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{SECT 0 {EXCHG {PARA 202 "" 0 "" {TEXT 206 9 "MTHBD 449" }}{PARA 203 "
" 0 "" {TEXT 207 16 "Lorenz Equations" }}{PARA 204 "" 0 "" }{PARA 204 
"" 0 "" {TEXT 208 54 "The third order nonlinear autonomous Lorenz equa
tions:" }}{PARA 204 "" 0 "" }{PARA 204 "" 0 "" {TEXT 208 5 "x' = " }
{XPPEDIT 18 0 "sigma*(-x+y)" "6#*&%&sigmaG\"\"\",&%\"xG!\"\"%\"yGF%F%"
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y' = " }{XPPEDIT 18 0 "r*x-y-x*z;" "6#,(*&%\"rG\"\"\"%\"xGF&F&%\"yG!\"
\"*&F'F&%\"zGF&F)" }{TEXT 209 1 " " }}{PARA 204 "" 0 "" }{PARA 204 "" 
0 "" {TEXT 208 5 "z' = " }{XPPEDIT 18 0 "-b*z+x*y;" "6#,&*&%\"bG\"\"\"
%\"zGF&!\"\"*&%\"xGF&%\"yGF&F&" }{TEXT 209 1 " " }}{PARA 204 "" 0 "" }
}{SECT 1 {PARA 205 "" 0 "" {TEXT 210 53 "1: Setting up the Lorenz equa
tions: Phase Portraits. " }}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 211 
8 "restart;" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 211 13 "sigma :=
 10: " }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 211 9 "b := 8/3:" }}}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 211 8 "r := 28:" }}}{EXCHG 
{PARA 204 "> " 0 "" {MPLTEXT 1 211 46 "diffeq1 := diff(x(t),t) = sigma
*(y(t) - x(t));" }}{PARA 206 "" 1 "" {XPPMATH 20 "6#>I(diffeq1G6\"/-I%
diffGI*protectedGF)6$-I\"xGF%6#I\"tGF%F.,&-I\"yGF%F-\"#5F+!#5" }{TEXT 
212 1 " " }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 211 52 "diffeq2 := \+
diff(y(t),t) = r*x(t) - y(t) - x(t)*z(t);" }}{PARA 206 "" 1 "" 
{XPPMATH 20 "6#>I(diffeq2G6\"/-I%diffGI*protectedGF)6$-I\"yGF%6#I\"tGF
%F.,(-I\"xGF%F-\"#GF+!\"\"*&F0\"\"\"-I\"zGF%F-F5F3" }{TEXT 212 1 " " }
}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 211 46 "diffeq3 := diff(z(t),t
) = -b*z(t) + x(t)*y(t);" }}{PARA 206 "" 1 "" {XPPMATH 20 "6#>I(diffeq
3G6\"/-I%diffGI*protectedGF)6$-I\"zGF%6#I\"tGF%F.,&F+#!\")\"\"$*&-I\"x
GF%F-\"\"\"-I\"yGF%F-F6F6" }{TEXT 212 1 " " }}}{EXCHG {PARA 204 "> " 0
 "" {MPLTEXT 1 211 14 "with(DEtools):" }}}{EXCHG {PARA 204 "> " 0 "" 
{MPLTEXT 1 211 127 "DEplot3d(\{diffeq1,diffeq2,diffeq3\},[x(t),y(t),z(
t)],t=0..20,[[x(0)=1,y(0)=1,z(0)=1]],x=-25..25,y=-25..30,z=0..60,steps
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2 6 1 2 2 1.0 72.0 -83.0 1 0 "Curve 1" }}}}}{SECT 1 {PARA 205 "" 0 "" 
{TEXT 210 34 "2: Graphs of x(t), y(t), and z(t)." }}{EXCHG {PARA 204 "
> " 0 "" {MPLTEXT 1 211 12 "with(plots):" }}{PARA 208 "" 1 "" {TEXT 
213 49 "Warning, the name changecoords has been redefined" }{TEXT 213 
1 "\n" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 211 8 "xo := 5:" }}}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 211 8 "yo := 5:" }}}{EXCHG 
{PARA 204 "> " 0 "" {MPLTEXT 1 211 8 "zo := 5:" }}}{EXCHG {PARA 204 ">
 " 0 "" {MPLTEXT 1 211 91 "soln := dsolve(\{diffeq1,diffeq2,diffeq3,x(
0)=xo,y(0)=yo,z(0)=zo\},\{x(t),y(t),z(t)\},numeric);" }}{PARA 206 "" 
1 "" {XPPMATH 20 "6#>I%solnG6\"f*6#I(x_rkf45GF%6)I$resGF%I%dataGF%I%va
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GF%FZ-I\"zGF%FZ>8'-_FIFL6#Q/soln_procedureF%@$4-I%typeGF=6$F?.I(numeri
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09!I(unknownGF%O-.F`vFDC$>8)-I(pointtoGF=6#&-F\\o6#Q0soln_proceduresF%
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{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 211 63 "odeplot(soln,[t,x(t)],0.
