It is telling you what you suspected, to wit, that lthe effect in a sense widens over time (very literally I suppose, since it is changing the frequency). You can see the same, in this simple example, from the analytic solution.

DSolve[{y''[t] + a y[t] == 0, y[0] == b, y'[0] == 0},  y[t], t]

(* Out[272]= {{y[t] -> b Cos[Sqrt[a] t]}} *)

One thing though is that using the first order change in your plot means it also strays over time. The variant below might be a better way to go.

Plot[Evaluate[{y[1.1, 1][t], y[1.0, 1][t], y[0.9, 1][t]} /. sol], {t, 
  0, 20}, Filling -> {2 -> {3}}]

Among other things, the peak magnitudes do not (incorrectly) change with variation in the a parameter.