  We develop a general compactification framework to facilitate analysis of
nonlinear nonautonomous ODEs where nonautonomous terms decay asymptotically.
The strategy is to compactify the problem: the phase space is augmented with a
bounded but open dimension and then extended at one or both ends by gluing in
flow-invariant subspaces that carry autonomous dynamics of the limit systems
from infinity. We derive the weakest decay conditions possible for the
compactified system to be continuously differentiable on the extended phase
space. This enables us to use equilibria and other compact invariant sets of
the limit systems from infinity to analyse the original nonautonomous problem
in the spirit of dynamical systems theory. Specifically, we prove that
solutions of interest are contained in unique invariant manifolds of saddles
for the limit systems when embedded in the extended phase space. The uniqueness
holds in the general case, that is even if the compactification gives rise to a
centre direction and the manifolds become centre or centre-stable manifolds. A
wide range of problems including pullback attractors, rate-induced critical
transitions (R-tipping) and nonlinear wave solutions fit naturally into our
framework.

Dynamical Systems