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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 108 "Here are the Lorenz Equations.  First we find the equibrium an
d classify them.  In more than two dimensions," }}{PARA 0 "" 0 "" 
{TEXT -1 126 "the classification is based on the \"eigenvalues\" of th
e Jacobian matrix.  It is very similar to what we did in two dimension
s." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "eqn1:=10*(y-x);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "eqn2:=28*x-y-x*z;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "eqn3:=x*y-(8/3)*z;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 50 "Equilibria:=solve(\{eqn1=0,eqn2=0,eqn3=0\},\{x,y,z\})
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "allvalues(Equilibria[2
]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "There are three equilibria
:  At the origin, and at (+/-6sqrt(2), +/-6sqrt(2), 27) " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Jacobian at the or
igin:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 45 "DF0:=matrix(3,3,[-10,10,0,28,-1,0,0,0,-8/3]);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "evalf(eigenvalues(DF0));" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Th
ese are real and mixed in sign -> The origin is a SADDLE" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "DF1:
=matrix(3,3,[-10,10,0,28-27,-1,-6*sqrt(2),6*sqrt(2),6*sqrt(2),-8/3]);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "evalf(eigenvalues(DF1))
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "DF2:=matrix(3,3,[-10,1
0,0,28-27,-1,6*sqrt(2),-6*sqrt(2),-6*sqrt(2),-8/3]);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 24 "evalf(eigenvalues(DF2));" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 251 "There is an unstable spiral in two dimen
sions, with a stable direction pulling the trajectory into the spiral.
  Let's plot the solution to the differential equations:  Be sure to r
otate the figure around so that you can see it's magnificent structure
!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "with(DEtools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 92 "deqns:=\{D(x)(t)=10*(y(t)-x(t)),D(y)(t)=28*x(t)-y(t)-x(t)*z(t)
,D(z)(t)=x(t)*y(t)-(8/3)*z(t)\};" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 93 "DEplot3d(deqns,[x(t),y(t),z(t)],t=0..50,[[x(0)=0,y(0)
=.5,z(0)=1]],stepsize=0.01,linecolor=t);" }}}{EXCHG {PARA 0 "> " 0 "" 
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