npg 18 545 2011

npg 18 545 2011
Nonlin. Processes Geophys., 18, 545–562, 2011
www.nonlin-processes-geophys.net/18/545/2011/
doi:10.5194/npg-18-545-2011
© Author(s) 2011. CC Attribution 3.0 License.
Nonlinear Processes
in Geophysics
Identification of dynamical transitions in marine palaeoclimate
records by recurrence network analysis
J. F. Donges1,2 , R. V. Donner1,3 , K. Rehfeld1,2 , N. Marwan1 , M. H. Trauth4 , and J. Kurths1,2
1 Potsdam
Institute for Climate Impact Research, P.O. Box 601203, 14412 Potsdam, Germany
of Physics, Humboldt University Berlin, Newtonstr. 15, 12489 Berlin, Germany
3 Institute for Transport and Economics, Dresden University of Technology, Würzburger Str. 35, 01187 Dresden, Germany
4 Department of Geosciences, University of Potsdam, Karl-Liebknecht-Str. 24, 14476 Potsdam, Germany
2 Department
Received: 25 May 2011 – Revised: 11 August 2011 – Accepted: 27 August 2011 – Published: 5 September 2011
Abstract. The analysis of palaeoclimate time series is usually affected by severe methodological problems, resulting
primarily from non-equidistant sampling and uncertain age
models. As an alternative to existing methods of time series
analysis, in this paper we argue that the statistical properties of recurrence networks – a recently developed approach
– are promising candidates for characterising the system’s
nonlinear dynamics and quantifying structural changes in its
reconstructed phase space as time evolves. In a first order
approximation, the results of recurrence network analysis are
invariant to changes in the age model and are not directly affected by non-equidistant sampling of the data. Specifically,
we investigate the behaviour of recurrence network measures
for both paradigmatic model systems with non-stationary parameters and four marine records of long-term palaeoclimate
variations. We show that the obtained results are qualitatively robust under changes of the relevant parameters of our
method, including detrending, size of the running window
used for analysis, and embedding delay. We demonstrate that
recurrence network analysis is able to detect relevant regime
shifts in synthetic data as well as in problematic geoscientific time series. This suggests its application as a general
exploratory tool of time series analysis complementing existing methods.
1
Introduction
Palaeoclimate proxy data representing past variations of environmental conditions can be obtained from various types of
geological archives distributed over the Earth’s surface. The
study of time series of such proxies, i.e. data that encode the
Correspondence to: J. F. Donges
([email protected])
temporal variability of physical, chemical, biological or sedimentological properties, is a major source of information fostering our understanding of the functioning of the complex
Earth system in the past, present, and future. However, nonequidistant sampling, uncertain age models, multi-scale, and
multi-stable state variability as well as relatively high noise
levels render the study of these proxy records a challenging
problem for time series analysis.
Methods used for time series analysis can be roughly classified as linear or nonlinear. On the one hand, linear methods are based on the evaluation of certain classical statistical characteristics and assume the presence of an underlying linear stochastic process with eventually some superimposed deterministic (e.g. periodic) components (Brockwell and Davis, 1991, 2002; Hamilton, 1994). Prominent
examples that are frequently used for the analysis of realworld time series, including such obtained from geological
archives (Schulz and Stattegger, 1997; Schulz and Mudelsee,
2002; Mudelsee et al., 2009; Rehfeld et al., 2011), are correlation functions and power spectra. On the other hand, nonlinear methods follow a dynamical systems point of view,
implicitly assuming the presence of certain types of deterministic behaviour (Abarbanel, 1996; Kantz and Schreiber,
1997; Donner and Barbosa, 2008).
The vast majority of existing linear or nonlinear methods
of time series analysis relies on the quantification of patterns
of temporal dependences between observations x(t) made at
different times t, i.e. aims to quantify functional relationships
of the form
X
x(t) =
f (x(t − τ ),τ,t) + η(t),
(1)
τ >0
where f (x,τ,t) is a general deterministic function, and
{η(t)} is a stochastic process (often assumed to be fully uncorrelated, i.e. δ-correlated, in time). For a stationary system, the functional dependence f does not explicitly depend
Published by Copernicus Publications on behalf of the European Geosciences Union and the American Geophysical Union.
546
J. F. Donges et al.: Identification of dynamical transitions in marine palaeoclimate records by RN analysis
on time t. In standard linear methods of time series analysis,
f is often assumed to be a linear function; in this case, the
parameters of f encode linear temporal correlations. More
generally, one may consider arbitrary (i.e. not explicitly specified) deterministic relationships f , which may be characterized using concepts such as mutual information (Kantz and
Schreiber, 1997).
In the following, we will refer to methods of time series
analysis that are based on the quantification of temporal interrelationships between observations, e.g. correlation and
mutual information functions or power spectra, as correlative methods. These clearly depend on how well the observation points are specified. In particular, in case of a nonuniform sampling of the considered time series, estimates
of even simple linear characteristics can often not be expressed in a straightforward analytical way. For example,
if one wishes to avoid interpolation (which leads to additional uncertainties), power spectra can be estimated using
harmonic regression of the data (e.g. by means of the LombScargle periodogram; Lomb, 1976; Scargle, 1982), projection methods (Foster, 1996a,b), or a variety of alternative
approaches (Babu and Stoica, 2010; Rehfeld et al., 2011).
However, in the specific case of palaeoclimate data where
typically not even the exact timing of the individual observations is sufficiently well known (Telford et al., 2004), correlative methods can have strong conceptual disadvantages.
In contrast to this large class of methods (which characterise time series from a more or less rigorous statistical
point of view), alternative concepts such as fractal dimensions and generalisations thereof have been first developed
in different mathematical disciplines and later applied to the
characterisation of the properties of certain dynamical systems (Sprott, 2003). Statistical estimates of such measures
can be obtained by a variety of different approaches, most of
which take into account the spatial arrangement of observations in the (possibly reconstructed) phase space. From this
perspective, the mentioned methods do not directly require
knowledge about the timing of observations, i.e. can be considered as non-correlative or geometric methods, since they
rely on geometric attractor properties in phase space rather
than on dynamical information. In the case of palaeoclimate
data with uncertain age models, geometric methods may provide a considerable alternative for statistical analysis. However, as a particular disadvantage, we note that the proper
estimation of fractal dimensions usually requires a considerably larger amount of data than necessary for most correlative methods (Sprott, 2003), which are typically not available
in palaeoclimatology.
Some fundamental relationships between the geometric
properties of attractors in phase space (e.g. Hausdorff and
box dimensions) and important invariants of the associated dynamics (e.g. Lyapunov exponents) are known to exist (Chlouverakis and Sprott, 2005). Note that certain measures of dimensionality include both geometric and dynamical information, i.e. all Rényi dimensions Dq for q > 1
Nonlin. Processes Geophys., 18, 545–562, 2011
including the information dimension D1 (Sprott, 2003).
However, besides fractal dimension estimates based on attractor topology there are only very few suitable and purely
geometric methods available. Recently, it has been suggested
to characterise the mutual proximity relationships of all pairs
of state vectors from the sampled attractor in phase space
by means of complex network methods (Zhang and Small,
2006; Yang and Yang, 2008; Xu et al., 2008; Marwan et al.,
2009; Donner et al., 2010a). Among others, the concept
of recurrence networks (RNs) (Marwan et al., 2009; Donner et al., 2010a,b) has been proven particularly useful for
this purpose. Since such complex network representations of
time series take only spatial information into account, they
can be considered as important examples of geometric methods of time series analysis. RNs provide a set of nonlinear
measures characterising the complexity of dynamical systems (Donner et al., 2010a, 2011a), e.g. allowing to distinguish periodic from chaotic dynamics. While recent findings demonstrate close interrelationships between certain RN
properties and fractal dimensions (Donner et al., 2011b), the
graph-theoretical measures can often be estimated with high
confidence from much shorter time series than fractal dimensions. This warrants their application as a tool for windowbased analysis of non-stationary data (Marwan et al., 2009;
Donner et al., 2011a). In contrast to the aforementioned approaches, transition networks (Nicolis et al., 2005) and visibility graphs (Lacasa et al., 2008) are correlative methods in
the sense that they depend explicitly on the temporal ordering
of observations.
