{VERSION 5 0 "IBM INTEL NT" "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 } {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 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier " 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 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 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 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 120 "Some solutions for Homewo rk Set 2 (to save memory, some of the plotting commands are left for y ou to evaluate in Maple)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "with(DEtools): with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warnin g, the name changecoords has been redefined\n" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 13 "Problem 2.7.6" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "f1 := r + x - x^3;\nV1 := - int(f1,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f1G,(%\"rG\"\"\"%\"xGF'*$)F(\"\"$F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#V1G,(*&%\"rG\"\"\"%\"xGF(!\"\"*&#F(\"\"#F (*$)F)F-F(F(F**&#F(\"\"%F(*$)F)F2F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Find bifurcation values:\n" }{MPLTEXT 1 0 33 "solve(\{f1= 0,diff(f1,x)=0\},\{x,r\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"rG ,$*&#\"\"#\"\"$\"\"\"-%'RootOfG6$,&F+!\"\"*&F*F+)%#_ZGF)F+F+/%&labelG% $_L1GF+F0/%\"xGF," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "This is not \+ very helpful; so let's try to do more work by hand, to help Maple:\n" }{MPLTEXT 1 0 41 "df1 := diff(f1,x);\nxcrit := solve(df1,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$df1G,&\"\"\"F&*&\"\"$F&)%\"xG\"\"#F&!\"\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&xcritG6$,$*&\"\"$!\"\"F(#\"\"\" \"\"#F),$*&F(F)F(F*F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "rc 1 := solve(subs(x=xcrit[1],f1),r); evalf(rc1);\nrc2 := solve(subs(x=xc rit[2],f1),r); evalf(rc2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$rc1G, $*(\"\"#\"\"\"\"\"*!\"\"\"\"$#F(F'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#$\"+&z,!\\Q!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$rc2G,$*(\"\"#\" \"\"\"\"*!\"\"\"\"$#F(F'F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+&z,! \\Q!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Now let's plot the pote ntial for different values of r:\n" }{MPLTEXT 1 0 551 "p1a := plot(sub s(r=-0.8,V1),x=-2..2,tickmarks=[0,0],labels=[\"x\",\"V\"]):\np1b := pl ot(subs(r=rc2,V1),x=-2..2,tickmarks=[0,0],labels=[\"x\",\"V\"]):\np1c \+ := plot(subs(r=0,V1),x=-2..2,tickmarks=[0,0],labels=[\"x\",\"V\"]):\np 1d := plot(subs(r=0.2,V1),x=-2..2,tickmarks=[0,0],labels=[\"x\",\"V\"] ):\np1e := plot(subs(r=rc1,V1),x=-2..2,tickmarks=[0,0],labels=[\"x\", \"V\"]):\np1f := plot(subs(r=0.8,V1),x=-2..2,tickmarks=[0,0],labels=[ \"x\",\"V\"]):\np1 := array(1..2,1..3):\np1[1,1] := p1a: p1[1,2] := p1 b: p1[1,3] := p1c: p1[2,1] := p1d: p1[2,2] := p1e: p1[2,3] := p1f:\ndi splay(p1);" }}{PARA 13 "" 1 "" {GLPLOT2D 1004 445 445 {PLOTDATA 2 "6=- %'CURVESG6$7U7$$!