Long time average of solution to ODE with almost periodic structure

Question

I encountered the following question in my studies: Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a Bohr almost periodic function such that $\inf_{\mathbb{R}} f = 0$ but $f(x) > 0$ for all $x\in \mathbb{R}$. An example is $$ f(x) = 2-\sin(2\pi x) - \sin(2\pi \sqrt{2}x).$$ If $\eta(\cdot)$ is the solution to the following ODE $$ \dot{\eta}(s) = f(\eta(s)), \qquad \eta(0) = 0.$$ Is there any tools that allow us to say something about the limit $$ \lim_{s\rightarrow +\infty} \frac{\eta(s)}{s}$$ and if the limit exists (I guess, by numerical implementations) can we say anything about the rate of convergence of $\frac{\eta(s)}{s}$ to that limit?