When considering network-based methods of time series
analysis, only RNs (Marwan et al., 2009; Donner et al.,
2011a; Hirata et al., 2011) and visibility graphs (Elsner et al.,
2009) have been used to analyse geoscientific data. So far
RN analysis is the only network-based technique that has
been applied to investigate palaeoclimate proxy records. In
this work, we discuss the application of RNs to studies of
palaeoclimate records, with a special focus on the identification of structural changes in the dynamics that are not easily
found when relying on simple linear statistics. As a benchmark example, we will mainly utilise three marine records of
aeolian dust flux from Northern Africa during the last 5 Myr
(million years) (Trauth et al., 2009; Marwan et al., 2009;
Donner et al., 2011a; Donges et al., 2011). In Sect. 2, we
present a detailed description of the considered data sets, the
necessary preprocessing steps, and the general idea of RNs
and their quantitative analysis. Application to typical nonlinear model systems with a systematic drift of the control
parameters in Sect. 3 suggests that network statistics are well
suited for identifying dynamical transitions from finite time
series. Finally, in Sect. 4, we describe the results of our investigations obtained for the different palaeoclimate time series
and discuss their robustness with respect to the fundamental
parameters of our method.
www.nonlin-processes-geophys.net/18/545/2011/
J.J.F.F.Donges
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indisplaying
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the
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sites
2004;
Larrasoaña
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considered
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(Tiedemann
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deMenocal,
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1995,
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2003).
2004;
2
Data and methods
Description
of the data
2 Data
Data
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and
Marine
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2.1 Description
Description
thedata
datadust flux from North Africa are
2.1
ofofthe
an important source of information on the long-term aridifiMarinerecords
records
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flux
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North Africa
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the continent
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theflux
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(Trauth
Marine
ofofterrigenous
dust
from
Africa
are
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pled
{ti }, where
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and N
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of the continent
during
Plio-Pleistocene
(Trauth
et al.,
2009).
Continuous
time series
{xthree
=
x(t
)}NN
the
number
of
samples,
are
available
from
sediment
et al., 2009). Continuous time series {xii = x(tii )}i=1
i=1
sampled
at 659
times(Atlantic
t , where
i isoffshore
an index
variable West
and
cores:
ODP
Ocean
subtropical
sampled at times ti ,i where i is an index variable and
N the number
of samples,
are available
from three
sediAfrica)
(Tiedemann
et al., 1994),
ODP 721/722
(Arabian
N the number of samples, are available from three sediment (deMenocal,
cores: ODP 659
(Atlantic
subtropical
Sea)
1995,
2004),Ocean
and offshore
ODP 967
(Eastern
ment
cores: ODP 659 (Atlantic
Ocean offshore
subtropical
West Africa) (Tiedemann
et al., 1994),
ODP (Fig.
721/722
(AraMediterranean
Sea) (Larrasoaña
et al., 2003)
1). In
adWest
Africa)
(Tiedemann
et
al.,
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the (deMenocal,
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and
ODP(Fig.
9671).
(Eastern
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Sea) (Larrasoaña
2003)
adsite
659 (Tiedemann
et al., 1994)etwill
be studied
as a In
proxy
Mediterranean
Sea)
(Larrasoaña
et
al.,
182003) (Fig. 1). In addition,
the benthic
oxygen
(δ18
O) record
from
ODP
for
variations
in global
iceisotope
volume,
which
can be
assumed
dition,
thea(Tiedemann
benthic
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(δ
O)
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from
ODP
sitehave
659
et al.,
1994)
studied
a proxy
to
considerable
impact
onwill
the be
continental
aridificasite
659
(Tiedemann
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will
be
studied
as
a
proxy
for variations
in global
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assumed
tion
via a southward
displacement
climatecan
andbevegetation
for
in series
globalare
ice
volume,
which
assumed
to variations
haveAll
a considerable
impact
onin the
continental
2. can be aridificazones.
time
shown
Fig.
totion
have
a
considerable
impact
on
the
continental
aridificavia a southward displacement of climate and vegetation
tion
viaDetrending
aAll
southward
displacement
climate
zones.
time series
are shown inofFig.
2. and vegetation
2.2
zones. All time series are shown in Fig. 2.
2.2 considered
Detrending
All
time series {xi } show a nonlinear trend of in2.2
Detrending
creasing
amplitude and variance towards the present. This
All considered
time
series {xiaridification
} show a nonlinear
trend
intrend
reflects the
successive
of North
andofEast
creasing
and variance
towards
the present.
This
All
considered
series
{xi }ofshow
a nonlinear
trendglacial
of
inAfrica
andamplitude
thetime
intensification
Northern
hemisphere
trend reflects
the
aridification
and East
creasing
amplitude
and variance
towards
the
present.
This
cycles
during
the successive
Plio-Pleistocene
(Trauthof
etNorth
al.,
2009).
To
Africa
and
the
intensification
of Northern
hemisphere
glacial
trend
reflects
the
successive
aridification
of North
East
prevent
corruption
of the results
of our analysis
andand
significyclesand
during
the
Plio-Pleistocene
(Trauth
et al., 2009).
To
Africa
the
of Northern
glacial
cance
test
dueintensification
to this
nonlinear
trend,
wehemisphere
attempt
to remove
prevent
corruption
of
the
results
of
our
analysis
and
significycles
during
thebyPlio-Pleistocene
et al.,of2009).
To
it to first
order
subtracting from(Trauth
xi the mean
a sliding
cance test
nonlinear
trend,
we
attempt
to
remove
prevent
corruption
of
the
results
of
our
analysis
and
signifiwindow
ofdue
sizeto
Wthis
(t
)
centered
at
t
for
all
time
points
t
,
i.e.
D i
i
i
it to first
ordertoby
subtracting
xiwe
theattempt
mean oftoa remove
sliding
cance
test due
this
nonlinearfrom
trend,
bW
(t
)/2c
D
iat t for all time points t ,
X
of sizebyW1subtracting
D (ti ) centered
i
it window
to first order
from
xii the mean of a sliding
x̂i.e.,
xi+j ,
(2)
i = xi −
window of2bW
sizeDW
(ti )+centered
(tiD)/2c
1 j =−bW at(t t)/2c
i for all time points ti ,
D i
i.e.,
bWD (ti )/2c
X
1
where
window sizexi+j
WD,,
(2)
x̂i = xifor
− a chosen detrendingbW
D (ti )/2c
2bWD (t1i )/2c + 1
X
j=−bWD (ti )/2c
xi+j ,
(2)
x̂i = xi −
2bWD (ti )/2c + 1
j=−bWD (ti )/2c
www.nonlin-processes-geophys.net/18/545/2011/
Fig. 2. Plio-Pleistocene variability of (A) δ 18 O at ODP site 659
18 atflux
(Tiedemann
et al., 1994),variability
and of terrigenous
from
North
Fig.
ofof(A)
δ 18δdust
O
ODP
sitesite
659659
Fig.2.2. Plio-Pleistocene
Plio-Pleistocene
variability
(A)
O at
ODP
et
al.,
1994),
(C)from
721/722
Africa
at
ODP
sites
(B)
659
(Tiedemann
(Tiedemann
et
al.,
1994),
and
of
terrigenous
dust
flux
from
North
(Tiedemann et al., 1994), and of terrigenous
dust
flux
North
(deMenocal,
1995,
2004),
and
(D) 967 et
(Larrasoaña
et (C)
al.,
2003).
