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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 26 "Iterating the Logistic \+
Map" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 241 "T
his worksheet iterates the function  f(x)=rx(1-x).  \nInteresting valu
es to try are   r = 1.5, r = 2.5,   r = 3.0,   and  r = 3.9.  For othe
r values of  r, you probably want to increase  n1 to a larger value, e
.g., 100 for r=3.9.  \nYou might" }}{PARA 0 "" 0 "" {TEXT -1 106 "also
 try changing the value of the intial condition  x0, and even the func
tion f, e.g. make f(x) = cos(x)." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(pl
ots):" }{TEXT -1 0 "" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name
 changecoords has been redefined\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 88 "(Note that the output of  lines ending in a coln are not displa
ced but are calculated.  " }}{PARA 0 "" 0 "" {TEXT -1 91 "The output o
f lines ending in a semicoln is displayed.  A line must end in either \+
a coln or" }}{PARA 0 "" 0 "" {TEXT -1 12 "a semicoln.)" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 40 "Summary of variables and functions used.
" }}{PARA 0 "" 0 "" {TEXT -1 92 "f := x -> r*x*(1-x):   Definition of \+
the function\nr:  parameter value\nx0:  initial condition" }}{PARA 0 "
" 0 "" {TEXT -1 86 "n0:  Initially there are  n0 iterates that are not
 plotted.  (Given the value of n0=0 " }}{PARA 0 "" 0 "" {TEXT -1 81 "t
his step is skipped. Change the value of n0 to display the values of i
terates.) " }}{PARA 0 "" 0 "" {TEXT -1 51 "n1:  Then n1 more iterates \+
are calculated with the " }}{PARA 0 "" 0 "" {TEXT -1 80 "stair steps o
r cobweb plotted on the graphical representation, and the values of" }
}{PARA 0 "" 0 "" {TEXT -1 70 "the the iterates are displayed.\na:  Lef
t endpoint of interval plotted." }}{PARA 0 "" 0 "" {TEXT -1 42 "b:  Ri
ght endpoint of the interval plotted" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Parameter and variable values.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "r  := 1.5:\nx0 := 0.1:
\nn0 := 0:\nn1 := 8:\na  := 0.0:\nb  := 1.0:\n" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 17 "Function iterated" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "f := x -> r*x*(1-x):" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 41 "Calculate iterates which are not plotted." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "for i from 1 to n0 do\n        x1:=
 f(x0):\n        x0:=x1:\nod:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "
Calculate iterate which are plotted, and display the outcome.  .  " }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "pic := \{plot(\{f(x),x\},x=
a..b, color=black)\}:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "for i fro
m 1 to n1 do\n        x1:= f(x0):\n        pic := pic union \{plot([[x
0,x0],[x0,x1],[x1,x1]], color=blue)\}:\n        printf(`\\t%d\\t%1.7f \+
\\n`,n0+i, x0);\n        x0:=x1:\nod:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 13 "display(pic);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "The follow
ing displays the values of the iterates rather the the graphical plot.
  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 403 "r  := 3.2:\nx0 := 0.
1:\nx00 := x0:\nn0 := 64:  #transient points not displayed\nn1 := 128:
  #number of iterates to print output\nfor i from 1 to n0 do\n        \+
x0:= f(x0):\nod:\nfor i from 1 to n1 do\n        x0:= f(x0):\nod:\npri
ntf(`\\n\\t r = %1.4f \\n\\n`, r);\nprintf(`\\t x0 = %1.4f \\n\\n`, x0
0);\nprintf(`\\t i \\t\\t f^(i)(x0) \\n`);\nfor j from 1 to n1 do\n   \+
x0 := f(x0):\n   printf(`\\t %d \\t %1.4f \\n`, j+n0, x0 );\nod:" }}}
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