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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Math 2280 - 2             \+
 FOURTH MAPLE ASSIGNMENT" }}{PARA 0 "" 0 "" {TEXT -1 56 "Treibergs    \+
                 Due Tuesday, April 2, 2002" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "This lab may be downloaded from
 our homepage as" }}{PARA 0 "" 0 "" {TEXT -1 60 "              http://
www.math.utah.edu/~treiberg/M2280L4.mws" }}{PARA 0 "" 0 "" {TEXT -1 
461 "We explore phase portraits of first order systems. The behavior o
f the linear systems is determined by the eigenvalues of the system. N
onlinear systems locally behave like linear ones, and the local behavi
or of the trajectories is determined by the linear approximations at t
he critical points. We end by examining Rayleigh's equation of the cla
rinet reed and van der Pol's equation for a circuit with nonlinear res
istor. These systems have stable limit cycles." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "1. Linear systems." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "wit
h(linalg):with(DEtools):with(plots):# " }}{PARA 7 "" 1 "" {TEXT -1 45 
"Warning, the name adjoint has been redefined\n" }}{PARA 7 "" 1 "" 
{TEXT -1 45 "Warning, the name adjoint has been redefined\n" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 171 "Enter a matrix. Compute its eigen
values, their algebraic multiplicity, and a basis for their eigenspace
. Pick off the eigenvectors and store them as rows in the matrix  P." 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "A:=matrix([[1,2/5],[1/5,4/5]]);ei
genvects(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7$7$
\"\"\"#\"\"#\"\"&7$#F*F-#\"\"%F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7%
#\"\"$\"\"&\"\"\"<#-%'vectorG6#7$F'!\"\"7%#\"\"'F&F'<#-F*6#7$\"\"#F'" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "P:=matrix([op(op(3,op(1,[
%]))),op(op(3,op(2,[%])))]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG
-%'matrixG6#7$7$\"\"\"!\"\"7$\"\"#F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 334 "Define a lin
ear system using the matrix. Since the matrix has distinct real eigenv
alues, its eigendirections are the principal directions of the matrix,
 and indicate the directions of expansion/contraction of the system. T
he matrix A gives the coefficients of the linear system.  IC1 is a lis
t of initial data for the integral curves." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 70 "ODE1:=\{diff(x(t),t)=(A[1,1]*x(t)+A[1,2]*y(t)),diff(y
(t),t)=A[2,1]*x(t)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "+A[2,2]*y(t)
\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ODE1G<$/-%%diffG6$-%\"xG6#%
\"tGF-,&F*\"\"\"*&#\"\"#\"\"&F/-%\"yGF,F/F//-F(6$F4F-,&F*#F/F3*&#\"\"%
F3F/F4F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "IC1:=[seq([x(
0)=evalf(cos(k*Pi/8)),y(0)=evalf(sin(k*Pi/8))],k=0..15)]:" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 368 "The instruction phaseportrait plots the \+
direction field, and integrates the system ODE1 whose variables are [x
(t),y(t)] and for which  t  ranges from 0.0..1.0, with initial data  I
C1, and plotting range -2<x<2, -2<y<2. It is stored as a plot structur
e in P1. P2  stores plots of straight lines in the eigendirections of \+
A. Display puts both P1 and P2 on the same plot." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 81 "P1:=phaseportrait(ODE1,[x(t),y(t)],0.0..1.0,IC1,x=-2.
.2,y=-2..2,linecolor=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 68 "P2:=plot(\{[P[1,1]*t,P[1,2]*t,t=-1..1],[P[2,1]*t,P[2,2]*t,t=-1..
1]\}):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "display(\{P1,P2\}
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{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 68 " Here we follow Korevaar's solution of homework problem #8, pag
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0 "> " 0 "" {MPLTEXT 1 0 39 "with(plots):with(linalg):with(DEtools):" 
}}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been
 redefined\n" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected na
mes norm and trace have been redefined and unprotected\n" }}{PARA 7 "
" 1 "" {TEXT -1 45 "Warning, the name adjoint has been redefined\n" }}
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45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 136 "The real parts are positive, so the local picture is t
hat of a spiral source. We can look at the original system near the po
int (-1,-1)." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 7 "ICs1:=[" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
39 "[x(0)=-1.1,y(0)=-1],[x(0)=-1,y(0)=-.9]," }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 39 "[x(0)=-.9,y(0)=-1],[x(0)=-1,y(0)=-1.1]," }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 42 "[x(0)=-1,y(0)=-.95],[x(0)=-1,y(0)=-1.05]]:" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "#look near the equilibrium
 point [-1,-1]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "DEplot(deqtn,[x(t),y(t)],t=
0..5,ICs1,x=-1.3..-.7,y=-1.3..-.7, arrows=small, stepsize=.1,color=bla
ck,linecolor=black,title=`Original system near (-1,-1)`);" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 42 "#solution to nonlinear system near [-1,-1]" 
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45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
22 "3. STABLE LIMIT CYCLES" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 433 "Finally, we illustrate by way of example, that a
 planar system may have a stable limit cycle, a closed periodic trajec
tory that attracts nearby trajectories. Consider the equation proposed
 by Lord Rayleigh for the vibration of a clarinet reed. (See Lab Proje
ct 6.4.)   mx'' = -kx + ax' -b(x')^3  which is stiffer than linear dra
g but soggier for small displacements. As in the text, we convert to a
 firsl order system with a=b=m=k=1." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
57 "ODE3:=\{diff(x(t),t)=y(t),diff(y(t),t)=-x(t)+y(t)-y(t)^3\};" }}
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