I'm relatively new to this literature and I'm wondering if there are any papers/books or hints that could help me out with this problem.
Let $A \subset \mathbb{R}^n$ be compact and convex, $\hat A \subset A$ be a finite discretization of $A$, with $|\hat A| = K$ and uniform step size, i.e. $\min_{x,x'\in \hat A, x\neq x'}|| x-x'||_{\infty} = {1\over K}$ . Define $\tau : A \mapsto \hat A$ as a projection from A onto $\hat A$, i.e. mapping points in $A$ to their nearest point in $\hat A$. Given a smooth, bounded and nonlinear $f: A \mapsto A$, I'm wondering whether I can make claims about rest points of a differential system with $f$ restricted to $\hat A$, given knowledge about rest points of the original, unrestricted $f$, as $K$ grows large.
In other words, consider the system \begin{equation}\label{one} \dot x = f(x) -x. \end{equation}
Suppose we know $x^* \in \hat A$ is an isolated rest point of the system that is asymptotically stable. Now consider $\tilde f(u) : A \mapsto \hat A$, such that $$\tilde f(u) = \tau(f(\tau(u)) $$ for all $u \in A$. Let $F_{\tilde f}$ be the Filippov regularization of $\tilde f(u) - u$. In my understanding, this will be an interval at every point where $\tilde f(u) - u$ jumps (with lower and upper bound corresponding to jump "start and end") (correct me if this is very wrong).
Now lets consider the differential inclusion $$ \dot u \in F_{\tilde f}(u) .$$ Since $x^* \in \hat A$, we know $\tau(x^*) =x^*$, and therefore $ F_{\tilde f}(x^*) = 0$. What I'm interested in is if I can make claims about the stability of $x^*$ with respect to solutions of the dynamic system inclusion on $u$. I'm especially interested in if there is a way to show that $x^*$, for some large enough $K$, would be an internally chain transitive set as defined in part 3.3 of