+++++5!\")$!+S3z9M!\"*7$$!+:p0k&*F-$!+d:kX^F-7$$!+q^u %=*F-$!+$3s#)*F- 7$$!+0siqmF-$!+)e![w**F-7$$!+!z$pZiF-$!+)*********F-7$$!+&yaE\"eF-$!+] r(*3**F-7$$!+!>s%HaF-$!+)Q*yZ(*F-7$$!+]$*4)*\\F-$!+&\\#4#\\*F-7$$!+]_& \\c%F-$!+L3`s\"*F-7$$!+]1aZTF-$!+;]i>))F-7$$!+/#)[oPF-$!+z5\"=Z)F-7$$! 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+&Q&odPF*7$Fdbn$!+z&phk$F*7$F^cn$!+%>LS^$F*7$Fbdn$!+!y\\9M$F*Fb[l-F\\ \\l6$7$FgdnFhaoF`\\l-F\\\\l6$7$F[enF\\boFe\\l-F$6$7fn7$Fj\\l$!+,+++?F* 7$F_]l$!+0U,c@F*7$Fd]l$!+aDV+BF*7$Fi]l$!+5?4vvDDF*7$$\"+ qU&G5#F*$!+TH&)QEF*7$Fc^l$!+BuxUFF*7$$\"+CKlX@F*$!+&)[hQGF*7$Fh^l$!+qZ 7EHF*7$F]_l$!+7FOxIF*7$Fb_l$!+mF$G>$F*7$Fg_l$!+UNB!H$F*7$F\\`l$!+#z2-P $F*7$Ff`l$!+yo!=V$F*7$F`al$!+aI " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 17 "Problem 2.8.2 (c)" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "Sketching the slope field and drawing solution curves is straig htforward with DEplot:\n" }}{PARA 0 "" 0 "" {TEXT -1 62 "In this probl em there is a half-stable fixed point at x = 1/2." }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 43 "deq2 := diff(x(t),t) = 1 - 4*x(t)*(1-x(t)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%deq2G/-%%diffG6$-%\"xG6#%\"tGF, ,&\"\"\"F.*(\"\"%F.F)F.,&F.F.F)!\"\"F.F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "DEplot(deq2,x(t),t=-1..10,x=-0.5..1.5,[[x(0)=-0.2],[ x(0)=0.5],[x(0)=0.6]],stepsize=0.05,linecolor=black);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 66 "Problems 2 .8.3, 2.8.4, 2.8.5 - implementation of numerical methods" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 316 "Here is one of many possible approaches. We first program general-purpose Euler, Improved Euler and Runge-Kut ta solvers, which will take any first-order ODE x'=f(t,x) as input, an d give the solution at the final time tf with step h, given x(t0) = x0 . Then we implement them on our particular, simple model problem." }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart; with(plots):" }} {PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been r edefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 298 "Euler := pro c(f,t0,tf,x0,h) \n local tt, xx, i, N; \n xx := eva lf(x0): tt := evalf(t0): N := evalf(ceil((tf-t0)/h)); \+ \n for i from 1 to N do \n xx := xx \+ + h*f(tt,xx);\n tt := tt + h; \n od;\n xx; \+ \n end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 338 "ImpEule r := proc(f,t0,tf,x0,h) \n local tt, xx, i, N, k; \n