Manipulate[With[{nsteps=num},
Module[{func},
func[x_]:=1-((Abs[2x-1])^(1/2));
Show[Plot[func[x],{x,0,1},PlotStyle->Directive[Red,Thin],PlotRange->{{0,1},{0,1}},Frame->True,PlotRangePadding->Scaled[0.02],PlotLabel->Style["(Subscript[z, n + 1]) = 1-|2(Subscript[z, n])-1((|)^(1/2))",FontSize->14],FrameTicksStyle->Directive[FontSize->14],GridLines->Automatic,FrameLabel->(Style[#,FontSize->16]&/@{"(Subscript[z, n])","(Subscript[z, n + 1])"})],Plot[x,{x,0,1},PlotStyle->Directive[Black,Thin]],Graphics@With[{ld=NestList[func[#]&,init,nsteps]},{PointSize[0.02],Point[{init,0}],Line[{{init,0},{init,ld[[2]]}}],({PointSize[0.02],Point[{ld[[#]],ld[[#+1]]}],Line[{{ld[[#]],ld[[#+1]]},{ld[[#+1]],ld[[#+1]]}}],Line[{{ld[[#+1]],ld[[#+1]]},{ld[[#+1]],ld[[#+2]]}}]})&/@Range[Length[ld]-2]}]]
]
]
,{{num,4,Style["No. of Steps",FontSize->14]},1,100,1},{{init,0.1,Style["Starting value",FontSize->14]},0,1,0.05},FrameLabel->{{None,None},{None,Style["A mathematical model of Lorenz's map",FontSize->16]}}]