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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Math 2280 - 1             \+
 FOURTH MAPLE ASSIGNMENT" }}{PARA 0 "" 0 "" {TEXT -1 61 "Treibergs    \+
                 Due Wednesday, November 5, 2003" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "This lab may be downloade
d from our homepage as" }}{PARA 0 "" 0 "" {HYPERLNK 17 "http://www.mat
h.utah.edu/~treiberg/M2281L4.mws" 1 "M2281L4.mws" "" }{TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 461 "We explore phase portraits of first orde
r systems. The behavior of the linear systems is determined by the eig
envalues of the system. Nonlinear systems locally behave like linear o
nes, and the local behavior of the trajectories is determined by the l
inear approximations at the critical points. We end by examining Rayle
igh's equation of the clarinet reed and van der Pol's equation for a c
ircuit with nonlinear resistor. These systems have stable limit cycles
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "1. L
inear systems." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "with(linalg):with(DEtools):with(plots):# " }}{PARA 7 
"" 1 "" {TEXT -1 80 "Warning, the protected names norm and trace have \+
been redefined and unprotected\n" }}{PARA 7 "" 1 "" {TEXT -1 45 "Warni
ng, the name adjoint has been redefined\n" }}{PARA 7 "" 1 "" {TEXT -1 
50 "Warning, the name changecoords has been redefined\n" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 171 "Enter a matrix. Compute its eigenvalues,
 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]]);eige
nvects(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$-%\"yG6#%
\"tGF-,&*&#\"\"\"\"\"&F1-%\"xGF,F1F1*&#\"\"%F2F1F*F1F1/-F(6$F3F-,&F3F1
*&#\"\"#F2F1F*F1F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "IC1:=
[seq([x(0)=evalf(0.2*cos(k*Pi/12)),y(0)=evalf(0.2*sin(k*Pi/12))],k=0..
25)]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 368 "The instruction phasepo
rtrait plots the direction field, and integrates the system ODE1 whose
 variables are [x(t),y(t)] and for which  t  ranges from 0.0..3.0, wit
h initial data  IC1, and plotting range -2<x<2, -2<y<2. It is stored a
s a plot structure in P1. P2  stores plots of straight lines in the ei
gendirections 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..3.0,IC1,x=-2..2,y=-2..2,linecolor=brown):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 100 "P2:=plot(\{[P[1,1]*t,P[1,2]*t,t=-2..2],[P[2,1]*t,P[2
,2]*t,t=-2..2]\},color=[maroon,gold],thickness=7):" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 37 "display(\{P1,P2\},scaling=constrained);" }}{PARA 
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f;*f1ALONFdo$\"3.X!\\5M;w?\"F07$$\"3@=(ybi`=%\\Fdo$\"3<*>)y'=n\\P\"F07
$$\"3s'*)4Z'3D\"p'Fdo$\"3/`s1mPlo:F07$$\"3,B#*=X;ie))Fdo$\"3a+!z.!>C$z
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