Limits of coefficeints in a system of O.D.E.s.

Question

Say that we have a system of O.D.E.s that depend on some real parameter $c$

\begin{equation*} \dot{x}_i = f_i(c,x_1,...,x_n) \ ,\ \ \ i=1,2,...,n \ . \end{equation*} I've not really seen what it means to take a limit of this system before solving it, e.g.

\begin{equation*} \lim_{c \to \infty} \dot{x}_i = f_i(c,x_1,...,x_n) = \ ? \end{equation*} Can this be well-defined? I could imagine something like taking the limit of solution to the original system $x^0_i(t) = \lim_{c \to \infty}x_i(t,c)$ gives you the solution to the limit equations, but could also be persuaded that this not true in general. For example, if we have

\begin{equation*} \dot{x}=ax \ , \ \ \ \dot{y}=y \end{equation*} then this works taking $a \to 0$, but what does taking $a \to \infty$ mean?

Answer 1

No it does not work always. Also, the limit to $∞$ is a bad frame of mind, it is better to formulate the problem in terms of a limit towards $0$.

Then if $f(a,x)$ has a limit $f(0,x)$, then one can use facilities such as the Gronwall lemma to estimate how much solutions to the same initial value differ in the time evolution. One should find that in general for $a\approx 0$ and bounded time intervals, the solutions remain close together.

One special effect that can happen is that $f(0,x)$ has lots of zeros, i.e., stationary points for the flow, that $f(a,x)$ does not have. However, the vector field will still be small in these regions, so that solutions for $a\approx 0$ stay for a good time in the neighborhood of these points.