Africa
sites
(Tiedemann
al.,al.,
1994),
(C)
721/722
AfricaatatODP
ODP
sites(B)
(B)659
659
(Tiedemann
et
1994),
721/722
(deMenocal,
2004),
and (D)
et al., 2003).
The horizontal
red bars
in panel
indicate
two consecutive
(deMenocal,1995,
1995,
2004),
and(A)
(D)967
967(Larrasoaña
(Larrasoaña
et al.,recur2003).
∗=
The
horizontal
in panel
consecutive
recurrence
windowsred
of bars
length
W ∗ =(A)
410indicate
kyr andtwo
mutual
offset 1W
The horizontal red bars in panel
(A) indicate two consecutive∗recur∗
rence
of the
length
W =
410
kyr
and
mutual
offset
∆W
=
41 kyrwindows
as used in
analysis
of
Sect.
4
and
in
Figs.
9–12.
rence windows of length W ∗ = 410 kyr and mutual offset ∆W ∗ =
41 kyr as used in the analysis of Sec. 4 and in Figs. 9–12.
41 kyr as used in the analysis of Sec. 4 and in Figs. 9–12.

 2(i − 1) for i < WD ,
WD for WD ≤ i ≤ N − WD ,
WD (ti ) =
where for achosen
detrending
size
WD ,
2(N
−
for iwindow
>
N −W
D. W
where for a choseni)detrending
window
size
,
(3)
D
That is, 
the effective detrending window size decreases to− 1) boundaries,
for i < W
 2(iseries’
wards the 
time
resulting
in x̂1 = x̂N = 0.
D,
2(i
−
1)
for
i
<
W

W
for
W
≤
iD
≤,N − Wof
(3)or
W
(t
)
=
This
approach
D avoids theDcomplication
D , locally
D simple
i

W
for
W
≤
i
≤
N
−
W
,
(3)
W
(t
)
=
D
D
D
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i
2(N
−
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−
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.
globally fitting
D performing high higher-order polynomials or
2(N
−
i)
for
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W
.
D
pass filtering given irregular sampling and uncertain dating of
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is, our
the method
effectiveofdetrending
size decreases
toysis as
choice is awindow
non-correlative
technique
That
is,
the
effective
detrending
window
size
decreases
wards
the
time
series’
boundaries,
resulting
in
x̂
=
x̂
=
0.to1
N
and its results are permutation invariant (Sect. 2.6), spuriwards
the
time
series’
boundaries,
resulting
in
x̂
=
x̂
= 0.
This
simple approachwhich
avoidsmay
the be
complication
of locally
1
N or
ous autocorrelations
introduced by
the sliding
This
simple
approach
avoids
the
complication
of
locally
globally
higher-order
polynomials
orproblem
performing
highwindow fitting
detrending
do not pose
a serious
here.
Weor
globally
higher-order
polynomials
or performing
highpass
filtering
irregular
sampling
dating
of
will
showfitting
in given
Sect.
4.2 that our
resultsand
are uncertain
robust
with
respect
measurements
to
remove
the
nonlinear
trend.
Since
RN
analpass
filtering
given
irregular
sampling
and
uncertain
dating
to a large range of reasonable choices of WD . Except of theof
ysis
as our method
of choice
a non-correlative
technique
measurements
to remove
the is
nonlinear
Since
RNdata.
analdetrending,
no further
preprocessing
wastrend.
applied
to
the
and
its
results
are
permutation
invariant
(Sec.
2.6),
ysis as our method
of resample
choice is the
a non-correlative
technique
Particularly,
we do not
time series to spurious
obtain
an
autocorrelations
which
introduced
thethe
sliding
winand itsspaced
resultsrecord
are
permutation
invariant
(Sec.
2.6),
spurious
evenly
inmay
the be
time
domain, by
since
necessary
dow
detrending
dowhich
not
pose
serious
problem
here.
We
will
autocorrelations
mayathe
be
introduced
the sliding
wininterpolation
would
corrupt
results
of thebyfurther
analysis
show
in
Sec.
4.2
that
our
results
are
robust
with
respect
dow
do not(see,
posee.g.
a serious
problem
here. Wetowill
Rehfeld
et al., 2011).
to
be detrending
performed below
ashow
largeinrange
choicesare
of robust
WD . Except
of the to
Sec.of4.2reasonable
that our results
with respect
detrending,
no
further
preprocessing
was
applied
to
the
a large
range of reasonable choices of WD . Exceptdata.
of the
2.3
Embedding
Particularly,
do not resample
the time
to obtain
detrending, we
no further
preprocessing
wasseries
applied
to the an
data.
evenly
spacedwe
record
in the
time domain,
since
the necessary
Particularly,
do not
resample
series
Univariate
time series
often
reflectthe
thetime
dynamics
oftoaobtain
higher-an
interpolation
would
corrupt
results of thesince
further
analysis
evenly spaced
record
in
thethe
time
thesome
necessary
dimensional
complex
system
as domain,
viewed through
obto be performed below (see, e.g., Rehfeld et al. (2011)).
interpolation
would corrupt
the results
of the
further
analysis
servation
function.
In typical
situations
it is
possible
to
to be performed below (see, e.g., Rehfeld et al. (2011)).
Nonlin. Processes Geophys., 18, 545–562, 2011
4 548
J.J.F.F.Donges
marine palaeoclimate
palaeoclimaterecords
recordsby
byRN
RNanalysis
analysis
Dongesetetal.:
al.:Identification
Identificationof
of dynamical
dynamical transitions in marine
reconstruct
the phase space trajectory using time-delay em2.3
Embedding
bedding, i.e. considering state vectors
the dynamics of a higherUnivariate
time series often reflect
(m,τ )
yi
= x̂i , x̂i+τ ,..., x̂i+(m−1)τ
(4)
dimensional
complex system as viewed through some obinstead of
the univariate
observations
x̂i themselves
(Packard
servation
function.
In typical
situations
it is possible
to reet al., 1980;
Takens,
1981).
Due to
the time-delay
finite lengthembedof the
construct
the phase
space
trajectory
using
available
time series, state
the index
i is now restricted to the range
ding,
i.e., considering
vectors
i = 1,...,N − (m − 1)τ . The embedding parameters embed to be appropriately de(m,τ
) dimension m and delay τ have
yiding =
x̂i , x̂i+τ ,..., x̂i+(m−1)τ
(4)
termined from the available data, e.g. using approaches such
et al., 1992)
and avas the of
false
(Kennel
instead
the nearest-neighbours
univariate observations
x̂i themselves
(Packard
and
Swinney,
1986)
metherage
mutual
information
(Fraser
et al., 1980; Takens, 1981). Due to the finite length of
the
ods,
respectively.
Although
there
are
good
reasons
for
apavailable time series, the index i is now restricted to the range
plying
embedding
techniques,
it
is
known
that
this
approach
i = 1,...,N − (m − 1)τ . The embedding parameters embedalsodimension
has conceptual
and
induce spurious
ding
m anddisadvantages
delay τ have to
bemay
appropriately
deterstructures
in
recurrence
plots
and
corresponding
misleading
mined from the available data, e.g., using approaches
such as
recurrence quantification
analysis
(RQA)
theresults
false of
nearest-neighbours
(Kennel et
al., 1992)
and (Thiel
averet
al.,
2006).
