Preamble

(* The following lines of code are based on Debra Lewis' notebooks found at http://people.ucsc.edu/~lewis/Math145/ *)

cobwebLast[f_,start_,range_,n_,plotColor_:Blue]:=With[{values=NestList[f,start,n]},
Show[{
Plot[{x,f[x]},Evaluate[Prepend[range,x]],PlotRange->{range,range}],
ListPlot[ Prepend[Join@@
({#,{Last[#],Last[#]}}&/@Partition[values,2,1]),{start,0}],PlotStyle -> {plotColor} ,Joined-> True,PlotStyle->Dashed,PlotRange->{range,range}],
Graphics[{PointSize[Large],plotColor,Point[{values[[-1]],values[[-1]]}]}]
},AspectRatio->1]
]

eventual[a_]:=Union[Drop[NestList[a #(1-#)&,Random[],1000],900],SameTest->(Chop[#1-#2]<(10^(-4))&)]

logbif[low_,high_,inc_]:=ListPlot[Join@@Table[{a,#}&/@eventual[a],{a,low,high,inc}],PlotStyle->PointSize[Small],PlotRange->{{low,high},{0,1}}]

Logistic Map

f:=Function[x,r x (1-x)]

Animate [cobwebLast[f/. r-> 3.05,.8,{0,1},s,Blue],{s,1,50,1}]

Animate [cobwebLast[f/. r-> 3.5,.8,{0,1},s,Blue],{s,1,50,1}]

Animate [cobwebLast[f/. r->3.9,.5,{0,1},s,Blue],{s,1,2000,1}]

Animate [Show[cobwebLast[f/. r->3.9,.98,{0,1},s,Orange],cobwebLast[f/. r->3.9,.98+0.000000001,{0,1},s,Blue]],{s,1,40,1}]

g1=logbif[1,4,.001]

g1=logbif[2.8,4,.0002];

g2=Plot[1-1/r,{r,1,4},PlotStyle -> {Interpreter["Color"]["orange"]}];

Show[g1,g2]

Plot[{x,Nest[f/. r-> 3,x,1]},{x,0,1}, AspectRatio -> 1]

Plot[{x,Nest[f/. r-> 2,x,2]},{x,0,1}, AspectRatio -> 1]

Plot[{x,Nest[f/. r-> 2.98,x,2]},{x,0,1}, AspectRatio -> 1]

Plot[{x,Nest[f/. r-> 3.5,x,2]},{x,.3,.9}, AspectRatio -> 1]

Plot[{x,Nest[f/. r-> 3.24,x,4]},{x,0,1}, AspectRatio -> 1]

Plot[{x,Nest[f/. r-> 3.4,x,4]},{x,0,1}, AspectRatio -> 1]