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{SECT 0 {EXCHG {PARA 261 "" 0 "" {TEXT 266 34 "A Maple Toolbox for Exp
loring the " }}{PARA 261 "" 0 "" {TEXT 269 34 "Behaviour of One-Dimens
ional Maps\n" }}}{EXCHG {PARA 262 "" 0 "" {TEXT 267 65 "Douglas Curran
 and Walter Middleton\nUniversity of Sunderland\nU.K." }{TEXT 268 1 "
\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 327 8 "Abstract" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 522 "This paper introduces \+
four Maple procedures which together form a toolbox for the exploratio
n of the properties of one-dimensional maps. The procedures enable stu
dents to  identify intervals within which  maps display convergent, cy
clic and chaotic behaviour. This enables students to obtain a \"feel\"
 for what is meant by these concepts. After describing each procedure \+
examples are given illustrating the use of the tool. A more extensive \+
example involving the exploration of an economic model by students is \+
also given. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
365 8 "Keywords" }{TEXT -1 56 ": one-dimensional maps, cycles, chaos, \+
Maple procedures." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
263 "" 0 "" {TEXT 256 12 "Introduction" }{TEXT 261 1 "\n" }}{PARA 0 "
" 0 "" {TEXT -1 131 "The paper outlines a series of four Maple procedu
res which can be used to explore the behaviour of one dimensional maps
 of the form" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" 
{XPPEDIT 18 0 "x[n+1]=g(x[n])" "6#/&%\"xG6#,&%\"nG\"\"\"\"\"\"F)-%\"gG
6#&F%6#F(" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
154 "Maps of this form are often used to introduce students to the stu
dy of chaotic dynamical systems (c.f. Drazin [1]).  Typically a map in
volves a parameter " }{TEXT 262 1 "a" }{TEXT -1 146 " which, when vari
ed leads to distinct types of dynamical behaviour.  The procedures ena
ble the student to explore this behaviour as the parameter " }{TEXT 
263 1 "a" }{TEXT -1 543 " is varied and to identify intervals within w
hich the map displays convergent, cyclic and chaotic behaviour.  The p
rocedures have been used with students at final year undergraduate and
 postgraduate levels.  Exploring the behaviour of a map is essential i
f students are to obtain a \"feel\" for what is meant by the concepts \+
of convergent, cyclic and chaotic behaviour.  This is in line with the
 view of staff at Sunderland who encourage students to use packages su
ch as Maple to underpin abstract concepts throughout their mathematica
l studies." }}{PARA 264 "" 0 "" {TEXT 258 1 "\n" }{TEXT 259 11 "The To
olbox" }{TEXT 260 260 "\n\nIn this section we describe the purpose of \+
the procedures, the input parameters and give the Maple  implementatio
n.  It should be noted that, although the code does contain error chec
ks on the input, care should be taken to ensure that the input is sens
ible." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 
"All four procedures utilise common parameters. The map under investig
ation is defined as a procedure " }{XPPEDIT 18 0 "g(x,a)" "6#-%\"gG6$%
\"xG%\"aG" }{TEXT -1 52 " which must be supplied by the user. The para
meters " }{TEXT 280 1 "x" }{TEXT -1 5 " and " }{TEXT 281 1 "a" }{TEXT 
-1 151 " represent the current value of the iteration and the value of
 the parameter which is varied to explore the behaviour of the map. Th
e initial value of " }{TEXT 282 1 "x" }{TEXT -1 19 " is represented by
 " }{XPPEDIT 18 0 "x0" "6#%#x0G" }{TEXT -1 48 " and the number of iter
ations is represented by " }{TEXT 283 1 "n" }{TEXT -1 80 ". The end po
ints of the interval over which the plot is rendered are denoted by " 
}{TEXT 285 2 "c " }{TEXT -1 4 "and " }{TEXT 286 1 "d" }{TEXT -1 13 ". \+
Procedures " }{TEXT 328 6 "cobweb" }{TEXT -1 5 " and " }{TEXT 329 6 "n
ewweb" }{TEXT -1 9 " include " }{TEXT 287 1 "a" }{TEXT -1 35 " as a pa
rameter. In the procedures " }{TEXT 330 5 "orbit" }{TEXT -1 5 " and " 
}{TEXT 331 7 "lyapexp" }{TEXT -1 98 " the transient behaviour of the m
ap needs to be suppressed. This is done by setting the parameter " }
{TEXT 284 1 "m" }{TEXT -1 109 " which determines the number of iterati
ons to be ignored. The latter two procedures also include a parameter \+
" }{TEXT 288 2 "r " }{TEXT -1 41 "which defines the resolution of the \+
plot." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 166 
"All four procedures utilise a common structure consisting of an outer
 procedure where input checks are performed and the plots are construc
ted and an inner procedure " }{TEXT 332 5 "steps" }{TEXT -1 45 " where
 the map is iterated. The results from " }{TEXT 333 5 "steps" }{TEXT 
-1 31 " are evaluated using a call to " }{TEXT 334 6 "evalhf" }{TEXT 
-1 49 " in order to maximise the speed of computation.  " }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 290 23 "1. The c
obweb procedure" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 4 "The " }{TEXT 335 6 "cobweb" }{TEXT -1 87 " procedure is us
ed to explore the convergence properties of a map for a fixed value of
 " }{TEXT 289 2 "a " }{TEXT -1 55 "which is provided as a parameter.  \+
The procedure plots " }{TEXT 291 1 "n" }{TEXT -1 23 " iterations of th
e map " }{TEXT 292 1 "g" }{TEXT -1 32 " starting from an initial value
 " }{TEXT 293 1 "x" }{TEXT -1 48 "0. The cobweb diagram is plotted ove
r the range " }{TEXT 294 1 "c" }{TEXT -1 4 " to " }{TEXT 295 1 "d" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 27 "It is good practice to use " }{TEXT 336 7 "restart" }
{TEXT -1 65 " to clear unwanted assignments. Since all the procedures \+
use the " }{TEXT 337 5 "plots" }{TEXT -1 28 " package we need to load \+
it." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 38 "The cobweb procedure is defined below." }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 603 "cobweb:=proc(g::procedure,x0::numeric,n::posint,c::numeric,d::n
umeric,a::numeric)\nlocal p1,p2,pts,steps;\nif c>=d then ERROR(`d shou
ld be greater than c`, d,c); fi;\npts:=array(1..4*n+2);\nsteps:=proc(g
::procedure,x0::numeric,n::posint,c::numeric,d::numeric,a::numeric,pts
::array)\nlocal x1,x2,i;\nx1:=x0;\npts[1]:=x0:pts[2]:=0:\nfor i from 1
 to n do\nx2:=g(x1,a);\npts[4*i-1]:=x1:pts[4*i]:=x2:pts[4*i+1]:=x2:pts
[4*i+2]:=x2:\nx1:=x2;\nod;\011\nend;\nevalhf(steps(g,x0,n,c,d,a,var(pt
s)));\np1:=plots[pointplot](op(pts),style=LINE,colour=red):\np2:=plot(
[g(x,a),x],x=c..d,colour=[blue,green]):\nplots[display](\{p1,p2\});\ne
nd:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 68 "The procedure accumulates the iterations to be plotted in the a
rray " }{TEXT 338 3 "pts" }{TEXT -1 151 ". At each iteration two point
s are calculated. The first is a point on the curve defined by the map
