{VERSION 6 0 "IBM INTEL NT" "6.0" } {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 "Helvetica" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "Helvetica" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 258 "Helvetica" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 259 "Helvetica" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 260 "Helvetica" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "Helvetica" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 262 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "Helvetica" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 " " 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "Helvetica" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 268 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 273 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple O utput" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Helveti ca" 1 14 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "with(PDEtools) : with(Optimization): with(LinearAlgebra): with(DEtools): with(plots): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "read \"PryceMethod.txt \": read \"run_phc_maple6.txt\":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "read \"MapletoPhc.txt\": " }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 45 "Test:=\"c:/my project/DAE+Phc+Pryce/TestSet/\";" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%TestGQEc:/my~project/DAE+Phc+Pryce/ TestSet/6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "phc:=\"phc\" ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$phcGQ$phc6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 15 " Mini tutorial " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "read(cat(Test,Pendulum)); declare(F);" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 101 "Pryce(F,vars): compute the parital pr olongation info to complete the DAE to ODE using Pryce's method" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 256 269 " input \n F: input DAE system; \n vars: the dependent variables; \n outp ut\n a list, first value is dimension of constraints (# of f ree variables)\n second element is a list containin g the values of c_i and d_j , " }}{PARA 0 "" 0 "" {TEXT 257 72 " \+ c[i] is the number of prolongation steps for F[i], " }}{PARA 0 "" 0 "" {TEXT 258 70 " d[j] is diff-order of vars[j] after porlongation. " }{TEXT -1 80 "\n warning : c,d , c[i] and d[j] CAN NOT appear in the input system !!! " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "sol:=Pryce(F,variables);" } {TEXT -1 0 "" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 130 "NewSys_b(F, cd List): the partial prolongation with block structure\n input\n \+ F: input DAE\n cdList: a list " }{TEXT 259 42 " con taining the values of c_i and d_j , " }}{PARA 256 "" 0 "" {TEXT -1 115 " output \n a list of lists, , the first list is th e top block (implicit ODE ), others are constraints" }}{PARA 0 "" 0 " " {TEXT -1 7 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "DF :=NewSys_b(F,sol[2]);" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 265 96 " JacM (F,vars, cdList): Jacobian matrix of the top block w.r.t highest orde r variables\n input" }{TEXT -1 10 "\n " }{TEXT 260 71 "F: i nput DAE system; \n vars: the dependent variables; \n \+ " }{TEXT -1 15 "cdList: a list " }{TEXT 261 68 " containing the value s of c_i and d_j , \n output\n a matrix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "M:=JacM(F,variables,sol[2]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 118 "---------------------------------------- ---------------------------------------------------------------------- --------" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 262 31 " to compute