xx := evalf(x0): tt := evalf(t0): N := evalf(ceil((tf-t0)/h)); \+ \n for i from 1 to N do\n k := f(tt,xx); \+ \n xx := xx + h*(k + f(tt+h,xx+h*k))/2;\n tt := \+ tt + h; \n od;\n xx; \n end:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 438 "RK4 := proc(f,t0,tf,x0,h) \+ \n local tt, xx, i, N, k1, k2, k3, k4; \n xx := eval f(x0): tt := evalf(t0): N := evalf(ceil((tf-t0)/h)); \n for i from 1 to N do\n k1 := f(tt,xx);\n k2 := f(tt+h/2,xx+h*k 1/2);\n k3 := f(tt+h/2,xx+h*k2/2);\n k4 := f(tt+h,xx+h*k3); \+ \n xx := xx + h*(k1 + 2*k2 + 2*k3 + k4)/6 ;\n tt := tt + h; \n od;\n xx; \n e nd:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "Now define the function, a nd find the exact solution:\n" }{MPLTEXT 1 0 97 "f := (t,x) -> -x;\nx0 := 'x0': soln := unapply(rhs(dsolve(\{diff(x(t),t) = f(t,x(t)),x(0)=x 0\})),t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"tG%\"xG6\"6$ %)operatorG%&arrowGF),$9%!\"\"F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%%solnGf*6#%\"tG6\"6$%)operatorG%&arrowGF(*&%#x0G\"\"\"-%$expG6#,$9 $!\"\"F.F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Exact value of \+ x(1), given x(0) = 1:\n" }{MPLTEXT 1 0 26 "evalf(subs(x0=1,soln(1))); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+7WzyO!#5" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 102 "Now compute the Euler, Improved Euler and Runge-K utta approximations to the solutions, and the errors." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 636 "t0 := 0: tf := 1.: x0 := 1:\nfor n from 1 to 4 do\n hval[n] := 10^(-n);\n eul[n] := Euler(f,t0,tf,x0,h val[n]);\n eul_err[n] := abs(soln(tf) - eul[n]);\n ie[n] := ImpEuler (f,t0,tf,x0,hval[n]);\n ie_err[n] := abs(soln(tf) - ie[n]);\n rk[n] \+ := RK4(f,t0,tf,x0,hval[n]);\n rk_err[n] := abs(soln(tf) - rk[n]);\n \+ if n = 1 then \n eul_errlist := [[hval[n],eul_err[n]]];\n ie_err list := [[hval[n],ie_err[n]]];\n rk_errlist := [[hval[n],rk_err[n]] ];\n else\n eul_errlist := [op(eul_errlist),[hval[n],eul_err[n]]]; \n ie_errlist := [op(ie_errlist),[hval[n],ie_err[n]]];\n rk_errl ist := [op(rk_errlist),[hval[n],rk_err[n]]];\n end if;\nod:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Now plot the errors:\n" }{MPLTEXT 1 0 43 "plot([eul_errlist,ie_errlist,rk_errlist]);\n" }{TEXT -1 172 "N otice that on this graph, the error for improved Euler is much smaller than that for the Euler method, and the RK error is not visible. So \+ we should plot a log-log graph:" }}{PARA 13 "" 1 "" {GLPLOT2D 320 320 320 {PLOTDATA 2 "6'-%'CURVESG6$7&7$$\"3/+++++++5!#=$\"3-+++5,5?>!#>7$$ \"3-+++++++5F-$\"3'***********4Z=!#?7$$F0F3$\"3,++++]:S=!#@7$$\"31++++ +++5F8$\"3-+++++qR=!#A-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!FGFF-F$6$7&7$F ($\"3F++++qV:mF87$F/$\"3%***********fxh!