{VERSION 2 3 "SUN SPARC SOLARIS" "2.3" } {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 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "2 D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 256 " " 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Helvetica" 1 12 0 0 0 0 0 2 2 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 6 6 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 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet \+ Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 } {PSTYLE "Author" 0 19 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 8 8 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 4" 5 20 1 {CSTYLE "" -1 -1 "" 1 10 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Helvetica " 1 12 0 0 0 0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 0 0 0 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 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 18 "" 0 "" {TEXT -1 20 "Phase-Plane Analysis" }}{PARA 18 "" 0 "" {TEXT 256 9 "phase.mws" }}{PARA 19 "" 0 "" {TEXT -1 36 "Douglas B. Meade\n(m eade@math.sc.edu)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 8 "Overview" }}{EXCHG {PARA 0 "" 0 " " {TEXT -1 250 "Initial value problems for nonlinear equations --- bot h first-order systems and higher-order equations --- are very difficul t to solve in closed form. However, significant amounts of information can often be obtained directly from the equations. The " }{TEXT 257 14 "phase portrait" }{TEXT -1 95 " is a simple, yet powerful, tool for uncovering some of this information. This introduction to " }{TEXT 258 20 "phase-plane analysis" }{TEXT -1 294 " will involve the examina tion of Examples 3, 4, and 5 from Section 5.7 of Nagle and Saff. While the worksheet follows the same general outline as the text, the works heet both includes additional information and omits other information. This worksheet is intended for parallel use with the text." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 10 "Objectives" }}{EXCHG {PARA 15 "" 0 "" {TEXT -1 35 "unde rstand what a phase portrait is" }}{PARA 15 "" 0 "" {TEXT -1 64 "be ab le to extract qualitative information from a phase portrait" }}{PARA 15 "" 0 "" {TEXT -1 8 "use the " }{HYPERLNK 17 "DEplot" 2 "Detools,DEp lot" "" }{TEXT -1 34 " command to create phase portraits" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 32 "New Maple Commands (and Options)" }}{EXCHG {PARA 15 "" 0 "" {HYPERLNK 17 "dirgrid=" 2 "DEtools,DEplot" "" }{TEXT -1 29 " opt ional argument for " }{TEXT 19 6 "DEplot" }{TEXT -1 66 ", used to spec ify a grid other than 20 x 20 (which is the default)" }}{PARA 15 "" 0 "" {HYPERLNK 17 "DEplot3d" 2 "DEtools,DEplot3d" "" }{TEXT -1 71 " c ommand for creating three-dimensional plots from a system of ODEs" }} {PARA 15 "" 0 "" {HYPERLNK 17 "linecolor=" 2 "DEtools,DEplot" "" } {TEXT -1 25 " optional argument for " }{TEXT 19 6 "DEplot" }{TEXT -1 59 ", used to give special coloring schemes for solution curves" }} {PARA 15 "" 0 "" {HYPERLNK 17 "piecewise" 2 "piecewise" "" }{TEXT -1 52 " command for creating a piecewise-defined function" }}{PARA 15 " " 0 "" {HYPERLNK 17 "scaling=" 2 "plot,options" "" }{TEXT -1 92 " \+ optional argument for any Maple plot, used to force equal scaling on a ll axes in a plot" }}{PARA 15 "" 0 "" {HYPERLNK 17 "scene=" 2 "DEtools ,DEplot" "" }{TEXT -1 29 " optional argument for " }{TEXT 19 6 " DEplot" }{TEXT -1 38 ", often used to create phase portraits" }}{PARA 15 "" 0 "" {HYPERLNK 17 "signum" 2 "signum" "" }{TEXT -1 92 " th e sign function ( 1 for positive numbers, -1 for negative numbers, and 0 for zero )" }}{PARA 15 "" 0 "" {HYPERLNK 17 "stepsize=" 2 "DEtools, DEplot" "" }{TEXT -1 25 " optional argument for " }{TEXT 19 6 "DEplo t" }{TEXT -1 76 ", controls the number of points plottted on a solutio n curve (default is 20)" }}{PARA 15 "" 0 "" {HYPERLNK 17 "view=" 2 "pl ot,options" "" }{TEXT -1 100 " optional argument for any Mapl e plot, specifies the viewing window to be used for the plot" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 32 "Example 3 ( Nagle/Saff, p. 290 )" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Determine the trajectories and corresponding equilib rium solutions of" }}}{EXCHG {PARA 258 "" 0 "" {XPPEDIT 18 0 "x" "I\"x G6\"" }{TEXT -1 4 "' = " }{XPPEDIT 18 0 "-y*(y-2)" ",$*&%\"yG\"\"\",&F $F%\"\"#!