$C^{k}$ extension and invariant manifolds for the compactification of nonautonomous systems with autonomous limits

We study the compactification of nonautonomous systems with autonomous limits and related dynamics. Although the $C^{1}$ extension of the compactification was well established, a great number of problems arising in bifurcation and stability analysis require the compactified systems with high-order smoothness. Inspired by this, we give a criterion for the $C^{k}$ ($k\geq 2$) extension of the compactification. After compactifying nonautonomous systems, the compactified systems may gain an additional center direction. We prove the existence and uniqueness of center or center-stable manifolds for general compact invariant sets including normally hyperbolic invariant manifolds and admissible sets.