 and the second is the equivalent point on the line " }{XPPEDIT 18 0 "
y=x" "6#/%\"yG%\"xG" }{TEXT -1 21 ". The plot structure " }{TEXT 339 
1 "p" }{TEXT -1 45 "1 contains a plot of the points in the array " }
{TEXT 340 3 "pts" }{TEXT -1 21 ". The plot structure " }{TEXT 341 1 "p
" }{TEXT -1 41 "2 contains plots of the map and the line " }{XPPEDIT 
18 0 "y=x" "6#/%\"yG%\"xG" }{TEXT -1 6 ". The " }{TEXT 342 7 "display
" }{TEXT -1 44 " command is used to render the overall plot." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "The first examp
le uses " }{TEXT 343 6 "cobweb" }{TEXT -1 82 " to illustrate convergen
t and non-convergent cases for the well-known logistic map" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {XPPEDIT 18 0 "x[n+1]=a*x[n]
*(1-x[n])" "6#/&%\"xG6#,&%\"nG\"\"\"\"\"\"F)*(%\"aGF)&F%6#F(F),&\"\"\"
F)&F%6#F(!\"\"F)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 55 "Before calling the procedure we need to define the map " 
}{TEXT 296 1 "g" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "g:=(x,a)->a*x*(1-x):" }}}
{EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 271 
62 "The results of applying the procedure for the convergent case " }
{MPLTEXT 1 0 0 "" }{XPPEDIT 18 0 "a=2" "6#/%\"aG\"\"#" }{TEXT 272 18 "
 are shown below. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 22 "cobweb(g,0.2,5,0,1,2);" }}{PARA 13 "" 1 "" 
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" {TEXT 297 9 "The case " }{XPPEDIT 18 0 "a=3.5" "6#/%\"aG$\"#N!\"\"" 
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{PARA 0 "" 0 "" {TEXT 298 23 "2. The newweb procedure" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 344 6 "new
web" }{TEXT -1 105 " procedure is used to explore the existence of sta
ble cycles of a map for a fixed value of the parameter " }{TEXT 270 1 
"a" }{TEXT -1 45 ".  The procedure plots iterations of the map " }
{TEXT 300 1 "g" }{TEXT -1 32 " starting from an initial value " }
{TEXT 301 1 "x" }{TEXT -1 39 "0. Nothing is plotted during the first \+
" }{TEXT 299 1 "m" }{TEXT -1 94 " iterations to allow for any transien
t behaviour present to die out. The procedure then plots " }{XPPEDIT 
18 0 "n-m" "6#,&%\"nG\"\"\"%\"mG!\"\"" }{TEXT -1 36 " iterations. In t
his case the array " }{TEXT 345 3 "pts" }{TEXT -1 101 " contains only \+
the co-ordinates of the point on the curve defined by the map . In the
 plot structure " }{TEXT 346 1 "p" }{TEXT -1 64 "1 these points are jo
ined by straight lines. The plot structure " }{TEXT 347 1 "p" }{TEXT 
-1 24 "2 is the same as before." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
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,c); fi;\nif m>=n then ERROR(`n should be greater than m`, n,m); fi;\n
pts:=array(1..2*(n-m));\nsteps:=proc(g::procedure,x0::numeric,c::numer
ic,d::numeric,a::numeric,m::posint,n::posint,pts::array)\nlocal x1,i; \+
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m do\nx1:=evalf(g(x1,a));\npts[2*i-1]:=x1:pts[2*i]:=g(x1,a):\nod;\011
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-1 0 "" }}{PARA 256 "" 0 "" {TEXT 273 101 "The following example based
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{PARA 0 "" 0 "" {TEXT 304 22 "A further increase in " }{TEXT 305 1 "a
" }{TEXT 306 74 " leads to a return to cyclic behaviour as shown in th
e three-cycle below.." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "newweb(g,0.2,0,1,0.94,250,400);" }}
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{PARA 0 "" 0 "" {TEXT 307 22 "3. The orbit procedure" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 348 5 "orbi
t" }{TEXT -1 102 " procedure is used to explore the long term behaviou
r of a map for a range of values of the parameter " }{TEXT 264 1 "a" }
{TEXT -1 66 ".  The procedure plots the behaviour of the iterations of
 the map " }{TEXT 308 1 "g" }{TEXT -1 32 " starting from an initial va
lue " }{TEXT 309 1 "x" }{TEXT -1 39 "0. Nothing is plotted during the \+
first " }{TEXT 310 1 "m" }{TEXT -1 94 " iterations to allow for any tr
ansient behaviour present to die out. The procedure then plots " }
{XPPEDIT 18 0 "n-m" "6#,&%\"nG\"\"\"%\"mG!\"\"" }{TEXT -1 82 "  iterat
ions. The procedure accumulates the iterations to be plotted in the ar
ray " }{TEXT 349 3 "pts" }{TEXT -1 68 ". At each iteration the point c
o-ordinates are the current value of " }{TEXT 311 1 "a" }{TEXT -1 25 "
 and the iteration value " }{TEXT 312 1 "x" }{TEXT -1 29 ". The struct
ure plotted is a " }{TEXT 350 9 "pointplot" }{TEXT -1 33 " of the arra
y elements using the " }{TEXT 351 12 "style=point " }{TEXT -1 7 "optio
n." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 686 " orbit:=proc(g::procedure,x0::numeric,c::numeric,d::
numeric,r::posint,m::posint,n::posint)\nlocal pts,steps;\nif c>=d then
 ERROR(`d should be greater than c`, d,c); fi;\nif m>=n then ERROR(`n \+
should be greater than m`, n,m); fi;\npts:=array(1..2*r*(n-m));\nsteps
:=proc(g::procedure,x0::numeric,c::numeric,d::numeric,r::posint,m::pos
int,n::posint,pts::array)\nlocal range,a,x1,i,j;\nrange:=d-c;\nj:=0;\n
for a from c by range/r while abs(d-a)>0.5e-6 do\nj:=j+1;\nx1:=x0;\nfo
r i from 1 to m do\nx1:=g(x1,a);\nod;\nfor i from 1 to n-m do\nx1:=g(x
1,a);\npts[(2*i-1)+2*(j-1)*(n-m)]:=a:pts[2*i+2*(j-1)*(n-m)]:=x1;\nod;
\nod;\nend;\nevalhf(steps(g,x0,c,d,r,m,n,var(pts)));\nplots[pointplot]
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"\n" }{TEXT 275 264 "This example shows the behaviour of the exponenti
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ore storing the points in the array. Note that the plots may take a co
nsiderable time. For example the plot below took 7.2 secs on a Sun Ult
ra 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "g:=(x,a)->x*exp(-a*(1-x)):" }}{PARA 0 "> " 0 "" 
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67L/AF07$F_^it$\"+FM'4x'F:7$F_^it$\"+pKv(4#F)7$F_^it$\"+&\\P$*[%F07$F_
^it$\"+8i#HF\"F)7$F_^it$\"+L#=X*[F:7$F_^it$\"+A[^DHF)7$F_^it$\"+w.'*[M
F,7$F_^it$\"+beoI6F:7$F_^it$\"+[43DDF)7$F_^it$\"+ds)*47F07$F_^it$\"++]
P[QF:7$F_^it$\"+'>4!=LF)7$F_^it$\"+LZ2&*)*Ffo7$F_^it$\"+*)3+sKF07$F_^i
t$\"+P#Q;o*F:7$F_^it$\"+h=N#3\"F)7$F_^it$\"+cQ%=6)F:7$F_^it$\"+c&39d\"
F)7$F_^it$\"+G+RC@F:7$F_^it$\"+Hf#*\\LF)7$F_^it$\"+ndwL*)Ffo7$F_^it$\"
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_^it$\"+!*4h\">$F)7$F_^it$\"+4F!=[\"F,7$F_^it$\"+I;V\"*[F07$F_^it$\"+]
Yan8F)7$F_^it$\"+auAvPF:7$F_^it$\"+>\"*RRLF)7$F_^it$\"+7i7S#*Ffo7$F_^i
t$\"+u,8cIF07$F_^it$\"+PA^6\"*F:7$F_^it$\"+TRrV7F)7$F_^it$\"+V#>tH&F:7
$F_^it$\"+^wq\\FF)7$F_^it$\"+t_3+gF,7$F_^it$\"+7c`\\>F:7$F_^it$\"+J'G$
oKF)7$F_^it$\"+&[:*f6F,7$F_^it$\"+Z-?LQF07$F_^it$\"+_c976F)7$F_^it$\"+
C&*H4vF:7$F_^it$\"+w#Gkz\"F)7$F_^it$\"+!e!R/6F:7$F_^it$\"+sT<*[#F)7$F_
^it$\"+yhg_8F07$F_^it$\"+1Ia!G%F:7$F_^it$\"+-QFsJF)7$F_^it$\"+=u,w:F,7
$F_^it$\"+K#=2?&F07$F_^it$\"+;6NQ9F)7$F_^it$\"+v?o)4$F:7$F_^it$\"+G_ut
MF)7$F_^it$\"+$Q_X+'Ffo7$F_^it$\"+q-B))>F07$F_^it$\"+Awc`hF:7$F_^it$\"
+u7jmBF)7$F_^it$\"+k+Dv>F07$F_^it$\"+$\\th6'F:7$F_^it$\"+jfD$Q#F)7$F_^
it$\"+deiw=F07$F_^it$\"+or*3$eF:7$F_^it$\"+l/\"3^#F)7$F_^it$\"+gW![E\"
F07$F_^it$\"+)e1],%F:7$F_^it$\"+'H`aE$F)7$F_^it$\"+&RB1<\"F,7$F_^it$\"
+l=WoQF07$F_^it$\"+[h)47\"F)7$F_^it$\"+r*\\#QtF:7$F_^it$\"+n'zQ'=F)7$F
_^it$\"+]\">y/*F07$F_^it$\"+Z/$p=#F)7$F_^it$\"+7]yCMF07$F_^it$\"+)3kz+
\"F)7$F_^it$\"+89S-)*F:7$F_^it$\"+JQZ]5F)7$F_^it$\"+xax-))F:7$F_^it$\"
+'Go(Q8F)7$F_^it$\"+ZJk(3%F:7$F_^it$\"+u)G5C$F)7$F_^it$\"+06il7F,7$F_^
it$\"+9,)4=%F07$F_^it$\"+^VO)>\"F)7$F_^it$\"+@Dp#)fF:7$F_^it$\"+gV!GW#
F)7$F_^it$\"+$z^9c\"F07$F_^it$\"+)4ga!\\F:7$F_^it$\"+U7%3#HF)7$F_^it$
\"+\\GF+NF,7$F_^it$\"+N;IZ6F:7$F_^it$\"+7@KZDF)7$F_^it$\"+$)*y\"H6F07$
F_^it$\"+'>T:g$F:7$F_^it$\"+mAb&Q$F)7$F_^it$\"+P?vpzFfo7$F_^it$\"+\"fK
rj#F07$F_^it$\"+.U`yzF:7$F_^it$\"+u0X>;F)7$F_^it$\"+AaJ]=F:7$F_^it$\"+
_%o;@$F)7$F_^it$\"+i5(**Q\"F,7$F_^it$\"+76x*e%F07$F_^it$\"+eV$oH\"F)7$
F_^it$\"+R.&fe%F:7$F_^it$\"+(fyQ0$F)7$F_^it$\"+)eEnH#F,7$F_^it$\"+H,*)
fvF0-%&STYLEG6#%&POINTG" 1 5 0 1 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 41 "The diagram above shows a stable 3-cycle." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT 313 24 "4. The lyapexp procedure" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 352 7 "lyapex
p" }{TEXT -1 117 " procedure is used to explore the behaviour of the L
yapunov exponent of a map for a range of values of the parameter " }
{TEXT 265 1 "a" }{TEXT -1 48 ".  The formula used to calculate the exp
onent is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" 
{XPPEDIT 18 0 "lambda=limit(sum(ln(abs(diff(g(x[n]),x))),n=0..N-1)/N,N
=infinity);" "6#/%'lambdaG-%&limitG6$*&-%$sumG6$-%#lnG6#-%$absG6#-%%di
ffG6$-%\"gG6#&%\"xG6#%\"nGF9/F;;\"\"!,&%\"NG\"\"\"\"\"\"!\"\"FAF@FC/F@
%)infinityG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 93 "The procedure plots an approximation to the Lyapunov exponent s
tarting from an initial value " }{TEXT 314 1 "x" }{TEXT -1 39 "0. Noth
ing is plotted during the first " }{TEXT 315 1 "m" }{TEXT -1 127 " ite
rations to allow for any transient behaviour present to die out. The p
rocedure then uses the above formula with N taken as " }{XPPEDIT 18 0 
"n-m" "6#,&%\"nG\"\"\"%\"mG!\"\"" }{TEXT -1 107 " to approximate the L
yapunov exponent. The procedure accumulates the iterations to be plott
ed in the array " }{TEXT 353 3 "pts" }{TEXT -1 68 ". At each iteration
 the point co-ordinates are the current value of " }{TEXT 316 1 "a" }
{TEXT -1 55 " and the Lyapunov exponent. The structure plotted is a " 
}{TEXT 354 9 "pointplot" }{TEXT -1 33 " of the array elements using th
e " }{TEXT 355 10 "style=line" }{TEXT -1 8 " option." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 757 "lyapexp:
=proc(g::procedure,x0::numeric,c::numeric,d::numeric,r::posint,m::posi
nt,n::posint)\nlocal dg,pts,steps;\nif c>=d then ERROR(`d should be gr
eater than c`, d,c); fi;\nif m>=n then ERROR(`n should be greater than
 m`, n,m); fi;\npts:=array(1..2*r);\ndg:=D[1](g);\nsteps:=proc(g::proc
edure,dg::procedure,x0::numeric,c::numeric,d::numeric,r::posint,m::pos
int,n::posint,pts::array)\nlocal range,a,x1,d1,i,j,sum,le;\nrange:=d-c
;\nj:=0;\nfor a from c by range/r while abs(d-a)>0.5e-6 do\011\nj:=j+1
;\nx1:=x0;\nfor i from 1 to m do\nx1:=g(x1,a);\nod;\nsum:=0;\nfor i fr
om 1 to n-m do\nx1:=g(x1,a);\nd1:=dg(x1,a);\011\011\011\nsum:=sum+ln(a
bs(d1));\nod;\nle:=sum/(n-m);\npts[2*j-1]:=a;\npts[2*j]:=le;\nod;\nend
;\nevalhf(steps(g,dg,x0,c,d,r,m,n,var(pts)));\nplots[pointplot](op(pts
),style=LINE);\nend:" }}}{EXCHG {PARA 265 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 277 91 "In the cases below, the behaviour of the
 Lyapunov exponent is shown for the same ranges of " }{TEXT 278 1 "a" 
}{TEXT 279 122 " as in the orbit diagrams. Note that these plots may a
lso take a long time. The plot below took 48 secs on a Sun Ultra 1. " 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 256 "> " 0 "" 
{MPLTEXT 1 0 26 "g:=(x,a)->x*exp(-a*(1-x)):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "lyapexp(g,0.2,-4,-2,100,300,5000);" }}{PARA 13 "" 1 "
" {GLPLOT2D 421 295 295 {PLOTDATA 2 "6$-%'CURVESG6#7`q7$$!\"%\"\"!$\"+
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\\CRfPF-7$$!++++!)GF1$\"++kUROF-7$$!++++gGF1$!+8N!R=#F-7$$!++++SGF1$\"
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\"+&)HI0hFhq7$$!++++gFF1$\"+ow*y?#F-7$$!++++SFF1$\"+Z!e7\"=F-7$$!++++?
FF1$!+@T$3i\"Fhq7$$!+++++FF1$\"+\"f#=C*)Fhq7$$!++++!o#F1$!+qFCd`Fhq7$$
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F-7$$!+++++BF1$!+CDJ6!*F-7$$!++++!G#F1$!+>+O27F17$$!++++gAF1$!++lA]@F1
7$$!++++SAF1$!+R`w&Q\"F17$$!++++?AF1$!+qiem)*F-7$$!+++++AF1$!+-ktowF-7
$$!++++!=#F1$!+!H4?9'F-7$$!++++g@F1$!+Pe\"*p\\F-7$$!++++S@F1$!+bmk<SF-
7$$!++++?@F1$!+Wc6:KF-7$$!+++++@F1$!+A#*=@DF-7$$!++++!3#F1$!+3Gl4>F-7$
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Fhq-%&STYLEG6#%%LINEG" 1 6 0 1 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 106 "This plot gives a
n overall impression of the behaviour of the Lyapunov exponent.  Note \+
in particular that " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{TEXT -1 
326 " becomes positive when stable cycles give way to chaos. Spikes in
 the graph correspond to the onset of a stable cycle.  The large spike
 which occurs in the right most part of the diagram corresponds to a s
table 2-cycle. Notice that not all cycles are represented in the plot \+
due to the resolution. If we increase the value of " }{TEXT 317 1 "r" 
}{TEXT -1 136 " to 200 we obtain additional spikes corresponding to th
e missing cycles. In addition the lengths of some of the spikes are in
creased.   " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 
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+++5AF1$!+fX#*[')F-7$$!+++++AF1$!+-ktowF-7$$!++++!>#F1$!+HP2[oF-7$$!++
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 135 "We can magnify a portion of the p
lot by choosing a smaller range to explore the behaviour near to the f
irst large spike as shown below." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "lyapexp(g,0.2,-3.7,-3.5,200,
300,5000);" }}{PARA 13 "" 1 "" {GLPLOT2D 375 375 375 {PLOTDATA 2 "6$-%
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cV\"y%F-7$$!++++#p$F*$\"+`o71]F-7$$!++++\"p$F*$\"+qlq&*\\F-7$$!++++!p$
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+++DNF*$\"+?7T?_F-7$$!++++CNF*$\"+Y%)pv\\F-7$$!++++BNF*$\"+6O+%3&F-7$$
!++++ANF*$\"+Jwz^_F-7$$!++++@NF*$\"+cAal^F-7$$!++++?NF*$\"+:BnS^F-7$$!
