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{SECT 0 {EXCHG {PARA 210 "" 0 "" }{PARA 210 "" 0 "" {TEXT 213 3 "   " 
}{TEXT 214 30 "Oscillating Chemical Reactions" }}{PARA 210 "" 0 "" }}
{EXCHG {PARA 210 "> " 0 "" {MPLTEXT 1 215 34 "restart:with(DEtools):wi
th(plots):" }}{PARA 211 "" 1 "" {TEXT 216 49 "Warning, the name change
coords has been redefined" }{TEXT 216 1 "\n" }}}{EXCHG {PARA 210 "" 0 
"" }{PARA 212 "" 0 "" {TEXT 217 129 "In the early1950's a Russian bioc
hemist Boris Belousov discovered a chemical reaction which would not c
onverge to an equilibrium," }}{PARA 213 "" 0 "" {TEXT 218 136 "but wou
ld oscillate periodically. The story turned dramatic, as no journal wo
uld agree to publish this discovery, as it went against the" }}{PARA 
214 "" 0 "" {TEXT 219 129 "established beliefs in Chemistry. In 1961 a
 graduate student in Moscow, named Zhabotinsky was asked by the adviso
r to look at the" }}{PARA 215 "" 0 "" {TEXT 220 106 "reaction. Zhaboti
nsky confirmed that Belousov was right. Below is a Java simulation of \+
B-Z reaction  from " }}{PARA 216 "" 0 "" {TEXT 221 54 "http://ccl.nort
hwestern.edu/netlogo/models/B-ZReaction" }{TEXT 222 65 " (Wilensky U, \+
2003 -- see the web page for the copyright notice)." }}{PARA 217 "" 0 
"" }{PARA 218 "" 0 "" {TEXT 223 81 "Click \"setup\" and then \"go\" to
 see the colour of the mixture change periodically." }}{PARA 210 "" 0 
"" }}{EXCHG {PARA 210 "" 0 "" {HYPERLNK 224 "Java simulation of B-Z re
action" 4 "ccl.northwestern.edu/netlogo/models/run.cgi?B-ZReaction.629
.476" "" }}}{EXCHG {PARA 210 "" 0 "" }{PARA 210 "" 0 "" }{PARA 219 "" 
0 "" {TEXT 225 144 "Next is a simpler example of an oscillating chemic
al reaction (Lengyel et al 1990, taken from Strogatz). In this example
, denoting x(t) and y(t)" }}{PARA 210 "" 0 "" {TEXT 226 106 "the conce
ntrations of two chemical agents, we have a system below describing th
e evolution of the reaction" }{TEXT 213 1 ":" }}{PARA 210 "" 0 "" }
{PARA 210 "" 0 "" }}{EXCHG {PARA 210 "> " 0 "" {MPLTEXT 1 215 89 "sys1
:=diff(x(t),t)=a-x(t)-4*x(t)*y(t)/(1+x(t)^2),diff(y(t),t)=b*x(t)*(1-y(
t)/(1+x(t)^2));" }}{PARA 220 "" 1 "" {XPPMATH 20 "6#>I%sys1G6\"6$/-I%d
iffGI*protectedGF*6$-I\"xGF%6#I\"tGF%F/,(I\"aGF%\"\"\"F,!\"\"*(F,F2-I
\"yGF%F.F2,&F2F2*$F,\"\"#F2F3!\"%/-F)6$F5F/*(I\"bGF%F2F,F2,&F2F2*&F5F2
F7F3F3F2" }{TEXT 227 1 " " }}}{EXCHG {PARA 210 "> " 0 "" {MPLTEXT 1 
215 17 "a:=10;b:=2;K:=50;" }}{PARA 220 "" 1 "" {XPPMATH 20 "6#>I\"aG6
\"\"#5" }{TEXT 227 1 " " }}{PARA 220 "" 1 "" {XPPMATH 20 "6#>I\"bG6\"
\"\"#" }{TEXT 227 1 " " }}{PARA 220 "" 1 "" {XPPMATH 20 "6#>I\"KG6\"\"
#]" }{TEXT 227 1 " " }}}{EXCHG {PARA 210 "" 0 "" }{PARA 210 "" 0 "" 
{TEXT 226 70 "Below is a plot of nullclines and the direction field fo
r the system. " }}{PARA 210 "" 0 "" }{PARA 210 "" 0 "" }}{EXCHG {PARA 
210 "> " 0 "" {MPLTEXT 1 215 79 "plot1:=plot([(a-x)*(1+x^2)/(4*x)],x=-
0.1..K/5,y=-0.1..K,color=green,axes=none):" }}}{EXCHG {PARA 210 "> " 0
 "" {MPLTEXT 1 215 66 "plot2:=plot([1+x^2],x=-0.1..K/5,y=-0.1..K,color
=violet,axes=none):" }}}{EXCHG {PARA 210 "> " 0 "" {MPLTEXT 1 215 74 "
plot3:=plot([0,t,t=-0.1..K],x=-0.1..K/5,y=-0.1..K,color=violet,axes=no
ne):" }{MPLTEXT 1 215 1 "\n" }}}{EXCHG {PARA 210 "> " 0 "" {MPLTEXT 1 
215 113 "plot4:=DEplot(\{sys1\},[x(t),y(t)],0..0.001,\{[0,0,2]\},x=-0.
