Suppose we have a parametrized family of ODEs given by
$\displaystyle \frac{dy}{dt} = \mu(y(t)) + \alpha f(t);\quad\quad y(0) = y_0$
where $\mu$ is a well-behaved nonlinear function, $f$ is some bounded oscillating function, say a finite sum of sinusoids, and $\alpha \in \mathbb{R}$. I'm interested in the behaviour of the solution at a fixed time, say $y_{\alpha}(t_1)$, regarded as a function of the scalar $\alpha$.
I must confess I don't know very much about ODEs. I think I took one short course on them as an undergraduate. I'm sure this kind of thing must have been studied quite extensively, but I don't know where to begin reading about it.
If someone could recommend some texts to look at or keywords to search for, I'd be very grateful.