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$!++++;NF*$\"+a7\\8_F-7$$!++++:NF*$\"+2\"=S7&F-7$$!++++9NF*$\"+!359C&F
-7$$!++++8NF*$\"+()3%H*\\F-7$$!++++7NF*$\"+>fr9[F-7$$!++++6NF*$\"+Je.8
ZF-7$$!++++5NF*$!+bl4X^F]fl7$$!++++4NF*$\"+]_i!*\\F-7$$!++++3NF*$\"+a^
+!=&F-7$$!++++2NF*$\"+rm)yK&F-7$$!++++1NF*$\"+6P<'>&F-7$$!++++0NF*$\"+
6:/YaF-7$$!++++/NF*$\"+LOVJ`F-7$$!++++.NF*$\"+n`A=aF-7$$!++++-NF*$\"+y
Tm&Q&F-7$$!++++,NF*$\"+$G\"*HK&F--%&STYLEG6#%%LINEG" 1 6 0 1 0 2 9 1 
4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 40 "This plot shows a stable cycle close to " }{XPPEDIT 18 0 "a=3.5
9" "6#/%\"aG$\"$f$!\"#" }{TEXT -1 53 " while the following plot shows \+
that it is a 3-cycle." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "newweb(g,0.2,0,4,-3.586,250,400);" 
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$\"\"!FF7$$\"1mmmT&)G\\a!#=$\"1:qP;-cG>!#;7$$\"1LLL3x&)*3\"!#<$\"1&3]%
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1+++0TN:CF]o$\"1@&>[ik\"4:FQ7$$\"1++]7RV'\\#F]o$\"1-0XLEGm6FQ7$$\"1+++
:#fke#F]o$\"1T!fD#4b\\()FJ7$$\"1LLL`4NnEF]o$\"1zian@?^nFJ7$$\"1+++],s`
FF]o$\"1hp\\BcT8^FJ7$$\"1mm;zM)>$GF]o$\"1vuA88)=(RFJ7$$\"1+++qfa<HF]o$
\"1)*)>y/<2,$FJ7$$\"1LL$eg`!)*HF]o$\"1([Go7&)zJ#FJ7$$\"1++]#G2A3$F]o$
\"13ivXcFi<FJ7$$\"1LLL$)G[kJF]o$\"1\"zkoiRqM\"FJ7$$\"1++]7yh]KF]o$\"1g
f:8<,;5FJ7$$\"1nmm')fdLLF]o$\"1v[XINIQx!#>7$$\"1nmm,FT=MF]o$\"1q3\"Rm'
y`eFdal7$$\"1LL$e#pa-NF]o$\"1]N0pMvNWFdal7$$\"1+++Sv&)zNF]o$\"15WK1F&f
V$Fdal7$$\"1LLLGUYoOF]o$\"1Hio?YaiDFdal7$$\"1nmm1^rZPF]o$\"1K#p#H#*Gq>
Fdal7$$\"1++]sI@KQF]o$\"1'QOWK]!)[\"Fdal7$$\"1++]2%)38RF]o$\"1/h1=2$p8
\"Fdal7$$\"\"%FF$\"1Puq!z6)4&)!#?-%'COLOURG6&%$RGBG$\"#5!\"\"FFFF-%+AX
ESLABELSG6$%\"xG%!G-%%VIEWG6$;FFFicl%(DEFAULTG" 1 2 0 1 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT 318 19 "An Economic Example" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 345 "This example arose ou
t of project work done by a student on the MSc (Decision Support Syste
ms) offered by the University of Sunderland. The project involved the \+
investigation of a cobweb model of an agricultural market with a non-l
inear supply function, Chiarella [2]. The mathematical model derived r
esulted in a one-dimensional map of the form" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 266 "" 0 "" {XPPEDIT 18 0 "g(x,a)=(1-gamma)*x+gamma/b*
(a-arctan(lambda*x))" "6#/-%\"gG6$%\"xG%\"aG,&*&,&\"\"\"\"\"\"%&gammaG
!\"\"F-F'F-F-*(F.F-%\"bGF/,&F(F--%'arctanG6#*&%'lambdaGF-F'F-F/F-F-" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }
{XPPEDIT 18 0 "gamma" "6#%&gammaG" }{TEXT -1 34 ", the supply paramete
r, satisfies " }{XPPEDIT 18 0 "gamma>0" "6#2\"\"!%&gammaG" }{TEXT -1 
5 " and " }{XPPEDIT 18 0 "gamma<1" "6#2%&gammaG\"\"\"" }{TEXT -1 17 ".
 The parameters " }{TEXT 321 1 "a" }{TEXT -1 5 " and " }{TEXT 322 1 "b
" }{TEXT -1 76 " arise from the assumption that the demand function is
 linear. The parameter" }{TEXT 319 1 " " }{XPPEDIT 18 0 "lambda" "6#%'
lambdaG" }{TEXT -1 60 " measures the slope of the demand function and \+
the variable " }{TEXT 320 1 "x" }{TEXT -1 62 " denotes the expected pr
ice. We shall choose parameter values " }{XPPEDIT 18 0 "gamma=0.3" "6#
/%&gammaG$\"\"$!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "b=0.25" "6#/%\"
bG$\"#D!\"#" }{TEXT -1 10 " and vary " }{XPPEDIT 18 0 "lambda" "6#%'la
mbdaG" }{TEXT -1 5 " and " }{TEXT 323 1 "a" }{TEXT -1 53 ". Note that \+
we need only consider positive values of " }{TEXT 324 1 "a" }{TEXT -1 
61 " since the diagrams exhibit rotational symmetry of order two." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "The plot \+
below shows an orbit diagram for " }{XPPEDIT 18 0 "lambda=3.846" "6#/%
'lambdaG$\"%YQ!\"$" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 40 "g:=(x,a)->0.7*x+1.2*(a-arctan(3.846*x)):" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "orbit(g,0.2,0,1.5,100,250,
400);" }}{PARA 13 "" 1 "" {GLPLOT2D 375 375 375 {PLOTDATA 2 "6$-%'POIN
TSG6dden7$\"\"!$!+rsWH\"*!#57$F'$\"+rsWH\"*F*F&F+F&F+F&F+F&F+F&F+F&F+F
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lFdcblFicblFdcblFicblFdcblFicblFdcblFicblFdcblFicblFdcblFicblFdcblFicb
lFdcblFicblFdcblFicblFdcblFicblFdcblFicblFdcblFicblFdcblFicblFdcblFicb
lFdcblFicblFdcblFicblFdcblFicblFdcblFicblFdcblFicblFdcblFicblFdcblFicb
lFdcblFicblFdcblFicblFdcblFicblFdcblFicblFdcblFicblFdcblFicblFdcblFicb
lFdcblFicblFdcblFicblFdcblFicblFdcblFicblFdcblFicblFdcblFicblFdcblFicb
lFdcblFicblFdcblFicblFdcblFicblFdcblFicblFdcblFicblFdcblFicblFdcblFicb
lFdcblFicblFdcblFicblFdcblFicblFdcblFicblFdcblFicblFdcblFicblFdcblFicb
lFdcblFicbl7$$FZF*$\"+o1W#z'F*7$F]dbl$\"+WX=#4\"F*F\\dblF`dblF\\dblF`d
blF\\dblF`dblF\\dblF`dblF\\dblF`dblF\\dblF`dblF\\dblF`dblF\\dblF`dblF
\\dblF`dblF\\dblF`dblF\\dblF`dblF\\dblF`dblF\\dblF`dblF\\dblF`dblF\\db
lF`dblF\\dblF`dblF\\dblF`dblF\\dblF`dblF\\dblF`dblF\\dblF`dblF\\dblF`d
blF\\dblF`dblF\\dblF`dblF\\dblF`dblF\\dblF`dblF\\dblF`dblF\\dblF`dblF
\\dblF`dblF\\dblF`dblF\\dblF`dblF\\dblF`dblF\\dblF`dblF\\dblF`dblF\\db
lF`dblF\\dblF`dblF\\dblF`dblF\\dblF`dblF\\dblF`dblF\\dblF`dblF\\dblF`d
blF\\dblF`dblF\\dblF`dblF\\dblF`dblF\\dblF`dblF\\dblF`dblF\\dblF`dblF
\\dblF`dblF\\dblF`dblF\\dblF`dblF\\dblF`dblF\\dblF`dblF\\dblF`dblF\\db
lF`dblF\\dblF`dblF\\dblF`dblF\\dblF`dblF\\dblF`dblF\\dblF`dblF\\dblF`d
blF\\dblF`dblF\\dblF`dblF\\dblF`dblF\\dblF`dblF\\dblF`dblF\\dblF`dblF
\\dblF`dblF\\dblF`dblF\\dblF`dblF\\dblF`dblF\\dblF`dblF\\dblF`dblF\\db
lF`dblF\\dblF`dblF\\dblF`dbl7$$\"++++]\"*F*$\"+_2b*['F*7$Fddbl$\"+=J.Y
7F*FcdblFhdblFcdblFhdblFcdblFhdblFcdblFhdblFcdblFhdblFcdblFhdblFcdblFh
dblFcdblFhdblFcdblFhdblFcdblFhdblFcdblFhdblFcdblFhdblFcdblFhdblFcdblFh
dblFcdblFhdblFcdblFhdblFcdblFhdblFcdblFhdblFcdblFhdblFcdblFhdblFcdblFh
dblFcdblFhdblFcdblFhdblFcdblFhdblFcdblFhdblFcdblFhdblFcdblFhdblFcdblFh
dblFcdblFhdblFcdblFhdblFcdblFhdblFcdblFhdblFcdblFhdblFcdblFhdblFcdblFh
dblFcdblFhdblFcdblFhdblFcdblFhdblFcdblFhdblFcdblFhdblFcdblFhdblFcdblFh
dblFcdblFhdblFcdblFhdblFcdblFhdblFcdblFhdblFcdblFhdblFcdblFhdblFcdblFh
