{VERSION 5 0 "IBM INTEL NT" "5.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0
1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0
0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1
257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0
0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0
0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1
262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0
0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0
0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1"
0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 6
6 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0
0 0 1 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0
257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1
-1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0
0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "
" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0
-1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1
0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }}
{SECT 0 {PARA 256 "" 0 "" {TEXT -1 38 "Cobwebs: A Visualization For It
eration" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT
-1 11 "Jan Medlock" }}{PARA 258 "" 0 "" {TEXT -1 21 "School of Mathema
tics" }}{PARA 259 "" 0 "" {TEXT -1 12 "Georgia Tech" }}{PARA 260 "" 0
"" {TEXT -1 22 "herod@ math.gatech.edu" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 96 " It often happens in science, mat
hematics, and engineering that it is of interest to find a " }{TEXT
256 11 "fixed point" }{TEXT -1 69 " of a function. That is, we might h
ave a function f and wish to find " }{XPPEDIT 18 0 "alpha" "6#%&alphaG
" }{TEXT -1 10 " such that" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0
"" 0 "" {TEXT -1 39 " " }
{XPPEDIT 18 0 "alpha=f(alpha)" "6#/%&alphaG-%\"fG6#F$" }{TEXT -1 1 ".
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 197 "In t
his worksheet, we investigate how the process of seeking such fixed po
ints through iteration can be visualized geometrically. The iteration \+
process we describe will start at some initial value " }{XPPEDIT 18 0
"x[0]" "6#&%\"xG6#\"\"!" }{TEXT -1 118 " and perform successive approx
imations with the hope of finding the fixed point. The iteration proce
ss goes like this:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 48 " Choose " }
{XPPEDIT 18 0 "x[0]" "6#&%\"xG6#\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0
"" {TEXT -1 11 "and compute" }}{PARA 0 "" 0 "" {TEXT -1 40 " \+
" }{XPPEDIT 18 0 "x[n+1]=f(x[n])" "6#/&%
\"xG6#,&%\"nG\"\"\"F)F)-%\"fG6#&F%6#F(" }{TEXT -1 1 "." }}{PARA 0 ""
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 363 " If the functio
n represents, for example, the reproductive rate for a population, it \+
might be as interesting to know how the iteration process progresses a
s it is to actually know what would be a stationary level for the popu
lation. After all, the iteration process in this case could represent \+
the successive population levels through successive generations." }}
{PARA 0 "" 0 "" {TEXT -1 122 " An example having nothing to do wit
h biology -- a toy problem -- used to introduce the ideas will be to s
eek numbers " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT -1 10 " such \+
that" }}{PARA 0 "" 0 "" {TEXT -1 41 " \+
" }{XPPEDIT 18 0 "alpha=cos(alpha)" "6#/%&alphaG-%$cosG6#F$" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 149 "One might begin to sear
ch for this number through various numerical methods, such as bisectio
n or Newton's method for finding a zero of the function " }{XPPEDIT
18 0 "g(alpha)=alpha-cos(alpha)" "6#/-%\"gG6#%&alphaG,&F'\"\"\"-%$cosG
6#F'!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 289 " The m
ethod of this worksheet is a different one. The method has become a pa
rt of the mathematics culture during the last decade of the twentieth \+
century. In part, the technique of iteration in f has led to such an i
nterest because of the richness of ideas associated with the process.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 247 " \+
Historically, one describes the process as finding a fixed point by s
imple iteration. More recently, with the appearance of a computer on t
he desktop, one is more likely to suggest an attempt at finding a fixe
d point by the construction of a " }{TEXT 257 6 "Cobweb" }{TEXT -1 1 "
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1
46 "The Computational Procedure for this Worksheet" }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 350 " We present here the
computational procedure for this worksheet. We use it to illustrate t
he ideas of the worksheet. The suggestion is that you execute the foll
owing portion of code, and use it in the development of the worksheet.