In
contrast,
many
important
dynamical
inage mutual information (Fraser and Swinney, 1986) methods,
variants can be estimated from non-embedded time series as
respectively. Although there are good reasons for applying
well, especially using recurrence plot-based methods (Thiel
embedding techniques, it is known that this approach also has
et al., 2004a). From here on we will use the simplified notaconceptual disadvantages and may induce spurious structures
tion y i for reconstructed state vectors and assign to them the
in recurrence
plots and corresponding misleading results of
ages ti , respectively.
recurrence
quantification analysis (RQA) (Thiel et al., 2006).
While the standard approaches for determining the optiIn contrast, many important dynamical invariants can be esmum embedding parameters typically provide feasible retimated
non-embedded
time series the
as well,
especially
sults infrom
the case
of many applications,
situation
is conusing
recurrence
plot-based
methods
(Thiel
et
al.,
2004a).
siderably more challenging for palaeoclimate records:
on
From
herehand,
on we
will useembedding
the simplified
notation
yi equally
for rethe one
traditional
methods
require
constructed
state vectors
to them of
thethe
ages
ti , respaced observations,
soand
thatassign
interpolation
available
spectively.
data might become necessary with all corresponding concepWhereas
the standard
approaches
forindetermining
optual
disadvantages.
On the
other hand,
the presencethe
of dattimum
embedding
parameters
typically
provide
feasible
reing uncertainties, even such interpolation is hardly possible
sults
the case
applications,
the situation
is conandin
would
lead of
to many
an enormous
enhancement
of uncertainty
siderably
more challenging
in the embedded
record. for palaeoclimate records: On
the one
hand,
traditional
embedding
methods
Given
these
methodological
difficulties
werequire
attemptequally
a comspaced
observations,
so that interpolation
promise:
(i) the embedding
dimension mof= the
3 isavailable
a tradedata
necessary
with
all series
corresponding
concepoff might
given become
the relatively
short
time
forbidding
larger
tual
disadvantages.
On the(Eckmann
other hand,etinal.,
the1992;
presence
of datembedding
dimensions
Kantz
and
ing
uncertainties,
interpolation
is hardly possible
Schreiber,
1997)even
and such
the underlying
high-dimensional
dynamics
estimated
by the false
nearest-neighbours
criteand
wouldaslead
to an enormous
enhancement
of uncertainty
al., 1992; Marwan et al., 2009). (ii) Us(Kennel etrecord.
inrion
the embedded
ing
a
Gaussian
kernel-based
of we
the attempt
autocorrelation
Given these methodologicalestimator
difficulties
a comfunction
adapted
to
irregularly
sampled
time
promise: (i) The embedding dimension m =series
3 is (Rehfeld
a tradeal., 2011),
we find that
the autocorrelation
of all fourlarger
time
offet given
the relatively
short
time series forbidding
series
has
decayed
markedly
after
10
kyr
(Fig.
3).
Hence,
embedding dimensions (Eckmann et al., 1992; Kantz and
we choose1997)
the delay
to cover
approximately
the same time
Schreiber,
and τthe
underlying
high-dimensional
dy∗ = 10 kyr for all considered records, i.e.
scale
τ
namics as estimated by the false nearest-neighbours criterion
τ =(Kennel
bτ ∗ /h1Tetic,al., 1992; Marwan et al., 2009). (ii) Us(5)
ing a Gaussian kernel-based estimator of the autocorrelation
where h1T i is the average sampling time (Table 1) and bxc
function adapted to irregularly sampled time series (Rehfeld
denotes the integer part of x. This yields τ1 = 2 for ODP
et al., 2011), we find that the autocorrelation of all four time
site 659, τ2 = 5 for site 721/722, and τ3 = 27 for site 967. A
series has decayed
markedly after 10 kyr (Fig. 3). Hence,
promising technique for consistent embedding of irregularly
we choose the delay τ to cover approximately the same time
sampled
time series is based on Legendre polynomials (Gibscale
τ ∗ = 10 kyr for all considered records, i.e.
son et al., 1992) and should be explored in future studies.
τ = bτ ∗ /h∆T ic,
Nonlin. Processes Geophys., 18, 545–562, 2011
(5)
Fig. 3. Linear autocorrelation functions C(τ ) for (A) the δ 18 O
Fig.
3. atLinear
autocorrelation
functions
C(τ ) for
(A)ODP
the sites
δ 18 O
record
ODP site
659 and the dust
flux records
from
record
at (C)
ODP
site 659and
and(D)
the967.
dustThe
fluxautocorrelation
records from functions
ODP sites
(B) 659,
721/722,
(B)
(C) 721/722,
(D) 967.
The autocorrelation
functions
were659,
estimated
using aand
Gaussian
kernel-based
estimator (Rehfeld
were
usingtoairregularly
Gaussian sampled
kernel-based
estimator
et al., estimated
2011) adapted
data (solid
line) (Rehfeld
and diet
al., 2011)
adapted
to irregularly
sampled data
and directly
from time
series
linearly interpolated
to a (solid
regularline)
sampling
h1T
i (dash-dotted
with sampling
line).to aFor
the Gaussian
rectly
from timetime
series
linearly
interpolated
regular
sampling
kernel-based
we iused
the recommended
optimum
bandwith
samplingestimator
time h∆T
(dash-dotted
line). For
the Gaussian
i/4 (Rehfeld
width h = h1T
et al.,
where h is
the standard
kernel-based
estimator
we used
the2011),
recommended
optimum
banddeviation
the i/4
Gaussian
kernel.
width
h =ofh∆T
(Rehfeld
et al., 2011), where h is the standard
deviation of the Gaussian kernel.
2.4 Windowed analysis
where h∆T i is the average sampling time (Tab. 1). This
For detecting
structural
changes
encoded by
yields
τ1 = 2 for
ODP site
659, in
τ2the
= 5dynamics
for site 721/722,
and
the
time
series,
we
slide
a
window
over
the
embedded
record
∗
τ3 = 27 for site 967 corresponding to τ = 10 kyr. A promis{y } and perform the subsequent analysis for the data coningi technique for consistent embedding of irregularly samtained in each window separately. However, the records unpled time series is based on Legendre polynomials (Gibson
der study are quite heterogeneous with respect to their basic
et al., 1992) and should be explored in future studies.
sampling properties (Table 1). The average sampling time
h1T iWindowed
differs widely
across the records. In order to assure
2.4
analysis
comparability of our results uncovered from the different
timedetecting
series, the
most natural
approach
to chooseencoded
windows
For
structural
changes
in theisdynamics
by
∗ in units of time. However, this approach
of atime
fixedseries,
size W
the
we slide
a window over the embedded record
has}two
disadvantages: The exact timing ti of the available
{y
i and perform the subsequent analysis for the data conobservations
not known
as is the However,
case for most
tained in eachiswindow
separately.
the geological
records unproxy
records,
and
due
to
the
non-uniform
sampling
rates,
der study are quite heterogeneous with respect to their
badifferent
windows
would
contain
different
amounts
of
data.
sic sampling properties (Tab. 1). The average sampling time
While
the latter
is not across
problematic
for statistical
teststoagainst
h∆T
i differs
widely
the records.