dblFcdblFhdblFcdblFhdblFcdblFhdblFcdblFhdblFcdblFhdblFcdblFhdblFcdblFh
dblFcdblFhdblFcdblFhdblFcdblFhdblFcdblFhdblFcdblFhdblFcdblFhdblFcdblFh
dblFcdblFhdblFcdblFhdblFcdblFhdblFcdblFhdblFcdblFhdblFcdblFhdblFcdblFh
dblFcdblFhdblFcdblFhdblFcdblFhdblFcdblFhdbl7$$\"+++++$*F*$\"+)3S)ohF*7
$F\\ebl$\"+$>e`T\"F*F[eblF`eblF[eblF`eblF[eblF`eblF[eblF`eblF[eblF`ebl
F[eblF`eblF[eblF`eblF[eblF`eblF[eblF`eblF[eblF`eblF[eblF`eblF[eblF`ebl
F[eblF`eblF[eblF`eblF[eblF`eblF[eblF`eblF[eblF`eblF[eblF`eblF[eblF`ebl
F[eblF`eblF[eblF`eblF[eblF`eblF[eblF`eblF[eblF`eblF[eblF`eblF[eblF`ebl
F[eblF`eblF[eblF`eblF[eblF`eblF[eblF`eblF[eblF`eblF[eblF`eblF[eblF`ebl
F[eblF`eblF[eblF`eblF[eblF`eblF[eblF`eblF[eblF`eblF[eblF`eblF[eblF`ebl
F[eblF`eblF[eblF`eblF[eblF`eblF[eblF`eblF[eblF`eblF[eblF`eblF[eblF`ebl
F[eblF`eblF[eblF`eblF[eblF`eblF[eblF`eblF[eblF`eblF[eblF`eblF[eblF`ebl
F[eblF`eblF[eblF`eblF[eblF`eblF[eblF`eblF[eblF`eblF[eblF`eblF[eblF`ebl
F[eblF`eblF[eblF`eblF[eblF`eblF[eblF`eblF[eblF`eblF[eblF`eblF[eblF`ebl
F[eblF`eblF[eblF`eblF[eblF`eblF[eblF`eblF[eblF`eblF[eblF`ebl7$$\"++++]
%*F*$\"+W<lCeF*7$Fdebl$\"+E'Qfg\"F*FceblFheblFceblFheblFceblFheblFcebl
FheblFceblFheblFceblFheblFceblFheblFceblFheblFceblFheblFceblFheblFcebl
FheblFceblFheblFceblFheblFceblFheblFceblFheblFceblFheblFceblFheblFcebl
FheblFceblFheblFceblFheblFceblFheblFceblFheblFceblFheblFceblFheblFcebl
FheblFceblFheblFceblFheblFceblFheblFceblFheblFceblFheblFceblFheblFcebl
FheblFceblFheblFceblFheblFceblFheblFceblFheblFceblFheblFceblFheblFcebl
FheblFceblFheblFceblFheblFceblFheblFceblFheblFceblFheblFceblFheblFcebl
FheblFceblFheblFceblFheblFceblFheblFceblFheblFceblFheblFceblFheblFcebl
FheblFceblFheblFceblFheblFceblFheblFceblFheblFceblFheblFceblFheblFcebl
FheblFceblFheblFceblFheblFceblFheblFceblFheblFceblFheblFceblFheblFcebl
FheblFceblFheblFceblFheblFceblFheblFceblFheblFceblFheblFceblFheblFcebl
Fhebl7$$\"+++++'*F*$\"+#=RsW&F*7$F\\fbl$\"+EfjF=F*F[fblF`fblF[fblF`fbl
F[fblF`fblF[fblF`fblF[fblF`fblF[fblF`fblF[fblF`fblF[fblF`fblF[fblF`fbl
F[fblF`fblF[fblF`fblF[fblF`fblF[fblF`fblF[fblF`fblF[fblF`fblF[fblF`fbl
F[fblF`fblF[fblF`fblF[fblF`fblF[fblF`fblF[fblF`fblF[fblF`fblF[fblF`fbl
F[fblF`fblF[fblF`fblF[fblF`fblF[fblF`fblF[fblF`fblF[fblF`fblF[fblF`fbl
F[fblF`fblF[fblF`fblF[fblF`fblF[fblF`fblF[fblF`fblF[fblF`fblF[fblF`fbl
F[fblF`fblF[fblF`fblF[fblF`fblF[fblF`fblF[fblF`fblF[fblF`fblF[fblF`fbl
F[fblF`fblF[fblF`fblF[fblF`fblF[fblF`fblF[fblF`fblF[fblF`fblF[fblF`fbl
F[fblF`fblF[fblF`fblF[fblF`fblF[fblF`fblF[fblF`fblF[fblF`fblF[fblF`fbl
F[fblF`fblF[fblF`fblF[fblF`fblF[fblF`fblF[fblF`fblF[fblF`fblF[fblF`fbl
F[fblF`fblF[fblF`fblF[fblF`fblF[fblF`fblF[fblF`fblF[fblF`fblF[fblF`fbl
F[fblF`fblF[fblF`fbl7$$\"++++](*F*$\"+WA1<]F*7$Fdfbl$\"+j()4+@F*FcfblF
hfblFcfblFhfblFcfblFhfblFcfblFhfblFcfblFhfblFcfblFhfblFcfblFhfblFcfblF
hfblFcfblFhfblFcfblFhfblFcfblFhfblFcfblFhfblFcfblFhfblFcfblFhfblFcfblF
hfblFcfblFhfblFcfblFhfblFcfblFhfblFcfblFhfblFcfblFhfblFcfblFhfblFcfblF
hfblFcfblFhfblFcfblFhfblFcfblFhfblFcfblFhfblFcfblFhfblFcfblFhfblFcfblF
hfblFcfblFhfblFcfblFhfblFcfblFhfblFcfblFhfblFcfblFhfblFcfblFhfblFcfblF
hfblFcfblFhfblFcfblFhfblFcfblFhfblFcfblFhfblFcfblFhfblFcfblFhfblFcfblF
hfblFcfblFhfblFcfblFhfblFcfblFhfblFcfblFhfblFcfblFhfblFcfblFhfblFcfblF
hfblFcfblFhfblFcfblFhfblFcfblFhfblFcfblFhfblFcfblFhfblFcfblFhfblFcfblF
hfblFcfblFhfblFcfblFhfblFcfblFhfblFcfblFhfblFcfblFhfblFcfblFhfblFcfblF
hfblFcfblFhfblFcfblFhfblFcfblFhfblFcfblFhfblFcfblFhfblFcfblFhfblFcfblF
hfblFcfblFhfblFcfblFhfblFcfblFhfbl7$$\"+++++**F*$\"+;pF\"[%F*7$F\\gbl$
\"+eOFwCF*F[gblF`gblF[gblF`gblF[gblF`gblF[gblF`gblF[gblF`gblF[gblF`gbl
F[gblF`gblF[gblF`gblF[gblF`gblF[gblF`gblF[gblF`gblF[gblF`gblF[gblF`gbl
F[gblF`gblF[gblF`gblF[gblF`gblF[gblF`gblF[gblF`gblF[gblF`gblF[gblF`gbl
F[gblF`gblF[gblF`gblF[gblF`gblF[gblF`gblF[gblF`gblF[gblF`gblF[gblF`gbl
F[gblF`gblF[gblF`gblF[gblF`gblF[gblF`gblF[gblF`gblF[gblF`gblF[gblF`gbl
F[gblF`gblF[gblF`gblF[gblF`gblF[gblF`gblF[gblF`gblF[gblF`gblF[gblF`gbl
F[gblF`gblF[gblF`gblF[gblF`gblF[gblF`gblF[gblF`gblF[gblF`gblF[gblF`gbl
F[gblF`gblF[gblF`gblF[gblF`gblF[gblF`gblF[gblF`gblF[gblF`gblF[gblF`gbl
F[gblF`gblF[gblF`gblF[gblF`gblF[gblF`gblF[gblF`gblF[gblF`gblF[gblF`gbl
F[gblF`gblF[gblF`gblF[gblF`gblF[gblF`gblF[gblF`gblF[gblF`gblF[gblF`gbl
F[gblF`gblF[gblF`gblF[gblF`gblF[gblF`gblF[gblF`gbl7$$\"++++05FO$\"+>')
H1NF*7$Fdgbl$\"+9]s@LF*7$Fdgbl$\"+`-J0NF*7$Fdgbl$\"+h=lALF*7$Fdgbl$\"+
lTL/NF*7$Fdgbl$\"+txcBLF*7$Fdgbl$\"+X,P.NF*7$Fdgbl$\"+CHZCLF*7$Fdgbl$
\"+!*zT-NF*7$Fdgbl$\"+'[n`K$F*7$Fdgbl$\"+,vZ,NF*7$Fdgbl$\"+C;DELF*7$Fd
gbl$\"+%[[0]$F*7$Fdgbl$\"+.b7FLF*7$Fdgbl$\"+]2j*\\$F*7$Fdgbl$\"+!G*)zK
$F*7$Fdgbl$\"+9Ts)\\$F*7$Fdgbl$\"+6J%)GLF*7$Fdgbl$\"+'RGy\\$F*7$Fdgbl$
\"+[roHLF*7$Fdgbl$\"+@M%p\\$F*7$Fdgbl$\"+S:_ILF*7$Fdgbl$\"+:!pg\\$F*7$
Fdgbl$\"+JkMJLF*7$Fdgbl$\"+7]?&\\$F*7$Fdgbl$\"+j>;KLF*7$Fdgbl$\"+[7N%
\\$F*7$Fdgbl$\"+v#oHL$F*7$Fdgbl$\"+iv]$\\$F*7$Fdgbl$\"+.bwLLF*7$Fdgbl$
\"+(ztE\\$F*7$Fdgbl$\"+zPbMLF*7$Fdgbl$\"+,)\\=\\$F*7$Fdgbl$\"+MKLNLF*7
$Fdgbl$\"+Da.\"\\$F*7$Fdgbl$\"+%*R5OLF*7$Fdgbl$\"+@0B!\\$F*7$Fdgbl$\"+
&=moL$F*7$Fdgbl$\"+Y\\V*[$F*7$Fdgbl$\"+F*>wL$F*7$Fdgbl$\"+h&[')[$F*7$F
dgbl$\"+T`OQLF*7$Fdgbl$\"+H7(y[$F*7$Fdgbl$\"+VD5RLF*7$Fdgbl$\"+9G5([$F
*7$Fdgbl$\"+[;$)RLF*7$Fdgbl$\"+)=Vj[$F*7$Fdgbl$\"+oFbSLF*7$Fdgbl$\"+>A
f&[$F*7$Fdgbl$\"+8gETLF*7$Fdgbl$\"+%y\\[[$F*7$Fdgbl$\"+#\\r>M$F*7$Fdgb
l$\"+ed6%[$F*7$Fdgbl$\"+4$pEM$F*7$Fdgbl$\"+@+R$[$F*7$Fdgbl$\"+o&fLM$F*
7$Fdgbl$\"+bCn#[$F*7$Fdgbl$\"+rB/WLF*7$Fdgbl$\"+WH'>[$F*7$Fdgbl$\"+=yr
WLF*7$Fdgbl$\"+u8E\"[$F*7$Fdgbl$\"+2gQXLF*7$Fdgbl$\"+Nwc![$F*7$Fdgbl$
\"+Kq/YLF*7$Fdgbl$\"+;;))zMF*7$Fdgbl$\"+!*4qYLF*7$Fdgbl$\"+7K?zMF*7$Fd
gbl$\"+rzMZLF*7$Fdgbl$\"+<B`yMF*7$Fdgbl$\"+n!))zM$F*7$Fdgbl$\"+G)oyZ$F
*7$Fdgbl$\"+n8i[LF*7$Fdgbl$\"+XE@xMF*7$Fdgbl$\"+dzC\\LF*7$Fdgbl$\"+pOc
wMF*7$Fdgbl$\"+Dz')\\LF*7$Fdgbl$\"+-=#fZ$F*7$Fdgbl$\"+a8[]LF*7$Fdgbl$
\"+\\pGvMF*7$Fdgbl$\"+F$)3^LF*7$Fdgbl$\"+<!fYZ$F*7$Fdgbl$\"+E*)o^LF*7$
Fdgbl$\"+9z.uMF*7$Fdgbl$\"+IKG_LF*7$Fdgbl$\"+]NUtMF*7$Fdgbl$\"+=8(GN$F
*7$Fdgbl$\"+Oe\"GZ$F*7$Fdgbl$\"+oKX`LF*7$Fdgbl$\"+&o9AZ$F*7$Fdgbl$\"+a
\"HSN$F*7$Fdgbl$\"+7+irMF*7$Fdgbl$\"+_!*faLF*7$Fdgbl$\"+M<.rMF*7$Fdgbl