Perhaps, after the ideas are established, you will come back and see \+
how this part of the procedure is constructed." }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 524 "cobweb:=proc(f,x0,a,b,Length) \n##Plots a spiderweb \+
diagram for f(x); \n##f is the function, x0 is the inital point, \n##
[a,b] is the interval for the plot, \n##Length is the number of iterat
ions##\n local L,i,x1,x2,p1,p2,p3;\n x1:=x0;\n L:=[[x1,0]]
;\n for i from 1 to Length do\n x2:=f(x1);\n L:=
[op(L),[x1,x2],[x2,x2]];\n x1:=x2;\n od;\n p1:=plot(L
,color=BLUE):\n p2:=plot(f,a..b,color=BLACK):\n p3:=plot(x,x=a
..b,color=BLACK,linestyle=4):\n plots[display](\{p1,p2,p3\});\nend
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 23 "The Cobweb Constr
uction" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90
" Suppose that f is a function for which we seek a fixed point. Th
at is we see a point " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT -1
11 " such that " }{XPPEDIT 18 0 "f(alpha) = alpha" "6#/-%\"fG6#%&alpha
GF'" }{TEXT -1 21 ". To do this, choose " }{XPPEDIT 18 0 "x[0]" "6#&%
\"xG6#\"\"!" }{TEXT -1 5 ". If " }{XPPEDIT 18 0 "x[0]=f(x[0])" "6#/&%
\"xG6#\"\"!-%\"fG6#&F%6#F'" }{TEXT -1 42 ", quit: you found a fixed po
int. Else let " }}{PARA 0 "" 0 "" {TEXT -1 26 " \+
" }{XPPEDIT 18 0 "x[1]=f(x[0])" "6#/&%\"xG6#\"\"\"-%\"fG6#&F%6#\"\"
!" }}{PARA 0 "" 0 "" {TEXT -1 12 "and compute " }{XPPEDIT 18 0 "f(x[1]
" "6#-%\"fG6#&%\"xG6#\"\"\"" }{TEXT -1 5 ". If " }{XPPEDIT 18 0 "x[1]=
f(x[1])" "6#/&%\"xG6#\"\"\"-%\"fG6#&F%6#F'" }{TEXT -1 42 ", quit: you \+
found a fixed point. Else, let" }}{PARA 0 "" 0 "" {TEXT -1 26 " \+
" }{XPPEDIT 18 0 "x[2]=f(x[1])" "6#/&%\"xG6#\"\"#-%
\"fG6#&F%6#\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 22 "Cont
inue this process." }}{PARA 0 "" 0 "" {TEXT -1 214 " This ideas se
ems very computational and perhaps does not suggest intuition. The cob
web construction illustrates this iteration in a geometric, intuitive \+
way. We illustrate with the toy problem suggested above." }}{PARA 0 "
" 0 "" {TEXT -1 27 " Let f(x) = cos(x) and " }{XPPEDIT 18 0 "x[0]
" "6#&%\"xG6#\"\"!" }{TEXT -1 40 " = -1. The next iterate can be compu
ted." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "f:=x->cos(x);\nx[0]:
=-1;\nx[1]:=evalf(f(x[0]));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
0 "" }}}{PARA 0 "" 0 "" {TEXT -1 12 "The initial " }{XPPEDIT 18 0 "x[0
]" "6#&%\"xG6#\"\"!" }{TEXT -1 92 " is not a fixed point. We have star
ted the iteration process. This process can be automated." }}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "x[0]:=-1;\nfor i from 1 to 5 do\n \+
x[i]:=evalf(cos(x[i-1]));\nod;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 156 " Perhaps the reader agre
es that this sequence of computations provides little geometric unders
tanding for the iteration process. Consider the following:" }}{PARA 0
"" 0 "" {TEXT -1 33 " Recognize that the value of " }{XPPEDIT 18
0 "x[1] = f(x[0])" "6#/&%\"xG6#\"\"\"-%\"fG6#&F%6#\"\"!" }{TEXT -1 48
" can be thought of as the mapping of the number " }{XPPEDIT 18 0 "x[0
]" "6#&%\"xG6#\"\"!" }{TEXT -1 74 " on the x-axis to one on the y-axis
. In the following picture, the number " }{XPPEDIT 18 0 "x[1]" "6#&%\"
xG6#\"\"\"" }{TEXT -1 50 " lies at the right end of the following blue
line." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 247 "p1:=plot([[-1,0],[-1,cos(-1)]],color=BLACK): p2:
=plot(\{[x,x,x=-2..2],[x,cos(x),x=-Pi..Pi]\},\n color=BLACK): \+
\np3:=plot([[-1,cos(-1)],[0,cos(-1)]],\n color=BLUE):\np4:=textp
lot([.25,cos(-1),`x[1]`]): \nplots[display](\{p1,p2,p3,p4\});" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT
-1 52 "Or, extending back down to the x-axis, the value of " }
{XPPEDIT 18 0 "x[1]=f(x[0]" "6#/&%\"xG6#\"\"\"-%\"fG6#&F%6#\"\"!" }
{TEXT -1 58 " lies at the bottom-right end of the following green line
." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 265 "p1:=plot([[-1,0],[-1,c
os(-1)]],color=BLACK): p2:=plot(\{[x,x,x=-2..2],[x,cos(x),x=-Pi..P
i]\},\n color=BLACK): p3:=plot([[-1,cos(-1)],[cos(-1),cos(-1)
],\n [cos(-1),0]],color=GREEN):\np4:=textplot([cos(-1),-.25,`x[1]`
]): \nplots[display](\{p1,p2,p3,p4\});" }}}{PARA 0 "" 0 "" {TEXT
-1 19 "Thus, the value of " }{XPPEDIT 18 0 "x[2] = f(x[1])" "6#/&%\"xG
6#\"\"#-%\"fG6#&F%6#\"\"\"" }{TEXT -1 70 " is the y-coordinate of the \+
point at the top of the following redline." }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 271 "p1:=plot([[-1,0],[-1,cos(-1)]],color=BLACK): p
2:=plot(\{[x,x,x=-2..2],[x,cos(x),x=-Pi..Pi]\},\n color=BLACK): \+
p3:=plot([[-1,cos(-1)],[cos(-1),cos(-1)],\n [cos(-1),cos(cos(-1
))]],color=RED):\np4:=textplot([cos(-1),-.25,`x[1]`]): \nplots[disp
lay](\{p1,p2,p3\});" }}}{PARA 0 "" 0 "" {TEXT -1 114 "We now continue \+
the process using the program initialized at the beginning of this wor
ksheet under the subheading " }{TEXT 258 45 "The Computaional Procedur
