  The celebrated Takens' embedding theorem concerns embedding an attractor of a
dynamical system in a Euclidean space of appropriate dimension through a
generic delay-observation map. The embedding also establishes a topological
conjugacy. In this paper, we show how an arbitrary sequence can be mapped into
another space as an attractive solution of a nonautonomous dynamical system.
Such mapping also entails a topological conjugacy and an embedding between the
sequence and the attractive solution spaces. This result is not a
generalization of Takens embedding theorem but helps us understand what exactly
is required by discrete-time state space models widely used in applications to
embed an external stimulus onto its solution space. Our results settle another
basic problem concerning the perturbation of an autonomous dynamical system. We
describe what exactly happens to the dynamics when exogenous noise perturbs
continuously a local irreducible attracting set (such as a stable fixed point)
of a discrete-time autonomous dynamical system.
