Let $X$ be a vector field on a compact manifold $M$ that has the form $$ X = \lambda_1 x^1 \partial_1 + \dots + \lambda_n x^n \partial_n + \dots$$ with respect to some chart $x$ around a point $p$. Also, let $\lambda_1, \dots, \lambda_n > 0$.

By the stable manifold theorem, there is an $n$-dimensional unstable submanifold $N$ of $M$ around $p$, i.e. a manifold with $\lim_{t \rightarrow -\infty} \Phi_t(p^\prime) = p$ for each $p^\prime \in N$, where $\Phi_t$ is the flow of $X$.

Thus, $N$ is just an open set in $M$. My question is: Is the boundary of $N$ smooth (I suspect yes) and if so, how to prove it or where to look it up?