the init ial points\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 263 318 "MapletoPhc_a(DA E, vars,indvar,ord) : to convert differential system to polynomials \+ \ninput \n DAE: a DAE system (could be list of lists)\n vars : dependent variables\n indvar: independent variables\n ord: the diff order of the DAE \noutput\n a list of polynomials with \+ the same structure as input DAE" }{TEXT -1 1 "\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "sys:= MapletoPhc_a(subs(L=1,DF[2..3]),variabl es,[t],3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "L1:= randpoly ([x,y],degree=1)/100.0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 " sol:= run_phc(phc,[op(sys[2]),L1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "subs(so l[1][1],sys[1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "L2:= ra ndpoly([xt,yt,u,v],degree=1)/100.0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "sol1:= run_phc(phc,[op(subs(sol[1][1],sys[1])),L2]); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "sol2:= run_phc(phc,[op( subs(sol[1][2],sys[1])),L2]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 264 536 "1. We need a more syst ematic way to compute the points of the constraints with block structu re.\n2. Subs the points back to check every Jacobian matrices,\n if we meet a singular matrix at a point p, then the whole irreducible co mponent passing p is a singular component, \n that will be very int eresting and more work need to do ...\n3. bottom up method may miss so me components, whose projection is embeded in the projection of others .\n these components have lower dim proj, how to compute the witnes s points on such components ?" }{TEXT -1 1 "\n" }{TEXT 266 52 " wha t is the physical meaning of such components?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "################## The Test Problems ##################" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 25 " Visconti's test problems" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PAGEBK }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "# A.11 \+ --------------------------------" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "J := 'J':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "read(\"c:/my project/DAE+Phc+Pryce/TestSet/Crane\"); declare(F):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%#p1G6#%\"tG,&\"\"\"F)*$)F'\"\"#F) !\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%#p2G6#%\"tG,&\"\"\"F)F'!\" \"" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"FG7.,&-%%diffG6$-%\"xG6#%\"t GF-\"\"\"-%#dxGF,!\"\",&-F(6$-%\"zGF,F-F.-%#dzGF,F1,&-F(6$-%\"dGF,F-F. -%#ddGF,F1,&-F(6$-%\"rGF,F-F.-%#drGF,F1,&*&%#M2GF.-F(6$F/F-F.F.*&-%$ta uGF,F.-%$sinG6#-%&thetaGF,F.F.,(*&FIF.-F(6$F7F-F.F.*&FMF.-%$cosGFQF.F. *&%\"mGF.%\"gGF.F1,**&%#M1GF.-F(6$F>F-F.F.*&%#C1GF.F:F.F.-%#u1GF,F1FLF 1,**&%\"JGF.-F(6$FEF-F.F.*&%#C2GF.FAF.F.*&%#C3GF.-%#u2GF,F.F.*&)Fio\" \"#F.FMF.F1,(*&FCF.FOF.F.F%*functionsG7\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*variablesG7.%\"xG%#dxG%\"zG%#dz G%\"dG%#ddG%\"rG%#drG%$tauG%&thetaG%#u1G%#u2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*constantsG<*/%#M1G\"\"\"/%#M2G\"\"#/%#C1GF(/%\"JGF(/ %#C2GF(/%#C3GF(/%\"mGF(/%\"gG\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %+minnumdiffG7.\"\"$F&F&F&\"\"#F'F'F'\"\"%F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(numdiffG7.