{VERSION 5 0 "IBM INTEL LINUX" "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 Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 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 "Maple Output" 0 11 1 {CSTYLE "" 
-1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 
-1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 
0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "
" -1 -1 "" 1 18 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 "" 0 257 1 {CSTYLE "" -1 -1 "" 1 18 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 "" 0 258 1 
{CSTYLE "" -1 -1 "" 1 18 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 "" 0 259 1 {CSTYLE "" -1 -1 "" 1 18 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 "" 0 
260 1 {CSTYLE "" -1 -1 "" 1 18 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 "" 0 261 1 {CSTYLE "" -1 -1 "" 1 18 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 "
" 0 262 1 {CSTYLE "" -1 -1 "" 1 18 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 "" 0 263 1 {CSTYLE "" -1 -1 "" 1 
18 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 "" 0 264 1 {CSTYLE "" -1 -1 "" 1 18 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 }}
{SECT 0 {EXCHG {PARA 264 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT -1 56 "This worksheet contains all the examples from the paper \+
" }}{PARA 257 "" 0 "" {TEXT -1 93 "       S. Cyganowski, L. Gruene and
 P.E. Kloeden: MAPLE for Stochastic Differential Equations" }}{PARA 
258 "" 0 "" {TEXT -1 149 "       (in: J.F. Blowey, J.P. Coleman, A.W. \+
Craig, eds., Theory and Numerics of Differential Equations, \n        \+
     Springer Verlag(2001), 127-178)" }}{PARA 259 "" 0 "" {TEXT -1 88 
"which can be obtained from http://www.math.uni-frankfurt.de/~numerik/
kloeden/maplesde/ ." }}{PARA 261 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 
0 "" {TEXT -1 39 "It assumes that you have installed the " }{TEXT 256 
10 "stochastic" }{TEXT -1 54 " package, which can be downloaded from t
he above link." }}{PARA 262 "" 0 "" {TEXT -1 66 "For an explanation of
 the examples please consult the above paper." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "Last Changed October 30,
 2003 by Lars Gruene (http://www.uni-bayreuth.de/departments/math/~lgr
uene/)" }}{PARA 263 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
19 "read `stochastic6`;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "
with(stochastic);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7B%&EulerG%#L0G%$
LFPG%#LJG%$MLJG%)MilsteinG%$SL0G%+Taylor1hlfG%(Taylor2G%#apG%$bpbG%*ch
ainruleG%'colourG%&comm1G%&comm2G%%convG%(correctG%)explicitG%+itoform
ulaG%*linearizeG%*linearsdeG%(milcommG%*momenteqnG%#paG%0pmatrix2pvect
orG%)positionG%0pvector2pmatrixG%*reducibleG%'sphereG%(wkeulerG%'wktay
2G%'wktay3G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "L0(x[2],[x[1
],x[2]],[[r,u],[s,v]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#&%\"xG6#\"
\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "LJ(x[2],[[r,u],[s,v]
],2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"vG" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 25 "LJ(x[2],[[r,u],[s,v]],1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%\"sG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "MLJ
(x[2],[x[2],x[1]],[[r,u],[s,v]],2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#%\"vG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "MLJ(x[2],[x[2],x[