\"\"F%F(" }{TEXT -1 6 ", \n" }{XPPEDIT 18 0 "y" "I\"yG6\" " }{TEXT -1 4 "' = " }{XPPEDIT 18 0 "(x-2)*(y-2)" "*&,&%\"xG\"\"\"\"\" #!\"\"F%,&%\"yGF%\"\"#F'F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Solution" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with( plots ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with( DEtoo ls ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "The Maple analysis begins by entering the syste m:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "SYS := [ diff( x(t), \+ t ) = -y(t) * ( y(t) - 2 )," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 " \+ diff( y(t), t ) = ( x(t) - 2 ) * ( y(t) - 2 ) ];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 169 "The critical points of this system were found, in Example 2, t o be the single point (2,0) and the line y=2. The trajectories for thi s system are the general solutions to" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "ODE := subs( SYS, x(t)=x, y(t)=y(x)," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 " diff( y(x), x ) = diff( y(t), t )/d iff( x(t), t ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "This equation, which is separable, is easy to solve by hand or with Maple:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "dsolve( ODE, y(x) );" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 15 "Point to Ponder" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "Do you see how this solution is equivalent to the one given in the text? " }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 136 "The stability of the critical points is determined \+ by the flow along each trajectory. One way in which this can be done i s to create a " }{TEXT 259 14 "phase portrait" }{TEXT -1 289 " for t his system. The phase portrait of a system is closely related to the d irection field of a single equation -- it shows how a particle moves f rom any point in space. The connection between direction fields and ph ase portraits is so close that the same Maple command is used for both : " }{TEXT 19 6 "DEplot" }{TEXT -1 11 " (from the " }{TEXT 19 7 "DEtoo ls" }{TEXT -1 10 " package)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "DEplot( SYS, [x,y], t=0..1, x=-2..6, y=-4..4, scene=[x,y]," }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 " title=`Example 3 (Section 5 .7): Phase Portrait` );" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 11 "Be Pa tient!" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 113 "This plot may take seve ral minutes to appear. Fortunately, the current CPU speeds significant ly reduce the delay." }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 54 "Improvi ng the Appearance of the Phase Portrait -- the " }{TEXT 19 7 "dirgrid " }{TEXT -1 7 " option" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 196 "Note th at the fact that all points on the line y=2 are equilibria is not cl ear from the default (20 x 20) grid selected by Maple. A better pict ure can be obtained using the optional argument " }{TEXT 19 15 "dirgr id=[21,21]" }{TEXT -1 10 "; try it!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 31 "Obsolete Fe ature from Release 3" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 107 "In Releas e 3 it was necessary to specify the system of ODEs in a special form b efore Maple would detect an " }{TEXT 260 10 "autonomous" }{TEXT -1 225 " system. The advantage of this was a drastic reduction in the tim e needed to create a phase portrait. It appears as though Release 4 is much better at detecting autonomous systems, and the special form is \+ no longer supported." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 186 "For those who might be interested -- or who might be n eeding to translate other Release 3 worksheets for use in Release 4 -- here are the commands that would have been used in Release 3:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 19 145 "SYSa := [ -y*(y-2), (x-2)*(y-2) ] ;\nDEplot( SYSa, [x,y], t=0..5, x=-2..6, y=-4..4, scene=[x,y], title=` Example 3 (Section 5.7): Phase Portrait` );" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 20 "Note that the above " }{TEXT 19 6 "DEplot" }{TEXT -1 41 " command generates an error in Release 4." }}}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 285 "Under standing the information presented in a phase portrait can be a little confusing --- until you know what to read the graph. The discussion i n the text should help with this -- READ IT!. It also helps to be able to explicitly see the dependent variables plotted against time. The \+ " }{TEXT 19 6 "scene=" }{TEXT -1 18 " argument of the " }{TEXT 19 6 " DEplot" }{TEXT -1 242 " command can be used to produce the plot of any combination of two or three variables for any initial condition. To i llustrate, we will create the phase portrait and plot each component o f the solution to this system with initial conditions " }{XPPEDIT 18 0 "x(0)=4.5" "/-%\"xG6#\"\"!$\"#X!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0)=0" "/-%\"yG6#\"\"!F&" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "The in itial condition can be specified as before, " }{TEXT 261 4 "i.e." } {TEXT -1 3 ", " }{TEXT 19 19 "IC := [ 0, 4.5, 0 ]" }{TEXT -1 241 ", \+ but the following form is somewhat more natural -- particularly if onl y one initial condition is being specified. (The extra square brackets are necessary as the initial conditions must be a list of points or l ist of lists of conditions.)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "IC := [ [ x(0)=4.5, y(0)=0 ] ];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 135 "The phase po rtrait is essentially the same as the one created before. Note that th e list of initial conditions replaces the ranges on " }{XPPEDIT 18 0 "x" "I\"xG6\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "y" "I\"yG6\"" } {TEXT -1 7 ", the " }{TEXT 19 8 "stepsize" }{TEXT -1 84 " option is a dded to produce a smoother approximation to the solution curve, and th e " }{TEXT 19 7 "dirgrid" }{TEXT -1 51 " option has been used as sugge sted previously. The " }{TEXT 19 9 "linecolor" }{TEXT -1 175 " option \+ is used to give an indication of time along the curve (early points wi ll be red, then moving through orange, yellow, green, cyan, blue, ...) ; see the on-line help for " }{HYPERLNK 17 "DEplot" 2 "DEtools,DEplot " "" }{TEXT -1 64 " for a full description of these, and other, option al arguments." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "Pxy := DEp lot( SYS, [x,y], t=0..5, IC, scene=[x,y], stepsize=0.1," }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 44 " dirgrid=[21,21], linecolor=t," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 " title=`Example 3 ( Section 5.7): Phase Portrait` ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "The correspon ding plots of " }{XPPEDIT 18 0 "x" "I\"xG6\"" }{TEXT -1 7 " vs. " } {XPPEDIT 18 0 "t" "I\"tG6\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "y" "I\"yG6\"" }{TEXT -1 5 " vs. " }{XPPEDIT 18 0 "t" "I\"tG6\"" }{TEXT -1 54 " are obtained by changing the right-hand side of the " }{TEXT 19 5 "scene" }{TEXT -1 10 " argument." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 16 "Bug in Release 4" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "T he need for the " }{TEXT 19 11 "arrows=NONE" }{TEXT -1 55 " argument is a result of the fact that, in Release 4, " }{TEXT 19 6 "DEplot" } {TEXT -1 7 " uses " }{TEXT 19 12 "arrows=SMALL" }{TEXT -1 98 " as it s default. This is a bug, since Maple should automatically use this as its default for any " }{TEXT 19 6 "DEplot" }{TEXT -1 207 " which incl udes the independent variable in the scene. This has been reported to \+ Maple's Technical Support; it has been fixed in the development versio n of Maple; the patch should be available in early 1997." }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "Ptx := DEplot( SYS, [x,y], t=0..5, \+ IC, scene=[t,x], stepsize=0.1," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 " \+ arrows=NONE, linecolor=t," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 " title=`Example 3 (Section 5.7): x vs. t` ):" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "Pty := DEplot( SYS, [x,y], \+ t=0..5, IC, scene=[t,y], stepsize=0.1," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 " arrows=NONE, linecolor=t," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 " title=`Example 3 (Section 5.7): y vs. \+ t` ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "The nicest way to view these plots is side-by-s ide. This can be achieved by " }{TEXT 19 7 "display" }{TEXT -1 37 "in g the plots as a 1 x 3 array; the " }{TEXT 19 19 "scaling=constrained " }{TEXT -1 59 " option ensures that the same scale is used for both \+ axes." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "display( array(1.. 3, [ Pxy, Ptx, Pty ] ), scaling=constrained );" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 36 "Impro ving the Appearance of the Plot" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 217 "It is usually necessary to expand the plot window within the Mapl e worksheet. To do this, click in the graphic region, then drag the an chors horizontally until the plot fill most of the horizontal width of the screen." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "Notice how, during the first stage, the solutions for bot h " }{XPPEDIT 18 0 "x" "I\"xG6\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "y" "I\"yG6\"" }{TEXT -1 23 " are decreasing with " }{XPPEDIT 18 0 "x" "I\"xG6\"" }{TEXT -1 9 ">0 and " }{XPPEDIT 18 0 "y" "I\"yG6\" " }{TEXT -1 65 "<0 -- this corresponds to the quadrant of the phase p lane with " }{XPPEDIT 18 0 "x" "I\"xG6\"" }{TEXT -1 9 ">2 and " } {XPPEDIT 18 0 "y" "I\"yG6\"" }{TEXT -1 53 "<0. The next stage of the \+ solution takes place for " }{XPPEDIT 18 0 "x" "I\"xG6\"" }{TEXT -1 9 "<2 and " }{XPPEDIT 18 0 "y" "I\"yG6\"" }{TEXT -1 11 "<0; here " } {XPPEDIT 18 0 "x" "I\"xG6\"" }{TEXT -1 21 " is decreasing and " } {XPPEDIT 18 0 "y" "I\"yG6\"" }{TEXT -1 23 " is increasing. When " } {XPPEDIT 18 0 "y" "I\"yG6\"" }{TEXT -1 5 ">0, " }{XPPEDIT 18 0 "x" "I \"xG6\"" }{TEXT -1 103 " increases. The last stage of the solution is its approach to one of the critical points on the line " }{XPPEDIT 18 0 "y" "I\"yG6\"" }{TEXT -1 131 "=2. Notice how the coloring of the curve indicates that the trajectory ``slows down'' as it approaches t he line of equilibria at " }{XPPEDIT 18 0 "y" "I\"yG6\"" }{TEXT -1 3 "=2." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 15 "Point to Ponder" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "How would the phase portrait and \+ trajectories look for the solution starting with initial conditions" } }}{EXCHG {PARA 15 "" 0 "" {XPPEDIT 18 0 "x(0)=3.5" "/-%\"xG6#\"\"!$\"# N!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y(0)=0" "/-%\"yG6#\"\"!F&" } {TEXT -1 1 "?" }}{PARA 15 "" 0 "" {XPPEDIT 18 0 "x(0)=3.5" "/-%\"xG6# \"\"!$\"#N!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y(0)=3" "/-%\"yG6#\" \"!\"\"$" }{TEXT -1 1 "?" }}{PARA 15 "" 0 "" {XPPEDIT 18 0 "x(0) = 4.5 " "/-%\"xG6#\"\"!$\"#X!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y(0) = 3 " "/-%\"yG6#\"\"!\"\"$" }{TEXT -1 1 "?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 32 "Example 4 ( Nagle/Saff, p. 293 ) " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "Use a software package to con struct a phase plane diagram about the critical point (0,0) for the li near system" }}}{EXCHG {PARA 259 "" 0 "" {XPPEDIT 18 0 "x" "I\"xG6\"" }{TEXT -1 4 "' = " }{XPPEDIT 18 0 "-2*x + y" ",&*&\"\"#\"\"\"%\"xGF%! \"\"%\"yGF%" }{TEXT -1 2 ",\n" }{XPPEDIT 18 0 "y" "I\"yG6\"" }{TEXT -1 4 "' = " }{XPPEDIT 18 0 "-5*x-4*y" ",&*&\"\"&\"\"\"%\"xGF%!