\"\"%F&F&F&F&F&F&F&F&F&F&F&" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%'tstartG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&tstopG\"\"\"" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(ICGuessG7br\" \"\"F&F&F&F&F&F&F&F&F&F&F&\"\"!F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F 'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F 'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F 'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(- %\"xG6#%\"tG\"\"\"%9will~now~be~displayed~asGF(F%F(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#*(-%#dxG6#%\"tG\"\"\"%9will~now~be~displayed~asGF(F%F (" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%\"zG6#%\"tG\"\"\"%9will~now~b e~displayed~asGF(F%F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%#dzG6#%\" tG\"\"\"%9will~now~be~displayed~asGF(F%F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%\"rG6#%\"tG\"\"\"%9will~now~be~displayed~asGF(F%F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%\"dG6#%\"tG\"\"\"%9will~now~be ~displayed~asGF(F%F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%#ddG6#%\"t G\"\"\"%9will~now~be~displayed~asGF(F%F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%#drG6#%\"tG\"\"\"%9will~now~be~displayed~asGF(F%F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%$tauG6#%\"tG\"\"\"%9will~now~be~displaye d~asGF(F%F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%&thetaG6#%\"tG\"\" \"%9will~now~be~displayed~asGF(F%F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #*(-%#u1G6#%\"tG\"\"\"%9will~now~be~displayed~asGF(F%F(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#*(-%#u2G6#%\"tG\"\"\"%9will~now~be~displayed~asG F(F%F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "#Jaco:= JacM(F, v ariables, sol[2]); Determinant(Jaco);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "G:= subs(constants,F);" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG7.,&&%\"xG6 #%\"tG\"\"\"%#dxG!\"\",&&%\"zGF)F+%#dzGF-,&&%\"dGF)F+%#ddGF-,&&%\"rGF) F+%#drGF-,&*&\"\"#F+&F,F)F+F+*&%$tauGF+-%$sinG6#%&thetaGF+F+,(*&FF-,*&F9F)F+F7F+% #u2GF+F?F-,(*&F8F+F@F+F+F4F+F(F-,&*&F8F+FHF+F+F0F-,(F(F+F+F-*$)F*F " 0 "" {MPLTEXT 1 0 8 "nops(G); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "G2:= [op(G[1..10]), x(t)^3-z(t)^2 -t^3 ,x(t)^2*z(t)-z (t)-2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#G2G7.,&&%\"xG6#%\"tG\"\" \"%#dxG!\"\",&&%\"zGF)F+%#dzGF-,&&%\"dGF)F+%#ddGF-,&&%\"rGF)F+%#drGF-, &*&\"\"#F+&F,F)F+F+*&%$tauGF+-%$sinG6#%&thetaGF+F+,(*&FF-,*&F9F)F+F7F+%#u2GF+F?F- ,(*&F8F+F@F+F+F4F+F(F-,&*&F8F+FHF+F+F0F-,(*$)F(\"\"$F+F+*$)F0F " 0 "" {MPLTEXT 1 0 54 "G1:= [op(G[1..10]), f1(x(t),z(t),t),f2(x(t),z(t),t) ] ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#G1G7.,&&%\"xG6#%\"tG\"\"\"%#dx G!\"\",&&%\"zGF)F+%#dzGF-,&&%\"dGF)F+%#ddGF-,&&%\"rGF)F+%#drGF-,&*&\" \"#F+&F,F)F+F+*&%$tauGF+-%$sinG6#%&thetaGF+F+,(*&FF-,*&F9F)F+F7F+%#u2GF+F?F-,(*&F 8F+F@F+F+F4F+F(F-,&*&F8F+FHF+F+F0F--%#f1G6%F(F0F*-%#f2GFW" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 267 66 " substitute the constants and replace s in cos by n1(t) and n2(t)" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 91 "H:= [op(subs(sin(theta(t))=n1(t),cos(theta(t))=n2(t ),G)), n1(t)^2+n2(t)^2-1]:declare(H): H:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%\"xG6#%\"tG\"\"\"%9will~now~be~displayed~asGF(F%F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%#dxG6#%\"tG\"\"\"%9will~now~be~displayed ~asGF(F%F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%\"zG6#%\"tG\"\"\"%9w ill~now~be~displayed~asGF(F%F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-% #dzG6#%\"tG\"\"\"%9will~now~be~displayed~asGF(F%F(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#*(-%\"rG6#%\"tG\"\"\"%9will~now~be~displayed~asGF(F%F (" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%\"dG6#%\"tG\"\"\"%9will~now~b e~displayed~asGF(F%F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%#ddG6#%\" tG\"\"\"%9will~now~be~displayed~asGF(F%F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%#drG6#%\"tG\"\"\"%9will~now~be~displayed~asGF(F%F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%$tauG6#%\"tG\"\"\"%9will~now~b