<s>
The	O
Lorenz	B-Architecture
system	I-Architecture
is	O
a	O
system	O
of	O
ordinary	O
differential	O
equations	O
first	O
studied	O
by	O
mathematician	O
and	O
meteorologist	O
Edward	O
Lorenz	O
.	O
</s>
<s>
In	O
particular	O
,	O
the	O
Lorenz	B-Architecture
attractor	I-Architecture
is	O
a	O
set	O
of	O
chaotic	O
solutions	O
of	O
the	O
Lorenz	B-Architecture
system	I-Architecture
.	O
</s>
<s>
In	O
popular	O
media	O
the	O
"	O
butterfly	O
effect	O
"	O
stems	O
from	O
the	O
real-world	O
implications	O
of	O
the	O
Lorenz	B-Architecture
attractor	I-Architecture
,	O
namely	O
that	O
in	O
a	O
chaotic	O
physical	O
system	O
,	O
in	O
the	O
absence	O
of	O
perfect	O
knowledge	O
of	O
the	O
initial	O
conditions	O
(	O
even	O
the	O
minuscule	O
disturbance	O
of	O
the	O
air	O
due	O
to	O
a	O
butterfly	O
flapping	O
its	O
wings	O
)	O
,	O
our	O
ability	O
to	O
predict	O
its	O
future	O
course	O
will	O
always	O
fail	O
.	O
</s>
<s>
The	O
shape	O
of	O
the	O
Lorenz	B-Architecture
attractor	I-Architecture
itself	O
,	O
when	O
plotted	O
in	O
phase	O
space	O
,	O
may	O
also	O
be	O
seen	O
to	O
resemble	O
a	O
butterfly	O
.	O
</s>
<s>
From	O
a	O
technical	O
standpoint	O
,	O
the	O
Lorenz	B-Architecture
system	I-Architecture
is	O
nonlinear	O
,	O
aperiodic	O
,	O
three-dimensional	O
and	O
deterministic	O
.	O
</s>
<s>
When	O
,	O
,	O
and	O
,	O
the	O
Lorenz	B-Architecture
system	I-Architecture
has	O
chaotic	O
solutions	O
(	O
but	O
not	O
all	O
solutions	O
are	O
chaotic	O
)	O
.	O
</s>
<s>
Almost	O
all	O
initial	O
points	O
will	O
tend	O
to	O
an	O
invariant	O
setthe	O
Lorenz	O
attractora	O
strange	O
attractor	O
,	O
a	O
fractal	B-Application
,	O
and	O
a	O
self-excited	O
attractor	O
with	O
respect	O
to	O
all	O
three	O
equilibria	O
.	O
</s>
<s>
The	O
Lorenz	B-Architecture
attractor	I-Architecture
is	O
difficult	O
to	O
analyze	O
,	O
but	O
the	O
action	O
of	O
the	O
differential	O
equation	O
on	O
the	O
attractor	O
is	O
described	O
by	O
a	O
fairly	O
simple	O
geometric	O
model	O
.	O
</s>
<s>
Example	O
solutions	O
of	O
the	O
Lorenz	B-Architecture
system	I-Architecture
for	O
different	O
values	O
of	O
Image:Lorenz	O
Ro14	O
20	O
41	O
20-200px.pngImage:Lorenz	O
Ro13-200px.png	O
(	O
Enlarge	O
)	O
(	O
Enlarge	O
)	O
Image:Lorenz	O
Ro15-200px.pngImage:Lorenz	O
Ro28-200px.png	O
(	O
Enlarge	O
)	O
(	O
Enlarge	O
)	O
For	O
small	O
values	O
of	O
,	O
the	O
system	O
is	O
stable	O
and	O
evolves	O
to	O
one	O
of	O
two	O
fixed	O
point	O
attractors	O
.	O
</s>
<s>
Sensitive	O
dependence	O
on	O
the	O
initial	O
conditionTime	O
(	O
Enlarge	O
)	O
Time	O
(	O
Enlarge	O
)	O
Time	O
(	O
Enlarge	O
)	O
Image:Lorenz	O
caos1-175.pngImage:Lorenz	O
caos2-175.pngImage:Lorenz	O
caos3-175.png	O
These	O
figures	O
—	O
made	O
using	O
,	O
and	O
—	O
show	O
three	O
time	O
segments	O
of	O
the	O
3-D	O
evolution	O
of	O
two	O
trajectories	O
(	O
one	O
in	O
blue	O
,	O
the	O
other	O
in	O
yellow	O
)	O
in	O
the	O
Lorenz	B-Architecture
attractor	I-Architecture
starting	O
at	O
two	O
initial	O
points	O
that	O
differ	O
only	O
by	O
10−5	O
in	O
the	O
-coordinate	O
.	O
</s>
<s>
The	O
resulting	O
plot	O
has	O
a	O
shape	O
very	O
similar	O
to	O
the	O
tent	B-Algorithm
map	I-Algorithm
.	O
</s>
<s>
Combining	O
this	O
with	O
the	O
chaos	O
known	O
to	O
be	O
exhibited	O
by	O
the	O
tent	B-Algorithm
map	I-Algorithm
,	O
he	O
showed	O
that	O
the	O
system	O
switches	O
between	O
the	O
two	O
lobes	O
chaotically	O
.	O
</s>
<s>
As	O
shown	O
in	O
Lorenz	O
's	O
original	O
paper	O
,	O
the	O
Lorenz	B-Architecture
system	I-Architecture
is	O
a	O
reduced	O
version	O
of	O
a	O
larger	O
system	O
studied	O
earlier	O
by	O
Barry	O
Saltzman	O
.	O
</s>
<s>
The	O
partial	O
differential	O
equations	O
modeling	O
the	O
system	O
's	O
stream	O
function	O
and	O
temperature	O
are	O
subjected	O
to	O
a	O
spectral	O
Galerkin	B-Algorithm
approximation	I-Algorithm
:	O
the	O
hydrodynamic	O
fields	O
are	O
expanded	O
in	O
Fourier	O
series	O
,	O
which	O
are	O
then	O
severely	O
truncated	O
to	O
a	O
single	O
term	O
for	O
the	O
stream	O
function	O
and	O
two	O
terms	O
for	O
the	O
temperature	O
.	O
</s>
<s>
Smale	O
's	O
14th	O
problem	O
says	O
,	O
'	O
Do	O
the	O
properties	O
of	O
the	O
Lorenz	B-Architecture
attractor	I-Architecture
exhibit	O
that	O
of	O
a	O
strange	O
attractor	O
?	O
'	O
.	O
</s>
<s>
To	O
prove	O
this	O
result	O
,	O
Tucker	O
used	O
rigorous	O
numerics	O
methods	O
like	O
interval	B-Algorithm
arithmetic	I-Algorithm
and	O
normal	O
forms	O
.	O
</s>