e for this Worksheet" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 13 "f:=x->cos(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 7 "x0:=-1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "cobweb(f,
x0,-2,2,10);" }}}{PARA 0 "" 0 "" {TEXT -1 194 "We see now what the num
erical calculations should do with this example: the iterates should c
onverge to a solution. Even more, they should be alternately larger an
d smaller than the fixed point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 64 "x[0]:=-1;\nfor i from 1 to 10 do\n x[i]:=evalf(cos(x[i-1]));
\nod;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 25 "Exercise for the student." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "Draw the \+
cobweb construction to find a fixed point for f(x) = sin(x)." }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 23 "A
pplications in biology" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 ""
0 "" {TEXT -1 188 " More than twenty years ago, Robert May capture
d the attention of the mathematics community in a stimulating article \+
titled Simple Mathematical Models with Very Complicated Dynamics ( " }
{TEXT 261 6 "Nature" }{TEXT -1 1 " " }{TEXT 262 3 "261" }{TEXT -1 290
" (1976), page 459). He presented with the clarity of a good teacher
the observation that even simple functions f could have interesting, \+
complicated dynamics. As much as any other method, the cobweb construc
tion gives insight into the dynamics. Recounting May's contribution in
his book, " }{TEXT 259 27 "Chaos, Making a New Science" }{TEXT -1
105 ", James Gleich writes a good summary of how this method arises in
mathematical biology. (See the chapter " }{TEXT 264 20 "Life's Ups an
d Downs" }{TEXT -1 2 ".)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 87 " Here is a suggestion for how the iteration pr
ocess arises in Mathematical Biology." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 206 " Suppose we have a population th
at grows at a rate proportioned to the amount present for small popula
tions, but reaches a limit of growth due to limited resources. Such an
equation is often modeled as " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 43 " growth rate \+
= " }{XPPEDIT 18 0 "r*x*(1-x)" "6#*(%\"rG\"\"\"%\"xGF%,&F%F%F&!\"\"F%
" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 138 "Here, r represents a rate of growth that might be determ
ined by environmental factors, as well as species specific informatio
n. The term " }{XPPEDIT 18 0 "1-x" "6#,&\"\"\"F$%\"xG!\"\"" }{TEXT -1
40 " limits the growth, for as x increases, " }{XPPEDIT 18 0 "1-x" "6#
,&\"\"\"F$%\"xG!\"\"" }{TEXT -1 428 " decreases. The construction of a
n iteration process could represent the level of the population in suc
ceeding years. The question would be, how does the term r affect the p
redicted changes in the population? One might guess that if r is large
, the population reaches its limit of growth quickly, and if it is sma
ll, the limit might be reached slowly. What actually happens is far mo
re interesting than this too simple analysis." }}{PARA 0 "" 0 ""
{TEXT -1 219 " To see the interesting possibilities, suppose the p
opulation size is between 0 and 1 in this simple model. Then we would \+
want each new iteration of the population level in that range too. Sin
ce the maximum value of" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 ""
0 "" {TEXT -1 31 " " }{XPPEDIT 18 0 "r*x
*(1-x)" "6#*(%\"rG\"\"\"%\"xGF%,&F%F%F&!\"\"F%" }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 " will occur when x = " }
{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 26 ", and achie
ves the value " }{XPPEDIT 18 0 "r/4" "6#*&%\"rG\"\"\"\"\"%!\"\"" }
{TEXT -1 133 " at that level, we restrict r to the values between 0 an
d 4. The following exercises gives the student an opportunity to explo
re the " }{TEXT 260 7 "cobwebs" }{TEXT -1 15 " for such r's. " }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 25 "E
xercise for the student." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 38 " Provide three cobweb diagrams for" }}{PARA 0
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 " \+
" }{XPPEDIT 18 0 "f(x) = r *x* (1-x)" "6#/-%\"fG6#%\"xG*(%\"rG\"\"\"F
'F*,&F*F*F'!\"\"F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 180 "Work out the cobweb diagram for r = 2/3,
for r = 2, and for r = 3 1/3. The structure will be strikingly differ
ent in all three. Describe what is different about the three examples.
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 201 "The
Maple code for this worksheet was constructed by Jan Medlock for semi
nar in Fall, 1996. With his permission, J. Herod wrote this worksheet \+
for placement in the collection of teaching models called " }{TEXT
263 64 "The First Two Years of Undergraduate Mathematics at Georgia Te
ch" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0" 0 }
{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }