\(\( (*\ bifurcation\ diagram\ of\ the\ quadratic\ map\ *) 


    f[x_] := 1 - a*x^2\)\)

aset = {{0, 1}}

Union[aset, {{1, 1}}]

Dimensions[{1, 4}][[1]]

\(\(nmin = 110; nmax = 150
\)\)

\(\(
aset = {{0, 1}}; dela = 0.01; 

    Do[a = ll*dela; x[0] = 0.234; Do[x[n + 1] = f[x[n]], {n, 0, nmax}]; 

      	\ \ \ \ \ \ \ set = {x[nmin]}; 
      Do[candidate = x[nmin + mm]; 

        		\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 

        		\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ nn = \(Dimensions[set]\)[\([1]\)]; 

        		\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ bb = 0; 

        		\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  (*
          if\ candidate\ is\ close\ enough\ to

            					\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ an\ elememt\ of\ 
            set, \ bb = 1\ *) 

        		\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
        Do[If[Abs[candidate - set[\([oo]\)]] < 0.01, 

            				\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ bb = 1], {oo, 1, nn}]; 

        		\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
        If[bb == 0, 

          			\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ set = Union[set, {candidate}]]	, 

        		\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {mm, 1, nmax - nmin}]; \ 
\ \ \ \ \ 
      pp = \(Dimensions[set]\)[\([1]\)]; 
	\ \ 
      temp = Table[{a, set[\([qq]\)]}, {qq, 1, pp}]; 
	
      aset = Union[aset, temp], 
{ll, 1, Floor[1.5/dela]}]; 

    ListPlot[aset, PlotRange -> All, Frame -> True]\)\)