{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 "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 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 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 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 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Headi ng 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 0 -1 0 0 0 0 0 0 0 0 -1 0 }{PSTYLE "List Item" 0 14 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 20 0 0 0 0 14 5 }{PSTYLE "List Subitem" 14 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 3 12 0 0 0 0 273 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 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 "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 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 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 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 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 "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 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 "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 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 "" 0 264 1 {CSTYLE "" -1 -1 "" 0 1 0 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 "" 0 265 1 {CSTYLE "" -1 -1 "" 0 1 0 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 "" 0 266 1 {CSTYLE "" -1 -1 "" 0 1 0 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 "" 0 267 1 {CSTYLE "" -1 -1 "" 0 1 0 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 "" 0 268 1 {CSTYLE "" -1 -1 "" 0 1 0 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 "" 0 269 1 {CSTYLE "" -1 -1 "" 0 1 0 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 "" 0 270 1 {CSTYLE "" -1 -1 "" 0 1 0 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 269 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 41 "ORDINARY DIFFERENTIAL EQUATIONS POWERTOOL" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 270 "" 0 "" {TEXT -1 22 "Unit 6 -- Bifurcat ions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 265 "" 0 "" {URLLINK 17 "Prof. Douglas B. Meade" 4 "http://www.math.sc.edu/~meade/" "" }} {PARA 258 "" 0 "" {URLLINK 17 "Industrial Mathematics Institute" 4 "ht tp://www.math.sc.edu/~IMI/" "" }}{PARA 259 "" 0 "" {URLLINK 17 "Depart ment of Mathematics" 4 "http://www.math.sc.edu/" "" }}{PARA 260 "" 0 " " {URLLINK 17 "University of South Carolina" 4 "http://www.sc.edu/" " " }}{PARA 261 "" 0 "" {TEXT -1 19 "Columbia, SC 29208\n" }}{PARA 263 " " 0 "" {TEXT -1 7 "URL: " }{URLLINK 17 "http://www.math.sc.edu/~mead e/" 4 "http://www.math.sc.edu/~meade/" "" }}{PARA 264 "" 0 "" {TEXT -1 25 "E-mail: meade@math.sc.edu" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 262 "" 0 "" {TEXT -1 38 "Copyright \251 2001 by Douglas B. Mea de" }}{PARA 266 "" 0 "" {TEXT -1 19 "All rights reserved" }}{PARA 268 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT -1 67 "--------------- ----------------------------------------------------" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 17 "Outline of Unit 6" }}{EXCHG {PARA 14 "" 0 "" {HYPERLNK 17 "6.A" 1 "" "6.A" }{TEXT -1 36 " Graphica l Approaches to Bifurcation" }}{PARA 256 "" 0 "" {HYPERLNK 17 "6.A-1" 1 "" "6.A-1" }{TEXT -1 42 " Animated Slope Fields and Solution Curves " }}{PARA 256 "" 0 "" {HYPERLNK 17 "6.A-2" 1 "" "6.A-2" }{TEXT -1 20 " Bifurcation Diagram" }}{PARA 14 "" 0 "" {HYPERLNK 17 "6.B" 1 "" "6.B " }{TEXT -1 45 " Analytic Determination of Bifurcation Points" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 " " {TEXT -1 14 "Initialization" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with( DEtools ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with( plots ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with( linalg ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "6.A" {TEXT -1 39 "6.A Grap hical Approaches to Bifurcation" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Consider the one-parameter family of functions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f := y^2 - \+ 2*y + mu;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "in the differential equation" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "ode \+ := diff( y(t), t ) = eval( f, y=y(t) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 "" 0 "6.A-1" {TEXT -1 47 "6.A-1 A nimated Slope Fields and Solution Curves" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 213 "One way to obtain an appreciation for the importance of \+ bifurcations is to examine the slope field and solution curves for dif ferent values of the parameter. Here, these plots will be put together as an animation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 33 "PARAM := [ seq( i/2, i=-8..8 ) ];" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 224 "To maximize the impact of this animation we want to be s ure that every equilibrium solution is displayed. One way to ensure th is is to include all equilibria for each value of the parameter in the list of initial conditions." }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "IC := \{ seq( [0,i], i=-5..5 ) \}:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "for mu in PARAM do" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 " yROOT := solve( f=0, y );" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 " yROOT2 := remove( has, \{yROOT\}, I );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 " IC := IC union \{ seq( [ 0,y0], y0 = yROOT2 ) \};" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "unassign( 'mu' ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "T he complete set of initial conditions is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "IC;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 14 " Maple Question" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Note that the IC are being kept as a " }{HYPERLNK 17 "set" 2 "set" "" }{TEXT -1 8 ", n ot a " }{HYPERLNK 17 "list" 2 "list" "" }{TEXT -1 90 ". Why is this i mportant? (Consult the on-line help for information about sets and lis ts.)