(*Created by Rafael Luís: rafaelluis@netmadeira.com
 2016-09-11: adapted for Mathematica 10 by Ross Moore *)

Clear[r,a];
lista={ };
R[r_,a_]:=a*r*(1-r);
r0=0.1;
ai=3.57;
af=4;
For [ a=ai,a<af,a+=(af-ai)/2000.0,
	     r=r0;
Do [    r=R[r,a];
	If[ j>80, AppendTo[ lista,{a,r}]] ,{j,1,120} ];
    ];
ListPlot[lista,AxesOrigin->{0,0},
PlotStyle->PointSize[0.001],Frame->True,FrameLabel->{"μ","(Subscript[F, n])"},RotateLabel->False,PlotRange->{{ai,af},{0,1}}]