#B7$F5$\"3=++++++!3'!#D7$F:$\" 3%)*************>#!#E-F@6&FBFFFCFF-F$6$7&7$F($\"3,++++++LL!#C7$F/FF7$F 5$\"3'***************f!#F7$F:$\"31+++++++;FY-F@6&FBFCFCFF-%+AXESLABELS G6$Q!6\"Fjo-%%VIEWG6$%(DEFAULTGF_p" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "loglogplot([eul_errlist,ie_errlist, rk_errlist]);" }}{PARA 13 "" 1 "" {GLPLOT2D 320 320 320 {PLOTDATA 2 "6 (-%'CURVESG6$7&7$$!+++++5!\"*$!+Ghn;!\\:$F*Fao/$!+8+\"p4$F*Fao/$!+\"\\d d/$F*Fao/$!\"$FDQ&.1e-2Fbo/$!+/+(*)p#F*Fao/$!+X(yG_#F*Fao/$!+4+%zR#F*F ao/$!+'**H5I#F*Fao/$!+]([=A#F*Fao/$!+g>!\\:#F*Fao/$!+8+\"p4#F*Fao/$!+ \"\\dd/#F*Fao/$!\"#FDQ&.1e-1Fbo/$!+/+(*)p\"F*Fao/$!+X(yG_\"F*Fao/$!+4+ %zR\"F*Fao/$!+'**H5I\"F*Fao/$!+]([=A\"F*Fao/$!+g>!\\:\"F*Fao/$!+8+\"p4 \"F*Fao/$!+\"\\dd/\"F*Fao/$FBFDQ#.1Fbo/$!+V+q*)p!#5Fao/$!+^uyG_FeuFao/ $!+(3+%zRFeuFao/$!+f**H5IFeuFao/$!+(\\([=AFeuFao/$!++'>!\\:FeuFao/$!*H ,5p*FeuFao/$!*2\\dd%FeuFao7fp/$FeuFDQ&1e-10Fbo/$!+/+(*)p*F*Fao/$!+X(yG _*F*Fao/$!+4+%zR*F*Fao/$!+'**H5I*F*Fao/$!+]([=A*F*Fao/$!+g>!\\:*F*Fao/ $!+8+\"p4*F*Fao/$!+\"\\dd/*F*Fao/$F*FDQ&1e-09Fbo/$!+/+(*)p)F*Fao/$!+X( yG_)F*Fao/$!+4+%zR)F*Fao/$!+'**H5I)F*Fao/$!+]([=A)F*Fao/$!+g>!\\:)F*Fa o/$!+8+\"p4)F*Fao/$!+\"\\dd/)F*Fao/$!\")FDQ&1e-08Fbo/$!+/+(*)p(F*Fao/$ !+X(yG_(F*Fao/$!+4+%zR(F*Fao/$!+'**H5I(F*Fao/$!+]([=A(F*Fao/$!+g>!\\:( F*Fao/$!+8+\"p4(F*Fao/$!+\"\\dd/(F*Fao/$!\"(FDQ&1e-07Fbo/$!+/+(*)p'F*F ao/$!+X(yG_'F*Fao/$!+4+%zR'F*Fao/$!+'**H5I'F*Fao/$!+]([=A'F*Fao/$!+g>! \\:'F*Fao/$!+8+\"p4'F*Fao/$!+\"\\dd/'F*Fao/$!\"'FDQ&1e-06Fbo/$!+/+(*)p &F*Fao/$!+X(yG_&F*Fao/$!+4+%zR&F*Fao/$!+'**H5I&F*Fao/$!+]([=A&F*Fao/$! +g>!\\:&F*Fao/$!+8+\"p4&F*Fao/$!+\"\\dd/&F*Fao/$!\"&FDQ&1e-05Fbo/$!+/+ (*)p%F*Fao/$!+X(yG_%F*Fao/$!+4+%zR%F*Fao/$!+'**H5I%F*Fao/$!+]([=A%F*Fa o/$!+g>!\\:%F*Fao/$!+8+\"p4%F*Fao/$!+\"\\dd/%F*FaoF[pF_pFbpFepFhpF[qF^ qFaqFdqFgqF[rF^rFarFdrFgrFjrF]sF`sFcsFgsFjsF]tF`tFctFftFitF\\uF_uFbuFf uFiuF\\vF_vFbvFevFhv" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 661 "By estimating the slopes of the lines (this could be d one more carefully) we see that the Euler method is first-order, and t he improved Euler method is second-order. For the Runge-Kutta method, however, the error does not behave as expected; with step size 0.01, \+ the error is computed as 0. (identically zero, to the precision used), and thus it does not show up on the log-log plot; while for smaller s tep size, the error increases again. This indicates that the Runge-Ku tta calculation is affected by round-off error. \n\nAt the cost of in creasing the computation time, we can reduce the round-off error by us ing more significant digits in our calculations:\n" }{MPLTEXT 1 0 13 " Digits := 20;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG\"#?" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 685 "t0 := 0: tf := 1.: x0 := 1: \nfor n from 1 to 4 do\n hval[n] := 10^(-n);\n eul[n] := Euler(f,t0, tf,x0,hval[n]);\n eul_err[n] := abs(soln(tf) - eul[n]);\n ie[n] := I mpEuler(f,t0,tf,x0,hval[n]);\n ie_err[n] := abs(soln(tf) - ie[n]);\n \+ rk[n] := RK4(f,t0,tf,x0,hval[n]);\n rk_err[n] := abs(soln(tf) - rk[n ]);\n if n = 1 then \n eul_errlist := [[hval[n],eul_err[n]]];\n \+ ie_errlist := [[hval[n],ie_err[n]]];\n rk_errlist := [[hval[n],rk_ err[n]]];\n else\n eul_errlist := [op(eul_errlist),[hval[n],eul_er r[n]]];\n ie_errlist := [op(ie_errlist),[hval[n],ie_err[n]]];\n \+ rk_errlist := [op(rk_errlist),[hval[n],rk_err[n]]];\n end if;\nod:\nl oglogplot([eul_errlist,ie_errlist,rk_errlist]);" }}{PARA 13 "" 1 "" {GLPLOT2D 320 320 320 {PLOTDATA 2 "6(-%'CURVESG6$7&7$$!5************** ******!#?$!5nW#)p9Ghn;7$$!5+++++++++?F-$!5&\\wH@ch4Nt#F-7$$!5+++++ ++++IF-$!5G>f1toM9NPF-7$$!5+++++++++SF-$!5I$*3gV\"Q1`t%F--%'COLOURG6&% $RGBG$\"#5!\"\"$\"\"!FEFD-F$6$7&7$F($!5li%>Xp[T%zJF-7$F/$!5M2,)3**4%=4 _F-7$F4$!5lZE.5n*>@@(F-7$F9$!5A%>_f(fJT7#*F--F>6&F@FDFAFD-F$6$7&7$F($! 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The other fixed po int is at x=ln r (which exists for r > 0, and is unstable for r < 1, s table for r > 1); there is a transcritical bifurcation at r=1.\n \nSom e representative numerical solutions:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "f7 := (x,r) -> x*(r - exp(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f7Gf*6$%\"xG%\"rG6\"6$%)operatorG%&arrowGF)*&9$\"\" \",&9%F/-%$expG6#F.!\"\"F/F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "deq7 := diff(x(t),t) = f7(x(t),r);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%%deq7G/-%%diffG6$-%\"xG6#%\"tGF,*&F)\"\"\",&%\"rGF. -%$expG6#F)!\"\"F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "Plot qualit atively different vector fields for r<1, r=1 and r>1:\n" }{MPLTEXT 1 0 305 "p7a := plot(f7(x,0.3),x=-3..1,tickmarks=[2,2],labels=[\"\",\"f \"]):\np7b := plot([f7(x,1),0,[[0,0.2],[0,-3]]],x=-3..1,tickmarks=[2,2 ],labels=[\"\",\"f\"],axes=none):\np7c := plot(f7(x,2),x=-2..1.5,tickm arks=[2,2],labels=[\"\",\"f\"]):\np7 := array(1..1,1..3):\np7[1,1] := \+ p7a: p7[1,2] := p7b: p7[1,3] := p7c: display(p7);" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 964 312 312 {PLOTDATA 2 "6:-%'CURVESG6$7W7$$!+++++5!\")$\"+\"=H*pH!\"*7$$!+?p0k&*F -$\"+c:NUKF-7$$!+q^u%=*F-$\"+U#p<[$F-7$$!+S%p\"e()F-$\"+R'QOv$F-7$$!+! 4m(G$)F-$\"+=*\\+.%F-7$$!+?OS,zF-$\"+M9y2VF-7$$!++R=0vF-$\"+$f0vc%F-7$ $!+S8#\\4(F-$\"+*)RaQ[F-7$$!++siqmF-$\"+j#**37&F-7$$!+!y$pZiF-$\"+H%[T S&F-7$$!+!yaE\"eF-$\"+DC-(p&F-7$$!+!>s%HaF-$\"+h#3f&fF-7$$!+]$*4)*\\F- $\"+8#3zC'F-7$$!+]_&\\c%F-$\"+$Q$4TlF-7$$!+]1aZTF-$\"+Y7#H#oF-7$$!+0#) [oPF-$\"+uodxqF-7$$!