In order
assure
homogeneity of the distribution of values in different wincomparability of our results uncovered from the different
dows, a quantitative comparison of statistical characteristics
time series, the most natural approach is to choose windows
of the associated RNs (see Sect. 2.5) is not possible. Thereof a fixed size W ∗ in units of time. However, this approach
fore, in the following, we will proceed in a different way by
has two disadvantages: The exact timing ti of the available
prescribing both the window size W and step size 1W for
observations is not known as is the case for most geological
RN analysis measured in units of sampling points. In order
proxy records, and due to the non-uniform sampling rates,
to derive W and 1W from the desired quantities in units of
different∗ windows∗would contain different amounts of data.
time, W and 1W , we divide by the average sampling time,
While the latter is not problematic for statistical tests against
ic,distribution of values in different winW = bW ∗ /h1T
(6)
homogeneity
of the
∗
dows,
a
quantitative
comparison
of
statistical
characteristics
1W = b1W /h1T ic.
(7)
of the associated RNs (see Sec. 2.5)
is not possible. There∗
In turn,
thefollowing,
actual window
sizeproceed
W (ti ) is
by the
fore,
in the
we will
in determined
a different way
by
average
sampling
time
in
the
size-W
window
centred
around
prescribing both the window size W and step size ∆W for
RN analysis measured in units of sampling points. In order
www.nonlin-processes-geophys.net/18/545/2011/
J. F. Donges et al.: Identification of dynamical transitions in marine palaeoclimate records by RN analysis
J. F. Donges et al.: Identification of dynamical transitions in marine palaeoclimate records by RN analysis
to derive W and ∆W from the∗ desired quantities in units
t . For a particular
choice of W and the associated step size
ofi time,
W ∗ and ∆W ∗ , we divide by the average sampling
1W ∗ in units of time, the resulting values of W and 1W ,
time,
the mean window widths, and step sizes as well as the corre∗
sponding
deviations are given in Table 1.
W = bWstandard
/h∆T ic,
(6)
∗ approach for determining the window size deThe
simple
∆W = b∆W /h∆T ic,
(7)
scribed above guarantees that the windows cover approxwhere
the integer
partallofrecords
x. In turn,
actual
imatelybxc
thedenotes
same time
span for
and the
positions
window
sizetime
W ∗ (t
the average
sampling
within the
series.
While mostby
sampling
intervals
take
i ) is determined
time
in the
centred
around outliers,
ti . For which
a parvalues
closesize-W
to the window
mean, there
are distinct
ticular
choicecorrespond
of W ∗ andtothe
associated
step to
size
∆W ∗ in
most likely
missing
data due
incomplete
units
time, the
resulting
values of of
Wthe
andsediment
∆W , the
mean
coreof
recovery,
hiata
or disturbances
such
as
4). Nevertheless,
the standard
deviation of
turbidites
(Fig. and
window
widths,
step sizes as well
as the corresponding
windowdeviations
size σ (W ∗are
) is given
still small
in comparison
to the averstandard
in Tab.
1.
hW ∗ i for
age
(Table
1), which the
suggests
thatsize
statisThewindow
simple size
approach
determining
window
detical characteristics
computed
windows
still
scribed
above guarantees
thatfor
thedifferent
windows
cover can
approxbe quantitatively
a reasonable
way.and positions
imately
the same compared
time spanin for
all records
µ
Formally,
theseries.
data series
}
within
the
µ-th
window,
within
the time
While{ymost
sampling
intervals
take
i
N−W
c,
is
given
by
µ
=
1,2,...,b
values close to 1W
the mean, there are distinct outliers, which
mostµ likely correspond to missing data due to incomplete
{y i recovery,
} = {y (µ−1)1W
core
hiata+ior}, disturbances of the sediment such(8)
as
turbidites (Fig. 4A). Nevertheless, the standard deviation of
where from here on i = 1,...,W . We use the window’s midwindow size σ(∆W ∗ ) is still small in comparison to the avpoint’s timing
erage window size h∆W ∗ i (Tab. 1), which suggests that statistical
characteristics
computed for different windows can
t µ = t(µ−1)1W
(9)
+bW/2c
still be quantitatively compared in a reasonable way.
toFormally,
attach an age
the series
scalar network
measures
g µ calculated
the to
data
{yiµ } within
the µ-th
window,
the data
withinby
the µ-th window.
µfrom
= 1,2,...,
is given
µ
{y2.5
{y(µ−1)∆W network
+i },
i } =Recurrence
analysis
(8)
where from here on i = 1,...,W . We use the window’s midRecurrence
point’s
timingin phase space is a basic property of complex dynamical systems. Since the seminal work of Poincaré (1890),
tµit=ist(µ−1)∆W
(9)
known that
under rather general conditions, dynamical
+bW/2c
systems tend to return arbitrarily close to their previous states
to attach an age to the scalar network measures g µ calculated
in the long-term limit. In the last decades, the recurrence
from the data within the µ-th window.
property has attracted considerable interest, since it has been
shown
that essential
information
about the main dynamical
2.5
Recurrence
network
analysis
properties is contained in the temporal pattern of mutual recurrences of
state (Thiel
et aal.,
2004b;
Robinson
and Thiel,
Recurrence
inaphase
space is
basic
property
of complex
dy2009).
Particularly,
the
visual
representations
of
recurrence
namical systems. Since the seminal work of Poincaré (1890),
(Eckmann
et al., 1987;
al., 2007) have
found
itplots
is known
that under
ratherMarwan
general etconditions,
dynamical
wide
use,
which
are
most
commonly
expressed
by
a
binary
systems tend to return arbitrarily close to their previous states
matrix
inrecurrence
the long-term
limit. In the last decades, the recurrence
µ
µ
µ
property
attracted
interest, since it has been
Rij (ε) =has
2(ε
− ky −considerable
y j k),
(10)
shown
that essentiali information
about the main dynamical
properties
in theε temporal
pattern threshold
of mutualand
rewhere foristhecontained
µ-th window,
is the recurrence
currences
of a state
(Thiel et al.,
2004b; In
Robinson
and Thiel,
2(·) denotes
the Heaviside
function.
the following
we
2009).
the visual
of recurrence
use theParticularly,
supremum norm
k · k torepresentations
measure distances
in the reconstructed
phase
space
of the
considered
(see
plots
(Eckmann
et al.,
1987;
Marwan
et al., observable
2007) haveyfound
Fig. use,
5 for which
examples).
The appropriate
of thebyimportant
wide
are most
commonly choice
expressed
a binary
parameter matrix
ε is discussed below.
recurrence
It
turned
out that
recurrence
plots show distinct line strucµ
Rtures,
Θ(ε −length
kyiµ − distribution
yjµ k),
(10)
ij (ε) =
whose
can be used for defining
suitable
of complexity
terms of threshold
RQA, or and
for
where
for measures
the µ-th window,
ε is theinrecurrence
Θ(·) denotes the Heaviside function. In the following we
www.nonlin-processes-geophys.net/18/545/2011/
549
5
Fig. 4. (A) Probability distribution (PDF) p(1T ) of the sampling
intervals of the three dust flux records according to their established
Fig. 4. (A) Probability distribution (PDF) p(∆T ) of the sampling
age models (ODP sites 659: solid line, 721/722: dash-dotted, 967:
intervals of the three dust flux records
according to their established
dashed). The distribution for the δ 18 O record at ODP site 659 is viage models (ODP sites 659: solid line, 721/722: dash-dotted, 967:
sually almost indistinguishable from18that of the corresponding dust
dashed). The distribution for the δ O record at ODP site 659 is viflux record and therefore not shown. The PDFs were estimated ussually
almost indistinguishable
fromhthat
the)(N
corresponding
dust
ing
a Gaussian
kernel with bandwidth
= σof
(1T
− 1)−1/5 (Taflux
record
and
therefore
not
shown.
