2o Σχημα

ClearAll["Global`*"]

f[{x_,y_}]:={y+1-a (x^2),b x}

a=1.4;
b=0.3;

orbit1=NestList[f,{0,0},2000000];

ListPlot[Take[orbit1,100000],AxesLabel->{"(Subscript[x, n])","(Subscript[y, n])"},AxesStyle->Thick,TicksStyle->Directive[Plain,10],LabelStyle->Directive[Italic,14],PlotStyle->PointSize[0.001]]

3o Σχημα

orbit2=Cases[orbit1,{x_/;0.55<x<0.75,y_/;0.15<y<0.21}];

ListPlot[orbit2,AxesLabel->{"(Subscript[x, n])","(Subscript[y, n])"},AxesStyle->Thick,TicksStyle->Directive[Plain,10],LabelStyle->Directive[Italic,14],PlotStyle->PointSize[0.0025]]

4o Σχημα

orbit3=Cases[orbit1,{x_/;0.62<x<0.64,y_/;0.185<y<0.191}];

ListPlot[orbit3,AxesLabel->{"(Subscript[x, n])","(Subscript[y, n])"},AxesStyle->Thick,TicksStyle->Directive[Plain,10],LabelStyle->Directive[Italic,14],PlotStyle->PointSize[0.004]]