1]],[[r,u],[s,v]],0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#&%\"xG6#\"\"
\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "itoformula([(x[1])^2]
,[a*x[1]],[[b*x[1]]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&TABLEG6#7
#/\"\"\"/%$dX1G,&*&,&*&%\"aGF()&%\"xG6#F(\"\"#F(F4*&)%\"bGF4F(F0F(F(F(
%#dtGF(F(**F4F(F7F(F0F(%$dW1GF(F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "LFP(p(s,x,t,y[1]),[a],[[b]]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,(-%%diffG6$-%\"pG6&%\"sG%\"xG%\"tG&%\"yG6#\"\"\"F,F0*&
%\"aGF0-F%6$F'F-F0F0*&#F0\"\"#F0*&)%\"bGF7F0-F%6$F'-%\"$G6$F-F7F0F0!\"
\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "p:=(s,x,t,y)->1/(sqrt
(2*pi*b^2*(t-s)))*exp(-(y-x-a*(t-s))^2/(2*b^2*(t-s)));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"pGf*6&%\"sG%\"xG%\"tG%\"yG6\"6$%)operatorG%&ar
rowGF+*&-%%sqrtG6#,$*(%#piG\"\"\")%\"bG\"\"#F6,&9&F69$!\"\"F6F9F=-%$ex
pG6#,$*(,(9'F69%F=*&%\"aGF6F:F6F=F9F8!\"#F:F=#F=F9F6F+F+F+" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "LFP(p(s,x,t,y[1]),[a],[[b]]);" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#,,*,\"\"##\"\"\"F%*(%#piGF')%\"bGF%F',
&%\"tGF'%\"sG!\"\"F'#!\"$F%-%$expG6#,$*(,(&%\"yG6#F'F'%\"xGF/*&%\"aGF'
F,F'F/F%F+!\"#F,F/#F/F%F'F)F'F+F%#F/\"\"%*,F&F'F%F&F(F?,&**F7F'F+F>F,F
/F=F'F'**F&F'F7F%F+F>F,F>F'F'F2F'F'*&#F'F%F'*0F=F'F%F&F(F?F7F'F+F>F,F/
F2F'F'F/*,#F'FAF'F%F&F(F?F,F/F2F'F'*&#F'FAF'*.F%F&F(F?F7F%F,F>F+F>F2F'
F'F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "correct([x[1],x[2]],[[r,0],[0,r]],2);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#&%\"xG6#\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "correct([x[1],x[2]],[[x[1],0],[0,r]],1);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#,$&%\"xG6#\"\"\"#F'\"\"#" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 50 "conv([-a^2*x[1]*(1-x[1]^2)],[[a*(1-x[1]^2)]],i
to);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&TABLEG6#7#/\"\"\"\"\"!" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "conv([0],[[a*(1-x[1]^2)]],st
rat);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&TABLEG6#7#/\"\"\"*()%\"aG
\"\"#F(&%\"xG6#F(F(,&!\"\"F(*$)F-F,F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 36 "SL0(x[2],[x[1],x[2]],[[r,u],[s,v]]);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#&%\"xG6#\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 42 "chainrule([(x[1])^2],[a*x[1]],[[b*x[1]]]);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%&TABLEG6#7#/\"\"\"/%$dX1G,&*(%\"aGF()&%\"x
G6#F(\"\"#F(%#dtGF(F2**F2F(%\"bGF(F.F(%%odW1GF(F(" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 16 "linearsde(-x,2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/&%\"XG6#%\"tG*&-%$expG6#,$F'!\"\"\"\"\",&&F%6#\"\"!F.-
%$IntG6$,$*&F.F.-F*6#,$%\"sGF-F-\"\"#/%\"WG;F2F'F.F." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 19 "reducible(x^3,x^2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/&%\"XG6#%\"tG,$*&&F%6#\"\"!\"\"\",&!\"\"F-*&%\"WGF-F*F
-F-F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "explicit(1/2*a^2
*x,a*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"XG6#%\"tG*&-%$expG6#*
&%\"aG\"\"\"%\"WGF.F.&F%6#\"\"!F." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 86 "linearize([x[2],-b*x[2]-sin(x[1])-c*sin(2*x[1])],[[0]
,[-a*(x[2])^2+sin(x[1])]],[0,0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/
%\"AG-%'matrixG6#7$7$\"\"!\"\"\"7$,&!\"\"F+*&\"\"#F+%\"cGF+F.,$%\"bGF.
