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{SECT 0 {PARA 258 "" 0 "" {TEXT 256 21 "Orbits and Flow Lines" }}
{PARA 257 "" 0 "" {TEXT 258 13 "Tommy Ratliff" }}{PARA 256 "" 0 "" 
{TEXT 257 17 "November 19, 1998" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 
12 "Introduction" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 62 "This worksheet contains a procedure orbit(A, v, n) that w
ill c" }{TEXT 259 0 "" }{TEXT -1 87 "alculate the first n states of th
e discrete dynamical system v_k+1 = Av_k where v_0=v. " }}{PARA 263 "
" 0 "" {TEXT -1 0 "" }}{PARA 262 "" 0 "" {TEXT -1 51 "The result is re
turned as a list that you can plot." }}{PARA 259 "" 0 "" {TEXT -1 0 "
" }}{PARA 260 "" 0 "" {TEXT -1 51 "Be sure to execute each command by \+
pressing return." }}{PARA 261 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 
4 "" 0 "" {TEXT -1 28 "Define the function orbit( )" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "with(linalg)
:" "6#-%%withG6#%'linalgG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
146 "orbit := proc(A,v, n)\n local L, i, v1; \nL := [v]; \nv1 := v; \n
for i to n do\n  v1 := multiply(A,v1);\n  L := [op(L), convert(v1,list
)];\n  od;\nL;\nend:" }}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 51 " Define A and use orbit( ) to do some coo
l plotting" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }{XPPEDIT 19 
1 "A := matrix([[3/8, 1/8], [1/24, 11/24]]);" "6#>%\"AG-%'matrixG6#7$7
$*&\"\"$\"\"\"\"\")!\"\"*&\"\"\"F,\"\")F.7$*&\"\"\"F,\"#CF.*&\"#6F,\"#
CF." }}}{EXCHG {PARA 264 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 260 34 "Plot the orbit of the point (3,5)." }}}{EXCHG {PARA 0 ">
 " 0 "" {XPPEDIT 19 1 "plot(orbit(A,[3, 5],10),scaling = constrained);
" "6#-%%plotG6$-%&orbitG6%%\"AG7$\"\"$\"\"&\"#5/%(scalingG%,constraine
dG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 263 62 "Now plot th
e orbits of several points on the same set of axes." }}}{EXCHG {PARA 
0 "> " 0 "" {XPPEDIT 19 1 "plot([orbit(A,[3, 5],6), orbit(A,[-3, 5],6)
, orbit(A,[-3, -5],6), orbit(A,[3, -5],6)],scaling = constrained);" "6
#-%%plotG6$7&-%&orbitG6%%\"AG7$\"\"$\"\"&\"\"'-F(6%F*7$,$\"\"$!\"\"\"
\"&\"\"'-F(6%F*7$,$\"\"$F4,$\"\"&F4\"\"'-F(6%F*7$\"\"$,$\"\"&F4\"\"'/%
(scalingG%,constrainedG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 261 30 "Now find the eigenvectors of A" }}}{EXCHG {PARA 0 "> " 
0 "" {XPPEDIT 19 1 "eigenvectors(A);" "6#-%-eigenvectorsG6#%\"AG" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 262 127 "We can plot the e
igenspaces of A on the same set of axes as the orbits using the displa
y[plots] command from the plots package." }}}{EXCHG {PARA 0 "> " 0 "" 
{XPPEDIT 19 1 "p1 := plot([-1*x/3, x],x = -5 .. 5,color = [black, blac
k]):" "6#>%#p1G-%%plotG6%7$,$*(\"\"\"\"\"\"%\"xGF,\"\"$!\"\"F/F-/F-;,$
\"\"&F/\"\"&/%&colorG7$%&blackGF8" }}}{EXCHG {PARA 0 "> " 0 "" 
{XPPEDIT 19 1 "p2 := plot([orbit(A,[3, 5],6), orbit(A,[-3, 5],6), orbi
t(A,[-3, -5],6), orbit(A,[3, -5],6)],scaling = constrained):" "6#>%#p2
G-%%plotG6$7&-%&orbitG6%%\"AG7$\"\"$\"\"&\"\"'-F*6%F,7$,$\"\"$!\"\"\"
\"&\"\"'-F*6%F,7$,$\"\"$F6,$\"\"&F6\"\"'-F*6%F,7$\"\"$,$\"\"&F6\"\"'/%
(scalingG%,constrainedG" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "pl
ots[display](p1,p2);" "6#-&%&plotsG6#%(displayG6$%#p1G%#p2G" }}}
{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "" "6#%#%?G" }}}}}{MARK "3 0 0
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