The Closing Lemma by Pugh states:
Let $\mathfrak{X}$ be the space of differentiable tangent vector fields on a differentiable manifold $M^n$ under the $C^1$ topology. Suppose that $X \in \mathfrak{X}$ has a nontrivial recurrent trajectory through $p \in M^n$ and suppose that $\mathcal{U}$ is a neighborhood of $X$ in $\mathfrak{X}$. Then there exists $Y \in \mathcal{U}$ such that $Y$ has a closed orbit through $p$.
Let's assume instead that there exists a equilibrium $p$ for $X$ with a corresponding homoclinic orbit 'around' $p$. Is there any results similar to Pugh's closing lemma that states that there is a vector field $Y$ close enough to $X$ (under an appropriate topology) that has a closed orbit through $p$?
Maybe there is an extension for chain recurrence?