1..K/5,y=-0.1..K,axes=none, arrows=small,stepsize=0.02):" }{MPLTEXT 1 
215 1 "\n" }}}{EXCHG {PARA 210 "> " 0 "" {MPLTEXT 1 215 35 "display(\{
plot1,plot2,plot3,plot4\});" }}{PARA 221 "" 1 "" {GLPLOT2D 509 509 
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\"2-+++++00\"F`o;$!%0EFccgl$\"1-++++]]_!#9Fhbgl" 1 2 2 0 10 1 2 6 1 1 
2 1.0 45.0 45.0 1 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" 
}}}}{EXCHG {PARA 222 "" 0 "" {TEXT 228 69 "Note that the rectangle in \+
the picture represents a trapping region. " }}{PARA 223 "" 0 "" {TEXT 
229 132 "However, there is a fixed point inside, so to use the Poincar
e-Bendixson theorem we first need to know that this point is repelling
." }}{PARA 210 "" 0 "" {TEXT 226 53 "It is easy to verify that this po
int is repelling for" }{TEXT 213 1 " " }{XPPEDIT 18 0 "b < b[c];" "6#2
%\"bG&F$6#%\"cG" }{TEXT 213 1 " " }{TEXT 213 1 "=" }{XPPEDIT 18 0 "3*a
/5-25/a;" "6#,&*(\"\"$\"\"\"%\"aGF&\"\"&!\"\"F&*&\"#DF&F'F)F)" }{TEXT 
213 1 " " }{TEXT 213 2 ". " }{TEXT 226 17 "In our case, when" }{TEXT 
213 1 " " }{XPPEDIT 18 0 "a = 10;" "6#/%\"aG\"#5" }{TEXT 213 1 " " }
{TEXT 213 2 ", " }{XPPEDIT 18 0 "b[c] = 3.5;" "6#/&%\"bG6#%\"cG-%&Floa
tG6$\"#N!\"\"" }{TEXT 213 1 " " }}{PARA 210 "" 0 "" {TEXT 226 21 "For \+
smaller values of" }{TEXT 213 1 " " }{XPPEDIT 18 0 "b;" "6#%\"bG" }
{TEXT 213 1 " " }{TEXT 213 1 " " }{TEXT 226 114 "we have a limit cycle
, for larger -- an attracting fixed point, that is we have an Andronov
-Hopf bifurcation here." }}{PARA 210 "" 0 "" {TEXT 226 34 " See the tw
o phase portraits below" }{TEXT 213 1 "." }}{PARA 210 "" 0 "" }}
{EXCHG {PARA 210 "> " 0 "" {MPLTEXT 1 215 107 "DEplot(\{sys1\},[x(t),y
(t)],0..100,\{[0,1,5]\},x=-0.1..5,y=-0.1..9, arrows=small,stepsize=0.0
3,linecolor=blue);" }}{PARA 221 "" 1 "" {GLPLOT2D 623 623 623 
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