<s>
The	O
Rössler	B-Algorithm
attractor	I-Algorithm
is	O
the	O
attractor	O
for	O
the	O
Rössler	B-Algorithm
system	I-Algorithm
,	O
a	O
system	O
of	O
three	O
non-linear	O
ordinary	O
differential	O
equations	O
originally	O
studied	O
by	O
Otto	O
Rössler	O
in	O
the	O
1970s	O
.	O
</s>
<s>
These	O
differential	O
equations	O
define	O
a	O
continuous-time	O
dynamical	O
system	O
that	O
exhibits	O
chaotic	O
dynamics	O
associated	O
with	O
the	O
fractal	B-Application
properties	O
of	O
the	O
attractor	O
.	O
</s>
<s>
Some	O
properties	O
of	O
the	O
Rössler	B-Algorithm
system	I-Algorithm
can	O
be	O
deduced	O
via	O
linear	O
methods	O
such	O
as	O
eigenvectors	O
,	O
but	O
the	O
main	O
features	O
of	O
the	O
system	O
require	O
non-linear	O
methods	O
such	O
as	O
Poincaré	O
maps	O
and	O
bifurcation	O
diagrams	O
.	O
</s>
<s>
The	O
original	O
Rössler	O
paper	O
states	O
the	O
Rössler	B-Algorithm
attractor	I-Algorithm
was	O
intended	O
to	O
behave	O
similarly	O
to	O
the	O
Lorenz	B-Architecture
attractor	I-Architecture
,	O
but	O
also	O
be	O
easier	O
to	O
analyze	O
qualitatively	O
.	O
</s>
<s>
An	O
orbit	B-Algorithm
within	O
the	O
attractor	O
follows	O
an	O
outward	O
spiral	O
close	O
to	O
the	O
plane	O
around	O
an	O
unstable	O
fixed	O
point	O
.	O
</s>
<s>
This	O
attractor	O
has	O
some	O
similarities	O
to	O
the	O
Lorenz	B-Architecture
attractor	I-Architecture
,	O
but	O
is	O
simpler	O
and	O
has	O
only	O
one	O
manifold	B-Architecture
.	O
</s>
<s>
Otto	O
Rössler	O
designed	O
the	O
Rössler	B-Algorithm
attractor	I-Algorithm
in	O
1976	O
,	O
but	O
the	O
originally	O
theoretical	O
equations	O
were	O
later	O
found	O
to	O
be	O
useful	O
in	O
modeling	O
equilibrium	O
in	O
chemical	O
reactions	O
.	O
</s>
<s>
The	O
defining	O
equations	O
of	O
the	O
Rössler	B-Algorithm
system	I-Algorithm
are	O
:	O
</s>
<s>
So	O
as	O
long	O
as	O
is	O
smaller	O
than	O
,	O
the	O
term	O
will	O
keep	O
the	O
orbit	B-Algorithm
close	O
to	O
the	O
plane	O
.	O
</s>
<s>
As	O
the	O
orbit	B-Algorithm
approaches	O
greater	O
than	O
,	O
the	O
-values	O
begin	O
to	O
climb	O
.	O
</s>
<s>
As	O
shown	O
in	O
the	O
general	O
plots	O
of	O
the	O
Rössler	B-Algorithm
Attractor	I-Algorithm
above	O
,	O
one	O
of	O
these	O
fixed	O
points	O
resides	O
in	O
the	O
center	O
of	O
the	O
attractor	O
loop	O
and	O
the	O
other	O
lies	O
relatively	O
far	O
from	O
the	O
attractor	O
.	O
</s>
<s>
The	O
last	O
eigenvalue/eigenvector	O
pair	O
is	O
attracting	O
along	O
an	O
axis	O
that	O
runs	O
through	O
the	O
center	O
of	O
the	O
manifold	B-Architecture
and	O
accounts	O
for	O
the	O
z	O
motion	O
that	O
occurs	O
within	O
the	O
attractor	O
.	O
</s>
<s>
The	O
blue	O
line	O
corresponds	O
to	O
the	O
standard	O
Rössler	B-Algorithm
attractor	I-Algorithm
generated	O
with	O
,	O
,	O
and	O
.	O
</s>
<s>
Note	O
that	O
the	O
magenta	O
line	O
nearly	O
touches	O
the	O
plane	O
of	O
the	O
attractor	O
before	O
being	O
pulled	O
upwards	O
into	O
the	O
fixed	O
point	O
;	O
this	O
suggests	O
that	O
the	O
general	O
appearance	O
and	O
behavior	O
of	O
the	O
Rössler	B-Algorithm
attractor	I-Algorithm
is	O
largely	O
a	O
product	O
of	O
the	O
interaction	O
between	O
the	O
attracting	O
and	O
the	O
repelling	O
and	O
plane	O
.	O
</s>
<s>
Although	O
these	O
eigenvalues	O
and	O
eigenvectors	O
exist	O
in	O
the	O
Rössler	B-Algorithm
attractor	I-Algorithm
,	O
their	O
influence	O
is	O
confined	O
to	O
iterations	O
of	O
the	O
Rössler	B-Algorithm
system	I-Algorithm
whose	O
initial	O
conditions	O
are	O
in	O
the	O
general	O
vicinity	O
of	O
this	O
outlier	O
fixed	O
point	O
.	O
</s>
<s>
Except	O
in	O
those	O
cases	O
where	O
the	O
initial	O
conditions	O
lie	O
on	O
the	O
attracting	O
plane	O
generated	O
by	O
and	O
,	O
this	O
influence	O
effectively	O
involves	O
pushing	O
the	O
resulting	O
system	O
towards	O
the	O
general	O
Rössler	B-Algorithm
attractor	I-Algorithm
.	O
</s>
<s>
An	O
example	O
would	O
be	O
plotting	O
the	O
value	O
every	O
time	O
it	O
passes	O
through	O
the	O
plane	O
where	O
is	O
changing	O
from	O
negative	O
to	O
positive	O
,	O
commonly	O
done	O
when	O
studying	O
the	O
Lorenz	B-Architecture
attractor	I-Architecture
.	O
</s>
<s>
In	O
the	O
case	O
of	O
the	O
Rössler	B-Algorithm
attractor	I-Algorithm
,	O
the	O
plane	O
is	O
uninteresting	O
,	O
as	O
the	O
map	O
always	O
crosses	O
the	O
plane	O
at	O
due	O
to	O
the	O
nature	O
of	O
the	O
Rössler	O
equations	O
.	O
</s>
<s>
For	O
example	O
,	O
with	O
a	O
value	O
of	O
4	O
,	O
there	O
is	O
only	O
one	O
point	O
on	O
the	O
Poincaré	O
map	O
,	O
because	O
the	O
function	O
yields	O
a	O
periodic	O
orbit	B-Algorithm
of	O
period	O
one	O
,	O
or	O
if	O
the	O
value	O
is	O
set	O
to	O
12.8	O
,	O
there	O
would	O
be	O
six	O
points	O
corresponding	O
to	O
a	O
period	O
six	O
orbit	B-Algorithm
.	O
</s>
<s>
In	O
the	O
original	O
paper	O
on	O
the	O
Lorenz	B-Architecture
Attractor	I-Architecture
,	O
Edward	O
Lorenz	O
analyzed	O
the	O
local	O
maxima	O
of	O
against	O
the	O
immediately	O
preceding	O
local	O
maxima	O
.	O
</s>
<s>
When	O
visualized	O
,	O
the	O
plot	O
resembled	O
the	O
tent	B-Algorithm
map	I-Algorithm
,	O
implying	O
that	O
similar	O
analysis	O
can	O
be	O
used	O
between	O
