F1[x_] = x+(x^2)+λ;

F1[F1[F1[x]]]

ContourPlot[{x+(x^2)+3 λ+(((x+(x^2)+λ))^2)+(((x+(x^2)+2 λ+(((x+(x^2)+λ))^2)))^2)==x},{λ,-3,-2},{x,-3,3},ContourStyle-> {{Red, Thickness[0.005]}},PlotRange-> All, Axes->True, Frame->False, AspectRatio->1,PlotPoints->200]

Solve[x+(x^2)+2 λ+(((x+(x^2)+λ))^2)==x,x]

Expand[F1[F1[-1-(Sqrt[-1-λ])]]]

Expand[F1[F1[-1+(Sqrt[-1-λ])]]]

DF1[x_] = D[F1[x],x]

Simplify[DF1[-1-(Sqrt[-1-λ])] DF1[-1+(Sqrt[-1-λ])] ]

F1m2[x_] = F1[x]/.λ->-9/4

F1m2[-1/2]

Plot[{x,F1m2[x]},{x,-5/2,3/2},PlotRange->{{-5/2,3/2},{-5/2,3/2}},AspectRatio->1]

Plot[{x,F1m2[x],F1m2[F1m2[F1m2[x]]]},{x,-5/2,3/2},PlotRange->{{-5/2,3/2},{-5/2,3/2}},AspectRatio->1]

Plot[{x,F1m2[x],F1m2[F1m2[x]]},{x,-5/2,3/2},PlotRange->{{-5/2,3/2},{-5/2,3/2}},AspectRatio->1]

Plot[{x,F1m2[x],F1m2[F1m2[x]],F1m2[F1m2[F1m2[x]]]},{x,-5/2,3/2},PlotRange->{{-5/2,3/2},{-5/2,3/2}},AspectRatio->1]

Join[{{1,2}},{3,4},2]

Join[{{1,2},{a,b}},{{3,4}},1]

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

Join[Pts,{{1.,1.}},1]

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_] = (x^2)+x+c;

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

ListPlot[PtsG]

ListPlot[PtsG,PlotRange->{{-1.75,-1.5},{-1,-0.3}}]

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