e~displayed~asGF(F%F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%#u1G6#%\" tG\"\"\"%9will~now~be~displayed~asGF(F%F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%#u2G6#%\"tG\"\"\"%9will~now~be~displayed~asGF(F%F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%#n1G6#%\"tG\"\"\"%9will~now~be ~displayed~asGF(F%F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%#n2G6#%\"t G\"\"\"%9will~now~be~displayed~asGF(F%F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "nops(H);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#8" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "variables:= [x, dx, z, dz, d , dd, r, dr, tau, u1, u2,n1,n2]; nops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*variablesG7/%\"xG%#dxG%\"zG%#dzG%\"dG%#ddG%\"rG%#drG %$tauG%#u1G%#u2G%#n1G%#n2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#8" }} }{EXCHG }{EXCHG {PARA 0 "" 0 "" {TEXT 268 42 " Using Pryce method to g et the c_i and d_j" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "t1:= \+ time(): sol:= Pryce(H,variables);print(`time for Pryce structure ana lysis = `, time()-t1);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$solG7$$!/ @dHT\\\"e#!#A7 " 0 "" {MPLTEXT 1 0 29 "MM:=SigMatrix(H,variables,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#MMG-%'RTABLEG6+\"*Kg#z9%)anythingG%'MatrixG%,rectang ularG%.Fortran_orderG7\"\"\"#;\"\"\"\"#8F/" }}}{EXCHG }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "H1:= [op(subs(sin(theta(t))=n1(t),cos(theta (t))=n2(t),G1)), n1(t)^2+n2(t)^2-1]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "MM1:=SigMatrix(H1,variables,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$MM1G-%'RTABLEG6+\"*c'G8:%)anythingG%'MatrixG%,rectan gularG%.Fortran_orderG7\"\"\"#;\"\"\"\"#8F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "H2:= [op(subs(sin(theta(t))=n1(t),cos(theta(t))=n2 (t),G2)), n1(t)^2+n2(t)^2-1]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "MM2:=SigMatrix(H2,variables,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$MM2G-%'RTABLEG6+\"*/sM[\"%)anythingG%'MatrixG%,rectangularG%.For tran_orderG7\"\"\"#;\"\"\"\"#8F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "MM-MM1; MM2-MM1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#- %'RTABLEG6+\"*/YQZ\"%)anythingG%'MatrixG%,rectangularG%.Fortran_orderG 7\"\"\"#;\"\"\"\"#8F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6+ \"*'fb*[\"%)anythingG%'MatrixG%,rectangularG%.Fortran_orderG7\"\"\"#; \"\"\"\"#8F-" }}}{EXCHG }{EXCHG }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "t1:= time(): sol:= Pryce(H1,variables);print(`time for Pryce s tructure analysis = `, time()-t1);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# >%$solG7$$!/(oxO\"*H;&!#A7 " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "t1:= time(): sol:= Pryce(H2,variables);print(`time for Pryce structure a nalysis = `, time()-t1);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$solG7$$ !/(oxO\"*H;&!#A7 " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG }{EXCHG } {EXCHG {PARA 0 "" 0 "" {TEXT 269 57 " compute the bolck structure of t he parital prolongation " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "DH:= NewSys_b(H,sol[2]) ; ode:= map(diff,DH[1],t): ode:=[seq(ode[i]=0,i=1..nops(ode))];" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%#DHG7'7/,&&%\"xG6&%\"tGF+F+F+\"\"\"& %#dxG6%F+F+F+!\"\",&&%\"zGF*F,&%#dzGF/F0,&&%\"dG6$F+F+F,&%#ddG6#F+F0,& &%\"rGF9F,&%#drGFF,FHF,F,*(FDF,FenF,FKF,F,*&F?F,FMF,F,F7F,&F)F9F0,**&F>F,FQF ,F,*(FDF,FenF,FSF,F,*&F?F,FUF,F,&F3F9F0F(F2,**&FDF,)FKFDF,F,*(FDF,FHF, FMF,F,*&FDF,)FSFDF,F,*(FDF,FQF,FUF,F,7-,&&F)F/F,&F.F9F0,&&F3F/F,&F5F9F 0,&FWF,F;F0,&FenF,FAF0,(*&FDF,F[pF,F,*&FJF,FHF,F,*&FGF,FKF,F,,(*&FDF,F ^pF,F,*&FJF,FQF,F,*&FGF,FSF,F,,**&FenF,FHF,F,*&F?F,FKF,F,FWF,&F)F%$odeG7//,&&%\"xG6'%\"tGF+F+F+F+\"\"\"&%#dxG6&F+F+F+F+ !