+&=exJ$F-$\"+pG$yP(F-7$$!+N2$f$HF-$\"+P<3HwF-7$$!+ SYx\"\\#F-$\"+6:W;zF-7$$!+N7i)4#F-$\"+(G/^;)F-7$$!+N'psm\"F-$\"+w;4I%) F-7$$!+54_c7F-$\"+k/)Hn)F-7$$!*5x%z#)F-$\"+IQO9*)F-7$$!*?PQM%F-$\"+C&z H7*F-7$$!(!zr)*F-$\"+L=EJ$*F-7$$\"*'zw5VF-$\"+o%od_*F-7$$\"*QQ#\\\")F- $\"+X(ePn*F-7$$\"+e\"*[H7F-$\"+n!Hv!)*F-7$$\"++dxd;F-$\"+k:r7**F-7$$\" +D0xw?F-$\"+2D\"z(**F-7$$\"+i&p@[#F-$\"+++++5F*7$$\"+vgHKHF-$\"+WB+q** F-7$$\"+mZvOLF-$\"+gU%o))*F-7$$\"+]2goPF-$\"+?9hI(*F-7$$\"+'R<*fTF-$\" +q8c?&*F-7$$\"+])Hxe%F-$\"+t#)31#*F-7$$\"+H!o-*\\F-$\"+Wa^=))F-7$$\"+7 k.6aF-$\"+Dih0$)F-7$$\"+;WTAeF-$\"+K_w$o(F-7$$\"+i!*3`iF-$\"+sKQ))oF-7 $$\"+M*zym'F-$\"+NGmkfF-7$$\"+3N1#4(F-$\"+Z3MR[F-7$$\"+HYt7vF-$\"+Hb-? 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[x(0)=1]],linecolor=black);\n" }{TEXT -1 55 "The lower fixed point is \+ unstable, the upper is stable." }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\" \"!F$$!+/G(R?\"!\"*" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "DEplot(subs(r=1,deq7),x(t),t=0..5,x=-3.. 3,[[x(0)=-0.3],[x(0)=0],[x(0)=0.7],[x(0)=2.4]],stepsize=0.05,linecolor =black);\n" }{TEXT -1 36 "One fixed point: x=0 is half-stable." }} {PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "solve(f7(x,2)=0,x);\nDEplot(subs(r=2,deq7),x(t),t=0..10,x=-3..3 ,[[x(0)=-0.3],[x(0)=0],[x(0)=0.2],[x(0)=2.4]],stepsize=0.05,linecolor= black);\n" }{TEXT -1 59 "The lower fixed point x=0 is unstable, the up per is stable." }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"!-%#lnG6#\"\"#" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "Bifurcation diagram: (exchange of stabilities: the upper branch i s always stable)\n" }{MPLTEXT 1 0 228 "p7a := plot([[-1,0],[1,0]],line style=1):\np7b := plot([[1,0],[3,0]],linestyle=3):\np7c := plot(ln(r), r=0.1..1,linestyle=2,numpoints=10):\np7d := plot(ln(r),r=1..3,linestyl e=1):\ndisplay(p7a,p7b,p7c,p7d,axes=none,view=[-1..3,-2..2]);" }} {PARA 13 "" 1 "" {GLPLOT2D 231 218 218 {PLOTDATA 2 "6)-%'CURVESG6%7$7$ $!\"\"\"\"!$F*F*7$$\"\"\"F*F+-%'COLOURG6&%$RGBG$\"#5F)F+F+-%*LINESTYLE G6#F.-F$6%7$F,7$$\"\"$F*F+F/-F66#F=-F$6%7;7$$\"3/+++++++5!#=$!3YXS*H4& e-B!#<7$$\"3(******\\#HyI6FF$!3S1+?Y[nz@FI7$$\"3))******\\ech7FF$!3)GC &\\09Bq?FI7$$\"3#******\\x[BR\"FF$!3IXIr0Ifr>FI7$$\"3)********pJJ_\"FF $!3mS`v[l\"=)=FI7$$\"33+++]vp%y\"FF$!35%*[*=8OLs\"FI7$$\"3=++++MEY?FF$ !3]Kyu%pple\"FI7$$\"39++]Z/&QF#FF$!3a&G*4m/6\"[\"FI7$$\"37+++&\\P9]#FF $!3o)*yOG&>dQ\"FI7$$\"3P++]UX-HFFF$!3wAt3e3k)H\"FI7$$\"31+++!f6m&HFF$! 3d$etM@T&=7FI7$$\"39+++lC]oMFF$!3QG#p%e@')e5FI7$$\"3A+++SLR!)RFF$!3H*[ _V\\W?@*FF7$$\"3]+++gtn&\\%FF$!3siZVXuo%*zFF7$$\"3!3+++Qh4,&FF$!3u%f14 /t&4pFF7$$\"3]+++XjzBbFF$!3Nb\"HVD(>NfFF7$$\"3?+++58jOgFF$!3#fw!pn'*QZ 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