<s>
The	O
logistic	B-Algorithm
map	I-Algorithm
is	O
a	O
polynomial	O
mapping	B-Algorithm
(	O
equivalently	O
,	O
recurrence	O
relation	O
)	O
of	O
degree	O
2	O
,	O
often	O
referred	O
to	O
as	O
an	O
archetypal	O
example	O
of	O
how	O
complex	O
,	O
chaotic	O
behaviour	O
can	O
arise	O
from	O
very	O
simple	O
nonlinear	O
dynamical	O
equations	O
.	O
</s>
<s>
The	O
case	O
of	O
the	O
logistic	B-Algorithm
map	I-Algorithm
is	O
a	O
nonlinear	O
transformation	O
of	O
both	O
the	O
bit-shift	B-Algorithm
map	I-Algorithm
and	O
the	O
case	O
of	O
the	O
tent	B-Algorithm
map	I-Algorithm
.	O
</s>
<s>
The	O
image	O
below	O
shows	O
the	O
amplitude	O
and	O
frequency	O
content	O
of	O
some	O
logistic	B-Algorithm
map	I-Algorithm
iterates	O
for	O
parameter	O
values	O
ranging	O
from	O
2to4	O
.	O
</s>
<s>
The	O
rate	B-Architecture
of	I-Architecture
convergence	I-Architecture
is	O
linear	O
,	O
except	O
for	O
,	O
when	O
it	O
is	O
dramatically	O
slow	O
,	O
less	O
than	O
linear	O
(	O
see	O
Bifurcation	O
memory	O
)	O
.	O
</s>
<s>
This	O
is	O
an	O
example	O
of	O
the	O
deep	O
and	O
ubiquitous	O
connection	O
between	O
chaos	O
and	O
fractals	B-Application
.	O
</s>
<s>
The	O
relative	O
simplicity	O
of	O
the	O
logistic	B-Algorithm
map	I-Algorithm
makes	O
it	O
a	O
widely	O
used	O
point	O
of	O
entry	O
into	O
a	O
consideration	O
of	O
the	O
concept	O
of	O
chaos	O
.	O
</s>
<s>
A	O
rough	O
description	O
of	O
chaos	O
is	O
that	O
chaotic	O
systems	O
exhibit	O
a	O
great	O
sensitivity	O
to	O
initial	O
conditions	O
—	O
a	O
property	O
of	O
the	O
logistic	B-Algorithm
map	I-Algorithm
for	O
most	O
values	O
of	O
between	O
about	O
3.57	O
and	O
4	O
(	O
as	O
noted	O
above	O
)	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
the	O
logistic	B-Algorithm
map	I-Algorithm
,	O
the	O
quadratic	O
difference	O
equation	O
describing	O
it	O
may	O
be	O
thought	O
of	O
as	O
a	O
stretching-and-folding	O
operation	O
on	O
the	O
interval	O
.	O
</s>
<s>
Figure	O
(	O
a	O
)	O
,	O
left	O
,	O
shows	O
a	O
two-dimensional	O
Poincaré	O
plot	O
of	O
the	O
logistic	B-Algorithm
map	I-Algorithm
's	O
state	O
space	O
for	O
,	O
and	O
clearly	O
shows	O
the	O
quadratic	O
curve	O
of	O
the	O
difference	O
equation	O
(	O
)	O
.	O
</s>
<s>
This	O
stretching-and-folding	O
does	O
not	O
just	O
produce	O
a	O
gradual	O
divergence	O
of	O
the	O
sequences	O
of	O
iterates	O
,	O
but	O
an	O
exponential	O
divergence	O
(	O
see	O
Lyapunov	O
exponents	O
)	O
,	O
evidenced	O
also	O
by	O
the	O
complexity	O
and	O
unpredictability	O
of	O
the	O
chaotic	O
logistic	B-Algorithm
map	I-Algorithm
.	O
</s>
<s>
This	O
quality	O
of	O
unpredictability	O
and	O
apparent	O
randomness	O
led	O
the	O
logistic	B-Algorithm
map	I-Algorithm
equation	O
to	O
be	O
used	O
as	O
a	O
pseudo-random	B-Algorithm
number	I-Algorithm
generator	I-Algorithm
in	O
early	O
computers	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
the	O
logistic	B-Algorithm
map	I-Algorithm
with	O
parameter	O
and	O
an	O
initial	O
state	O
in	O
,	O
the	O
attractor	O
is	O
also	O
the	O
interval	O
and	O
the	O
probability	O
measure	O
corresponds	O
to	O
the	O
beta	O
distribution	O
with	O
parameters	O
and	O
.	O
</s>
<s>
Hence	O
,	O
and	O
fortunately	O
,	O
even	O
if	O
we	O
know	O
very	O
little	O
about	O
the	O
initial	O
state	O
of	O
the	O
logistic	B-Algorithm
map	I-Algorithm
(	O
or	O
some	O
other	O
chaotic	O
system	O
)	O
,	O
we	O
can	O
still	O
say	O
something	O
about	O
the	O
distribution	O
of	O
states	O
arbitrarily	O
far	O
into	O
the	O
future	O
,	O
and	O
use	O
this	O
knowledge	O
to	O
inform	O
decisions	O
based	O
on	O
the	O
state	O
of	O
the	O
system	O
.	O
</s>
<s>
The	O
Bifurcation	O
diagram	O
for	O
the	O
