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

TL;DR

This paper develops criteria for high-order smooth compactifications of nonautonomous systems with autonomous limits, enabling advanced bifurcation and stability analysis through invariant manifold theory.

Contribution

It introduces a criterion for $C^{k}$ compactification of nonautonomous systems, extending previous $C^{1}$ results, and proves the existence and uniqueness of invariant manifolds in the compactified setting.

Findings

Established $C^{k}$ extension criteria for compactification

Proved existence and uniqueness of invariant manifolds

Enhanced tools for bifurcation and stability analysis

Abstract

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.