Compactification for asymptotically autonomous dynamical systems: theory, applications and invariant manifolds

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