<s>
In	O
mathematics	O
,	O
particularly	O
in	O
dynamical	O
systems	O
,	O
Arnold	B-Algorithm
tongues	I-Algorithm
(	O
named	O
after	O
Vladimir	O
Arnold	O
)	O
are	O
a	O
pictorial	O
phenomenon	O
that	O
occur	O
when	O
visualizing	O
how	O
the	O
rotation	O
number	O
of	O
a	O
dynamical	O
system	O
,	O
or	O
other	O
related	O
invariant	O
property	O
thereof	O
,	O
changes	O
according	O
to	O
two	O
or	O
more	O
of	O
its	O
parameters	O
.	O
</s>
<s>
The	O
regions	O
of	O
constant	O
rotation	O
number	O
have	O
been	O
observed	O
,	O
for	O
some	O
dynamical	O
systems	O
,	O
to	O
form	O
geometric	O
shapes	O
that	O
resemble	O
tongues	O
,	O
in	O
which	O
case	O
they	O
are	O
called	O
Arnold	B-Algorithm
tongues	I-Algorithm
.	O
</s>
<s>
Arnold	B-Algorithm
tongues	I-Algorithm
are	O
observed	O
in	O
a	O
large	O
variety	O
of	O
natural	O
phenomena	O
that	O
involve	O
oscillating	O
quantities	O
,	O
such	O
as	O
concentration	O
of	O
enzymes	O
and	O
substrates	O
in	O
biological	O
processes	O
and	O
cardiac	B-Application
electric	I-Application
waves	I-Application
.	O
</s>
<s>
Sometimes	O
the	O
frequency	O
of	O
oscillation	B-Algorithm
depends	O
on	O
,	O
or	O
is	O
constrained	O
(	O
i.e.	O
,	O
phase-locked	B-Algorithm
or	O
mode-locked	O
,	O
in	O
some	O
contexts	O
)	O
based	O
on	O
some	O
quantity	O
,	O
and	O
it	O
is	O
often	O
of	O
interest	O
to	O
study	O
this	O
relation	O
.	O
</s>
<s>
For	O
instance	O
,	O
the	O
outset	O
of	O
a	O
tumor	O
triggers	O
in	O
the	O
area	O
a	O
series	O
of	O
substance	O
(	O
mainly	O
proteins	O
)	O
oscillations	B-Algorithm
that	O
interact	O
with	O
each	O
other	O
;	O
simulations	O
show	O
that	O
these	O
interactions	O
cause	O
Arnold	B-Algorithm
tongues	I-Algorithm
to	O
appear	O
,	O
that	O
is	O
,	O
the	O
frequency	O
of	O
some	O
oscillations	B-Algorithm
constrain	O
the	O
others	O
,	O
and	O
this	O
can	O
be	O
used	O
to	O
control	O
tumor	O
growth	O
.	O
</s>
<s>
Other	O
examples	O
where	O
Arnold	B-Algorithm
tongues	I-Algorithm
can	O
be	O
found	O
include	O
the	O
inharmonicity	O
of	O
musical	O
instruments	O
,	O
orbital	O
resonance	O
and	O
tidal	O
locking	O
of	O
orbiting	O
moons	O
,	O
mode-locking	O
in	O
fiber	B-Architecture
optics	I-Architecture
and	O
phase-locked	B-Algorithm
loops	O
and	O
other	O
electronic	O
oscillators	B-Algorithm
,	O
as	O
well	O
as	O
in	O
cardiac	B-Algorithm
rhythms	I-Algorithm
,	O
heart	B-Application
arrhythmias	I-Application
and	O
cell	O
cycle	O
.	O
</s>
<s>
Mode	O
locking	O
occurs	O
when	O
the	O
freely-spinning	O
disk	O
turns	O
at	O
a	O
frequency	O
that	O
is	O
a	O
rational	O
multiple	O
of	O
that	O
of	O
the	O
driven	B-Algorithm
rotator	I-Algorithm
.	O
</s>
<s>
Arnold	B-Algorithm
tongues	I-Algorithm
appear	O
most	O
frequently	O
when	O
studying	O
the	O
interaction	O
between	O
oscillators	B-Algorithm
,	O
particularly	O
in	O
the	O
case	O
where	O
one	O
oscillator	O
drives	O
another	O
.	O
</s>
<s>
This	O
is	O
a	O
particular	O
case	O
of	O
driven	O
oscillators	B-Algorithm
,	O
with	O
a	O
driving	O
force	O
that	O
has	O
a	O
periodic	O
behaviour	O
.	O
</s>
<s>
As	O
a	O
practical	O
example	O
,	O
heart	B-Algorithm
cells	I-Algorithm
(	O
the	O
external	O
oscillator	O
)	O
produce	O
periodic	O
electric	O
signals	O
to	O
stimulate	O
