Following the previous post Asymptotic behaviour of the solution to some delayed stochastic differential equation I consider a deterministic version (as no answer is received) :
$$\frac{d x^\theta}{d t} =F\big(x^\theta((t-\theta)^+)\big),\quad \forall t>0$$
$$\frac{d x}{d t}=F\big(x(t)\big),\quad \forall t>0,$$
where $\theta\ge 0$, $x^\theta, x$ are functions taking values in $\mathbb R^d$ and $F:\mathbb R^d\to \mathbb R^d$ is bounded and Lipschitz. For any $z\in \mathbb R^d$, the above ODEs are well posed with $x^\theta(0):=z$ and $x(0):=z$. Assume further the solutions $x^\theta, x$ are bounded.
If $\lim_{t\to\infty}x(t)$ exists, under which condition one has $\lim_{t\to\infty}x(t)=\lim_{t\to\infty}x^\theta(t)$? Can we prove the desired equality when $\theta>0$ is small enough or find an counterexample?
