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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 99 "This worksheet is just for fun-  Take a look at some interestin
g direction fields and phase planes!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 106 "The first example shows what is called
 a \"Limit Cycle\"- rather than converging to a fixed point, solutions
" }}{PARA 0 "" 0 "" {TEXT -1 92 "are converging to a periodic solution
!  There is also a repelling equilibrium at the origin." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(D
Etools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "ode1:=[diff(y(t
),t)=y(t)+x(t)-y(t)*(x(t)^2+y(t)^2),diff(x(t),t)=x(t)-y(t)-x(t)*(x(t)^
2+y(t)^2)];" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 106 "ic:=[[x(0)=0.1,y(0)=0.1],[x(0)=-1.5,y(0)=-1.5],
[x(0)=-0.1,y(0)=0.1],[x(0)=1.5,y(0)=1.5],[x(0)=-2,y(0)=2]];" }}{PARA 
11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "
DEplot(ode1,[x(t),y(t)],t=-2..9,ic,x=-3..3,y=-3..3,linecolor=black,ste
psize=0.01);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "The next example is the
 Van der Pol equation,  x'' + m (x^2-1)x'+ x = 0, m>0. " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 109 "To show the behavio
r, we re-write as a system:  u=x, v=x'  so that u'=v and v'=-u+m(1-u^2
)v.  Try plugging in" }}{PARA 0 "" 0 "" {TEXT -1 61 "different values \+
of M to see what the resulting limit set is!" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "ode2:=[diff(u(t),
t)=v(t), diff(v(t),t)=-u(t)+M*(1-u(t)^2)*v(t)];" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%ode2G7$/-%%diffG6$-%\"uG6#%\"tGF--%\"vGF,/-F(6$F.F-,
&F*!\"\"*(%\"MG\"\"\",&F7F7*$)F*\"\"#F7F4F7F.F7F7" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 57 "ic2:=[[u(0)=0,v(0)=1],[u(0)=3,v(0)=-1],[u(0)
=-2,v(0)=2]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ic2G7%7$/-%\"uG6#
\"\"!F+/-%\"vGF*\"\"\"7$/F(\"\"$/F-!\"\"7$/F(!\"#/F-\"\"#" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "ode2A:=subs(M=0.2,ode2);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%&ode2AG7$/-%%diffG6$-%\"uG6#%\"tGF--%\"vGF
,/-F(6$F.F-,&F*!\"\"*($\"\"#F4\"\"\",&F8F8*$)F*F7F8F4F8F.F8F8" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "DEplot(ode2A,[u(t),v(t)],t=-
1..9,ic2,u=-4..4,v=-4..4,linecolor=black,stepsize=0.01);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "Here's ju
st a fun phase plane!  Before you plot it, think about where the equil
ibria are..." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 55 "ode3:=[diff(u(t),t)=sin(v(t)), diff(v(t),t)=cos(
u(t))];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ode3G7$/-%%diffG6$-%\"uG
6#%\"tGF--%$sinG6#-%\"vGF,/-F(6$F1F--%$cosG6#F*" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 115 "DEplot(ode3,[u(t),v(t)],t=-1..9,u=-9..9,v=-6.
.6,linecolor=black,stepsize=0.01,dirgrid=[50,50],color=sin(u)*cos(v));
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "13 0 0" 0 }{VIEWOPTS 1 1 0 1 1 
1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }