<s>
The	O
Hénon	B-Algorithm
map	I-Algorithm
,	O
sometimes	O
called	O
Hénon	O
–	O
Pomeau	O
attractor/map	O
,	O
is	O
a	O
discrete-time	O
dynamical	O
system	O
.	O
</s>
<s>
The	O
map	O
depends	O
on	O
two	O
parameters	O
,	O
a	O
and	O
b	O
,	O
which	O
for	O
the	O
classical	O
Hénon	B-Algorithm
map	I-Algorithm
have	O
values	O
of	O
a	O
=	O
1.4	O
and	O
b	O
=	O
0.3	O
.	O
</s>
<s>
For	O
the	O
classical	O
values	O
the	O
Hénon	B-Algorithm
map	I-Algorithm
is	O
chaotic	O
.	O
</s>
<s>
The	O
map	O
was	O
introduced	O
by	O
Michel	O
Hénon	O
as	O
a	O
simplified	O
model	O
of	O
the	O
Poincaré	O
section	O
of	O
the	O
Lorenz	B-Architecture
model	I-Architecture
.	O
</s>
<s>
The	O
Hénon	B-Algorithm
attractor	I-Algorithm
is	O
a	O
fractal	B-Application
,	O
smooth	O
in	O
one	O
direction	O
and	O
a	O
Cantor	O
set	O
in	O
another	O
.	O
</s>
<s>
The	O
Hénon	B-Algorithm
map	I-Algorithm
maps	O
two	O
points	O
into	O
themselves	O
:	O
these	O
are	O
the	O
invariant	O
points	O
.	O
</s>
<s>
For	O
the	O
classical	O
values	O
of	O
a	O
and	O
b	O
of	O
the	O
Hénon	B-Algorithm
map	I-Algorithm
,	O
one	O
of	O
these	O
points	O
is	O
on	O
the	O
attractor	O
:	O
</s>
<s>
The	O
unstable	O
manifold	O
of	O
the	O
fixed	O
point	O
in	O
the	O
attractor	O
is	O
contained	O
in	O
the	O
strange	O
attractor	O
of	O
the	O
Hénon	B-Algorithm
map	I-Algorithm
.	O
</s>
<s>
The	O
Hénon	B-Algorithm
map	I-Algorithm
does	O
not	O
have	O
a	O
strange	O
attractor	O
for	O
all	O
values	O
of	O
the	O
parameters	O
a	O
and	O
b	O
.	O
</s>
<s>
For	O
example	O
,	O
by	O
keeping	O
b	O
fixed	O
at	O
0.3	O
the	O
bifurcation	O
diagram	O
shows	O
that	O
for	O
a	O
=	O
1.25	O
the	O
Hénon	B-Algorithm
map	I-Algorithm
has	O
a	O
stable	O
periodic	O
orbit	O
as	O
an	O
attractor	O
.	O
</s>
<s>
If	O
multiple	O
Hénon	B-Algorithm
maps	I-Algorithm
are	O
plotted	O
,	O
for	O
each	O
map	O
varying	O
the	O
value	O
of	O
b	O
,	O
then	O
stacking	O
all	O
maps	O
together	O
,	O
a	O
Bifurcation	O
diagram	O
is	O
produced	O
.	O
</s>
<s>
The	O
Hénon	B-Algorithm
map	I-Algorithm
may	O
be	O
decomposed	O
into	O
the	O
composition	O
of	O
three	O
functions	O
acting	O
on	O
the	O
domain	O
one	O
after	O
the	O
other	O
.	O
</s>
<s>
The	O
Hénon	B-Algorithm
map	I-Algorithm
may	O
also	O
be	O
deconstructed	O
into	O
a	O
one-dimensional	O
map	O
,	O
defined	O
similarly	O
to	O
the	O
Fibonacci	B-Algorithm
Sequence	I-Algorithm
.	O
</s>
<s>
Although	O
the	O
Hénon	B-Algorithm
map	I-Algorithm
can	O
be	O
plotted	O
on	O
the	O
x	O
-	O
and	O
y-axes	O
,	O
by	O
varying	O
a	O
and	O
b	O
,	O
we	O
obtain	O
two	O
