This article establishes the foundation for a new theory of
invariant/integral manifolds for non-autonomous dynamical systems. Current
rigorous support for dimensional reduction modelling of slow-fast systems is
limited by the rare events in stochastic systems that may cause escape, and
limited in many applications by the unbounded nature of PDE operators. To
circumvent such limitations, we initiate developing a backward theory of
invariant/integral manifolds that complements extant forward theory. Here, for
deterministic non-autonomous ODE systems, we construct a conjugacy with a
normal form system to establish the existence, emergence and exact construction
of center manifolds in a finite domain for systems `arbitrarily close' to that
specified. A benefit is that the constructed invariant manifolds are known to
be exact for systems `close' to the one specified, and hence the only error is
in determining how close over the domain of interest for any specific
application. Built on the base developed here, planned future research should
develop a theory for stochastic and/or PDE systems that is useful in a wide
range of modelling applications