the	O
map	O
and	O
attractor	O
.	O
</s>
<s>
For	O
the	O
Rössler	B-Algorithm
attractor	I-Algorithm
,	O
when	O
the	O
local	O
maximum	O
is	O
plotted	O
against	O
the	O
next	O
local	O
maximum	O
,	O
,	O
the	O
resulting	O
plot	O
(	O
shown	O
here	O
for	O
,	O
,	O
)	O
is	O
unimodal	O
,	O
resembling	O
a	O
skewed	O
Hénon	B-Algorithm
map	I-Algorithm
.	O
</s>
<s>
Knowing	O
that	O
the	O
Rössler	B-Algorithm
attractor	I-Algorithm
can	O
be	O
used	O
to	O
create	O
a	O
pseudo	O
1-d	O
map	O
,	O
it	O
then	O
follows	O
to	O
use	O
similar	O
analysis	O
methods	O
.	O
</s>
<s>
Rössler	B-Algorithm
attractor	I-Algorithm
's	O
behavior	O
is	O
largely	O
a	O
factor	O
of	O
the	O
values	O
of	O
its	O
constant	O
parameters	O
,	O
,	O
and	O
.	O
</s>
<s>
In	O
general	O
,	O
varying	O
each	O
parameter	O
has	O
a	O
comparable	O
effect	O
by	O
causing	O
the	O
system	O
to	O
converge	O
toward	O
a	O
periodic	O
orbit	B-Algorithm
,	O
fixed	O
point	O
,	O
or	O
escape	O
towards	O
infinity	O
,	O
however	O
the	O
specific	O
ranges	O
and	O
behaviors	O
induced	O
vary	O
substantially	O
for	O
each	O
parameter	O
.	O
</s>
<s>
Periodic	B-Algorithm
orbits	I-Algorithm
,	O
or	O
"	O
unit	O
cycles	O
,	O
"	O
of	O
the	O
Rössler	B-Algorithm
system	I-Algorithm
are	O
defined	O
by	O
the	O
number	O
of	O
loops	O
around	O
the	O
central	O
point	O
that	O
occur	O
before	O
the	O
loops	O
series	O
begins	O
to	O
repeat	O
itself	O
.	O
</s>
<s>
Bifurcation	O
diagrams	O
are	O
a	O
common	O
tool	O
for	O
analyzing	O
the	O
behavior	O
of	O
dynamical	O
systems	O
,	O
of	O
which	O
the	O
Rössler	B-Algorithm
attractor	I-Algorithm
is	O
one	O
.	O
</s>
<s>
Comparative	O
to	O
the	O
other	O
parameters	O
,	O
varying	O
generates	O
a	O
greater	O
range	O
when	O
period-3	O
and	O
period-6	O
orbits	B-Algorithm
will	O
occur	O
.	O
</s>
<s>
This	O
pattern	O
repeats	O
itself	O
as	O
increases	O
–	O
there	O
are	O
sections	O
of	O
periodicity	O
interspersed	O
with	O
periods	O
of	O
chaos	O
,	O
and	O
the	O
trend	O
is	O
towards	O
higher-period	O
orbits	B-Algorithm
as	O
increases	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
period	O
one	O
orbit	B-Algorithm
only	O
appears	O
for	O
values	O
of	O
around	O
4	O
and	O
is	O
never	O
found	O
again	O
in	O
the	O
bifurcation	O
diagram	O
.	O
</s>
<s>
The	O
same	O
phenomenon	O
is	O
seen	O
with	O
period	O
three	O
;	O
until	O
,	O
period	O
three	O
orbits	B-Algorithm
can	O
be	O
found	O
,	O
but	O
thereafter	O
,	O
they	O
do	O
not	O
appear	O
.	O
</s>
<s>
The	O
above	O
set	O
of	O
images	O
illustrates	O
the	O
variations	O
in	O
the	O
post-transient	O
Rössler	B-Algorithm