logistic	B-Algorithm
map	I-Algorithm
can	O
be	O
visualized	O
with	O
the	O
following	O
Python	B-Language
code	I-Language
:	O
</s>
<s>
Although	O
exact	O
solutions	O
to	O
the	O
recurrence	O
relation	O
are	O
only	O
available	O
in	O
a	O
small	O
number	O
of	O
cases	O
,	O
a	O
closed-form	O
upper	O
bound	O
on	O
the	O
logistic	B-Algorithm
map	I-Algorithm
is	O
known	O
when	O
.	O
</s>
<s>
There	O
are	O
two	O
aspects	O
of	O
the	O
behavior	O
of	O
the	O
logistic	B-Algorithm
map	I-Algorithm
that	O
should	O
be	O
captured	O
by	O
an	O
upper	O
bound	O
in	O
this	O
regime	O
:	O
the	O
asymptotic	O
geometric	O
decay	O
with	O
constant	O
,	O
and	O
the	O
fast	O
initial	O
decay	O
when	O
is	O
close	O
to	O
1	O
,	O
driven	O
by	O
the	O
term	O
in	O
the	O
recurrence	O
relation	O
.	O
</s>
<s>
We	O
can	O
exploit	O
the	O
relationship	O
of	O
the	O
logistic	B-Algorithm
map	I-Algorithm
to	O
the	O
dyadic	B-Algorithm
transformation	I-Algorithm
(	O
also	O
known	O
as	O
the	O
bit-shift	B-Algorithm
map	I-Algorithm
)	O
to	O
find	O
cycles	O
of	O
any	O
length	O
.	O
</s>
<s>
The	O
reason	O
that	O
the	O
dyadic	B-Algorithm
transformation	I-Algorithm
is	O
also	O
called	O
the	O
bit-shift	B-Algorithm
map	I-Algorithm
is	O
that	O
when	O
is	O
written	O
in	O
binary	O
notation	O
,	O
the	O
map	O
moves	O
the	O
binary	O
point	O
one	O
place	O
to	O
the	O
right	O
(	O
and	O
if	O
the	O
bit	O
to	O
the	O
left	O
of	O
the	O
binary	O
point	O
has	O
become	O
a	O
"	O
1	O
"	O
,	O
this	O
"	O
1	O
"	O
is	O
changed	O
to	O
a	O
"	O
0	O
"	O
)	O
.	O
</s>
<s>
The	O
iterate	O
001001001	O
...	O
maps	O
into	O
010010010	O
...	O
,	O
which	O
maps	O
into	O
100100100	O
...	O
,	O
which	O
in	O
turn	O
maps	O
into	O
the	O
original	O
001001001	O
...	O
;	O
so	O
this	O
is	O
a	O
3-cycle	O
of	O
the	O
bit	B-Algorithm
shift	I-Algorithm
map	I-Algorithm
.	O
</s>
<s>
Using	O
the	O
above	O
translation	O
from	O
the	O
bit-shift	B-Algorithm
map	I-Algorithm
to	O
the	O
logistic	B-Algorithm
map	I-Algorithm
gives	O
the	O
corresponding	O
logistic	O
cycle	O
611260467	O
...	O
→	O
950484434	O
...	O
→	O
188255099	O
...	O
→	O
611260467	O
....	O
We	O
could	O
similarly	O
translate	O
the	O
other	O
bit-shift	O
3-cycle	O
into	O
its	O
corresponding	O
logistic	O
cycle	O
.	O
</s>
<s>
Likewise	O
,	O
cycles	O
of	O
any	O
length	O
can	O
be	O
found	O
in	O
the	O
bit-shift	B-Algorithm
map	I-Algorithm
and	O
then	O
translated	O
into	O
the	O
corresponding	O
logistic	O
cycles	O
.	O
</s>
<s>
However	O
,	O
since	O
almost	O
all	O
numbers	O
in	O
are	O
irrational	O
,	O
almost	O
all	O
initial	O
conditions	O
of	O
the	O
bit-shift	B-Algorithm
map	I-Algorithm
lead	O
to	O
the	O
non-periodicity	O
of	O
chaos	O
.	O
</s>
<s>
The	O
number	O
of	O
cycles	O
of	O
(	O
minimal	O
)	O
length	O
for	O
the	O
logistic	B-Algorithm
map	I-Algorithm
with	O
(	O
tent	B-Algorithm
map	I-Algorithm
with	O
)	O
is	O
a	O
known	O
integer	O
sequence	O
:	O
2	O
,	O
1	O
,	O
2	O
,	O
3	O
,	O
6	O
,	O
9	O
,	O
18	O
,	O
30	O
,	O
56	O
,	O
99	O
,	O
186	O
,	O
335	O
,	O
630	O
,	O
1161	O
....	O
</s>
<s>
This	O
tells	O
us	O
that	O
the	O
logistic	B-Algorithm
map	I-Algorithm
with	O
has	O
2	O
fixed	O
points	O
,	O
1	O
cycle	O
of	O
length	O
2	O
,	O
2	O
cycles	O
of	O
length	O
3	O
and	O
so	O
on	O
.	O
</s>
<s>
Since	O
this	O
case	O
of	O
the	O
logistic	B-Algorithm
map	I-Algorithm
is	O
chaotic	O
for	O
almost	O
all	O
initial	O
conditions	O
,	O
all	O
of	O
these	O
finite-length	O
cycles	O
are	O
unstable	O
.	O
</s>
<s>
The	O
gradual	O
increase	O
of	O
at	O
interval	O
changes	O
dynamics	O
from	O
regular	O
to	O
chaotic	O
one	O
with	O
qualitatively	O
the	O
same	O
bifurcation	O
diagram	O
as	O
those	O
for	O
logistic	B-Algorithm
map	I-Algorithm
.	O
</s>
