<s>
In	O
descriptive	O
statistics	O
and	O
chaos	O
theory	O
,	O
a	O
recurrence	B-Application
plot	I-Application
(	O
RP	O
)	O
is	O
a	O
plot	O
showing	O
,	O
for	O
each	O
moment	O
in	O
time	O
,	O
the	O
times	O
at	O
which	O
the	O
state	O
of	O
a	O
dynamical	O
system	O
returns	O
to	O
the	O
previous	O
state	O
at	O
,	O
</s>
<s>
Moreover	O
,	O
the	O
recurrence	O
of	O
states	O
,	O
in	O
the	O
meaning	O
that	O
states	O
are	O
again	O
arbitrarily	O
close	O
after	O
some	O
time	O
of	O
divergence	B-Application
,	O
is	O
a	O
fundamental	O
property	O
of	O
deterministic	O
dynamical	O
systems	O
and	O
is	O
typical	O
for	O
nonlinear	O
or	O
chaotic	O
systems	O
(	O
cf	O
.	O
</s>
<s>
One	O
way	O
to	O
visualize	O
the	O
recurring	O
nature	O
of	O
states	O
by	O
their	O
trajectory	O
through	O
a	O
phase	O
space	O
is	O
the	O
recurrence	B-Application
plot	I-Application
,	O
introduced	O
by	O
Eckmann	O
et	O
al	O
.	O
</s>
<s>
However	O
,	O
making	O
a	O
recurrence	B-Application
plot	I-Application
enables	O
us	O
to	O
investigate	O
certain	O
aspects	O
of	O
the	O
m-dimensional	O
phase	O
space	O
trajectory	O
through	O
a	O
two-dimensional	O
representation	O
.	O
</s>
<s>
The	O
recurrence	B-Application
plot	I-Application
represents	O
the	O
collection	O
of	O
pairs	O
of	O
times	O
such	O
recurrences	O
,	O
i.e.	O
,	O
the	O
set	O
of	O
with	O
,	O
with	O
and	O
discrete	O
points	O
of	O
time	O
and	O
the	O
state	O
of	O
the	O
system	O
at	O
time	O
(	O
location	O
of	O
the	O
trajectory	O
at	O
time	O
)	O
.	O
</s>
<s>
The	O
recurrence	B-Application
plot	I-Application
visualises	O
with	O
coloured	O
(	O
mostly	O
black	O
)	O
dot	O
at	O
coordinates	O
if	O
,	O
with	O
time	O
at	O
the	O
-	O
and	O
-axes	O
.	O
</s>
<s>
Phase	O
space	O
reconstruction	O
is	O
not	O
essential	O
part	O
of	O
the	O
recurrence	B-Application
plot	I-Application
(	O
although	O
often	O
stated	O
in	O
literature	O
)	O
,	O
because	O
it	O
is	O
based	O
on	O
phase	O
space	O
trajectories	O
which	O
could	O
be	O
derived	O
from	O
the	O
system	O
's	O
variables	O
directly	O
(	O
e.g.	O
,	O
from	O
the	O
three	O
variables	O
of	O
the	O
Lorenz	B-Architecture
system	I-Architecture
)	O
.	O
</s>
<s>
The	O
visual	O
appearance	O
of	O
a	O
recurrence	B-Application
plot	I-Application
gives	O
hints	O
about	O
the	O
dynamics	O
of	O
the	O
system	O
.	O
</s>
<s>
Caused	O
by	O
characteristic	O
behaviour	O
of	O
the	O
phase	O
space	O
trajectory	O
,	O
a	O
recurrence	B-Application
plot	I-Application
contains	O
typical	O
small-scale	O
structures	O
,	O
as	O
single	O
dots	O
,	O
diagonal	O
lines	O
and	O
vertical/horizontal	O
lines	O
(	O
or	O
a	O
mixture	O
of	O
the	O
latter	O
,	O
which	O
combines	O
to	O
extended	O
clusters	O
)	O
.	O
</s>
<s>
In	O
contrast	O
to	O
the	O
heuristic	O
approach	O
of	O
the	O
recurrence	O
quantification	O
analysis	O
,	O
which	O
depends	O
on	O
the	O
choice	O
of	O
the	O
embedding	O
parameters	O
,	O
some	O
dynamical	O
invariants	O
as	O
correlation	O
dimension	O
,	O
K2	O
entropy	O
or	O
mutual	O
information	O
,	O
which	O
are	O
independent	O
on	O
the	O
embedding	O
,	O
can	O
also	O
be	O
derived	O
from	O
recurrence	B-Application
plots	I-Application
.	O
</s>
<s>
Close	B-Application
returns	I-Application
plots	I-Application
are	O
similar	O
to	O
recurrence	B-Application
plots	I-Application
.	O
</s>
<s>
The	O
main	O
advantage	O
of	O
recurrence	B-Application
plots	I-Application
is	O
that	O
they	O
provide	O
useful	O
information	O
even	O
for	O
short	O
and	O
non-stationary	O
data	O
,	O
where	O
other	O
methods	O
fail	O
.	O
</s>
<s>
Multivariate	O
extensions	O
of	O
recurrence	B-Application
plots	I-Application
were	O
developed	O
as	O
cross	B-Application
recurrence	I-Application
plots	I-Application
and	O
joint	B-Application
recurrence	I-Application
plots	I-Application
.	O
</s>
<s>
Cross	B-Application
recurrence	I-Application
plots	I-Application
consider	O
the	O
phase	O
space	O
trajectories	O
of	O
two	O
different	O
systems	O
in	O
the	O
same	O
phase	O
space	O
(	O
Marwan	O
&	O
Kurths	O
2002	O
)	O
:	O
</s>
<s>
Cross	B-Application
recurrence	I-Application
plots	I-Application
compare	O
the	O
occurrences	O
of	O
similar	O
states	O
of	O
two	O
systems	O
.	O
</s>
<s>
Joint	B-Application
recurrence	I-Application
plots	I-Application
are	O
the	O
Hadamard	O
product	O
of	O
the	O
recurrence	B-Application
plots	I-Application
of	O
the	O
considered	O
sub-systems	O
(	O
Romano	O
et	O
al	O
.	O
</s>
<s>
In	O
contrast	O
to	O
cross	B-Application
recurrence	I-Application
plots	I-Application
,	O
joint	B-Application
recurrence	I-Application
plots	I-Application
compare	O
the	O
simultaneous	O
occurrence	O
of	O
recurrences	O
in	O
two	O
(	O
or	O
more	O
)	O
systems	O
.	O
</s>
<s>
Joint	B-Application
recurrence	I-Application
plots	I-Application
can	O
be	O
used	O
in	O
order	O
to	O
detect	O
phase	O
synchronisation	O
.	O
</s>
