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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Math 2280 - 1             \+
FOURTH MAPLE ASSIGNMENT" }}{PARA 0 "" 0 "" {TEXT -1 56 "Treibergs     \+
                Due Wednesday, Nov 9, 2005" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "This lab may be downloaded fro
m our homepage as            http://www.math.utah.edu/~treiberg/M2282L
4.mws" }}{PARA 0 "" 0 "" {TEXT -1 457 "We explore phase portraits of f
irst order systems. The behavior of the linear systems is determined b
y the eigenvalues of the system. Nonlinear systems locally behave like
 linear ones, and the local behavior of the trajectories is determined
 by the linear approximations at the critical points. We end by examin
ing Rayleigh's equation of the clarinet reed and the Brusselator equat
ion for an exotic chemical reaction. These systems have stable limit c
ycles." }}{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 "with(linalg):with(DEtools):with(plots):# " }}
{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and tra
ce have been redefined and unprotected\n" }}{PARA 7 "" 1 "" {TEXT -1 
45 "Warning, 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 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$\"\"#F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 333 "Define a linear system using the matrix. Since the matrix has \+
distinct real eigenvalues, its eigendirections are the principal direc
tions of the matrix, and indicate the directions of expansion/contract
ion of the system. The matrix A gives the coefficients of the linear s
ystem. IC1 is a list of initial data for the integral curves." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "ODE1:=\{diff(x(t),t)=(A[1,1]*x(t)+A
[1,2]*y(t)),diff(y(t),t)=A[2,1]*x(t)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 14 "+A[2,2]*y(t)\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ODE1G<$/-%
%diffG6$-%\"xG6#%\"tGF-,&F*\"\"\"*&#\"\"#\"\"&F/-%\"yGF,F/F//-F(6$F4F-
,&*&#F/F3F/F*F/F/*&#\"\"%F3F/F4F/F/" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 70 "IC1:=[seq([x(0)=evalf(0.3*cos(k*Pi/8)),y(0)=evalf(0.3
*sin(k*Pi/8))],k=" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "0..15)]:" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 368 "The instruction phaseportrait plo
ts 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  IC1, and plotting range -2<x<2, -2<y<2. It is stored as a plot s
tructure in P1. P2  stores plots of straight lines in the eigendirecti
ons of A. Display puts both P1 and P2 on the same plot." }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 80 "P1:=phaseportrait(ODE1,[x(t),y(t)],-2..3.0,IC1
,x=-2..2,y=-2..2,linecolor=brown):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 67 "P2:=plot(\{[P[1,1]*t,P[1,2]*t,t=-1..1],[P[2,1]*t,P[2,
2]*t,t=-3..3]\}," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "color=[gold,mar
oon], thickness=4):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "disp
lay(\{P1,P2\},scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 213 
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