\"\"\"\"!/,&&%\"zGF*F,&%#dzGF/F0F1/,&&%\"dG6%F+F+F+F,&%#ddG6$F+F+F0F1 /,&&%\"rGF " 0 "" {MPLTEXT 1 0 2 "H;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7/,&&% \"xG6#%\"tG\"\"\"%#dxG!\"\",&&%\"zGF'F)%#dzGF+,&&%\"dGF'F)%#ddGF+,&&% \"rGF'F)%#drGF+,&*&\"\"#F)&F*F'F)F)*&%$tauGF)%#n1GF)F),(*&F:F)&F/F'F)F )*&F=F)%#n2GF)F)\"#5F+,*&F3F'F)F1F)%#u1GF+FF)F)F2F)F&F+,&*&F6F)FCF)F)F.F+,(F&F)F)F+*$)F(F:F)F),(F.F)F)F +F(F),(*$)F>F:F)F)*$)FCF:F)F)F)F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "H1:= [op(H[1..10]), x(t)^5- t*x(t), z(t)^3*x(t) - t^4 , H[13]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#H1G7/,&&%\"xG6#%\"tG\" \"\"%#dxG!\"\",&&%\"zGF)F+%#dzGF-,&&%\"dGF)F+%#ddGF-,&&%\"rGF)F+%#drGF -,&*&\"\"#F+&F,F)F+F+*&%$tauGF+%#n1GF+F+,(*&FF-,*&F9F)F+F7F+%#u2GF+F?F-,(*&F8F+F@F+F+ F4F+F(F-,&*&F8F+FEF+F+F0F-,&*$)F(\"\"&F+F+*&F*F+F(F+F-,&*&)F0\"\"$F+F( F+F+*$)F*\"\"%F+F-,(*$)F@F \+ " 0 "" {MPLTEXT 1 0 12 "rifsimp(H1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "nops(DH); for i from 1 to 5 do print(map(degree, DH[i ])); od: " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#7/\"\"\"F$F$F$\"\"#F%F%F$F%F%F$F$F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7-\"\"\"F$F$F$\"\"#F%F%F%F$F$F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+\"\"\"F$\"\"#F%F%F%F$F$F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"\"\"F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\" \"#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "DH[2]; map(deg ree,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7-,&&%\"xG6%%\"tGF(F(\"\"\" &%#dxG6$F(F(!\"\",&&%\"zGF'F)&%#dzGF,F-,&&%\"dG6#F(F)%#ddGF-,&&%\"rGF6 F)%#drGF-,(*&\"\"#F)F*F)F)*&&%$tauGF6F)%#n1GF)F)*&FAF)&FBF6F)F),(*&F>F )F1F)F)*&F@F)%#n2GF)F)*&FAF)&FHF6F)F),**&F9F)FBF)F)*&F:F)FDF)F)F4F)&F& F6F-,(*&F9F)FHF)F)*&F:F)FJF)F)&F0F6F-F%F/,&*(F>F)FBF)FDF)F)*(F>F)FHF)F JF)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7-\"\"\"F$F$F$\"\"#F%F%F%F$F$ F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "init_t:=stats[random, uniform](1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'init_tG$\"+0')=dR!# 5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 270 66 " set the initial time and convert maple notation to phc notation" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "sys:= subs(t=init_t, MapletoPhc_a(DH,variables,[t],4) );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$sysG7'7/,&%&xttttG\"\"\"%&dxt ttG!\"\",&%&zttttGF)%&dztttGF+,&%$dttGF)%$ddtGF+,&%$rttGF)%$drtGF+,**& \"\"#F)F*F)F)*&%&tauttGF)%#n1GF)F)*(F7F)%%tautGF)%$n1tGF)F)*&%$tauGF)% %n1ttGF)F),**&F7F)F.F)F)*&F9F)%#n2GF)F)*(F7F)F " 0 "" {MPLTEXT 1 0 23 "solu:=solve_b(p hc,sys):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "nops(solu); " } }{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "mapledifflist := (DEtools[checkrank])([seq(variables[ i](t),i=1..nops(variables))], degree = 4):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 55 "phclist := MapletoPhc_a(mapledifflist,variables,[t] ,4);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(phclistG7]o%&dttttG%'ddtttt G%'drttttG%'dxttttG%'dzttttG%'n1ttttG%'n2ttttG%&rttttG%(tauttttG%'u1tt ttG%'u2ttttG%&xttttG%&zttttG%%dtttG%&ddtttG%&drtttG%&dxtttG%&dztttG%&n 1tttG%&n2tttG%%rtttG%'tautttG%&u1tttG%&u2tttG%%xtttG%%ztttG%$dttG%%ddt tG%%drttG%%dxttG%%dzttG%%n1ttG%%n2ttG%$rttG%&tauttG%%u1ttG%%u2ttG%$xtt G%$zttG%#dtG%$ddtG%$drtG%$dxtG%$dztG%$n1tG%$n2tG%#rtG%%tautG%$u1tG%$u2 tG%#xtG%#ztG%\"dG%#ddG%#drG%#dxG%#dzG%#n1G%#n2G%\"rG%$tauG%#u1G%#u2G% \"xG%\"zG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "DRing := (diff alg[differential_ring])(derivations = [t], ranking = variables, notati on = diff);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&DRingG%)ODE_ringG" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "jetlist := map(diffalg[den ote], mapledifflist, jet, DRing)[1 .. nops(mapledifflist)];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(jetlistG7]o&%\"dG6&%\"tGF)F)F)&%#ddGF(&%# drGF(&%#dxGF(&%#dzGF(&%#n1GF(&%#n2GF(&%\"rGF(&%$tauGF(&%#u1GF(&%#u2GF( &%\"xGF(&%\"zGF(&F'6%F)F)F)&F+FC&F-FC&F/FC&F1FC&F3FC&F5FC&F7FC&F9FC&F; FC&F=FC&F?FC&FAFC&F'6$F)F)&F+FQ&F-FQ&F/FQ&F1FQ&F3FQ&F5FQ&F7FQ&F9FQ&F;F Q&F=FQ&F?FQ&FAFQ&F'6#F)&F+Fin&F-Fin&F/Fin&F1Fin&F3Fin&F5Fin&F7Fin&F9Fi n&F;Fin&F=Fin&F?Fin&FAFin&F'6\"&F+Fgo&F-Fgo&F/Fgo&F1Fgo&F3Fgo&F5Fgo&F7 Fgo&F9Fgo&F;Fgo&F=Fgo&F?Fgo&FAFgo" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 272 79 " to generate the initial condition automatically using one (re al) witness point" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "read \+ \"PryceMethod.txt\": ic:= ic(diffconvert(solu[4],phclist,jetlist),init _t);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#icG7I/-%\"xG6#$\"+0')=dR!#5 $\"+Me1M%)F,/-%\"zGF)$\"+&R6G/'F,/--%\"DG6#F(F)$!*@xV\"z!\"*/-%#dxGF)F 9/--F76#F1F)$F,!\"\"/-%#dzGF)FC/---%#@@G6$F7\"\"#F8F)$!#?FD/--F76#F>F) FO/--FKFBF)\"\"!/--F76#FGF)FX/-%$tauGF)$\"/pUhH.x5!#7/-%#n1GF)$\"0/TNw 1Rr$!#:/-%#n2GF)$\"0f_)3pw%G*Fbo/-%\"rGF)$\"02,@02$3lFbo/-%\"dGF)$\"+w 7%p,'F,/---FL6$F7\"\"$F8F)FX/--FKFTF)FX/--FepFBF)FX/--FKFfnF)FX/--F76# F_pF)$!*@xV\"RF;/-%#ddGF)Feq/--F76#FjoF)$!/pUhH.x5!#8/-%#drGF)F^r/--F7 6#FinF)$FXFX/--F76#F_oF)$!+')Qm'p)!#g/--F76#FeoF)$\"0)o.%)e5p5!#k/---F L6$F7\"\"%F8F)FX/--FepFTF)FX/--FjsFBF)FX/--FepFfnF)FX/--FKFdqF)FO/--F7 6#FiqF)FO/--FKF]rF)$\"0C$=]cK![#Ffs/--F76#FcrF)F`u/--FKFgrF)$\"01ZfPdK z\"!#w/--FKF\\sF)$!/pxKp-r8!$N\"/--FKFcsF)$!+rEV:7!#s/-%#u1GF)$!*sP9R' !\")/-%#u2GF)$\"+dit%=\"F]w" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "variables;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7/%\"x G%#dxG%\"zG%#dzG%\"dG%#ddG%\"rG%#drG%$tauG%#u1G%#u2G%#n1G%#n2G" }}} {EXCHG {PARA 258 "" 0 "" {TEXT -1 43 " using dsolve to compute the int egral curve" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "p:= dsolve( \{op(ode), op(ic)\}, type=numeric, range=init_t..1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "pp:= dsolve(\{op(ode), op(ic)\}, type=num eric, implicit = true, range=init_t..1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "#p:= dsolve(\{diff(y(x),x)+ diff(z(x),x) = y(x), dif f(y(x),x) - diff(z(x),x) = z(x), z(0)=0, y(0)=1\}, type=numeric, range =-2..2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "?line" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "P1:= odeplot(pp,[[t, x(t),1 -z(t), color = blue],[t, r(t), 0.5, color = green],\n \+ [t,sqrt((x(t)-d(t))^2+z(t)^2),0.6,color=black],\n [t, d(t), 1, \+ color = red]], frames = 100):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "#odeplot(pp,[u1(t),1], frames = 20);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 38 "plots[display](P1, insequence = true);" }}{PARA 13 "" 1 "" {GLPLOT3D 567 567 567 {PLOTDATA 3 "6%-%(ANIMATEG6`q7&-%'CURVES G6$X,%)anythingG6\"6\"[gl'!%\"!!#$\"\"\"$3FD95375331081353FEAFD2FCD9DE 41E3FD9537533108135-%'COLOURG6&%$RGBG$\"\"!F3F2$\"*++++\"!\")-F(6$X,F+ F,F-[gl'!%\"!!#$\"\"\"$3FD95375331081353FE4D39AEA4385713FE000000000000 0-F/6&F1F2F4F2-F(6$X,F+F,F-[gl'!%\"!!#$\"\"\"$3FD95375331081353FE4D39A EA4385713FE3333333333333-F/6&F1F3F3F3-F(6$X,F+F,F-[gl'!%\"!!#$\"\"\"$3 FD95375331081353FE341140B07978F3FF0000000000000-F/6&F1F4F2F27&-F(6$X,F +F,F-[gl'!%\"!!#'\"#\"$3FD95375331081353FEAFD2FCD9DE41E3FD953753310813 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