additional	O
dimensions	O
for	O
plotting	O
.	O
</s>
<s>
The	O
Hénon	B-Algorithm
map	I-Algorithm
therefore	O
,	O
can	O
be	O
plotted	O
in	O
four-dimensional	O
space	O
.	O
</s>
<s>
If	O
one	O
solves	O
the	O
one-dimensional	O
Hénon	B-Algorithm
map	I-Algorithm
for	O
the	O
special	O
case	O
:	O
</s>
<s>
In	O
1976	O
France	O
,	O
the	O
Lorenz	B-Architecture
attractor	I-Architecture
is	O
analyzed	O
by	O
the	O
physicist	O
Yves	O
Pomeau	O
who	O
performs	O
a	O
series	O
of	O
numerical	O
calculations	O
with	O
J.L.	O
</s>
<s>
It	O
is	O
the	O
Lorenz	B-Architecture
attractor	I-Architecture
,	O
that	O
is	O
to	O
say	O
,	O
the	O
one	O
corresponding	O
to	O
the	O
original	O
differential	O
equations	O
,	O
and	O
its	O
geometric	O
structure	O
that	O
interest	O
them	O
.	O
</s>
<s>
Stretching	O
,	O
folding	O
,	O
sensitivity	O
to	O
initial	B-Algorithm
conditions	I-Algorithm
are	O
naturally	O
brought	O
in	O
this	O
context	O
in	O
connection	O
with	O
the	O
Lorenz	B-Architecture
attractor	I-Architecture
.	O
</s>
<s>
If	O
the	O
analysis	O
is	O
ultimately	O
very	O
mathematical	O
,	O
Pomeau	O
and	O
Ibanez	O
follow	O
,	O
in	O
a	O
sense	O
,	O
a	O
physicist	O
approach	O
,	O
experimenting	O
with	O
the	O
Lorenz	B-Architecture
system	I-Architecture
numerically	O
.	O
</s>
<s>
They	O
make	O
it	O
possible	O
to	O
highlight	O
a	O
singular	O
behavior	O
of	O
the	O
Lorenz	B-Architecture
system	I-Architecture
:	O
there	O
is	O
a	O
transition	O
,	O
characterized	O
by	O
a	O
critical	O
value	O
of	O
the	O
parameters	O
of	O
the	O
system	O
,	O
for	O
which	O
the	O
system	O
switches	O
from	O
a	O
strange	O
attractor	O
position	O
to	O
a	O
configuration	O
in	O
a	O
limit	O
cycle	O
.	O
</s>
<s>
In	O
dynamical	O
system	O
,	O
the	O
Koopman	B-Algorithm
operator	I-Algorithm
is	O
a	O
natural	O
linear	O
operator	O
on	O
the	O
space	O
of	O
scalar	O
fields	O
.	O
</s>
<s>
For	O
general	O
nonlinear	O
systems	O
,	O
the	O
eigenfunctions	B-Algorithm
of	O
this	O
operator	O
cannot	O
be	O
expressed	O
in	O
any	O
nice	O
form	O
.	O
</s>
<s>
These	O
modes	O
can	O
give	O
insight	O
into	O
the	O
symbolic	O
dynamics	O
of	O
chaotic	O
maps	O
like	O
the	O
Hénon	B-Algorithm
map	I-Algorithm
.	O
</s>
<s>
A	O
3-D	O
generalization	O
for	O
the	O
Hénon	B-Algorithm
map	I-Algorithm
was	O
proposed	O
by	O
Hitz	O
and	O
Zele	O
.	O
</s>
<s>
For	O
and	O
it	O
can	O
be	O
shown	O
that	O
almost	O
all	O
initial	B-Algorithm
conditions	I-Algorithm
inside	O
the	O
unit	O
sphere	O
generate	O
chaotic	O
signals	O
with	O
largest	O
Lyapunov	O
exponent	O
.	O
</s>
