I am simulating the behavior of a dynamical system, say $$\dot{x} = f(Ax; \lambda), $$ with an Euler update, where $x\in \mathbb{R}^n$ and $\lambda$ are some parameters. In my scenario, $A\in \mathbb{R^{n \times n}}$ is in fact a function of $x(0)$ and $n$ is large enough that computing $A$ is prohibitively expensive. Instead, at each step, I sample a subset of $m\ll n$ indices, $\sigma \subset \{1, \ldots, n \}$, uniformly at random and only compute couplings $A_{\sigma}$ for updating $(x_{\sigma_1}, \ldots, x_{\sigma_m})$.
By very casual visual inspection, the long-term behavior $\lim_{t\to \infty} x(t)$ I get from this updating scheme looks like what I'd expect from the deterministic, synchronous ($m=n$) setting, even when $m/n \approx 1 \times 10^{-3}$. For instance, the dependency of this long-term behavior on the parameters $\lambda$ seems intact, though naturally convergence is much slower when $m$ is small.
My questions are, 1) for which $m$ (or $f, A$) should I expect the long-term behaviors in the synchronous and asynchronous schemes to be similar and in what sense will they be similar? and 2) assuming the similarity is real, is there a body of literature I can read on stochastic, asynchronous methods for solving very high-dimensional ODEs?
