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

Pts = {{0,0}};
For[k=-1.31068, k≥-1.31072,k=k-0.0000001,
{y = 0.0;
For[r=1,r<600,r++,
{y=F[y,k];
If[r>550, 
Pts = Join[Pts,{{k,y}},1];
];}
]}
]

Plot1 = ListPlot[Pts,PlotStyle->{{Red}}]

ListPlot[Pts,PlotStyle->{{Red}}, PlotRange->{{-1.3107028,-1.3107025},{-0.000001,0.000001}}]

Pts2 = {{0,0}};
For[k=-1.305, k≥-1.315,k=k-0.00001,
{y = 0.0;
For[r=1,r<600,r++,
{y=F[y,k];
If[r>550, 
Pts2 = Join[Pts2,{{k,y}},1];
];}
]}
]

ListPlot[Pts2,PlotStyle->{{Red}}, PlotRange->{{-1.3108,-1.3106},{-0.001,0.001}}]

ListPlot[Pts,PlotStyle->{{Red}}, PlotRange->{{-1.39,-1.38},{-0.1,0.1}}]

Pts3 = {{0,0}};
For[k=-1.381, k≥-1.383,k=k-0.000001,
{y = 0.0;
For[r=1,r<600,r++,
{y=F[y,k];
If[r>550, 
Pts3 = Join[Pts3,{{k,y}},1];
];}
]}
]

ListPlot[Pts3,PlotStyle->{{Red}}, PlotRange->{{-1.381549,-1.381547},{-0.0001,0.0001}}]

1/(-1+1.3107)

(-1+1.3107)/(-1.3107+1.381548)

Plot2 = ContourPlot[{(x^2)+c==x},{c,-2,0},{x,-2,2},ContourStyle-> {{Blue, Thickness[0.01]}}, Axes->True, Frame->False, AspectRatio->1, AxesLabel->{"c","x"}, PlotRange->{{-2,0},{-2,2}}]

Show[Plot1,Plot2]

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}}, AxesOrigin->{1.1,1/2}]