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "The first frame of the animation is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "DEp lot( eval(ode,mu=PARAM[1]), \{y(t)\}, t=0..5, y=-10..10, IC, arrows=ME DIUM );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 136 "Since there are 17 frames in the full an imation it may take a considerable amount of time to complete the exec ution of the next command." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "PLOTseq := seq( DEplot( ode, \{y(t) \}, t=0..5, y=-10..10, IC," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 " \+ arrows=MEDIUM ), mu=PARAM ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "display( PLOTseq, insequence=true );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 232 "The fac t that there is a bifurcation somewhere in this range of parameters is evident from the animation. To identify the value of the parameter wh ere the bifurcation occurs, one typically examines a bifurcation diagr am (see below)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "6.A-2" {TEXT -1 25 "6.A-2 Bifurcation Diagram" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Cont inuing the previous example, recall that" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "f;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "ode;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "where " } {XPPEDIT 18 0 "mu" "6#%#muG" }{TEXT -1 174 " is the parameter. The bif urcation diagram displays the equilibria of the ODE as a function of t he parameter. In general, this means plotting the implicitly defined f unction " }{XPPEDIT 18 0 "y=y(mu)" "6#/%\"yG-F$6#%#muG" }{TEXT -1 7 " \+ where " }{XPPEDIT 18 0 "f[mu](y)=0" "6#/-&%\"fG6#%#muG6#%\"yG\"\"!" } {TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "implicitplot( f=0, mu=-8..8, y=-4..4, style=POIN T, view=[ -8..2, -4..4 ] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "The bifurcation value is \+ " }{XPPEDIT 18 0 "mu=1" "6#/%#muG\"\"\"" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "For a second example, consider the one-parameter family of ODEs determined by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 19 "f := y^3 - alpha*y:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "ode := diff( y(t), t ) = eval( f, y=y(t) );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT -1 18 " is the parameter." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "The bifurcation diagram for this example is the familiar \+ \"pitchfork\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 67 "implicitplot( f=0, alpha=-4..4, y=-2..2, style =POINT, axes=BOXED );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "6.B" {TEXT -1 48 "6.B Analytic Determination of Bifurcation Points " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "Not every equilibrium solution is a bifurcation point. For a given value of the parameter, " } {XPPEDIT 18 0 "mu" "6#%#muG" }{TEXT -1 87 ", a necessary condition for an equilibrium solution to be a bifurcation point is that \n" } {XPPEDIT 18 0 "diff( f[mu](y), y ) = 0" "6#/-%%diffG6$-&%\"fG6#%#muG6# %\"yGF-\"\"!" }{TEXT -1 14 ". That is, if " }{XPPEDIT 18 0 "f[mu](y)=0 " "6#/-&%\"fG6#%#muG6#%\"yG\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 " diff( f[mu](y), y ) = 0" "6#/-%%diffG6$-&%\"fG6#%#muG6#%\"yGF-\"\"!" } {TEXT -1 7 ", then " }{XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT -1 45 " is a \+ possible bifurcation point for the ODE " }{XPPEDIT 18 0 "diff( y(t), t ) = f[mu](y(t))" "6#/-%%diffG6$-%\"yG6#%\"tGF*-&%\"fG6#%#muG6#-F(6#F* " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f := y*(1-y)^2 + mu;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "bi f_eq1 := f = 0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "bif_eq2 \+ := diff( f, y ) = 0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "bif _sol := solve( \{ bif_eq1, bif_eq2 \}, \{ mu, y \} );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "bif_pt := seq( eval([mu,y],BP), BP= [bif_sol] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "bif_diag := implicitplot( f=0, mu=- 2..2, y=-2..4, style=POINT ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "bi f_pt_P := plot( [bif_pt], style=POINT, color=BLUE," }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 48 " symbol=CROSS, symbolsize=16 ):" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "display( bif_diag, bif_pt_P, axes= BOXED );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "[Back to " }{HYPERLNK 17 "ODE Powertool Table of Cont ents" 1 "unit00.mws" "" }{TEXT -1 1 "]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 1 0" 10 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }