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

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