Function Repository Resource:

CobwebPlot

Source Notebook

Visualize one-dimensional iterated functions

Contributed by: Paul Abbott

ResourceFunction["CobwebPlot"][f,x0,n,{a,b}]

plots the real function f along with the diagonal line over the range a to b and, starting from x0, computes n iterates of the function, reflecting each iteration of the function in the diagonal line.

Details and Options

ResourceFunction["CobwebPlot"] has the same options as Graphics.
ResourceFunction["CobwebPlot"] is useful for visualizing the logistic map and other functions.

Examples

Basic Examples (1) 

Display a cobweb plot of 10 iterations of the map starting at x0=2:

In[1]:=
ResourceFunction["CobwebPlot"][x |-> 3 Sin[x], 2, 10, {0, Pi}]
Out[1]=

Options (1) 

Show the cobweb as a dashed red line:

In[2]:=
ResourceFunction["CobwebPlot"][x |-> Abs[x - 2], -7, 60, {-8, 11}, CobwebPlotStyle -> Directive[Dashed, Red]]
Out[2]=

Applications (1) 

For the logistic map, xn+1:=α xn(1-xn), display the cobweb plot of , varying the parameter α, the starting value x0 and the number of iterations n:

In[3]:=
Manipulate[
 ResourceFunction["CobwebPlot"][x |-> \[Alpha] x (1 - x), Subscript[x,
   0], n, {0, 1}, AspectRatio -> Automatic, Frame -> True, Axes -> True, PlotRange -> {{0, 1}, {-0.06, 1.05}}, GridLines -> Automatic, Epilog -> Text["\!\(\*SubscriptBox[\(x\), \(0\)]\)", {Subscript[x, 0], -0.05}]], {\[Alpha], 3, 4, 0.01}, {{Subscript[x, 0], 0.2}, 0.0001, 1}, {{n, 60}, 1, 200, 1}]
Out[3]=

Properties and Relations (1) 

Display the logistic map along with the associated Feigenbaum diagram:

In[4]:=
Manipulate[
 GraphicsColumn[
  {ResourceFunction["CobwebPlot"][x |-> \[Alpha] x (1 - x), Subscript[
    x, 0], n, {0, 1}, AspectRatio -> Automatic, Frame -> True, Axes -> True,
    PlotRange -> {{0, 1}, {-0.06, 1.05}},
    ImageSize -> 300,
    GridLines -> Automatic, Epilog -> Text["\!\(\*SubscriptBox[\(x\), \(0\)]\)", {Subscript[x, 0], -0.05}]],
   Graphics[
    {Raster[ImageData@\!\(\*
GraphicsBox[
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"], {{0, 225}, {450, 0}}, {0, 255},
ColorFunction->GrayLevel],
BoxForm`ImageTag[
           "Byte", ColorSpace -> "Grayscale", Interleaving -> None, Magnification -> Automatic],
Selectable->False],
(ExpressionUUID -> "8ef0f7fd-b9a9-4ecf-bdf5-5aef5c484f96"),
DefaultBaseStyle->"ImageGraphics",
ImageSize->Automatic,
ImageSizeRaw->{450, 225},
PlotRange->{{0, 450}, {0, 225}}]\), {{2, 1}, {4, 0}}],
     {Point[{Dynamic[\[Alpha]], Subscript[x, 0]}], Red, Line[{{Dynamic[\[Alpha]], 0}, {Dynamic[\[Alpha]], 1}}]}},
    ImageSize -> 450,
    Axes -> True,
    AxesLabel -> {"\[Alpha]", "x"}
    ]}
  ],
 {{\[Alpha], 3.7}, 2, 4},
 {{Subscript[x, 0], 0.2}, 0.0001, 1},
 {{n, 60}, 1, 200, 1}
 ]
Out[4]=

Neat Examples (1) 

For integer values the map corresponds to the (reduced) Collatz (or 3X+1) function: if x is even, return x/2, but if x is odd, return (3x+1)/2. For small starting integers n, the Collatz map iterates to the cycle 1↔2 and the cobweb plot converges to the yellow square:

In[5]:=
Manipulate[
 ResourceFunction["CobwebPlot"][
  x |-> x/2 + (x/2 + 1/4) (1 - Cos[\[Pi] x]), n, m, {0, 20}, Frame -> True, Axes -> True, Epilog -> {Text[n, {n, -1/2}], Yellow, Opacity[0.5], Rectangle[{1, 1}]}], {{n, 7}, 1, 20, 1},
 {{m, 5}, 1, 20, 1}]
Out[5]=

Publisher

Paul Abbott

Version History

  • 1.0.0 – 08 February 2021

License Information