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{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 211 16 "# end of section" }}}}
{SECT 1 {PARA 205 "" 0 "" {TEXT 210 36 "3: Sensitivity to Initial Cond
itions" }}{EXCHG {PARA 204 "" 0 "" {TEXT 208 133 "Here we show the sen
sitivity to initial conditions by combining plots of solutions x(t) wi
th a slightly different initial conditions." }}}{EXCHG {PARA 204 "> " 
0 "" {MPLTEXT 1 211 8 "xo := 5:" }}}{EXCHG {PARA 204 "> " 0 "" 
{MPLTEXT 1 211 8 "yo := 5:" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 
211 8 "zo := 5:" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 211 91 "soln
 := dsolve(\{diffeq1,diffeq2,diffeq3,x(0)=xo,y(0)=yo,z(0)=zo\},\{x(t),
y(t),z(t)\},numeric);" }}{PARA 206 "" 1 "" {XPPMATH 20 "6#>I%solnG6\"f
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Fip.I&arrayGF=O-I%evalGF=6$Fip\"\"\"0FipQ)procnameF%OFip-Fio6$FE7&Q%la
stF%.I%lastGF%Q(initialF%.I(initialGF%C$Fhp@$-Fbo6$Fip.I%listGF=O7#-I$
seqGF=6$/&FU6#,&8*FjqFjqFjq&FipFds/Ffs;\"\"!\"\"$3-Fbo6$F?I\"=GF=-Fio6
$-I$lhsGF=6#F?7$FdrFerC$@%-Fbo6$-I$rhsGF=FdtF[s>Fip-Fjn6#/Fdr7$Fjs-I#o
pGF=6#FjtYQKinitial~values~must~be~specified~in~a~listF%Fhr/F?Q.solnpr
ocedureF%O-Fhq6#Fjn/F?Q(sysvarsF%OFU@%09!I(unknownGF%O-.F`vFDC$>8)-I(p
ointtoGF=6#&-F\\o6#Q0soln_proceduresF%6#FjsO-.FgvFDZ%C$>Fip-FjnFdtF^sF
%YF%F%F%F%" }{TEXT 212 1 " " }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 
211 83 "graph1 := odeplot(soln,[t,x(t)],0..20,numpoints=200,labels=[\"
t\",\"x(t)\"],color=red):" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 
211 11 "xo := 5.01:" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 211 8 "y
o := 5:" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 211 8 "zo := 5:" }}}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 211 91 "soln := dsolve(\{diffeq1
,diffeq2,diffeq3,x(0)=xo,y(0)=yo,z(0)=zo\},\{x(t),y(t),z(t)\},numeric)
;" }}{PARA 206 "" 1 "" {XPPMATH 20 "6#>I%solnG6\"f*6#I(x_rkf45GF%6)I$r
esGF%I%dataGF%I%varsGF%I)solnprocGF%I)outpointGF%I&ndsolGF%I\"iGF%6#Ii
nCopyright~(c)~2000~by~Waterloo~Maple~Inc.~All~rights~reserved.GF%F%C*
>I8_EnvDSNumericSaveDigitsGF%I'DigitsGF%>F6\"#9@%/I-_EnvInFsolveGF%I%t
rueGI*protectedGF=>8(-&I&evalfGF=6#F56#9$>F?-FBFD>8%`6$I$GetGF%I$SetGF
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%-I\"xGF%6#FW-I\"yGF%FZ-I\"zGF%FZ>8'-_FIFL6#Q/soln_procedureF%@$4-I%ty
peGF=6$F?.I(numericGF=C$@,-I'memberGF=6$FE7+Q&startF%.I&startGF%Q'meth
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F%C$>8$-Fjn6#-I(convertGF=6$FE.I'stringGF=@&-Fbo6$Fip.I&arrayGF=O-I%ev
alGF=6$Fip\"\"\"0FipQ)procnameF%OFip-Fio6$FE7&Q%lastF%.I%lastGF%Q(init
ialF%.I(initialGF%C$Fhp@$-Fbo6$Fip.I%listGF=O7#-I$seqGF=6$/&FU6#,&8*Fj
qFjqFjq&FipFds/Ffs;\"\"!\"\"$3-Fbo6$F?I\"=GF=-Fio6$-I$lhsGF=6#F?7$FdrF