The
PDFs
were
estimated
ble 1) following Scott’s rule (Scott, 1982). (B, C, D) Temporal
using a Gaussian
kerneltimes
with for
bandwidth
= σ(∆T
)(N − 1)−1/5
variation
of the sampling
the three hdust
flux records.
(Tab. 1) following Scott’s rule (Scott, 1982). (B,C,D) Temporal
variation of the sampling times for the three dust flux records.
estimating dynamical invariants such as correlation dimension, 2nd-order Rényi entropy, or generalised mutual inforuse the(Marwan
supremum
norm
k · kIntothe
measure
in the remation
et al.,
2007).
contextdistances
of palaeoclimate
constructed
phase space
considered
observable
y (see
research,
recurrence
plots of
andtheRQA
have been
successfully
Fig. 5 for
The appropriate
applied
forexamples).
tracing dynamical
changes choice
(Trauthofetthe
al.,important
2003;
parameter
ε is2003)
discussed
below. records with different ageMarwan
et al.,
and aligning
It turned
that recurrence
plots show
line strucdepth
modelsout
(Marwan
et al., 2002).
RQAdistinct
is a correlative
tures, whose
distribution
be used
forondefining
method
of time length
series analysis,
as it can
explicitly
relies
temsuitable
measures
complexity
in terms
of RQA,
poral
structures
in theofform
of diagonal
and vertical
lines.or for
estimating
as correlation
dimenRecently, dynamical
it has beeninvariants
suggestedsuch
to approach
recurrence
sion, 2nd-order
Rényi entropy,
generalised
informatrices
from a complex
networkor
perspective
by mutual
identifying
mation
(Marwan
et
al.,
2007).
In
the
context
of
palaeoclimate
µ
µ
Aresearch,
(11)
ij (ε) − δij plots and RQA have been successfully
ij (ε) = Rrecurrence
for tracing dynamical changes (Trauth et al., 2003;
(δapplied
ij denoting Kronecker’s delta) with the adjacency matrix
Marwan
et al.,
2003) and
aligningtorecords
with different
ageof
a complex
network
associated
the underlying
time sedepth
models et(Marwan
et al.,
2002).
RQA
is a 1correlative
ries
(Marwan
al., 2009;
Donner
et al.,
2010a)
. This
method of time series analysis, as it explicitly relies on tem1 Note that similar approaches can also be found in other
poral
structures in the form of diagonal and vertical lines.
Recently, it has been suggested to approach recurrence
Nonlin. Processes Geophys., 18, 545–562, 2011
550
J. F. Donges et al.: Identification of dynamical transitions in marine palaeoclimate records by RN analysis
6
J. F. Donges et al.: Identification of dynamical transitions in marine palaeoclimate records by RN analysis
Table 1. Basic properties of the analysed palaeoclimate time series. N is the number of samples contained in the time series, h1T i
the mean sampling interval, and σ (1T ) the standard deviation of sampling intervals. For a desired window size W ∗ = 410 kyr and step
Table
1. ∗Basic
properties
of theinanalysed
time W
series.
N isgive
the the
number
of samples
contained
thesize
time
h∆T
size 1W
= 41
Sect. 4.1palaeoclimate
for RN analysis),
and 1W
corresponding
window
and in
step
(inseries,
numbers
ofi
kyr
(as chosen
∗
∗ the
∗ ) the
the
mean sampling
σ(∆T
) theeffective
standardwindow
deviation
sampling
intervals.
desired
window
size W
= 410deviations
kyr and step
observations),
W ∗ interval,
and 1Wand
average
andofstep
size, and
σ (W ∗ ) For
and aσ (1W
associated
standard
(in
∗
size
= 41 kyr (as chosen in Sec. 4.1 for RN analysis), W and ∆W give the corresponding window and step size (in numbers of
units∆W
of time).
observations), hW ∗ i and h∆W ∗ i the average effective window and step size, and σ(W ∗ ) and σ(∆W ∗ ) the associated standard deviations
∗
(in units of time).
h1T i σ (1T )
N
W
1W
W
σ (W ∗ )
1W ∗
σ (1W ∗ )
A
(kyr)
(kyr)
σ(∆T ) (kyr)
4.282.88 2.88
4.102.69 2.69
1.811.52 1.52
0.360.31 0.31
B
1400
Time (kyr)
Time (kyr)
1300
1200
1100
1000
1000
(kyr)
W
∆W hW ∗ i (kyr)
400.37
95 95 9 9
400.37
100
10
408.16
100
10
408.16
401.97
226226 22 22
401.97
1139
113
409.10
1139
113
409.10
C
2350
3350
2250
3300
2200
2150
1400
3250
3200
3150
3100
2050
1200 1300
Time (kyr)
σ(∆W ∗ ) (kyr)
4.36
3.22
6.09
7.51
3400
2300
2100
1100
(kyr)
(kyr)
(kyr)
σ(W ∗ ) (kyr) h∆W ∗ i (kyr)
46.5846.5838.37
4.36
38.37
33.0933.0941.25
3.22
41.25
62.6662.6639.29
6.09
39.29
78.0478.0440.67
7.51
40.67
Time (kyr)
N 18 h∆T i (kyr)
ODP 659 δ 18 O ODP 659
1170δ O 4.281170
ODP 659 dust ODP 659
1221dust
4.101221
ODP
721/722
dust
ODP 721/722 dust 2757
1.812757
ODP
967
dust
ODP 967 dust
8417
0.368417
3050
2100
2200
Time (kyr)
2300
3100
3200 3300
Time (kyr)
3400
Fig. 5. Recurrence plots (equivalently adjacency matrices of the RNs shown in Fig. 6) obtained from the dust flux record at ODP site 659,
∗ 41 kyr, and embedding parameters
centred
around (A)plots
1.2, (B)
2.2, and (C)
3.2 Myr BP,
usingof
window
sizeshown
W ∗ =in410
size 1W
Fig. 5. Recurrence
(equivalently
adjacency
matrices
the RNs
Fig.kyr,
6) step
obtained
from =
the dust flux record at ODP site 659,
∗
∗
m = 3, around
τ = 10(A)
kyr.1.2,
ε was
a data-adaptive
way towindow
yield a size
fixedW
edge
0.05∆W
for ∗each
centred
(B) chosen
2.2, andin(C)
3.2 Myr BP, using
=density
410 kyr,ρ(ε)
step=size
= 41window.
kyr, and embedding parameters
m = 3, τ ∗ = 10 kyr. ε was chosen in a data-adaptive way to yield a fixed edge density ρ(ε) = 0.05 for each window.
analogy implies that each sampled state vector is assigned a
vertex infrom
the aRN,
wherenetwork
two vertices
are linked
if the cormatrices
complex
perspective
by identifying
responding state vectors are recurrent, i.e. mutually close,
µ
Ainµij (ε)
= Rspace
δij 6). According to the conventions(11)
phase
(Fig.
of
ij (ε) −
Sect. 2.3, each vertex i in the µ-th window has an age
(δµij denoting Kronecker’s delta) with the adjacency matrix
t = t(µ−1)1W +i attached to it. To simplify the notation
ofi a complex
network associated to the underlying time sewhen defining network measures, we will drop the1window
ries (Marwan et al., 2009; Donner et al., 2010a) . This
index µ in the following.
analogy implies that each sampled state vector is assigned
The edge density
a vertex in the RN, where two vertices are linked if the corresponding state
X are recurrent, i.e., mutually close,
1 vectors
= space (Fig. 6).