$\"+OI;bLF*7$Fdgbl$\"+o(\\/Z$F*7$Fdgbl$\"+w6sbLF*7$Fdgbl$\"+LS()pMF*7$
Fdgbl$\"+XNFcLF*7$Fdgbl$\"+\\WIpMF*7$Fdgbl$\"+7-#oN$F*7$Fdgbl$\"+R4uoM
F*7$Fdgbl$\"+X7OdLF*7$Fdgbl$\"+DM=oMF*7$Fdgbl$\"+7n*yN$F*7$Fdgbl$\"+L=
jnMF*7$Fdgbl$\"+!oE%eLF*7$Fdgbl$\"+(3'3nMF*7$Fdgbl$\"+97&*eLF*7$Fdgbl$
\"+;hamMF*7$Fdgbl$\"+y.ZfLF*7$Fdgbl$\"+Y=,mMF*7$Fdgbl$\"+NU)*fLF*7$Fdg
bl$\"+4K[lMF*7$Fdgbl$\"+\\G\\gLF*7$Fdgbl$\"+L,'\\Y$F*7$Fdgbl$\"+zi*4O$
F*7$Fdgbl$\"+_DWkMF*7$Fdgbl$\"+(e%\\hLF*7$Fdgbl$\"+)RIRY$F*7$Fdgbl$\"+
Ly)>O$F*7$Fdgbl$\"+/OUjMF*7$Fdgbl$\"+ugZiLF*7$Fdgbl$\"+2@#HY$F*7$Fdgbl
$\"+o$fHO$F*7$Fdgbl$\"+TeUiMF*7$Fdgbl$\"+txVjLF*7$Fdgbl$\"+WZ$>Y$F*7$F
dgbl$\"+V8\"RO$F*7$Fdgbl$\"+a([9Y$F*7$Fdgbl$\"+N,QkLF*7$Fdgbl$\"+5y'4Y
$F*7$Fdgbl$\"+.U%[O$F*7$Fdgbl$\"+_=\\gMF*7$Fdgbl$\"++OIlLF*7$Fdgbl$\"+
@3-gMF*7$Fdgbl$\"+y$edO$F*7$Fdgbl$\"+eYbfMF*7$Fdgbl$\"+\"f3iO$F*7$Fdgb
l$\"+1L4fMF*7$Fdgbl$\"+)GamO$F*7$Fdgbl$\"+4njeMF*7$Fdgbl$\"+@b4nLF*7$F
dgbl$\"+7[=eMF*7$Fdgbl$\"+SB`nLF*7$Fdgbl$\"+evtdMF*7$Fdgbl$\"+$zkzO$F*
7$Fdgbl$\"+&*[HdMF*7$Fdgbl$\"+HHRoLF*7$Fdgbl$\"+pn&oX$F*7$Fdgbl$\"+&z;
)oLF*7$$\"++++?5FO$\"+#)*QR]$F*7$Fhcdl$\"+c(QR]$F*7$Fhcdl$\"+o*QR]$F*7
$Fhcdl$\"+p(QR]$F*7$Fhcdl$\"+c*QR]$F*7$Fhcdl$\"+!yQR]$F*7$Fhcdl$\"+X*Q
R]$F*7$Fhcdl$\"+!zQR]$F*7$Fhcdl$\"+O*QR]$F*7$Fhcdl$\"+*zQR]$F*7$Fhcdl$
\"+G*QR]$F*7$Fhcdl$\"+2)QR]$F*7$Fhcdl$\"+?*QR]$F*7$Fhcdl$\"+9)QR]$F*7$
Fhcdl$\"+9*QR]$F*7$Fhcdl$\"+?)QR]$F*7$Fhcdl$\"+3*QR]$F*7$Fhcdl$\"+D)QR
]$F*7$Fhcdl$\"+.*QR]$F*7$Fhcdl$\"+I)QR]$F*7$Fhcdl$\"+**)QR]$F*7$Fhcdl$
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*7$Fhcdl$\"+T)QR]$F*7$Fhcdl$\"+)))QR]$F*7$Fhcdl$\"+W)QR]$F*7$Fhcdl$\"+
&))QR]$F*7$Fhcdl$\"+Z)QR]$F*7$Fhcdl$\"+$))QR]$F*7$Fhcdl$\"+\\)QR]$F*7$
Fhcdl$\"+\"))QR]$F*7$Fhcdl$\"+^)QR]$F*7$Fhcdl$\"+z)QR]$F*7$Fhcdl$\"+`)
QR]$F*7$Fhcdl$\"+x)QR]$F*7$Fhcdl$\"+a)QR]$F*7$Fhcdl$\"+w)QR]$F*7$Fhcdl
$\"+b)QR]$F*7$Fhcdl$\"+v)QR]$F*7$Fhcdl$\"+d)QR]$F*7$Fhcdl$\"+u)QR]$F*7
$Fhcdl$\"+e)QR]$F*7$Fhcdl$\"+t)QR]$F*7$Fhcdl$\"+f)QR]$F*7$Fhcdl$\"+s)Q
R]$F*F`\\el7$Fhcdl$\"+r)QR]$F*7$Fhcdl$\"+g)QR]$F*7$Fhcdl$\"+q)QR]$F*7$
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lFe]el7$Fhcdl$\"+o)QR]$F*7$Fhcdl$\"+j)QR]$F*Fh]elF[^elFh]elF[^el7$Fhcd
l$\"+n)QR]$F*F[^elF^^el7$Fhcdl$\"+k)QR]$F*F^^elFa^elF^^elFa^elF^^elFa^
elF^^elFa^el7$Fhcdl$\"+m)QR]$F*7$Fhcdl$\"+l)QR]$F*Fd^elFg^elFd^elFg^el
Fd^elFg^elFd^elFg^elFd^elFg^elFd^elFg^elFd^elFg^elFd^elFg^elFd^elFg^el
Fd^elFg^elFd^elFg^elFd^elFg^elFd^elFg^elFd^elFg^elFd^elFg^elFd^elFg^el
Fd^elFg^elFd^elFg^elFd^elFg^elFd^elFg^elFg^elFg^elFg^elFg^elFg^elFg^el
Fg^elFg^elFg^elFg^elFg^elFg^elFg^elFg^elFg^elFg^elFg^elFg^elFg^elFg^el
Fg^elFg^elFg^elFg^elFg^elFg^elFg^elFg^elFg^elFg^elFg^elFg^el7$$\"++++N
5FO$\"+Qp8)f$F*Fj^elFj^elFj^elFj^elFj^elFj^elFj^elFj^elFj^elFj^elFj^el
Fj^elFj^elFj^elFj^elFj^elFj^elFj^elFj^elFj^elFj^elFj^elFj^elFj^elFj^el
Fj^elFj^elFj^elFj^elFj^elFj^elFj^elFj^elFj^elFj^elFj^elFj^elFj^elFj^el
Fj^elFj^elFj^elFj^elFj^elFj^elFj^elFj^elFj^elFj^elFj^elFj^elFj^elFj^el
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celFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFh
celFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFh
celFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFh
celFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFh
celFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFh
celFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFh
celFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFh
celFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFh
celFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFh
celFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcelFhcel7$
$\"++++!H\"FO$\"+?'G9v&F*F]delF]delF]delF]delF]delF]delF]delF]delF]del
F]delF]delF]delF]delF]delF]delF]delF]delF]delF]delF]delF]delF]delF]del
F]delF]delF]delF]delF]delF]delF]delF]delF]delF]delF]delF]delF]delF]del
F]delF]delF]delF]delF]delF]delF]delF]delF]delF]delF]delF]delF]delF]del
F]delF]delF]delF]delF]delF]delF]delF]delF]delF]delF]delF]delF]delF]del
F]delF]delF]delF]delF]delF]delF]delF]delF]delF]delF]delF]delF]delF]del
F]delF]delF]delF]delF]delF]delF]delF]delF]delF]delF]delF]delF]delF]del
F]delF]delF]delF]delF]delF]delF]delF]delF]delF]delF]delF]delF]delF]del
F]delF]delF]delF]delF]delF]delF]delF]delF]delF]delF]delF]delF]delF]del
F]delF]delF]delF]delF]delF]delF]delF]delF]delF]delF]delF]delF]delF]del
F]delF]delF]delF]delF]delF]delF]delF]delF]delF]delF]delF]delF]delF]del
7$$\"++++08FO$\"+'*\\_?fF*FbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbde
lFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbde
lFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbde
lFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbde
lFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbde
lFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbde
lFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbde
lFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbde
lFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbde
lFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbde
lFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbdelFbde
l7$$\"++++?8FO$\"+G/g&4'F*FgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgde
lFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgde
lFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgde
lFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgde
lFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgde
lFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgde
lFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgde
lFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgde
lFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgde
lFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgde
lFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgdelFgde
l7$$\"++++N8FO$\"+(QpoF'F*F\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF
\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\ee
lF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\
eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF
\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\ee
lF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\
eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF
\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\ee
lF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\
eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF
\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\ee
lF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\
eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF\\eelF
\\eelF\\eel7$$F\\pFO$\"+cdakkF*FaeelFaeelFaeelFaeelFaeelFaeelFaeelFaee
lFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaee
lFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaee
lFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaee
lFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaee
lFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaee
lFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaee
lFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaee
lFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaee
lFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaee
lFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaeelFaee
lFaeel7$$\"++++l8FO$\"+i:%)emF*FeeelFeeelFeeelFeeelFeeelFeeelFeeelFeee
lFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeee
lFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeee
lFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeee
lFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeee
lFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeee
lFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeee
lFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeee
lFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeee
lFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeee
lFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeeelFeee
lFeeel7$$\"++++!Q\"FO$\"+(el*foF*FjeelFjeelFjeelFjeelFjeelFjeelFjeelFj
eelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFj
eelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFj
eelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFj
eelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFj
eelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFj
eelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFj
eelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFj
eelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFj
eelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFj
eelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFjeelFj
eelFjeel7$$\"++++&R\"FO$\"+#*=7oqF*F_felF_felF_felF_felF_felF_felF_fel
F_felF_felF_felF_felF_felF_felF_felF_felF_felF_felF_felF_felF_felF_fel
F_felF_felF_felF_felF_felF_felF_felF_felF_felF_felF_felF_felF_felF_fel
F_felF_felF_felF_felF_felF_felF_felF_felF_felF_felF_felF_felF_felF_fel
F_felF_felF_felF_felF_felF_felF_felF_felF_felF_felF_felF_felF_felF_fel
F_felF_felF_felF_felF_felF_felF_felF_felF_felF_felF_felF_felF_felF_fel
F_felF_felF_felF_felF_felF_felF_felF_felF_felF_felF_felF_felF_felF_fel
F_felF_felF_felF_felF_felF_felF_felF_felF_felF_felF_felF_felF_felF_fel
F_felF_felF_felF_felF_felF_felF_felF_felF_felF_felF_felF_felF_felF_fel
F_felF_felF_felF_felF_felF_felF_felF_felF_felF_felF_felF_felF_felF_fel
F_felF_felF_felF_felF_felF_felF_felF_felF_felF_felF_felF_felF_felF_fel
F_felF_fel7$$\"++++59FO$\"+Q#3NG(F*FdfelFdfelFdfelFdfelFdfelFdfelFdfel
FdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfel
FdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfel
FdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfel
FdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfel
FdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfel
FdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfel
FdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfel
FdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfel
FdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfel
FdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfelFdfel
FdfelFdfel7$$\"++++D9FO$\"+dWJ1vF*FifelFifelFifelFifelFifelFifelFifelF
ifelFifelFifelFifelFifelFifelFifelFifelFifelFifelFifelFifelFifelFifelF
ifelFifelFifelFifelFifelFifelFifelFifelFifelFifelFifelFifelFifelFifelF
ifelFifelFifelFifelFifelFifelFifelFifelFifelFifelFifelFifelFifelFifelF
ifelFifelFifelFifelFifelFifelFifelFifelFifelFifelFifelFifelFifelFifelF
ifelFifelFifelFifelFifelFifelFifelFifelFifelFifelFifelFifelFifelFifelF
ifelFifelFifelFifelFifelFifelFifelFifelFifelFifelFifelFifelFifelFifelF
ifelFifelFifelFifelFifelFifelFifelFifelFifelFifelFifelFifelFifelFifelF
ifelFifelFifelFifelFifelFifelFifelFifelFifelFifelFifelFifelFifelFifelF
ifelFifelFifelFifelFifelFifelFifelFifelFifelFifelFifelFifelFifelFifelF
ifelFifelFifelFifelFifelFifelFifelFifelFifelFifelFifelFifelFifelFifelF
ifelFifel7$$\"++++S9FO$\"+n2sOxF*F^gelF^gelF^gelF^gelF^gelF^gelF^gelF^
gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^
gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^
gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^
gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^
gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^
gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^
gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^
gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^
gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^
gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^gelF^
gelF^gel7$$\"++++b9FO$\"+'3'*[(zF*FcgelFcgelFcgelFcgelFcgelFcgelFcgelF
cgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelF
cgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelF
cgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelF
cgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelF
cgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelF
cgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelF
cgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelF
cgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelF
cgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelF
cgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelFcgelF
cgelFcgel7$$\"++++q9FO$\"+^j*4A)F*FhgelFhgelFhgelFhgelFhgelFhgelFhgelF
hgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelF
hgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelF
hgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelF
hgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelF
hgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelF
hgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelF
hgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelF
hgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelF
hgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelF
hgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelFhgelF
hgelFhgel7$$\"++++&[\"FO$\"+'*G;v%)F*F]helF]helF]helF]helF]helF]helF]h
elF]helF]helF]helF]helF]helF]helF]helF]helF]helF]helF]helF]helF]helF]h
elF]helF]helF]helF]helF]helF]helF]helF]helF]helF]helF]helF]helF]helF]h
elF]helF]helF]helF]helF]helF]helF]helF]helF]helF]helF]helF]helF]helF]h
elF]helF]helF]helF]helF]helF]helF]helF]helF]helF]helF]helF]helF]helF]h
elF]helF]helF]helF]helF]helF]helF]helF]helF]helF]helF]helF]helF]helF]h
elF]helF]helF]helF]helF]helF]helF]helF]helF]helF]helF]helF]helF]helF]h
elF]helF]helF]helF]helF]helF]helF]helF]helF]helF]helF]helF]helF]helF]h
elF]helF]helF]helF]helF]helF]helF]helF]helF]helF]helF]helF]helF]helF]h
elF]helF]helF]helF]helF]helF]helF]helF]helF]helF]helF]helF]helF]helF]h
elF]helF]helF]helF]helF]helF]helF]helF]helF]helF]helF]helF]helF]helF]h
elF]helF]hel-%&STYLEG6#%&POINTG" 1 5 0 1 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 119 "The Lyapunov exponents are always negati
ve and chaos does not occur. The negative exponents are shown by the f
ollowing " }{TEXT 356 7 "lyapexp" }{TEXT -1 9 " diagram." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "lyap
exp(g,0.2,0.4,1,200,300,5000);" }}{PARA 13 "" 1 "" {GLPLOT2D 375 375 
375 {PLOTDATA 2 "6$-%'CURVESG6#7dw7$$\"+++++S!#5$!+\\qr89F*7$$\"++++IS
F*$!+C_(*G8F*7$$\"++++gSF*$!+sIXX7F*7$$\"++++!4%F*$!+)3MJ;\"F*7$$\"+++
+?TF*$!+GC+#3\"F*7$$\"++++]TF*$!+rG/-5F*7$$\"++++!=%F*$!+$)zSK#*!#67$$
\"++++5UF*$!+>;#eX)FK7$$\"++++SUF*$!+UY`!p(FK7$$\"++++qUF*$!+FtTOpFK7$
$\"+++++VF*$!+\"*\\M$>'FK7$$\"++++IVF*$!+)y(>haFK7$$\"++++gVF*$!+?0')R
ZFK7$$\"++++!R%F*$!+TCAHSFK7$$\"++++?WF*$!+1q<HLFK7$$\"++++]WF*$!+#4@'
REFK7$$\"++++![%F*$!+P5Xg>FK7$$\"++++5XF*$!+`'\\:H\"FK7$$\"++++SXF*$!+
`$*oKj!#77$$\"++++qXF*$!+V=,!))*!#87$$\"+++++YF*$!+ntII8FK7$$\"++++IYF
*$!+7<j]EFK7$$\"++++gYF*$!+T8m&*RFK7$$\"++++!p%F*$!+3xnn`FK7$$\"++++?Z
F*$!+R>PqnFK7$$\"++++]ZF*$!+e/i3#)FK7$$\"++++!y%F*$!+!4g()o*FK7$$\"+++
+5[F*$!+6K*=7\"F*7$$\"++++S[F*$!+^2&4G\"F*7$$\"++++q[F*$!+uBOZ9F*7$$\"
+++++\\F*$!+#=)yA;F*7$$\"++++I\\F*$!+L#G$4=F*7$$\"++++g\\F*$!+3Jl4?F*7
$$\"++++!*\\F*$!+^M>FAF*7$$\"++++?]F*$!+wpUmCF*7$$\"++++]]F*$!+VHLLFF*
7$$\"++++!3&F*$!+i'[h.$F*7$$\"++++5^F*$!+.`q'Q$F*7$$\"++++S^F*$!+3w'H!
QF*7$$\"++++q^F*$!+G%eVJ%F*7$$\"+++++_F*$!+kNcu\\F*7$$\"++++I_F*$!+Yg]
**eF*7$$\"++++g_F*$!+YCDJuF*7$$\"++++!H&F*$!+F\"QIC\"!\"*7$$\"++++?`F*
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "Th
is example has indicated how the tools " }{TEXT 361 5 "orbit" }{TEXT 
-1 1 "," }{TEXT 362 8 " lyapexp" }{TEXT -1 5 " and " }{TEXT 363 6 "new
web" }{TEXT -1 108 " have been used by students to investigate and con
firm properties of maps occurring in mathematical models. " }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 11 "Conclusi
on\n" }}{PARA 0 "" 0 "" {TEXT -1 666 "The above work demonstrates the \+
application of Maple to the exploration of certain properties of one-d
imensional maps.  Using Maple as a programming language has the advant
age of using Maple's built-in graph plotting procedures.  The tools ha
ve proved useful as a prelude to a mathematical analysis of a map wher
e this is tractable, e.g. the logistic map,  and as an essential aid t
o a numerical investigation in cases where the mathematics is intracta
ble, e.g. the exponential map. Although it has not been considered in \+
this paper it is also possible to integrate the graphical approach wit
h an analytical investigation of a map using Maple's symbolic capabili
ty. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 364 10 "
References" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 49 "[1]  P.G. Drazin: Nonlinear Systems, CUP, (1992);" }}{PARA 0 "
" 0 "" {TEXT -1 115 "[2] C. Chiarella: The Cobweb Model: its instabili
ty and the onset of chaos, Economic Modelling, pp 377-384, (1983)." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 326 11 "Biograph
ies" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 695 "W
alter Middleton obtained his BSc degree from the University of Hull in
 1963 and his\nMSc from the CNAA in 1978. Since 1975 he has been a Sen
ior Lecturer at\nthe University of Sunderland (formerly Sunderland Pol
ytechnic) and\npresently holds the post of Reader in Mathematics Learn
ing Systems in\nthe School of Computing and Information Systems. He ha
s teaching and\nresearch interests in the use of computer algebra syst
ems in teaching,\ntraining for research students using the WWW and CAL
. He has been\ninvolved in a number of government funded projects aime
d at bringing\ntechnology into the teaching of mathematics and in  TEM
PUS European\nprojects with Bulgaria, Hungary and currently with Eston
ia." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 539 "D
ouglas Curran obtained his BSc degree from the University of Dundee in
 1969 and his\nPhD from the CNAA in 1983. Since 1971 he has been a Lec
turer/Senior\nlecturer at the University of Sunderland (formerly Sunde
rland\nPolytechnic). He has teaching and research interests in the use
 of\ncomputer algebra systems in teaching, dynamical systems models in
\neconomics and the numerical solution of differential equations. He h
as\nbeen involved in a number of TEMPUS European projects with Bulgari
a,\nHungary and currently with Estonia and the Ukraine." }}}}{MARK "23
 3 2" 35 }{VIEWOPTS 1 1 0 1 1 1803 }