F[x_, c_] = (x^2)+c;

Pts = {{0,0}};
For[k=0, k≥-2,k=k-0.001,
{y = 0.0;
For[r=1,r<600,r++,
{y=F[y,k];
If[r>550, 
Pts = Join[Pts,{{k,y}},1];
];}
]}
]

ListPlot[Pts,PlotStyle->{{Red}}]

G[x_, c_] = c x (1-x);

PtsG = {{0,0}};
For[k=1, k≤4,k=k+0.002,
{y = 0.5;
For[r=1,r<600,r++,
{y=G[y,k];
If[r>550, 
PtsG = Join[PtsG,{{k,y}},1];
];}
]}
]

ListPlot[PtsG,PlotRange->{{1,4},{0,1}}]

H[x_, c_] = c(2(x^2)-4 x + 1);

Solve[H[(1+4c)/(4c),c]==(1+4c)/(4c),c]

N[{{c->(1/4) (-2-(Sqrt[2]))},{c->(1/4) (-2+(Sqrt[2]))}}]

PtsH = {{0,0}};
For[k=N[(1/4) (-2-(Sqrt[2]))], k≥   -1.618,k=k-0.0002,
{y = 1;
For[r=1,r<600,r++,
{y=H[y,k];
If[r>550, 
PtsH = Join[PtsH,{{k,y}},1];
];}
]}
]

ListPlot[PtsH, PlotRange->{{-1.618,-0.853},{0,2}}]

K[x_, c_] = (E^(-8 (x^2)))+c;

PtsK = {{0,0}};
For[k=1., k≥   -1.,k=k-0.001,
{y = 0;
For[r=1,r<600,r++,
{y=N[K[y,k]];
If[r>550, 
PtsK = Join[PtsK,{{k,y}},1];
];}
]}
]

ListPlot[PtsK]