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Notebook[{

Cell[CellGroupData[{
Cell["The Logistic Map", "Title"],

Cell["R.G. Palmer, 9/10/99", "Subsubtitle"],

Cell[CellGroupData[{

Cell["Background and Copyright", "Subsection"],

Cell[TextData[{
  "This ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " Notebook was written by Richard G. Palmer (Physics Department, Duke \
University) for use in a course he taught.  As of 1999, it has been made \
available for general non-profit use under the following copyright \
provision."
}], "Text"],

Cell[TextData[{
  StyleBox["This Mathematica Notebook is Copyright Richard G. Palmer, 1997",
    FontWeight->"Bold"],
  ".  It may be freely used by individuals, and by classes at academic \
institutions, provided:\n1. Credit is given to Richard Palmer as the original \
author; and\n2. It is not bought or sold or exchanged for profit, or \
incorporated into material that is bought or sold or exchanged for profit.\n\
Any other use requires the written permission of Richard Palmer, Dept. of \
Physics, Box 90305, Duke University, Durham, NC 27708, USA.",
  "See ",
  StyleBox["http://www.phy.duke.edu/~palmer",
    FontFamily->"Courier",
    FontSize->12,
    FontWeight->"Bold"],
  " for the email address."
}], "Text"],

Cell[TextData[{
  "See ",
  StyleBox["http://www.phy.duke.edu/~palmer/notebooks/",
    FontFamily->"Courier",
    FontSize->12,
    FontWeight->"Bold"],
  " for other ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " noteboks by Richard Palmer."
}], "Text"],

Cell["I would like to thank Joshua Socolar.", "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["Introduction", "Section"],

Cell["The logistic map is a simple equation", "Text"],

Cell["y := a y (1-y)", "Text",
  TextAlignment->Center],

Cell["\<\
that computes a new value of \"y\" from a previous one.  \"a\" is \
just a parameter.  If you fix \"a\" and start from an initial value of \"y\", \
you can use the logistic map to produce a whole sequence of values of \
subsequent \"y\" values.\
\>", "Text"],

Cell[TextData[{
  "That sounds pretty boring, but in fact is rather important because the \
sequence is ",
  StyleBox["chaotic",
    FontSlant->"Italic"],
  " for certain values of \"a\".  \"chaotic\" is a technical word with a \
precise meaning that we'll come to later."
}], "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["Time Series", "Section"],

Cell["\<\
Let's try it.  Here's one way to compute and print out 20 \
successive \"y\" values.  I set \"a\" to 3.5 and started off \"y\" at 0.4.  \
If you've done the Mandelbrot notebook you should have no trouble \
understanding this program.  Evaluate it.\
\>", "Text"],

Cell[BoxData[{
    \(\(a\  = \ 3.5;\)\), "\n", 
    \(\(y\  = \ 0.4;\)\), "\n", 
    \(Do[\n\ty\  = \ a\ y\ \((1 - y)\); \n\tPrint[y], \n\t{20}\n]\)}], "Input"],

Cell[TextData[{
  "Hmm, what's happening?  It's a bit hard to tell.  Let's make a picture, a \
graph of the \"y\" values against the iteration number.  Thinking of the \
iterations as happening through time, this is called a ",
  StyleBox["time series",
    FontSlant->"Italic"],
  "."
}], "Text"],

Cell[TextData[{
  "First we need to modify the above program to make a list of values instead \
of printing them out.  Then we can plot the list.  We can make a list by \
changing ",
  StyleBox["Do[]",
    FontWeight->"Bold"],
  " to ",
  StyleBox["Table[]",
    FontWeight->"Bold"],
  ", and removing the ",
  StyleBox["Print[]",
    FontWeight->"Bold"],
  ":"
}], "Text"],

Cell[BoxData[{
    \(\(a\  = \ 3.5;\)\), "\n", 
    \(\(y\  = \ 0.4;\)\), "\n", 
    \(Table[\n\ty\  = \ a\ y\ \((1 - y)\), \n\t{20}\n]\)}], "Input"],

