# Parametrized ODEs

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.

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You haven't said what aspects of the "behaviour" you want to know about, so it's rather hard to answer in this generality. One approach that is sometimes useful: the partial derivative $p(t) = \frac{\partial}{\partial \alpha} y(t)$ satisfies an initial value problem which is linear (given $y(t)$): $p'(t) = \mu'(y(t)) p(t) + f(t)$, $p(0) = 0$. So e.g. with appropriate bounds on $\mu'$ and $f$ you might get useful bounds on $p(t)$. –  Robert Israel May 12 '11 at 22:45
Thanks, that's a good point. I was a little vague about what I'm looking for because I don't know, exactly. I just want to understand this thing better, and I thought there may be some systematic treatment in the literature. –  Simon May 12 '11 at 22:56

You could try to use sub and supersolutions. Let $M=\sup\{f(t):0\le t\le t_1\}$ and $m=\inf\{f(t):0\le t\le t_1\}$, and let $y^+$, $y^-$ be the solutions of the differential equations $z'=\mu(z)+\alpha M$, $z'=\mu(z)+\alpha m$, with initial condition $z(0)=y_0$. Then $y^-(t_1)\le y_\alpha(t_1)\le y^+(t_1)$.
You can obtain estimates on $y^+(t_1)$ and $y^-(t_1)$ from the identities $$\int_{y_0}^{y^+(t_1)}\frac{dz}{\mu(z)+\alpha M}=\int_{y_0}^{y^-(t_1)}\frac{dz}{\mu(z)+\alpha m}=t_1,$$ obtained by integration of the equations $z'=\mu(z)+\alpha M$ and $z'=\mu(z)+\alpha m$ from $t=0$ to $t=t_1$.