erC$@%-Fbo6$-I$rhsGF=FdtF[s>Fip-Fjn6#/Fdr7$Fjs-I#opGF=6#FjtYQKinitial~
values~must~be~specified~in~a~listF%Fhr/F?Q.solnprocedureF%O-Fhq6#Fjn/
F?Q(sysvarsF%OFU@%09!I(unknownGF%O-.F`vFDC$>8)-I(pointtoGF=6#&-F\\o6#Q
0soln_proceduresF%6#FjsO-.FgvFDZ%C$>Fip-FjnFdtF^sF%YF%F%F%F%" }{TEXT 
212 1 " " }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 211 84 "graph2 := o
deplot(soln,[t,x(t)],0..20,numpoints=200,labels=[\"t\",\"x(t)\"],color
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21A128FFC8BF30-%&COLORG6)%$RGBG$\"#5!\"\"$\"\"!F1F2F2F2F/-%+AXESLABELS
G6'Q\"tF)Q%x(t)F)-%%FONTG6$%*HELVETICAGF0%+HORIZONTALGF=F9" 1 2 2 0 
10 1 2 6 1 4 2 1.0 45.0 45.0 1 0 "Curve 1" }}}}{EXCHG {PARA 204 "" 0 "
" {TEXT 208 135 "Here a slight change in initial conditions xo goes fr
om 5 to 5.01 causes drastic changes in the solution. Typical of chaoti
c behavior. " }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 211 16 "# end o
f section" }}}}{SECT 1 {PARA 205 "" 0 "" {TEXT 210 16 "4: Phase planes
 " }}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 211 8 "xo := 5:" }}}{EXCHG 
{PARA 204 "> " 0 "" {MPLTEXT 1 211 8 "yo := 5:" }}}{EXCHG {PARA 204 ">
 " 0 "" {MPLTEXT 1 211 8 "zo := 5:" }}}{EXCHG {PARA 204 "> " 0 "" 
{MPLTEXT 1 211 91 "soln := dsolve(\{diffeq1,diffeq2,diffeq3,x(0)=xo,y(
0)=yo,z(0)=zo\},\{x(t),y(t),z(t)\},numeric);" }}{PARA 206 "" 1 "" 
{XPPMATH 20 "6#>I%solnG6\"f*6#I(x_rkf45GF%6)I$resGF%I%dataGF%I%varsGF%
I)solnprocGF%I)outpointGF%I&ndsolGF%I\"iGF%6#IinCopyright~(c)~2000~by~
Waterloo~Maple~Inc.~All~rights~reserved.GF%F%C*>I8_EnvDSNumericSaveDig
itsGF%I'DigitsGF%>F6\"#9@%/I-_EnvInFsolveGF%I%trueGI*protectedGF=>8(-&
I&evalfGF=6#F56#9$>F?-FBFD>8%`6$I$GetGF%I$SetGF%b6#I+thismoduleGF%6#I%
DataGF%F%FKF%F%F%F%I+_m24438404GF%FQ>8&7&I\"tGF%-I\"xGF%6#FW-I\"yGF%FZ
-I\"zGF%FZ>8'-_FIFL6#Q/soln_procedureF%@$4-I%typeGF=6$F?.I(numericGF=C
$@,-I'memberGF=6$FE7+Q&startF%.I&startGF%Q'methodF%.I'methodGF%Q%leftF
%Q&rightF%Q)leftdataF%Q*rightdataF%Q+enginedataF%C$>8$-Fjn6#-I(convert
GF=6$FE.I'stringGF=@&-Fbo6$Fip.I&arrayGF=O-I%evalGF=6$Fip\"\"\"0FipQ)p
rocnameF%OFip-Fio6$FE7&Q%lastF%.I%lastGF%Q(initialF%.I(initialGF%C$Fhp
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!\"\"$3-Fbo6$F?I\"=GF=-Fio6$-I$lhsGF=6#F?7$FdrFerC$@%-Fbo6$-I$rhsGF=Fd
tF[s>Fip-Fjn6#/Fdr7$Fjs-I#opGF=6#FjtYQKinitial~values~must~be~specifie
d~in~a~listF%Fhr/F?Q.solnprocedureF%O-Fhq6#Fjn/F?Q(sysvarsF%OFU@%09!I(
unknownGF%O-.F`vFDC$>8)-I(pointtoGF=6#&-F\\o6#Q0soln_proceduresF%6#Fjs
O-.FgvFDZ%C$>Fip-FjnFdtF^sF%YF%F%F%F%" }{TEXT 212 1 " " }}}{EXCHG 
{PARA 204 "> " 0 "" {MPLTEXT 1 211 79 "odeplot(soln,[x(t),y(t)],0..10,
numpoints=400,labels=[\"x(t)\",\"y(t)\"],color=red);" }}{PARA 207 "" 
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