system	I-Algorithm
as	O
is	O
varied	O
over	O
a	O
range	O
of	O
values	O
.	O
</s>
<s>
,	O
period-1	O
orbit	B-Algorithm
.	O
</s>
<s>
,	O
period-2	O
orbit	B-Algorithm
.	O
</s>
<s>
,	O
period-4	O
orbit	B-Algorithm
.	O
</s>
<s>
,	O
period-8	O
orbit	B-Algorithm
.	O
</s>
<s>
,	O
period-3	O
orbit	B-Algorithm
.	O
</s>
<s>
,	O
period-6	O
orbit	B-Algorithm
.	O
</s>
<s>
The	O
attractor	O
is	O
filled	O
densely	O
with	O
periodic	B-Algorithm
orbits	I-Algorithm
:	O
solutions	O
for	O
which	O
there	O
exists	O
a	O
nonzero	O
value	O
of	O
such	O
that	O
.	O
</s>
<s>
Periodic	B-Algorithm
orbits	I-Algorithm
are	O
the	O
roots	O
of	O
the	O
function	O
,	O
where	O
is	O
the	O
evolution	O
by	O
time	O
and	O
is	O
the	O
identity	O
.	O
</s>
<s>
As	O
the	O
majority	O
of	O
the	O
dynamics	O
occurs	O
in	O
the	O
x-y	O
plane	O
,	O
the	O
periodic	B-Algorithm
orbits	I-Algorithm
can	O
then	O
be	O
classified	O
by	O
their	O
winding	O
number	O
around	O
the	O
central	O
equilibrium	O
after	O
projection	O
.	O
</s>
<s>
It	O
seems	O
from	O
numerical	O
experimentation	O
that	O
there	O
is	O
a	O
unique	O
periodic	O
orbit	B-Algorithm
for	O
all	O
positive	O
winding	O
numbers	O
.	O
</s>
<s>
The	O
attractor	O
can	O
be	O
dissected	O
into	O
easier	O
to	O
digest	O
invariant	O
manifolds	B-Architecture
:	O
1D	O
periodic	B-Algorithm
orbits	I-Algorithm
and	O
the	O
2D	O
stable	O
and	O
unstable	O
manifolds	B-Architecture
of	O
periodic	B-Algorithm
orbits	I-Algorithm
.	O
</s>
<s>
These	O
invariant	O
manifolds	B-Architecture
are	O
a	O
natural	O
skeleton	O
of	O
the	O
attractor	O
,	O
just	O
as	O
rational	O
numbers	O
are	O
to	O
the	O
real	O
numbers	O
.	O
</s>
<s>
For	O
the	O
purposes	O
of	O
dynamical	O
systems	O
theory	O
,	O
one	O
might	O
be	O
interested	O
in	O
topological	O
invariants	O
of	O
these	O
manifolds	B-Architecture
.	O
</s>
<s>
Periodic	B-Algorithm
orbits	I-Algorithm
are	O
copies	O
of	O
embedded	O
in	O
,	O
so	O
their	O
topological	O
properties	O
can	O
be	O
understood	O
with	O
knot	O
theory	O
.	O
</s>
<s>
The	O
periodic	B-Algorithm
orbits	I-Algorithm
with	O
winding	O
numbers	O
1	O
and	O
2	O
form	O
a	O
Hopf	O
link	O
,	O
showing	O
that	O
no	O
diffeomorphism	O
can	O
separate	O
these	O
orbits	B-Algorithm
.	O
</s>
<s>
The	O
banding	O
evident	O
in	O
the	O
Rössler	B-Algorithm
attractor	I-Algorithm
is	O
similar	O
to	O
a	O
Cantor	O
set	O
rotated	O
about	O
its	O
midpoint	O
.	O
</s>
<s>
Additionally	O
,	O
the	O
half-twist	O
that	O
occurs	O
in	O
the	O
Rössler	B-Algorithm
attractor	I-Algorithm
only	O
affects	O
a	O
part	O
of	O
the	O
attractor	O
.	O
</s>