/%\"BG-%&TABLEG6#7#/F+-F&6#7$7$F*F*7$F+F*" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 34 "A:=array(1..2,1..2,[[2,2],[1,1]]);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7$7$\"\"#F*7$\"\"\"F," }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "A := matrix([[2, 2], [1, 1]]
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7$7$\"\"#F*7$
\"\"\"F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "B1:=array(1..2,
1..2,[[3,1],[4,2]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#B1G-%'matri
xG6#7$7$\"\"$\"\"\"7$\"\"%\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 8 "B:=[B1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG7#%#B1G" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "sphere(A,B);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#,(\"\"'\"\"\"*&,&&%\"sG6#F%\"\"$*&\"\"%F%&F)6#\"\"
#F%F%F%F(F%!\"\"*&,&F(F%*&F0F%F.F%F%F%F.F%F1" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 2 "q;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&,&&%\"s
G6#\"\"\"\"\"#&F'6#F*F)F)F&F)F)*&F%F)F+F)F)*($\"+++++]!#5F),&F&$\"\"'
\"\"!*&$\"\"&F5F)F+F)F)F)F&F)F)*(F/F),&F&F7*&$\"\"%F5F)F+F)F)F)F+F)F)*
&$F)F5F)),&*&,&F&$\"\"$F5*&F<F)F+F)F)F)F&F)F)*&,&F&F)*&$F*F5F)F+F)F)F)
F+F)F)F*F)!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "q0;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,&&%\"sG6#\"\"\"\"\"#&F'6#F*F)F)F&
F)F)*&F%F)F+F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "print(q
k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&TABLEG6#7#/\"\"\",&*&,&&%\"s
G6#F(\"\"$*&\"\"%F(&F-6#\"\"#F(F(F(F,F(F(*&,&F,F(*&F4F(F2F(F(F(F2F(F(
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "print(h);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#-%&TABLEG6#7$/\"\"\",(\"\"%F(*&,&&%\"sG6#F(\"\"#
&F.6#F0F(F(F-F(!\"\"*&F,F(F1F(F3/F0,(F0F(F+F3F4F3" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 10 "print(hk);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#-%&TABLEG6#7#/\"\"\"-F$6#7$/F(,(\"\"%F(*&,&&%\"sG6#F(\"\"$*&F.F(&F26
#\"\"#F(F(F(F1F(!\"\"*&,&F1F(*&F8F(F6F(F(F(F6F(F9/F8,(\"\"'F(F/F9F:F9
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "A:=array(1..2,1..2,[[a1
1,a12],[a21,a22]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrix
G6#7$7$%$a11G%$a12G7$%$a21G%$a22G" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 42 "B:=array(1..2,1..2,[[b11,b12],[b21,b22]]);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'matrixG6#7$7$%$b11G%$b12G7$%$b21G%$
b22G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "E:=[B];" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%\"EG7#%\"BG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "momenteqn(A,E);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%
'matrixG6#7%7%,&%$a11G\"\"#*$)%$b11GF*\"\"\"F.,&%$a12GF**(F*F.%$b12GF.
F-F.F.*$)F2F*F.7%,&%$a21GF.*&F-F.%$b21GF.F.,*F)F.%$a22GF.*&F2F.F9F.F.*
&F-F.%$b22GF.F.,&F0F.*&F2F.F>F.F.7%*$)F9F*F.,&F7F**(F*F.F>F.F9F.F.,&F;
F**$)F>F*F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "P:=array(
