So, you are asking "what's the point in investigating chaos dynamics"?
Ok some facts. When you solve f(y)=0, you come up with denegerate solutions y(t)=y0. These solutions are important, because they give us information about the flow associated with the DE - the behaviour of non-degenerate solutions.
As you say, if f(y)>0, then dy/dt>0; So y(t) is increasing, and so the limit of (t,y(t)) when t->t0 exists (maybe infinite, depends on the domain for t.) If y0 is asymptotically stable, then (t,y(t)) will tend to (t0,y0); Isn't it nice to know all solutions tend to a fixed one as time passes? Warms my heart. Too bad "stable points" are really rare :P
If you need to see some examples, a classic one is the logistic equation, and here's some nice stuff concerning its chaotic behaviour.
Actually, I personally enjoy the behaviour of the sexier () Van Der Pol equation.