  This paper studies the concept of stable perturbation
$B\in\mathbb{C}^{n\times n}$ for the core-EP inverse of a matrix
$A\in\mathbb{C}^{n\times n}$ with index $k$. For a given stable perturbation
$B$ of $A$, explicit expressions of its core-EP inverse
$B^{\scriptsize\textcircled{\tiny $\dagger$}}$ and its projection at zero
$B^\pi$ are presented. Then, the perturbation bounds of $\parallel
B^{\scriptsize\textcircled{\tiny $\dagger$}}-A^{\scriptsize\textcircled{\tiny
$\dagger$}}\parallel/\parallel A^{\scriptsize\textcircled{\tiny
$\dagger$}}\parallel$ and $\parallel B^{\pi}-A^{\pi}\parallel$ are given
provided that $B$ is a stable perturbation of $A$. In addition, we investigate
the concept of acute perturbation of $A$. We give a perturbation analysis with
respect to core-EP inverses. We provide a condition under which the acute
perturbation coincides with the stable perturbation for core-EP inverses.