\"\"*&\" \"%F%%\"yGF%F'" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Solution" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with( plots ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 " with( DEtools ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "The Maple version of this system o f ODEs is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "SYS := [ diff( x(t), t ) = -2*x(t) + y(t)," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 " \+ diff( y(t), t ) = -5*x(t) - 4*y(t) ];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 127 "The dis cussion in the text begins with a view of the phase portrait for a var iety of initial conditions on the rectangle -4 <= " }{XPPEDIT 18 0 "x " "I\"xG6\"" }{TEXT -1 13 " <= 4, -5 <= " }{XPPEDIT 18 0 "y" "I\"yG6\" " }{TEXT -1 6 " <= 5." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "IC := [ [0, 4, -5], [0, 4, 0], [0, 4, 5], [0, -4, 5]," }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 58 " [0, -4, -5], [0, -4, 0], [0, 0, -5], [ 0, 0, 5] ];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "DEplot( SYS , [ x, y ], t=0..1, IC, scene=[x,y], stepsize=0.1," }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 58 " title=`Example 4 (Section 5.7): Phase Portr ait` );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 16 "Points to Ponder" }}{EXCHG {PARA 0 "" 0 " " {TEXT -1 70 "The trajectories, in phase space, appear to spiral towa rds the origin." }}}{EXCHG {PARA 15 "" 0 "" {TEXT -1 58 "How would the trajectories look when plotted against time?" }}{PARA 15 "" 0 "" {TEXT -1 70 "Use the techniques from the previous example to check you r conjecture." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "To zoom in on the behavior close t o the origin, all that is needed is to restrict the " }{TEXT 19 4 "vie w" }{TEXT -1 30 " to a rectangle, say, -0.1 <= " }{XPPEDIT 18 0 "x" "I \"xG6\"" }{TEXT -1 17 " <= 0.1, -0.1 <= " }{XPPEDIT 18 0 "y" "I\"yG6\" " }{TEXT -1 72 " <= 0.1. (Note that it is also necessary to increase t he time interval.)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "DEplo t( SYS, [ x, y ], t=0..10, IC, scene=[x,y], view=[-0.1..0.1,-0.1..0.1] ," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 80 " stepsize=0.1, title=`E xample 4 (Section 5.7): Phase Portrait -- ZOOM` );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "Not e that the view option must change the default arrows setting to " } {TEXT 19 11 "arrows=NONE" }{TEXT -1 10 " (specify " }{TEXT 19 12 "arro ws=SMALL" }{TEXT -1 98 " to include the direction field in the plot). \+ This process can be continued (almost) indefinitely." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 32 "Example 5 ( Nagle/Saff, p. 294 )" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "The motion of a spring--mass system with ``sticky friction'' ( see Nagle and Saff, p. 294) can be modelled by" }}}{EXCHG {PARA 260 " " 0 "" {XPPEDIT 18 0 "m*x" "*&%\"mG\"\"\"%\"xGF$" }{TEXT -1 3 "''(" } {XPPEDIT 18 0 "t" "I\"tG6\"" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "-k*x" ",$*&%\"kG\"\"\"%\"xGF%!\"\"" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "t" "I\"t G6\"" }{TEXT -1 4 ") + " }{XPPEDIT 18 0 "F[friction]" "&%\"FG6#%)frict ionG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "where the frictional forc e depends on the instantaneous velocity, " }{XPPEDIT 18 0 "v(t)=diff( x(t),t)" "/-%\"vG6#%\"tG-%%diffG6$-%\"xG6#F&F&" }{TEXT -1 26 ", and t he spring force, " }{XPPEDIT 18 0 "F[spring] = -k*x(t)" "/&%\"FG6#%'s pringG,$*&%\"kG\"\"\"-%\"xG6#%\"tGF*!\"\"" }{TEXT -1 44 ", relative t o the maximum static friction, " }{XPPEDIT 18 0 "mu" "I#muG6\"" } {TEXT -1 22 ", that can be exerted." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "Note: Specific from of " }{TEXT 19 11 "F[friction]" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 167 "[ Since there is no simpl e means of getting Maple to produce a typeset form of F[friction], th e specific formula for F[friction] will be given only in the solution. ]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 39 "Describe the motion for the case where " } {XPPEDIT 18 0 "m" "I\"mG6\"" }{TEXT -1 1 "=" }{XPPEDIT 18 0 "mu" "I#mu G6\"" }{TEXT -1 1 "=" }{XPPEDIT 18 0 "k" "I\"kG6\"" }{TEXT -1 1 "=" } {XPPEDIT 18 0 "1" "\"\"\"" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Solution" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 14 "with( plots ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with( DEtools ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "The Maple representation of this \+ differential equation is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "ODE := m*diff( x(t), t$2 ) = -k * x(t) + F[friction];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "where" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "F[friction] := piecewise( diff( x(t), t ) <> 0, -mu*s ignum( diff( x(t),t ) )," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 " \+ abs(k*x(t)) " 0 " " {MPLTEXT 1 0 67 " abs(k*x(t))>=mu, mu *signum(x(t)) );" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 29 "Simplificati on -- moving mass" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "Note, as indi cated in the text, that the friction reduces to " }{XPPEDIT 18 0 "-mu *signum( diff(x(t),t) )" ",$*&%#muG\"\"\"-%'signumG6#-%%diffG6$-%\"xG6 #%\"tGF/F%!\"\"" }{TEXT -1 124 " assuming the mass is always moving. \+ To implement this simplification, replace the previous command with (t he much simpler)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 19 47 "F[friction] : = -mu * signum( diff( x(t), t ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 19 6 "DEplot" }{TEXT -1 301 " will accept second- (and higher-)order equations in the first argument, but will \+ only create plots of solution curves -- not phase portraits. To obtain a phase portrait, the equation must be converted to a first-order sys tem of ODEs. In this example this is accomplished by introducing the v elocity " }{XPPEDIT 18 0 "v" "I\"vG6\"" }{TEXT -1 1 "=" }{XPPEDIT 18 0 "x" "I\"xG6\"" }{TEXT -1 75 "' as the second dependent variable. Th us, the problem can be re-written as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "SYS := [ diff( x(t), t ) = v(t)," }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 69 " m * diff( v(t), t ) = subs( diff(x(t),t)=v (t), rhs(ODE) ) ];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "The specific parameters of interes t at this time are" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "PARAM := \{ m=1, mu=1, k=1 \};" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "so t he corresponding system is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "SYS2 := subs( PARAM, SYS );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "The direction field fo r this problem is now easily obtained" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "DEplot( SYS2, [ x, v ], t=0..10, x=-5..5, v=-5..5, sc ene=[x,v]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 " title=`Exampl e 5 (Section 5.7) -- Direction Field` );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "IC := [ [ 0, 7.5, 0 ], [ 0, \+ -7.5, 0 ], [ 0, 2.5, 0 ], [ 0, -2.5, 0 ] ];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "DEpl ot( SYS2, [ x, v ], t=0..10, IC, scene=[x,v]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 " arrows=NONE, linecolor=[GREEN,BLUE,MAGENTA,CY AN]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 " title=`Example 5 (S ection 5.7) -- Direction Field` );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 16 "Points to Po nder" }}{EXCHG {PARA 15 "" 0 "" {TEXT -1 85 "The discussion in the tex t shows that the point ( -1/2, 0 ) is an equilibrium point." }}{PARA 15 "" 0 "" {TEXT -1 56 "Does the phase portrait exhibit this feature? \+ (Explain.)" }}{PARA 15 "" 0 "" {TEXT -1 76 "Do the trajectories ever r each the equilibrium, or are they only asymptotic?" }}{PARA 15 "" 0 " " {TEXT -1 87 "How would the direction field and trajectories change f or different sets of parameters?