heart	O
contractions	O
(	O
the	O
driven	O
oscillator	O
)	O
;	O
here	O
,	O
it	O
could	O
be	O
useful	O
to	O
determine	O
the	O
relation	O
between	O
the	O
frequency	O
of	O
the	O
oscillators	B-Algorithm
,	O
possibly	O
to	O
design	O
better	O
artificial	B-Device
pacemakers	I-Device
.	O
</s>
<s>
This	O
map	O
displays	O
very	O
diverse	O
behavior	O
depending	O
on	O
the	O
parameters	O
and	O
;	O
if	O
we	O
fix	O
and	O
vary	O
,	O
the	O
bifurcation	O
diagram	O
around	O
this	O
paragraph	O
is	O
obtained	O
,	O
where	O
we	O
can	O
observe	O
periodic	B-Algorithm
orbits	I-Algorithm
,	O
period-doubling	O
bifurcations	O
as	O
well	O
as	O
possible	O
chaotic	O
behavior	O
.	O
</s>
<s>
Since	O
the	O
sine	O
oscillates	O
at	O
frequency	O
1Hz	O
,	O
the	O
number	O
of	O
oscillations	B-Algorithm
of	O
the	O
sine	O
per	O
cycle	O
of	O
will	O
be	O
,	O
thus	O
characterizing	O
a	O
phase-locking	O
of	O
.	O
</s>
<s>
This	O
also	O
means	O
that	O
if	O
is	O
a	O
periodic	O
orbit	B-Algorithm
for	O
parameter	O
,	O
then	O
it	O
is	O
also	O
a	O
periodic	O
orbit	B-Algorithm
for	O
any	O
.	O
</s>
<s>
For	O
small	O
to	O
intermediate	O
values	O
of	O
K	O
(	O
that	O
is	O
,	O
in	O
the	O
range	O
of	O
K	O
=	O
0	O
to	O
about	O
K	O
=	O
1	O
)	O
,	O
and	O
certain	O
values	O
of	O
Ω	O
,	O
the	O
map	O
exhibits	O
a	O
phenomenon	O
called	O
mode	O
locking	O
or	O
phase	B-Algorithm
locking	I-Algorithm
.	O
</s>
<s>
In	O
a	O
phase-locked	B-Algorithm
region	O
,	O
the	O
values	O
θn	O
advance	O
essentially	O
as	O
a	O
rational	O
multiple	O
of	O
n	O
,	O
although	O
they	O
may	O
do	O
so	O
chaotically	O
on	O
the	O
small	O
scale	O
.	O
</s>
<s>
The	O
phase-locked	B-Algorithm
regions	O
,	O
or	O
Arnold	B-Algorithm
tongues	I-Algorithm
,	O
are	O
illustrated	O
in	O
yellow	O
in	O
the	O
figure	O
to	O
the	O
right	O
.	O
</s>
<s>
This	O
ability	O
to	O
"	O
lock	O
on	O
"	O
in	O
the	O
presence	O
of	O
noise	O
is	O
central	O
to	O
the	O
utility	O
of	O
the	O
phase-locked	B-Algorithm
loop	O
electronic	O
circuit	O
.	O
</s>
<s>
Fixing	O
K	O
and	O
taking	O
a	O
cross-section	O
through	O
this	O
image	O
,	O
so	O
that	O
ω	O
is	O
plotted	O
as	O
a	O
function	O
of	O
Ω	O
,	O
gives	O
the	O
"	O
Devil	O
's	O
staircase	O
"	O
,	O
a	O
shape	O
that	O
is	O
generically	O
similar	O
to	O
the	O
Cantor	B-Algorithm
function	I-Algorithm
.	O
</s>
<s>
For	O
the	O
circle	O
map	O
it	O
can	O
be	O
shown	O
that	O
in	O
this	O
region	O
,	O
no	O
more	O
than	O
two	O
stable	O
mode	O
locking	O
regions	O
can	O
overlap	O
,	O
but	O
if	O
there	O
is	O
any	O
limit	O
to	O
the	O
number	O
of	O
overlapping	O
Arnold	B-Algorithm
tongues	I-Algorithm
for	O
general	O
synchronised	O
systems	O
is	O
not	O
known	O
.	O
</s>
<s>
In	O
essence	O
,	O
the	O
standard	B-Algorithm
map	I-Algorithm
introduces	O
a	O
momentum	O
pn	O
which	O
is	O
allowed	O
to	O
dynamically	O
vary	O
,	O
rather	O
than	O
being	O
forced	O
fixed	O
,	O
as	O
it	O
is	O
in	O
the	O
circle	O
map	O
.	O
</s>
<s>
The	O
standard	B-Algorithm
map	I-Algorithm
is	O
studied	O
in	O
physics	O
by	O
means	O
of	O
the	O
kicked	B-Algorithm
rotor	I-Algorithm
Hamiltonian	O
.	O
</s>
<s>
thumb|upright|Circle	O
map	O
showing	O
mode-locked	O
regions	O
or	O
Arnold	B-Algorithm
tongues	I-Algorithm
in	O
black	O
.	O
</s>
