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Backwards theory supports modelling via invariant manifolds for non-autonomous dynamical systems

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.