Betti numbers under small perturbations

We study how Betti numbers of ideals in a local ring change under small perturbations. Given $p\in \mathbb{N}$ and given an ideal $I$ of a Noetherian local ring $(R,\mathfrak{m})$, our main result states that there exists $N>0$ such that if $J$ is an ideal with $I\equiv J{\mathfrak{m}}^N$ and with the same Hilbert function as $I$, then the Betti numbers ${\beta }_i^R(R\mathrm{/}I)$ and ${\beta }_i^R(R\mathrm{/}J)$ coincide for $0\le i\le p$. Moreover, we present several cases in which an ideal $J$ such that $I\equiv J{\mathfrak{m}}^N$ is forced to have the same Hilbert function as $I$, and therefore the same Betti numbers.