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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 575 "The purpose of this proje
ct is to study a model considered by the meterologist Ed Lorenz.  (In \+
the Maple manual the name Lorenz is incorrectly spelled Lorentz, the n
ame of a mathematician [G.G. Lorentz, presently at UT Austin].  See th
e problem on page 162.)  Lorenz came up with these equations by simpli
fying a mathematical model for weather.  (Actually, I changed the numb
er 28 to 25 to avoid the square root; it doesn't change the qualitive \+
behavoir of the equatios.)  Rather than try to give the equations in s
tandard math notation, let me just give everything in Maple:" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 40 "restart:d1:=diff(x(t),t)=10*(y(t)-x(t));
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "d2:=diff(y(t),t)=25*x(t)-y(t)-x
(t)*z(t);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "d3:=diff(z(t),t)=-8/3*
z(t)+x(t)*y(t);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "init:=x(0)=0,y(0
)=1,z(0)=0;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 175 "Recall that an eq
uilibrium point of a system of first order equations is a point where \+
all derivatives are zero.  We found this by hand in class; however, Ma
ple can do it too:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "chg:=
x(t)=X,y(t)=Y,z(t)=Z;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "e1
:=subs(chg,rhs(d1)); e2:=subs(chg,rhs(d2)); e3:=subs(chg,rhs(d3));" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "solve(\{e1,e2,e3\},\{X,Y,Z
\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "You can solve the system \+
numerically and look at a few values." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 61 "sol:=dsolve(\{d1,d2,d3,init\}, [x(t),y(t),z(t)], type
=numeric);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "sol(0);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "sol(20);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 74 "Plot the equation with various t-ranges.  This is \+
best done interactively." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
126 "with(DEtools):DEplot3d(\{d1,d2,d3\}, \{x(t),y(t),z(t)\}, t=20..40
, [[init]],\nstepsize=.01, linecolor=t/10-2, scaling=constrained);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 84 "The project is to animate this by showing a point in 3D t
hat moves along the path.  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
64 "with(plots):pointplot3d(\{[0,1,1],[1,-1,2],[3,0,5] \},axes=BOXED);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "pointplot3d(points,colo
r=red);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 \+
0" 3 }{VIEWOPTS 1 1 0 1 1 1803 }