1..5,1..5,[[2,5,10,17,26],[5,6,11,18,27],[10,11,12,19,28],[17,18,19,20
,29],[26,27,28,29,30]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG-%'m
atrixG6#7'7'\"\"#\"\"&\"#5\"#<\"#E7'F+\"\"'\"#6\"#=\"#F7'F,F1\"#7\"#>
\"#G7'F-F2F6\"#?\"#H7'F.F3F7F:\"#I" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "pmatrix2pvector(P);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#-%'vectorG6#71\"\"#\"\"&\"#5\"#<\"#E\"\"'\"#6\"#=\"#F\"#7\"#>\"#G\"#
?\"#H\"#I" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "P:=[[1,2],[2,4
]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG7$7$\"\"\"\"\"#7$F(\"\"%
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "pmatrix2pvector(P);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%\"\"\"\"\"#\"\"%" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "p:=[1,2,3,4,5,6,7,8,9,10,11,
12,13,14,15];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG71\"\"\"\"\"#\"
\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#6\"#7\"#8\"#9\"#:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "pvector2pmatrix(p);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#-%'matrixG6#7'7'\"\"\"\"\"#\"\"$\"\"%\"\"&7'F)\"\"
'\"\"(\"\")\"\"*7'F*F/\"#5\"#6\"#77'F+F0F4\"#8\"#97'F,F1F5F8\"#:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "comm1([1,0],[0,x[1]]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#Q\\oCommutative~noise~of~the~first~kin
d~doesn't~exist~for~this~system6\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 42 "comm2([1,0],[(x[2])^2*(x[1])^4,(x[1])^2]);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#Q]oCommutative~noise~of~the~second~kind~does
n't~exist~for~this~system6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 32 "colour([a(x[1],t)],[b(x[1],t)]);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%&TABLEG6#7$/\"\"\"/&&%#dxG6#F(6#%\"tG,&*&-%\"aG6$&%\"xGF-F/F(%
#dtGF(F(*(-%\"bGF4F(&%\"zGF.F(F7F(F(/\"\"#/&%#dzGF.,&*(%&gammaGF(F;F(F
7F(!\"\"*&%%betaGF(&%#dWGF.F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 55 "colour([x[2],x[1]*(alpha-x[1]^2)-x[2]],[0,sigma*x[1]]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&TABLEG6#7%/\"\"\"/&&%#dxG6#F(6#%\"t
G*&&%\"xG6#\"\"#F(%#dtGF(/F4/&&F,F3F.,&*&,&*&&F2F-F(,&%&alphaGF(*$)F>F
4F(!\"\"F(F(F1FCF(F5F(F(**%&sigmaGF(F>F(&%\"zGF.F(F5F(F(/\"\"$/&%#dzGF
.,&*(%&gammaGF(FFF(F5F(FC*&%%betaGF(&%#dWGF.F(F(" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 38 "Euler([x[2],x[1]],[[r,u],[s,v*x[1]]]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&TABLEG6#7$/\"\"\"/&%#Y1G6#,&%\"nGF(
F(F(,*&F+6#F.F(*&&%#Y2GF1F(&%&DeltaGF1F(F(*&%\"rGF(&%(DeltaW1GF1F(F(*&
%\"uGF(&%(DeltaW2GF1F(F(/\"\"#/&F4F,,*F3F(*&F0F(F5F(F(*&%\"sGF(F9F(F(*
(%\"vGF(F0F(F=F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "Milst
ein([x[2],x[1]],[[r,u],[s,v*x[1]]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#-%&TABLEG6#7$/\"\"\"/&%#Y1G6#,&%\"nGF(F(F(,*&F+6#F.F(*&&%#Y2GF1F(&%&
DeltaGF1F(F(*&%\"rGF(&%(DeltaW1GF1F(F(*&%\"uGF(&%(DeltaW2GF1F(F(/\"\"#
