Consider a $C^r$ vector field $f \colon \mathbb{R}^n \to \mathbb{R}^n$ with $r \geq 1$. Let $\bar x$ be a critical point of $f$, that is, $f(\bar x) = 0$. Suppose that the spectrum of $\mathrm{D}f(\bar x)$ has $k$ eigenvalues with zero real part and $n - k$ eigenvalues with non-zero real part. Then we know that there exists at least one $k$-dimensional center manifold at $\bar x$.
My question is: what regularity can we guarantee for this center manifold?
In the book "Differential equations and dynamical systems" by Perko, Theorem 1 in Section 2.12 ensures that there exists a $C^r$ center manifold. However the book does not provide a proof of this statement and it cites "Applications of centre manifold theory" by Carr instead. But in that article the theorem requires $r \geq 2$ so it does not apply when $f$ is only $C^1$.
Moreover, in the Wikipedia page center manifold and all references therein, the theorems only guarantee a $C^{r - 1}$ center manifold. That is: we lose 1 order of regularity.
Where is this mismatch coming from? I have read in some other references that there are many center manifolds, but only one of them is $C^r$. Is it indeed the case?
Thanks for your time reading this question.
