong connection between the
spatially delocalized state of the condensate and its temporal chaos.
