In[]:=
x0=0.3;f[x_]:=3.2*x*(1-x);iterates=NestList[f,x0,40];Show[Plot[{f[x],x},{x,0,1}],Graphics[Line[Prepend[Riffle[Transpose[{iterates,iterates}],Partition[iterates,2,1]],{x0,0}]]]]
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ListPlot[ParallelTable[Thread[{r,Nest[r#(1-#)&,Range[0,1,0.001],1000]}],{r,0,2.5,0.02}],PlotStylePointSize[0.005]]
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