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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 50 "Sequences -- Period-Doub
ling and the Road to Chaos" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 80 "The following two cells generate two movies study
ing logistic models of the form" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 29 "f(p) = \+
a * (1 - 0.01 * p) * p" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 259 
"" 0 "" {TEXT -1 18 "p(n + 1) = f(p(n))" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 5 "for  " }{TEXT 256 31 "a = 1.00, 1.05,
 1.10, ... 4.00." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 117 "The first cell generates a movie showing the first 20 ge
nerations of each model starting with the initial condition  " }{TEXT 
257 8 "p(1) = 5" }{TEXT -1 58 ".  The second cell generates a movie sh
owing the following" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 24 "1.   The diagonal line  " }{TEXT 258 5 "y = p" }{TEXT 
-1 12 "  (in black)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 16 "2.   The curve  " }{TEXT 259 8 "y = f(p)" }{TEXT -1 10 
"  (in red)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 15 "3.   The curve " }{TEXT 260 12 " y = f(f(p))" }{TEXT -1 11 "  (
in blue)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
47 "Equilibrium points are points where the curve  " }{TEXT 261 8 "y =
 f(p)" }{TEXT -1 20 "  crosses the line  " }{TEXT 262 5 "y = p" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 46 "Points in 2-cycles are points where the curve " }{TEXT 
263 12 " y = f(f(p))" }{TEXT -1 19 " crosses the line  " }{TEXT 264 5 
"y = p" }{TEXT -1 33 " that are not equilibrium points." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 59 "Evaluate the next t
wo cells now.  This will take some time." }{TEXT -1 1 " " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 366 "wit
h(plots):\nwith(plottools):\n\nf := p -> a * (1 - 0.01 * p) * p:\n\nIn
itialValue := 5:\n\npop := proc(n)\n  if n = 1 then InitialValue else \+
f(pop(n-1)) fi\nend:\n\nMovieList := [seq(i, i = 0..60)]:\n\nfor i fro
m 0 to 60 do\n    a := 1.0 + 0.05 * i:\n    MovieList[i + 1] :=\n     \+
  plot([seq([n, pop(n)], n = 1..20)], 1..20, 0..100):\nod:\n\ndisplay(
MovieList, insequence = true);\n\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 838 "with(plots):\nwith(plottools):\n\nf := p -> a * (1 -
 0.01 * p) * p:\n\nInitialValue := 5:\n\npop := proc(n)\n  if n = 1 th
en InitialValue else f(pop(n-1)) fi\nend:\n\nAnimationList := [seq(i, \+
i = 0..60)]:\n\nfor i from 0 to 60 do\n    a := 1.0 + 0.05 * i:\n    b
lk := display(\{line([0, 100], [100, 100]),\n                    line(
[0, 0], [100, 0]),\n                    line([0, 0], [0, 100]),\n     \+
               line([100, 0], [100, 100]),\n                    line([
0, 0], [100, 100])\},\n                   color=black, scaling=CONSTRA
INED):\n    rd := plot(f(p), p = 0..100, y = 0..100,\n               c
olor=RED, scaling=CONSTRAINED, axes=NONE):\n    blu := plot(f(f(p)), p
 = 0..100, y = 0..100,\n               color=BLUE, scaling=CONSTRAINED
, axes=NONE):\n    AnimationList[i + 1] := display([blk, rd, blu]):\no
d:\n\ndisplay(AnimationList, insequence=true);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "The following cell g
enerates a movie studying the logistic models" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" 
{TEXT -1 29 "f(p) = a * (1 - 0.01 * p) * p" }}{PARA 261 "" 0 "" {TEXT 
-1 0 "" }}{PARA 262 "" 0 "" {TEXT -1 18 "p(n + 1) = f(f(p))" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 5 "for  " }{TEXT 266 34 "a = 3.400, 3.405, 3.410, ... 3.
600" }{TEXT -1 33 ".  Each frame shows the following" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "1.  The diagonal line  \+
" }{TEXT 267 5 "y = p" }{TEXT -1 13 "  (in black)." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "2.  The curve  " }{TEXT 
268 10 "y = f(f(p)" }{TEXT -1 12 ")  (in red)." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "3.  The curve " }{TEXT 
269 17 " y = f(f(f(f(p)))" }{TEXT -1 13 ")  (in blue)." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "" {TEXT -1 51 "Evaluate the next \+
cell now.  It may take some time." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 858 "with(plots):\nwith(plottool
s):\n\nf := p -> a * (1 - 0.01 * p) * p:\n\nInitialValue := 5:\n\npop \+
:= proc(n)\n  if n = 1 then InitialValue else f(pop(n-1)) fi\nend:\n\n
AnimationList := [seq(i, i = 0..40)]:\n\nfor i from 0 to 40 do\n    a \+
:= 3.400 + 0.005 * i:    \n    blk := display(\{line([0, 100], [100, 1
00]),\n                    line([0, 0], [100, 0]),\n                  \+
  line([0, 0], [0, 100]),\n                    line([100, 0], [100, 10
0]),\n                    line([0, 0], [100, 100])\},\n               \+
     color=black, scaling=CONSTRAINED):\n    rd := plot(f(f(p)), p = 0
..100, y = 0..100,\n               color=RED, scaling=CONSTRAINED, axe
s=NONE):\n    blu := plot(f(f(f(f(p)))), p = 0..100, y = 0..100,\n    \+
           color=BLUE, scaling=CONSTRAINED, axes=NONE):\n    Animation
List[i + 1] := display([blk, rd, blu]):\nod:\n\ndisplay(AnimationList,
 insequence=true); i:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 129 "The next cell is identical to the preced
ing cell except that it shows a close-up view of one of the points in \+
the two 2-cycle.   " }{TEXT 270 16 "Evaluate it now." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 
0 858 "with(plots):\nwith(plottools):\n\nf := p -> a * (1 - 0.01 * p) \+
* p:\n\nInitialValue := 5:\n\npop := proc(n)\n  if n = 1 then InitialV
alue else f(pop(n-1)) fi\nend:\n\nAnimationList := [seq(i, i = 0..40)]
:\n\nfor i from 0 to 40 do\n    a := 3.400 + 0.005 * i:    \n    blk :
= display(\{line([32, 56], [56, 56]),\n                    line([32, 3
2], [56, 32]),\n                    line([32, 32], [32, 56]),\n       \+
             line([56, 32], [56, 56]),\n                    line([32, \+
32], [56, 56])\},\n                    color=black, scaling=CONSTRAINE
D):\n    rd := plot(f(f(p)), p = 32..56, y = 32..56,\n               c
olor=RED, scaling=CONSTRAINED, axes=NONE):\n    blu := plot(f(f(f(f(p)
))), p = 32..56, y = 32..56,\n               color=BLUE, scaling=CONST
RAINED, axes=NONE):\n    AnimationList[i + 1] := display([blk, rd, blu
]):\nod:\n\ndisplay(AnimationList, insequence=true); i:" }}}}{MARK "0 \+
0 0" 50 }{VIEWOPTS 1 1 0 1 1 1803 }