Cell[TextData[{
  "In general ",
  StyleBox["Table[",
    FontWeight->"Bold"],
  StyleBox["expression",
    FontWeight->"Bold",
    FontSlant->"Italic"],
  StyleBox[", {n}]",
    FontWeight->"Bold"],
  " evaluates ",
  StyleBox["expression",
    FontWeight->"Bold",
    FontSlant->"Italic"],
  " n times and puts the n results in a list.  A list in ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " is a set of items separated by commas and enclosed in curly brackets."
}], "Text"],

Cell[TextData[{
  "To plot a list we can use ",
  StyleBox["ListPlot[]",
    FontWeight->"Bold"],
  ":"
}], "Text"],

Cell[BoxData[
    \(\(?ListPlot\)\)], "Input"],

Cell[TextData[{
  "Read that carefully.  We're going to use the first version, ",
  StyleBox["ListPlot[{y1, y2, ... }]",
    FontWeight->"Bold"],
  " for now, and the other version,  ",
  StyleBox["ListPlot[{{x1, y1}, {x2, y2}, ... }]",
    FontWeight->"Bold"],
  " later."
}], "Text"],

Cell[TextData[{
  "I'll just put the main part of the list-making program inside ",
  StyleBox["ListPlot[]",
    FontWeight->"Bold"],
  ".  I'll also add a ",
  StyleBox["PlotJoined->True",
    FontWeight->"Bold"],
  " option to make ",
  StyleBox["ListPlot",
    FontWeight->"Bold"],
  " join up the points with lines:"
}], "Text"],

Cell[BoxData[{
    \(\(a\  = \ 3.5;\)\), "\n", 
    \(\(y\  = \ 0.4;\)\), "\n", 
    \(ListPlot[Table[y\  = \ a\ y\ \((1 - y)\), {20}], 
      PlotJoined -> True]\)}], "Input"],

Cell["\<\
Do you see what's happening?  It settles down to a four-cycle, in \
which every fourth \"y\" value is approximately equal.  Look back at the \
values we printed out above to see that there too.  But it's much easier to \
notice with a picture.\
\>", "Text"],

Cell[TextData[{
  "The above four-cycle behavior is ",
  StyleBox["not",
    FontSlant->"Italic"],
  " chaotic.  It's regular and periodic."
}], "Text"],

Cell["\<\
Here's a slightly improved version of the program.  I made it do \
100 initial iterations before making the list, just so that the initial \
settling-down transients don't confuse us.  I also changed from 20 to 40 \
points. \
\>", "Text"],

Cell[BoxData[{
    \(\(a\  = \ 3.5;\)\), "\n", 
    \(\(y\  = \ 0.4;\)\), "\n", 
    \(\(Do[y\  = \ a\ y\ \((1 - y)\), {100}];\)\), "\n", 
    \(ListPlot[Table[y\  = \ a\ y\ \((1 - y)\), {40}], 
      PlotJoined -> True]\)}], "Input"],

Cell["\<\
Now let's try some different values of \"a\".  This should give a \
two-cycle:\
\>", "Text"],

Cell[BoxData[{
    \(\(a\  = \ 3.2;\)\), "\n", 
    \(\(y\  = \ 0.4;\)\), "\n", 
    \(\(Do[y\  = \ a\ y\ \((1 - y)\), {100}];\)\), "\n", 
    \(ListPlot[Table[y\  = \ a\ y\ \((1 - y)\), {40}], 
      PlotJoined -> True]\)}], "Input"],

Cell["This should give an 8-cycle -- look carefully:", "Text"],

Cell[BoxData[{
    \(\(a\  = \ 3.56;\)\), "\n", 
    \(\(y\  = \ 0.4;\)\), "\n", 
    \(\(Do[y\  = \ a\ y\ \((1 - y)\), {100}];\)\), "\n", 
    \(ListPlot[Table[y\  = \ a\ y\ \((1 - y)\), {40}], 
      PlotJoined -> True]\)}], "Input"],

Cell["And now, dah dah, dah dah, a chaotic case:", "Text"],

Cell[BoxData[{
    \(\(a\  = \ 3.9;\)\), "\n", 
    \(\(y\  = \ 0.4;\)\), "\n", 
    \(\(Do[y\  = \ a\ y\ \((1 - y)\), {100}];\)\), "\n", 
    \(ListPlot[Table[y\  = \ a\ y\ \((1 - y)\), {40}], 
      PlotJoined -> True]\)}], "Input"],