Aij (ε)
inρ(ε)
phase
According to the conventions(12)
of
W (W − 1) i,j
Sec. 2.3, each vertex i in the µ-th window has an age tµi =
t(µ−1)∆W +i attached to it. To simplify the notation when
measures which fraction of the maximum theoretically posdefining network measures, we will drop the window index
sible number W (W − 1)/2 of undirected edges is present in
µ in the following.
the RN, where the number of vertices W is determined by
The edge density
the chosen recurrence window size. ρ(ε) is equivalent to the
1 in X
recurrence rate
traditional RQA.
Aij (ε)
(12)
ρ(ε) =
W (W − 1)
i,j applications of data analysis, such as dengeoscientifically relevant
drograms
in agglomerative cluster analysis, or nonlinear decompo1
Note that similar approaches can also be found in other geosition of multivariate data using isometric feature mapping (Gámez
scientifically relevant applications of data analysis, such as dendroet al., 2004).
grams in agglomerative cluster analysis, or nonlinear decomposition of multivariate data using isometric feature mapping (Gámez
etNonlin.
al., 2004).
Processes Geophys., 18, 545–562, 2011
The properties of the resulting RNs (parameterised by
the
single which
parameter
ε) have
been
shown totheoretically
trace structures
measures
fraction
of the
maximum
posin
phase
space
corresponding
to
dynamically
ob-in
sible number W (W − 1)/2 of undirected edgesinvariant
is present
jects
(Donner
et
al.,
2010a,
2011b)
as
well
as
changes
in
the RN, where the number of vertices W is determinedthe
by
dynamical
arbitrary
timeρ(ε)
series
(Marwan ettoal.,
the
chosen behaviour
recurrenceofwindow
size.
is equivalent
the
2009; Donner
2011a). RQA.
For detecting
bifurcations
in
recurrence
rateetinal.,
traditional
The properties
of the retime
series,
global-scale
network
characteristics
of
complex
sulting RNs (parameterised by the single parameter ε) have
Bocnetwork
theory
are ofstructures
main interest
(Newman,
been
shown
to trace
in phase
space 2003;
correspondcaletti
et
al.,
2006;
Costa
et
al.,
2007).
Here
we
will
focus
ing to dynamically invariant objects (Donner et al., 2010a,onin
the following
measures:
press)
as well four
as changes
in the dynamical behaviour of arbi-
trary time series (Marwan et al., 2009; Donner et al., 2011).
i. Transitivity T : the transitivity
For detecting bifurcations in time series, global-scale netP
work characteristics
of complex network theory are of main
Aj k Aki
i,j,k Aij 2003;
interest
(Newman,
Boccaletti et al., 2006; da Costa
T = P
(13)
A
A
ki
kj
et al., 2007). Here
we
will
focus on the following four meai,j,k
sures:
of an unweighted and undirected network characterises
(i) the
Transitivity
T : The transitivity
overall probability
that two randomly chosen neighP
bours of an also
randomly
chosen vertex are connected
i,j,k Aij Ajk Aki
T = P 2003). In case of RNs, T serves as a mea(13)
(Newman,
Aki Akj
i,j,k
sure for the
regularity
of the dynamics as encoded in
the
RN’s
mesoscopic
(Donner
etcharacterises
al., 2010a).
of an
unweighted
and structure
undirected
network
the overall probability that two randomly chosen neighbourswww.nonlin-processes-geophys.net/18/545/2011/
of an also randomly chosen vertex are connected
J. F. Donges
et al.:et Identification
of of
dynamical
marinepalaeoclimate
palaeoclimate
records
by RN
analysis
J. F. Donges
al.: Identification
dynamicaltransitions
transitions in
in marine
records
by RN
analysis
B
A
7
551
C
Fig. 6. Recurrence networks obtained from the dust flux record at ODP site 659, centred around (A) 1.2, (B) 2.2, and (C) 3.2 Myr BP and
Fig. 6. Recurrence networks obtained from the dust flux record at ODP site 659, centred
around (A) 1.2, (B) 2.2, and (C) 3.2 Myr BP and
µ
corresponding to the recurrence plots of Fig. 5. Vertex color indicates the age
ti associated to single state vectors µ (from blue [= old]
corresponding to the recurrence plots of Fig. 5. Vertex color indicates the age tµi associated
to single state vectors µ (from blue [=old] to red
to red [=young]).
[= young]).TheThe
two-dimensional
visualisation
been with
obtained
with the
software
package
using placement
a force-directed
two-dimensional
graph graph
visualisation
has beenhas
obtained
the software
package
GUESS
using aGUESS
force-directed
placement
algorithm
(http://graphexploration.cond.org).
It
is
important
to
note
that
in
this
visualisation,
node
positions
are
determined
algorithm (http://graphexploration.cond.org). It is important to note that in this visualisation, node positions are determined by the
aforementioned
algorithm and
do notand
correspond
to a projection
of the of
node
coordinates
in the
reconstructed
three-dimensional
phase space.
by the aforementioned
algorithm
do not correspond
to a projection
the node
coordinates
in the
reconstructed
three-dimensional phase
space.
Specifically, regular dynamics (e.g. on a periodic orbit)
(Newman,
2003). In case
of RNs,values
T serves
is typically
characterised
by higher
of as
thea meatransure for the regularity of the dynamics as encoded in
sitivity T than chaotic dynamics. T can furthermore
the RN’s mesoscopic structure (Donner et al., 2010a).
be interpreted
as aregular
globaldynamics
measure(e.g.,
of theonunderlying
atSpecifically,
a periodic ortractive
set’s
effective
dimensionality
d
(Donner
et
al.,
bit) is typically characterised by higher values of the
d
2011b),
i.e. the theoretical
resultdynamics.
is T = (3/4)
ustransitivity
T than chaotic
T canwhen
furthering themore
supremum
norm as
in aphase
Forofcontinuousbe interpreted
globalspace.
measure
the underlying attractive
effective
dimensionality
d (Donner
time systems,
this set’s
implies
T = 3/4
for a periodic
orbit
d
in for
press),
i.e., the
theoreticalHowever,
result is T for
= (3/4)
and T et<al.,
3/4
chaotic
dynamics.
small
when
using
the
supremum
norm
in
phase
space.
For
numbers of vertices (state vectors) W as used in this
systems,
implies
T =from
3/4 these
for a
work continuous-time
the estimated values
of Tthiswill
deviate
periodic orbit and T < 3/4 for chaotic dynamics. Howtheoretical expectations (Donner et al., 2011b).
ever, for small numbers of vertices (state vectors) W as
in thiswith
workshort
the estimated
values
of T will deviWhenused
dealing
time series
(segments)
as it
from
expectations
et al.,
is theate
case
in these
this theoretical
work, transitivity
is a(Donner
more robust
in press).
measure
than the related global clustering coefficient
C (Watts and Strogatz, 1998; Newman, 2003), since
the latter gives relatively more weight to sparsely sampled regions
in phase
low degree
When dealing
withspace
short (vertices
time serieswith
(segments)
as it
k) (Donner
al.,in2010a,
2011a).
is the et
case
this work,
transitivity is a more robust
measure than the related global clustering coefficient
ii. Average
path length
L: the average
path length
C (Watts
and Strogatz,
1998; Newman,
2003), since
the latter gives relatively more weight to sparsely samL = lpled
(14)
ij i,j regions in phase space (vertices with low degree
k) (Donner et al., 2010a, 2011).
