{VERSION 3 0 "SUN SPARC SOLARIS" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 } {CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 256 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 260 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 1 10 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE " " -1 265 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "New Page" -1 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 1 0 -1 0 }{PSTYLE "" 3 257 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 258 1 {CSTYLE "" -1 -1 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 264 10 "BDH2-5.mws" }{TEXT -1 21 " -- The Lorenz System" }}}{SECT 1 {PARA 257 "" 0 "" {TEXT 256 7 "Getting" }{TEXT -1 1 " " } {TEXT 257 7 "started" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 140 "Every Map le worksheet should begin by re-initializing the Maple \"kernel\" and \+ loading the additional packages that we are most likely to use." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 14 "with( plots ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with( DEtools ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 258 18 "Defining t he model" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 178 "All differential equa tion models begin with a differential equation. If the problem is an i ntial value problem, an initial condition is also needed. Replace the \+ question marks ( " }{TEXT 0 2 "%?" }{TEXT -1 67 " ) in the following i nput regions to define the relevant ODE model." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 42 "ode1 := diff( x(t), t ) = 10*(y(t)-x(t)) :" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "ode2 := diff( y(t), t ) = 28*x(t)-y (t)-x(t)*z(t) :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "ode3 := diff( z( t), t ) = -8/3*z(t) + x(t)*y(t) :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "MODEL := [ ode1, ode2, ode3 ] ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "VAR := \{ x(t), y(t), z(t) \} ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 259 21 "Eq uilibrium Solutions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "EQUIL eqns := \{ rhs( ode1 ) = 0, rhs( ode2 ) = 0, rhs( ode3 ) = 0 \} ;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "EQUILsoln := solve( EQUILeqn s, VAR ) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "allvalues( %[ 2] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 260 52 "Direction Field, Phase Portrait, and Sol ution Curves" }{TEXT 263 2 " (" }{HYPERLNK 17 "DEplot3d" 2 "DEplot3d" "" }{TEXT -1 1 ")" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Replace the q uestion marks ( " }{TEXT 19 3 "%? " }{TEXT -1 79 ") in the following i nput regions to create the direction field for your model." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "DOMAIN := t = 0 .. 5.3 ; # specify reasonable interval for indep var" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 24 "WINDOW := x = -20 .. 20," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 " y = -25 .. 25," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 " z = 0 .. 50;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 262 14 "Phase Por trait" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "IC := [ x(0) = 10, \+ y(0) = 7, z(0)=7 ] ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "PHA SEplot := DEplot3d( MODEL, VAR, DOMAIN, [ IC ], WINDOW," }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 48 " arrows=NONE, stepsize=0.0 1," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 " scene=[ \+ x, y, z ], linecolor=BLUE," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 " \+ title = `Phase Portrait for Lorenz System` ):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "PHASEplot;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 261 15 "Solut ion Curves" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "IC;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "P1:= DEplot( MODEL, VAR, DOMAIN, [ \+ IC ], stepsize=0.01," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 " \+ scene=[ t, x ], arrows=NONE, linecolor=BLUE ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "P2:= DEplot( MODEL, VAR, DOMAIN, [ IC ], stepsize=0.0 1," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 " scene=[ t, y ], \+ arrows=NONE, linecolor=GREEN ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 " P3:= DEplot( MODEL, VAR, DOMAIN, [ IC ], stepsize=0.01," }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 58 " scene=[ t, z ], arrows=NONE, line color=RED ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "display( [ \+ P1, P2, P3 ] , title = `Solution Curves for Lorenz System` ); # combin ed solution curves" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} }{SECT 1 {PARA 258 "" 0 "" {TEXT -1 59 "Animated Visualization of Sens itivity to Initial Conditions" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "R eplace the question marks ( " }{TEXT 19 3 "%? " }{TEXT -1 79 ") in the following input regions to create the direction field for your model ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "DOMAIN := t = 0 .. 30 \+ ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "WINDOW := x = -20 .. 2 0," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 " y = -25 .. 25," }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 " z = 0 .. 50;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 266 33 "Phase Portrait -- with Animation!" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 39 "IC1 := [ x(0) = 0, y(0) = 1, z(0)=0 ] ;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "PHASEplot1 := DEplot3d( MODE L, VAR, DOMAIN, [ IC1 ], WINDOW," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 " arrows=NONE, stepsize=0.01," }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 55 " scene=[ x, y, z ], linecolor=B LUE," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 " title \+ = `Phase Portrait for Lorenz System (dy=0)` ):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 " IC2 := [ x(0) = 0, y(0) = 1.1, z(0)=0 ] ;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 60 "PHASEplot2 := DEplot3d( MODEL, VAR, DOMAIN, [ IC2 ] , WINDOW," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 " a rrows=NONE, stepsize=0.01," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 " \+ scene=[ x, y, z ], linecolor=BLUE," }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 75 " title = `Phase Portrait for Lo renz System (dy=0.1)` ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "IC3 := [ x(0) = 0, y(0) = 1.01, z(0)=0 ] ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "PHAS Eplot3 := DEplot3d( MODEL, VAR, DOMAIN, [ IC3 ], WINDOW," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 " arrows=NONE, stepsize=0. 01," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 " scene=[ x, y, z ], linecolor=BLUE," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 76 " \+ title = `Phase Portrait for Lorenz System (dy=0.01)` \+ ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "IC4 := [ x(0) = 0, y(0) = 1.001, z(0)=0 ] ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "PHASEplot4 := DEplot3d( M ODEL, VAR, DOMAIN, [ IC4 ], WINDOW," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 " arrows=NONE, stepsize=0.01," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 " scene=[ x, y, z ], linecolo r=BLUE," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 77 " tit le = `Phase Portrait for Lorenz System (dy=0.001)` ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "display( [ seq( PHASEplot.i, i=1..4 ) ], insequence=t rue ) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 265 34 "Solution Curves -- with Animation!" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "IC1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "P1:= DEplot( MODEL, VAR, DOMAIN, [ IC1 ], stepsi ze=0.01," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 " scene=[ t, x ], arrows=NONE, linecolor=BLUE ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "P2:= DEplot( MODEL, VAR, DOMAIN, [ IC1 ], stepsize=0.01," }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 " scene=[ t, y ], arrows =NONE, linecolor=GREEN ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "P3:= D Eplot( MODEL, VAR, DOMAIN, [ IC1 ], stepsize=0.01," }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 58 " scene=[ t, z ], arrows=NONE, linecolor =RED ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "LS1 := display( [ P1, P2, P3 ] , title = `Solution Curves for Lorenz System (dy=0)` ) : # combined solution curves" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "LS1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "IC2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "P1:= DEplot( MODEL, VAR, DOMAIN, [ IC2 ], stepsize=0. 01," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 " scene=[ t, x ], arrows=NONE, linecolor=BLUE ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 " P2:= DEplot( MODEL, VAR, DOMAIN, [ IC2 ], stepsize=0.01," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 " scene=[ t, y ], arrows=NONE, lin ecolor=GREEN ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "P3:= DEplot( MOD EL, VAR, DOMAIN, [ IC2 ], stepsize=0.01," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 " scene=[ t, z ], arrows=NONE, linecolor=R ED ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "LS2 := display( [ P1, P2, P3 ] , title = `Solution Curves for Lorenz System (dy=0.1)` ) : # combined solution curves" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "display( LS2 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "IC3;" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 56 "P1:= DEplot( MODEL, VAR, DOMAIN, [ IC3 ], step size=0.01," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 " scene=[ \+ t, x ], arrows=NONE, linecolor=BLUE ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "P2:= DEplot( MODEL, VAR, DOMAIN, [ IC3 ], stepsize=0.01," }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 " scene=[ t, y ], arrows =NONE, linecolor=GREEN ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "P3:= D Eplot( MODEL, VAR, DOMAIN, [ IC3 ], stepsize=0.01," }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 58 " scene=[ t, z ], arrows=NONE, linecolor =RED ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "LS3 := display( [ P1, P2, P3 ] , title = `Solution Curves for Lorenz System (dy=0.01) ` ): # combined solution curves" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "LS3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "IC4;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "P1:= DEplot( MODEL, VAR, DOMAIN, [ IC4 ], stepsize=0. 01," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 " scene=[ t, x ], arrows=NONE, linecolor=BLUE ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 " P2:= DEplot( MODEL, VAR, DOMAIN, [ IC4 ], stepsize=0.01," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 " scene=[ t, y ], arrows=NONE, lin ecolor=GREEN ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "P3:= DEplot( MOD EL, VAR, DOMAIN, [ IC4 ], stepsize=0.01," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 " scene=[ t, z ], arrows=NONE, linecolor=R ED ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "LS4 := display( [ P1, P2, P3 ] , title = `Solution Curves for Lorenz System (dy=0.001)` ): # combined solution curves" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "LS4;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "display( [ seq( LS.i, i=1..4 ) ], i nsequence=true );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "displa y( [ seq( display( LS.i, view=[ 0.. 5,-25..50] ), i=1..4 ) ], insequen ce=true );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "display( [ se q( display( LS.i, view=[10..15,-25..50] ), i=1..4 ) ], insequence=true );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "display( [ seq( disp lay( LS.i, view=[20..25,-25..50] ), i=1..4 ) ], insequence=true );" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}}{MARK "0 0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 }