Cell[TextData[{
  "It's not regular at all, and ",
  StyleBox["never",
    FontSlant->"Italic"],
  " settles down to be so.  That's one of the characteristics of chaos."
}], "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["Sensitive Dependence on Initial Conditions", "Section"],

Cell[TextData[{
  "The characteristic, one that's often used to ",
  StyleBox["define",
    FontSlant->"Italic"],
  " chaos, is ",
  StyleBox["sensitive dependence on initial conditions, ",
    FontSlant->"Italic"],
  "sometimes called the",
  " \"butterfly effect\".  That means that if you change the starting point \
just slightly (as a butterfly flapping its wings changes the wind pattern) \
you get a totally different sequence after a while."
}], "Text"],

Cell["\<\
Also try the same change on one of the periodic cases above; \
they're not chaotic, and nearby starting values should give nearby sequences:\
\
\>", "Text"],

Cell["\<\
Try it.  Here are two sequences starting from slightly different \
initial values of y, with the parameter \"a\" chosen to yield a periodic \
sequence.\
\>", "Text"],

Cell[BoxData[
    \(a\  = \ 3.5; \ny\  = \ 0.4; \n
    ListPlot[Table[y\  = \ a\ y\ \((1 - y)\), {50}], PlotJoined -> True]\)], 
  "Input"],

Cell[BoxData[
    \(a\  = \ 3.5; \ny\  = \ 0.400001; \n
    ListPlot[Table[y\  = \ a\ y\ \((1 - y)\), {50}], PlotJoined -> True]\)], 
  "Input"],

Cell["Note that the sequences look very similar.", "Text"],

Cell["\<\
Now let's try the same thing, but with \"a\" set so the behavior is \
chaotic.  \
\>", "Text"],

Cell[BoxData[
    \(a\  = \ 3.9; \ny\  = \ 0.4; \n
    ListPlot[Table[y\  = \ a\ y\ \((1 - y)\), {50}], PlotJoined -> True]\)], 
  "Input"],

Cell[BoxData[
    \(a\  = \ 3.9; \ny\  = \ 0.400001; \n
    ListPlot[Table[y\  = \ a\ y\ \((1 - y)\), {50}], PlotJoined -> True]\)], 
  "Input"],

Cell["You get a totally different sequence after a while!", "Text"],

Cell[TextData[{
  "To see what is happening more clearly, we can plot the difference between \
the two sequences instead of the actual values separately.  The following \
function does this.  Evaluate the next cell, but don't try to understand it \
the ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  ".  It won't produce any output."
}], "Text"],

Cell[BoxData[
    \(timeerror[a_, y_, de_] := 
      Module[{r = y, s = y + de}, \n
        ListPlot[
          Table[r\  = \ a\ r\ \((1 - r)\), {50}] - 
            Table[s\  = \ a\ s \((1 - s)\), {50}], PlotJoined -> True, 
          PlotRange -> {\(-1\), 1}]]\)], "Input"],

Cell["\<\
We can now run the function, generating a plot of the difference \
between sequences starting at 0.4 and 0.4001.  Evaluate the next cell.\
\>", \
"Text"],

Cell[BoxData[
    \(timeerror[3.9, 0.4, 0.001]\)], "Input"],

Cell[TextData[{
  StyleBox["Change the 0.001 in the previous cell to smaller values ",
    FontColor->RGBColor[1, 0, 0]],
  "(0.0001, 0.00001, ...). What do you observe?  ExponentialGrowth.nb \
explores this issue in more detail."
}], "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["Graphical Representation", "Section"],

Cell["\<\
How can we understand what's happening?  At one level we could just \
say, well, that's what this equation does!  But it would be nice to get some \
deeper understanding of why it does what it does -- why it can have 2-cycles, \
4-cycles, etc, and why it can go chaotic.  To gain a full (deep) \
understanding of all this would take too long, but we can at least get a \
better feeling for it by making a graphical representation of the \
iteration.\
\>", "Text"],