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 121 "The phase portrait, together with the solution curves, can be produced exactly as in the first example \+ of this worksheet." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "Pxv : = DEplot( SYS2, [x,v], t=0..10, IC, scene=[x,v], arrows=NONE," }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 " linecolor=[GREEN,BLU E,MAGENTA,CYAN], stepsize=0.25," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 " title=`Example 5 (Section 5.7): Phase Portrait` ):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "Ptx := DEplot( SYS2, [x,v], t=0..10 , IC, scene=[t,x], arrows=NONE," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 " linecolor=[GREEN,BLUE,MAGENTA,CYAN], stepsize=0.25," }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 " title=`Example 5 (Se ction 5.7): x vs. t` ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "Ptv := D Eplot( SYS2, [x,v], t=0..10, IC, scene=[t,v], arrows=NONE," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 " linecolor=[GREEN,BLUE,MAGENT A,CYAN], stepsize=0.25," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 " \+ title=`Example 5 (Section 5.7): v=x' vs. t` ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "display( array( 1..3, [ Pxv, Ptx, Ptv ] ), scali ng=constrained );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "For additional discussion of these solutions, please consult the text (p. 296 in Nagle and Saff)." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 43 "Three-Dimensional Plots for ODEs (optional)" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "To conclude, " }{TEXT 19 8 "DEplot3d" } {TEXT -1 184 " can be used to produce a three-dimensional plot that, w hen viewed with different orientations, can display either the phase p lane or either of the two solution spaces. The syntax for " }{TEXT 19 8 "DEplot3d" }{TEXT -1 45 " is essentially the same as the one used fo r " }{TEXT 19 6 "DEplot" }{TEXT -1 26 ", with the exception that " } {TEXT 19 5 "scene" }{TEXT -1 29 " now has three elements; see " } {HYPERLNK 17 "?DEplot3d" 2 "DEtools,DEplot3d" "" }{TEXT -1 41 " for a \+ full description of this command.." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "DEplot3d( SYS2, [x,v], t=0..10, IC, scene=[t,x,v], st epsize=0.25," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 " linecolor =[GREEN,BLUE,MAGENTA,CYAN]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 " \+ title=`Example 5 (Section 5.7): 3D View of Trajectories` );" }}} {SECT 1 {PARA 20 "" 0 "" {TEXT -1 15 "Point to Ponder" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "Can you rotate this plot obtain each of the thr ee views displayed in the previous plot." }}}{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 9 "Copyright" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "phase.m ws -- a Maple worksheet for use with phase-plane analysis" }}{PARA 0 "" 0 "" {TEXT -1 81 " (for use with secti on 5.7, p. 287, of Nagle and Saff)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 45 "Prepared by: Douglas B. Meade (7 March 1 996)" }}{PARA 0 "" 0 "" {TEXT -1 88 " as part \+ of the Maple Supplement for Nagle/Saff (Addison-Wesley)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "Updated to Maple V, \+ Release 4, by Douglas B. Meade (27 December, 1996)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "For more information, ple ase contact the author at one of following addresses:" }}{PARA 0 "" 0 "" {TEXT -1 90 " Department of Mathematics, Univ. o f South Carolina, Columbia, SC 29208" }}{PARA 0 "" 0 "" {TEXT -1 35 " \+ (803) 777-6183" }}{PARA 0 "" 0 "" {TEXT -1 38 " \+ meade@math.sc.edu" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 35 "(copyright, Douglas B. Meade, 1996)" }}} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 0 } {VIEWOPTS 1 1 0 3 4 1802 }