Period Doubling and Renormalization in the Logistic Family

Example of Period Doubling Bifurcation

Here is the function we will be looking at.

F[μ_,x_]:=μ x (1-x)

A period doubling bifurcation occurs at (Subscript[μ, 0])=3.

μ =1
Panel[Column[{Row[{Slider[Dynamic[μ],{1,4}]," μ="Dynamic[μ]}],Dynamic[Plot[{x,F[μ,x]},{x,0,1},AspectRatio->Automatic,PlotRange->{0,1},ImageSize->{500,500}]]}]]

Same diagram, but now with the squared map drawn:

μ=1;
Panel[Column[{Row[{Slider[Dynamic[μ],{1,4}]," μ="Dynamic[μ]}],Dynamic[Plot[{x,F[μ,x],F[μ,F[μ,x]]},{x,0,1},AspectRatio->Automatic,PlotRange->{0,1},ImageSize->{500,500}]]}]]

P[μ_]:=((μ-1)/μ)

Rect[μ_]:=Graphics[{EdgeForm[Directive[Blue]],Line[{{1-P[μ],1-P[μ]},{1-P[μ],P[μ]},{P[μ],P[μ]},{P[μ],1-P[μ]},{1-P[μ],1-P[μ]}}]}]
μ=0;
Panel[Column[{Row[{Slider[Dynamic[μ],{0,4}]," μ="Dynamic[μ]}],Dynamic[{
Show[
Plot[{x,F[μ,x],F[μ,F[μ,x]]},{x,0,1},AspectRatio->Automatic,PlotRange->{0,1},ImageSize->{500,500}],
Rect[μ]
]}
]}]]

Renormalization:

When μ>3, we can consider the action of (Subscript[F, μ]^2) on the interval [1-(Subscript[p, μ]),(Subscript[p, μ])]. We conjugate it under the linear map which sends (Subscript[p, μ]) to zero.

L[μ_,x_]:=P[μ]+x*(1-2*P[μ])
Linv[μ_,y_]:=((-1+μ-y μ)/(-2+μ))

RF[μ_,x_]:=Linv[μ,F[μ,F[μ,L[μ,x]]]]

This is the graph of the renormalized map:

μ =2.01;
Panel[Column[{Row[{Slider[Dynamic[μ],{2.01,4}]," μ="Dynamic[μ]}],Dynamic[Plot[{x,RF[μ,x]},{x,0,1},AspectRatio->Automatic,PlotRange->{0,1},ImageSize->{500,500}]]}]]

μ =2.01;
Panel[Column[{Row[{Slider[Dynamic[μ],{2.01,4}]," μ="Dynamic[μ]}],Dynamic[Plot[{x,RF[μ,x],RF[μ,RF[μ,x]]},{x,0,1},AspectRatio->Automatic,PlotRange->{0,1},ImageSize->{500,500}]]}]]