Cell[TextData[{
  "First let's look at the effect of taking one step in the iteration.  \
Here's a picture of the function ",
  StyleBox["a y (1-y)",
    FontFamily->"Courier",
    FontSize->12,
    FontWeight->"Bold"],
  ", for a=2.8 (a smaller value than we tried before), and one iteration \
starting from y=0.2"
}], "Text"],

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  ".  The vertical line goes from the initial y value of 0.2 up to the curve, \
so the height at which it intersects the curve, shown by the horizontal line, \
is the next value in the iteration.  That is"
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If we want to see the effect of taking two steps, we first need to \
get the new y value (0.448) back onto the x-axis.  We can do that by \
\"reflecting\" it in a 45 degree line:\
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Cell["\<\
The second intercept with the curve (near the top) is the result of \
the second iteration, i.e.,\
\>", "Text"],

Cell[BoxData[
    \(a\  = \ 2.8; \ny\  = \ 0.2; \ny\  = \ a\ y\ \((1 - y)\); \n
    a\ y\ \((1 - y)\)\)], "Input"],

Cell["\<\
OK, so now you should be able to see how to continue the process; \
we alternately go vertically to the curve and horizontally to the 45 degree \
line.  I wrote a function to do \"n\" steps of that process.  Just evaluate \
the following (it won't produce any output) to define the function.  Don't \
even try to understand it!\
\>", "Text"],

Cell[BoxData[
    \(logplot[y0_, a_, n_]\  := \ 
      Module[{}, \n\t\t
        Plot[a\ y \((1 - y)\), \ {y, 0, 1}, 
          Epilog -> {Line[{{0, 0}, {1, 1}}], 
              Apply[\(({Line[{{#1, #2}, {#2, #2}}], 
                      Line[{{#2, #2}, {#2, #3}}]})\)&, 
                Partition[NestList[\((a  # \((1 - #)\))\)&, y0, n], 3, 1], {
                  1}]}]]\)], "Input"],

Cell[TextData[{
  "The ",
  StyleBox["logplot[]",
    FontFamily->"Courier",
    FontSize->12,
    FontWeight->"Bold"],
  " function takes three arguments: the initial y value, the value of a, and \
the number of steps (n).  So the following reproduces the diagram above \
except for one vertical line:"
}], "Text"],

Cell[BoxData[
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Cell["And now we can do more steps:", "Text"],

Cell[BoxData[
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Cell["\<\
It should be pretty clear that in this case the iteration is \
converging on a limiting number, given by the point where the curve and the \
45 degree line intersect.  Indeed that's what happens at this value of a, as \
we can see in a time series plot:\
\>", "Text"],

Cell[BoxData[
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    ListPlot[Table[y\  = \ a\ y\ \((1 - y)\), {20}], PlotJoined -> True]\)], 
  "Input"],

Cell["\<\
Now let's go back to the cyclic and chaotic cases.  Here's a \
2-cycle, at a=3.2:\
\>", "Text"],

Cell[BoxData[
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Cell["\<\
Notice the initial stepladder and spiral, leading into the heavy \
rectangle that is the 2-cycle itself.  If we start from a different initial y \
value, the ladder and spiral may be different, but the final cycle is the \
same.  In that sense it's an \"attractor\":\
\>", "Text"],

Cell[BoxData[
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Cell["\<\
If we just want to see the final attractor itself, we can do (say) \
100 iterations ourselves first to generate the starting value for the plot:\
\
\>", "Text"],

Cell[BoxData[
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    logplot[y, a, 25]\)], "Input"],

Cell["Here's a 4-cycle (without the transient spiral):", "Text"],

Cell[BoxData[
    \(a\  = \ 3.5; \ny\  = \ 0.1; \nDo[y\  = \ a\ y\ \((1 - y)\), {100}]; \n
    logplot[y, a, 25]\)], "Input"],

Cell["And here's a chaotic case:", "Text"],

Cell[BoxData[
    \(a\  = \ 3.9; \ny\  = \ 0.1; \nDo[y\  = \ a\ y\ \((1 - y)\), {100}]; \n
    logplot[y, a, 25]\)], "Input"],

Cell["\<\
It just never settles down.  Try running it for more steps, by \
changing the 25 in the last line above to (say) 200.  You should see that it \
covers (almost?) every y value in a wide range, but never exactly repeats \
itself.\
\>", "Text"],