/&F4F,,.F3F(*&F0F(F5F(F(*&%\"sGF(F9F(F(*(%\"vGF(F0F(F=F(F(*(F8F(FHF(&^
#F(6$F(F@F(F(*(F<F(FHF(&FK6$F@F@F(F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "comm1([b*x[1],c*x[1]]);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#QYThis~system~exhibits~commutative~noise~of~the~first~kind6\"" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "milcomm([a*x[1]], [[b*x[1]
,c*x[1]]]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%&TABLEG6#7#/\"\"\"/&%
#Y1G6#,&%\"nGF(F(F(,0&F+6#F.F(*(%\"aGF(F0F(&%&DeltaGF1F(F(*(%\"bGF(F0F
(&%(DeltaW1GF1F(F(*(%\"cGF(F0F(&%(DeltaW2GF1F(F(**#F(\"\"#F()F7F@F(F0F
(,&*$)F8F@F(F(F4!\"\"F(F(*,F7F(F0F(F;F(F<F(F8F(F(**F?F()F;F@F(F0F(,&*$
)F<F@F(F(F4FEF(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Taylor
1hlf([x[2],x[1]],[[r,u],[s,v]]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%
&TABLEG6#7$/\"\"\"/&%#Y1G6#,&%\"nGF(F(F(,0&F+6#F.F(*&&%#Y2GF1F(&%&Delt
aGF1F(F(*(#F(\"\"#F(F0F()F5F9F(F(*&%\"rGF(&%(DeltaW1GF1F(F(*&%\"sGF(&^
#F(6$F(\"\"!F(F(*&%\"uGF(&%(DeltaW2GF1F(F(*&%\"vGF(&FB6$F9FDF(F(/F9/&F
4F,,0F3F(*&F0F(F5F(F(*(F8F(F3F(F:F(F(*&F@F(F=F(F(*&F<F(FAF(F(*&FJF(FGF
(F(*&FFF(FKF(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Taylor2(
[x[2],x[1]],[[r,u],[s,v]]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%&TABL
EG6#7$/\"\"\"/&%#Y1G6#,&%\"nGF(F(F(,0&F+6#F.F(*&&%#Y2GF1F(&%&DeltaGF1F
(F(*(#F(\"\"#F(F0F()F5F9F(F(*&%\"rGF(&%(DeltaW1GF1F(F(*&%\"sGF(&%\"JG6
$F(\"\"!F(F(*&%\"uGF(&%(DeltaW2GF1F(F(*&%\"vGF(&FB6$F9FDF(F(/F9/&F4F,,
0F3F(*&F0F(F5F(F(*(F8F(F3F(F:F(F(*&F@F(F=F(F(*&F<F(FAF(F(*&FJF(FGF(F(*
&FFF(FKF(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "wkeuler([x[1
],x[2]],[[r,u],[s,v]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&TABLEG6#
7$/\"\"\"/&%#Y1G6#,&%\"nGF(F(F(,*&F+6#F.F(*&F0F(&%&DeltaGF1F(F(*&%\"rG
F(&%)DeltaWs1GF1F(F(*&%\"uGF(&%)DeltaWs2GF1F(F(/\"\"#/&%#Y2GF,,*&FAF1F
(*&FCF(F3F(F(*&%\"sGF(F7F(F(*&%\"vGF(F;F(F(" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 34 "wktay2([x[1],x[2]],[[r,u],[s,v]]);" }}{PARA 12 "" 
1 "" {XPPMATH 20 "6#-%&TABLEG6#7$/\"\"\"/&%#Y1G6#,&%\"nGF(F(F(,,&F+6#F
.F(*&F0F(&%&DeltaGF1F(F(*(#F(\"\"#F(F0F()F3F7F(F(*&,&%\"rGF(*(F6F(F3F(
F;F(F(F(&%)DeltaWs1GF1F(F(*&,&%\"uGF(*(F6F(F3F(FAF(F(F(&%)DeltaWs2GF1F
(F(/F7/&%#Y2GF,,,&FHF1F(*&FJF(F3F(F(*(F6F(FJF(F8F(F(*&,&%\"sGF(*(F6F(F
3F(FOF(F(F(F=F(F(*&,&%\"vGF(*(F6F(F3F(FSF(F(F(FCF(F(" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 34 "wktay3([x[2],x[1]],[[r,u],[s,v]]);" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#-%&TABLEG6#7$/\"\"\"/&%#Y1G6#,&%\"nGF(
F(F(,6&F+6#F.F(*&&%#Y2GF1F(&%&DeltaGF1F(F(*&%\"rGF(&%(DeltaW1GF1F(F(*&
%\"uGF(&%(DeltaW2GF1F(F(*&F0F(&^#F(6$\"\"!FCF(F(*&%\"sGF(&FA6$F(FCF(F(
*&%\"vGF(&FA6$\"\"#FCF(F(*&F3F(&FA6%FCFCFCF(F(*&F8F(&FA6%F(FCFCF(F(*&F
<F(&FA6%FLFCFCF(F(/FL/&F4F,,6F3F(*&F0F(F5F(F(*&FEF(F9F(F(*&FIF(F=F(F(*
&F3F(F@F(F(*&F8F(FFF(F(*&F<F(FJF(F(*&F0F(FNF(F(*&FEF(FQF(F(*&FIF(FTF(F
(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 9 0" 53 
}{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }