r=2;
f[x_]:=r x(1-x);
u1=0.9;
u2=f[u1];
u3=f[u2];
u4=f[u3];
u5=f[u4];
u6=f[u5];
abb0a=Plot[u,{u,0,0.6},DisplayFunction->Identity];
abb0b=Plot[f[u],{u,0,1},DisplayFunction->Identity];
abb1=ParametricPlot[{u1,u},{u,0,u2},DisplayFunction->Identity];
abb1a=ParametricPlot[{u,u2},{u,u2,u1},DisplayFunction->Identity];
abb1b=ParametricPlot[{u2,u},{u,u2,u3},DisplayFunction->Identity];
abb2a=ParametricPlot[{u,u3},{u,u3,u2},DisplayFunction->Identity];
abb2b=ParametricPlot[{u3,u},{u,u3,u4},DisplayFunction->Identity];
abb3a=ParametricPlot[{u,u4},{u,u4,u3},DisplayFunction->Identity];
abb3b=ParametricPlot[{u4,u},{u,u4,u5},DisplayFunction->Identity];
abb4a=ParametricPlot[{u,u5},{u,u5,u4},DisplayFunction->Identity];
abb4b=ParametricPlot[{u5,u},{u,u5,u6},DisplayFunction->Identity];
abb=Show[{abb0a,abb0b,abb1,abb1a,abb1b,abb2a,abb2b,abb3a,abb3b,abb4a,abb4b},
PlotRange->{{0,1},{0,0.6}},ImageSize->{400,300},DisplayFunction->$DisplayFunction];