  A novel type of trajectory on semiflows, called asymptotically unpredictable,
was proposed by Fen and Tokmak Fen [15]. The presence of sensitivity, which is
an indispensable feature of chaotic dynamics, is a crucial property that arises
from such trajectories. In the present paper, we show the existence and
uniqueness of asymptotically unpredictable solutions for quasilinear systems
with delay making benefit of the contraction mapping principle. Additionally,
we introduce the notion of an asymptotically unpredictable sequence. It is
verified that there exist asymptotically unpredictable sequences which are not
unpredictable. Discrete-time equations possessing asymptotically unpredictable
orbits are also under investigation. Examples of continuous-time and
discrete-time systems with asymptotically unpredictable solutions are provided.
