a=0.2;
modell={r'[t]==r[t] (a-r[t]) (r[t]-1),
        w'[t]==1,
        r[0]==r0, w[0]==0};
tmax=50;
p:={loesung=NDSolve[modell,{r,w},{t,0,tmax},MaxSteps->500];
bild=Plot[Evaluate[{r[t] Cos[w[t]]}/.loesung],{t,0,tmax},
PlotRange->All, DisplayFunction->Identity,
AxesLabel->{"t","x"}]};
r0=0.199; p; bild1=bild;
r0=0.201; p; bild2=bild;
Show[bild1];
Show[bild2];
bild3=Show[bild1,bild2,DisplayFunction->$DisplayFunction,
ImageSize->{400,300}];