Cell["\<\
Do you understand any better now?  A little bit, maybe?  By playing \
with these plots, and seeing just exactly what happens to a stable point or \
cycle as you change \"a\", it's possible to get a deeper and deeper \
understanding of this rich range of behavior.\
\>", "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["Bifurcation Diagram", "Section"],

Cell[TextData[{
  "To get a picture of what happens as a function of the parameter \"a\", \
let's plot a bunch of 100 \"y\" values vertically against a sequence of \"a\" \
values.  This is called a ",
  StyleBox["bifurcation diagram",
    FontSlant->"Italic"],
  ".  The following program does the job, varying \"a\" from 2.5 to 4 in \
increments of 0.01 (see the line that says ",
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    FontWeight->"Bold"],
  ").   Don't worry too much about how the program works, unless you're \
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    FontSlant->"Italic"],
  " interested you'll need to figure out what ",
  StyleBox["Flatten[list, 1]",
    FontWeight->"Bold"],
  " does to a list of lists of lists!]"
}], "Text"],

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            x\  = \ a\ x\ \((1 - x)\), \ {100}]; \n\t\t\tTable[{a, 
              x\  = \ a\ x\ \((1 - x)\)}, \ {100}], \n\t\t\t{a, 2.5, 4, 
            0.01}\n\t\t], \n\t\t1\n\t]]\)], "Input"],

Cell["\<\
What are you seeing?  Well, remember that the horizontal axis is \
the parameter \"a\", and for each \"a\" we're seeing 100 \"y\" values plotted \
as dots vertically above that \"a\".\
\>", "Text"],

Cell[TextData[{
  "On the left, for \"a\" less than about 3, all the \"y\" values fall on top \
of each other -- the sequence converges to a single value.  This is a \
1-cycle, and ",
  StyleBox["very",
    FontSlant->"Italic"],
  " boring.  It corresponds to the \"spiralling in to a point\" behavior we \
first saw in the graphical representation of the process.  Then there's a \
region of 2-cycles, from about a = 3 to about a = 3.45.  Then a region of \
4-cycles, which is where we first started (at a = 3.5)."
}], "Text"],

Cell["\<\
It's hard to see what happens next, but actually there are \
8-cycles, 16-cycles, 32-cycles, etc, etc, doubling without limit, but \
occupying more and more tiny intervals of \"a\".\
\>", "Text"],

Cell["\<\
Then suddenly we get chaos, seen here by the \"y\" points being \
spread out vertically over a whole range of the vertical axis, instead of \
just falling on a few lines.  The system is chaotic for most values of a \
above about 3.57.\
\>", "Text"],

Cell[TextData[{
  "But there are also gaps, called ",
  StyleBox["windows",
    FontSlant->"Italic"],
  ", in which the behavior is periodic again.  The most prominent one, which \
you can see pretty clearly above, is at around a = 3.84, where's there are \
3-cycles.  Here's a blowup of that region (note the line that sets the a \
values):"
}], "Text"],

Cell[BoxData[
    \(ListPlot[\n\t
      Flatten[\n\t\t
        Table[\n\t\t\tx\  = \ 0.4; \n\t\t\t
          Do[x\  = \ a\ x\ \((1 - x)\), \ {100}]; \n\t\t\t
          Table[{a, x\  = \ a\ x\ \((1 - x)\)}, \ {400}], \n
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Cell[TextData[{
  "The 3-cycle branches (or ",
  StyleBox["bifurcates",
    FontSlant->"Italic"],
  ") into a 6-cycle, then a 12-cycle, and so on, but then returns to chaos."
}], "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["Exercise", "Section"],

Cell["\<\
Make a similar picture of the 5-cycle window that's near a = 3.74.  \
Print out:
\t1. a bifurcation diagram;
\t2. a time-series picture of a 5-cycle;
\t3. a graphical representation of it (without transients).
You can just about see the 5-cycle window on the first bifurcation diagram, \
but you'll need to pin down the appropriate range of \"a\" for a nice \
picture, and modify the previous program accordingly.\
\>", "Text"]
}, Closed]]
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