Chapter 3 of James Gleick's book on chaos (''Chaos - Making a New 
Science'', 1987, [http://www.around.com/chaos.html]) discusses how very simple equations can exhibit
chaotic behavior when particular parameters are manipulated. The
example equation he provides is a model for population growth in
a fish pond, which looks like this:

[http://jrankin.ath.cx/tclerswiki/equation.png]

Meaning that the population for the next iteration (year, month,
day, whatever) equals the rate of population growth ''r'' multiplied 
by the current population ''x'' multiplied by a term that keeps
the population within bounds (e.g. as the population increases, food
becomes increasingly scarce and some of our fish die or move to
less crowded ponds).

The parameter manipulated is the rate of growth ''r'', and Mr.
Gleick discusses in depth what happens to the population as ''r'' is
increased. He also provides several graphs that show how the
fish population oscillates in regular patterns over multiple 
iterations and eventually becomes chaotic, i.e., no regular pattern
is readily apparent.

For a fun exercise, and to help explain these concepts to myself, I
scripted this model in Tcl to see if I could
get similar results. I also wanted to output the data in such a way
that I could import it into [Excel] to create graphs.

Well, fortunately the math was within my reach so generating the data 
was easy, and instead of using [Excel] I was able to use the [emu_graph] 
package to generate the graph on the fly.

Here's the script:

 #!/usr/local/ActiveTcl/bin/wish

 # Rate of population growth
 # Nice values 2, 2.75, 3, 3.25, 3.5, 4 (chaos ensues!)
 set r 2.75

 # Starting population
 set x .4
 set init_x $x

 # Number of iterations
 set iterations 50


 for {set i 1} {$i <= $iterations} {incr i} {
     set x [expr $r * $x * [expr 1 - $x]]
     lappend data $i
     lappend data $x
 }

 package require emu_graph
 wm title . "Simple Chaos Theory (r=$r, x=$init_x, $iterations\
         iterations)"

 canvas .c -width 500 -height 300
 pack .c 

 emu_graph::emu_graph graph -canvas .c -width 400 -height 225
 graph data d2 -colour red -points 0 -lines 1 -coords $data

Setting ''r'' to 2.75 results in the following graph, where the
population oscillates for a while before settling down to a steady 
state.

[http://jrankin.ath.cx/tclerswiki/2.75.png]

Setting ''r'' to 3.25 results in a regular pattern that oscillates
between levels each iteration, known as Period 2.

[http://jrankin.ath.cx/tclerswiki/3.25.png]

Setting ''r'' to 3.5 results in more complex, yet still regular,
behavior known as Period 4.

[http://jrankin.ath.cx/tclerswiki/3.5.png]

Finally, increasing ''r'' to 4 results in apparent chaos. However, 
as Gleick writes, regularities will still appear now and then, only 
to give way to more chaotic behavior. My impression from the book
thus far is that these regularities would not necessarily be
visualized in graphs like these, but would be apparent through other
means of visualizations (fractals, perhaps?).

[http://jrankin.ath.cx/tclerswiki/4.png]

'''Notes:'''

On my workstation, setting ''r'' to anything above 4 results in this error:

 Error in startup script: floating-point value too large to represent
     while executing
 "expr $r * $x * [expr 1 - $x]"

[AM] This is because then the iterate will grow in size indefinitely. 
Another chaotic function is presented at [The Q function]

[DKF]: The classic way of visualising the Simplified Malthusian fractal is to plot ''r'' (on a scatter graph) against the points in the "cycle", running the thing for some number of iterations (perhaps 100?) to let it settle.  In general, this produces ''x'' points for each value of ''r'', where ''x'' is the cycle length.  Look out for cycles of length 3...

[WJR] The above as a Tcl script:

 #!/usr/local/ActiveTcl/bin/wish

 # Rate of population growth
 # Nice values 2, 2.75, 3, 3.25, 3.5, 4 (chaos ensues!)
 set r [list 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4]

 # Starting population
 set x .4
 set init_x $x

 # Number of iterations
 set iterations 100

 foreach r_value $r {
     for {set i 1} {$i <= $iterations} {incr i} {
         set x [expr $r_value * $x * [expr 1 - $x]]
         lappend data $r_value
         lappend data $x
     }
 }

 package require emu_graph
 wm title . "Simple Chaos Theory"

 canvas .c -width 500 -height 300
 pack .c

 emu_graph::emu_graph graph -canvas .c -width 400 -height 225
 graph data d2 -colour red -points 1 -lines 0 -coords $data

The results are shown below. Again, as ''r'' increases we see the data points split as they alternate between different levels, and eventually become chaotic.

[http://jrankin.ath.cx/tclerswiki/r-scatter.png]

[Lars H]: It would probably be better to record values 101-200 than 1-100. What one is interested in in this latter kind of plot are the points on the ''attractor'' (set towards which the values of the process are attracted), and the first couple of values are very "tainted" by the (completely irrelevant) $init_x value. Also, to get plots like the ones in Gleick's ''Chaos'' (where one can actually see the period-doubling), one would need a much smaller r step size than the 0.25 in the above script.

Concerning the book ''Chaos'', I'd like to share a quaint observation of mine: Next to it (on the science shelf in my home-town library), one finds a book entitled ''Cosmos'' (when I first observed this combination, it was a book by George Gamow, but right now it is a book by Stephen Hawking), which is kind of fun considering that Cosmos (at least in Greek mythology) was the direct opposite of Chaos. None of which has anything to do with Tcl, though.
----

Here's a slightly improved version of the script above:

 #!/usr/local/ActiveTcl/bin/wish

 # Starting population
 set x .4

 # Number of iterations
 set iterations 100

 for {set r_value 1} {$r_value <= 4} {set r_value [expr $r_value + .01]} {
     for {set i 1} {$i <= $iterations} {incr i} {
         set x [expr $r_value * $x * [expr 1 - $x]]
 	 lappend data $r_value
 	 lappend data $x
     }
 }

 package require emu_graph
 wm title . "Simple Chaos Theory"

 canvas .c -width 500 -height 300
 pack .c 

 emu_graph::emu_graph graph -canvas .c -width 400 -height 225
 graph data d2 -colour red -points 1 -lines 0 -coords $data

The bifurcations in the resulting graph are much clearer and strongly resemble the graph on page 71 of the Gleick book:

[http://jrankin.ath.cx/tclerswiki/r-scatter-2.png]

----
[[
[Category Mathematics]
]]