is defined as the mean value of the shortest path lengths
lij between all mutually reachable pairs of vertices (i,j )
(measured in terms of geodesic graph distance, i.e. the
minimum number of edges that have to be traversed on
any path connecting the vertices i and j ) (Watts and
www.nonlin-processes-geophys.net/18/545/2011/
Strogatz, 1998; Newman, 2003). A pair of vertices (i,j )
(ii) Average
length L:
The average
path length
is calledpath
mutually
reachable
if there
exists at least one
path connecting i and j . Since for comparable values of
ε, the average distances along different types of orbits
L = hlij ii,j
(14)
typically differ significantly, changes in L can be used
as sensitive indicators of dynamical transitions (Maret al.,
2009;
Donner
etthe
al.,shortest
2010a).path lengths
iswan
defined
as the
mean
value of
lij between all mutually reachable pairs of vertices
iii. (i,j)
Assortativity
complex
networkgraph
is called
assortative
(measuredR:
in aterms
of geodesic
distance,
if
vertices
tend
to
connect
preferentially
to
vertices
i.e., the minimum number of edges that have to be traP with
a similar
number
connections
(degreei kand
i = j) j Aij ).
versed
on any
path of
connecting
the vertices
On theand
other
hand, 1998;
it is called
disassortative
if vertices
of
(Watts
Strogatz,
Newman,
2003). A pair
of
vertices
(i,j) isprefer
called to
mutually
high degree
link toreachable
vertices if
ofthere
low exists
degree, and
atvice
leastversa
one path
connecting
i andThis
j. Since
for compa(Newman,
2002).
assortativity
property
rable
values
of
ε,
the
average
distances
along
different
can be quantified by the Pearson correlation
coefficient
types of orbits typically differ significantly, changes in
D
E2
L can be used
as sensitive indicators
of dynamical
tran1P
1
k
k
A
−
(k
+
k
)
i
j
ij
i
j
sitions (Marwan
et
al.,
2009;
Donner
et
al.,
2010a).
j
>i
L
2
i,j
R=
E2
1 2
2 )A − 1 (k + k )
(k
+
k
ij
i
j
j >i
j
L
2 is calledi,jassor(iii) Assortativity
R:2 Ai complex
network
D
1P
(15)
tative if vertices tend to connect preferentially to verbetween
the
,kjconnections
of the vertices
onkboth
tices
of a similar
numberkiof
(degree
i = ends
Pdegrees
P
). On
other
called
disassortative if
ofj A
allijL
= thej >i
Aijhand,
edgesit is
(i,j
), where
vertices of high degree prefer to link to vertices of low
X1
degree,
2002). This assorta1 and vice versa1 (Newman,
2
(ki + kj )
=
i,j
L j >i 2
(ki + kj )Aij
(16)
is the mean of the average edge end-point degree (ki +
kj )/2 (Costa et al., 2007). In the RN context, R can
be considered as a measure for the local continuity of
Nonlin. Processes Geophys., 18, 545–562, 2011
552
J. F. Donges et al.: Identification of dynamical transitions in marine palaeoclimate records by RN analysis
the phase space density of state vectors (Donner et al.,
2010a).
be viewed as a test against stationarity of the higher-order geometrical properties of the time series that are quantified by
qualitatively different RN measures.
iv. Network diameter D: the network diameter
2.7
D = max lij
i,j
(17)
is the maximum geodesic (shortest-path) distance between all mutually reachable pairs of vertices in the network (Newman, 2003). From this definition, there are
obvious relationships with the average path length L,
which are expected to lead to strong correlations between both measures (Donner et al., 2010a).
In order to apply RNs in a sliding window analysis, a reference framework is necessary. Here, we consider a dataadaptive choice of ε that guarantees for a fixed edge density
ρ of 5 %, which has been found a reasonable choice in previous studies (Donner et al., 2010b). One should note, however, that even with this choice the characteristics of RNs
can only be compared in a meaningful way if the network
size W is kept fixed (see Sect. 2.1). Among the considered
complex network measures, T and R are mainly affected by
finite-sample problems otherwise, whereas L and D explicitly depend on ε and W (Donner et al., 2010a).
2.6
Significance test
We perform a relatively simple statistical test of whether the
network characteristics in a certain time interval differ significantly from the general network characteristics expected
given the phase space distribution of state vectors yi from
the whole detrended and embedded record and a certain recurrence window size W . The corresponding null hypothesis
is that the network measures observed for a certain window
are consistent with being calculated from a random draw of
W state vectors from the prescribed phase space distribution
induced by the complete detrended time series. We can justly
assume a thus randomised embedded time series without
losing essential information, because all network measures
g(·) considered here are permutation-invariant when considering a fixed subset of state vectors y 1 ,...,y W . More formally, g(y 1 ,...,y W ) = g(y π(1) ,...,y π(W ) ) for arbitrary permutations π. A similar test for RQA measures requires a
more advanced method (Schinkel et al., 2009). In order to
create an appropriate null-model, we use the following approach: (i) draw randomly W state vectors from the embedded time series (corresponding to the window size chosen for
the original data), (ii) construct a RN from this set of state
vectors, and (iii) calculate the network measures of interest.
Repeating this procedure sufficiently many times, we obtain
a test distribution for each of the network measures and estimate its 0.05 and 0.95 quantiles that can be interpreted as
90 % confidence bounds. The proposed significance test can
Nonlin. Processes Geophys., 18, 545–562, 2011
Implementation
We implemented the above described methods using the
programming language Python (van Rossum and Drake,
2006), the packages NumPy (Oliphant, 2006) and SciPy
(Jones et al., 2011) as well as embedded C++ code. Complex network measures have been calculated employing the
Python package igraph (Csárdi and Nepusz, 2006).
3
Dynamical transitions in model systems
To validate the proposed methodology for detecting transitions in time series based on RNs, we apply it to the logistic
map and the Lorenz system with drifting bifurcation parameter as paradigmatic examples of discrete and continuous-time
dynamical systems, respectively. While step-like changes of
bifurcation parameters have already been studied for discrete
(Marwan et al., 2009) and continuous-time dynamical systems (Zou et al., 2010; Donner et al., 2011a), here we are
particularly interested in the effect of transients, which are
expected to be present in real-world systems and, hence, data
extracted from them. We will check whether the global network quantifiers described above are able to detect transitions in the system’s dynamics induced by bifurcations due
to a slowly changing control parameter. For this purpose
we are specifically looking for time intervals (or equivalently
values of the bifurcation parameter) where one or more of
the considered network quantifiers undergo sudden changes.
This requires taking into account the measures’ interpretation in terms of dynamical systems theory (Sect. 2.5). Furthermore, we will study how their performance and the level
of resolved detail depend on the window size W . This analysis particularly shows that the window sizes W chosen for
the RN analysis of terrigenous dust flux records (Table 1) are
indeed appropriate for detecting bifurcations.
3.1
Logistic map
We iterate the logistic map
xi+1 = ri xi (1 − xi )
ri+1 = ri + 1r
(18)
while varying the bifurcation parameter linearly from r1 =
3.8 to rM = 3.9 in M = 10 000 equidistant steps setting
1r = 1 × 10−5 (Fig. 7), similar to Trulla et al. (1996). We
analyse the resulting time series {xi } without embedding or
detrending. The transition from chaotic
√ to 3-periodic dynamics after an interior crisis at r = 1 + 8 ≈ 3.8284 (Wackerbauer et al., 1994) is clearly displayed by all four measures.
As expected from theoretical considerations for discrete-time
www.nonlin-processes-geophys.net/18/545/2011/
J. F. Donges et al.: Identification of dynamical transitions in marine palaeoclimate records by RN analysis
10
J. F. Donges et al.: Identification of dynamical transitions in marine palaeoclimate records by RN analysis
Transitivity
Average path length
Assortativity
Diameter
553
Fig. 7. (A) Transitivity
T Transitivity
, (B) average
length
L, (C)
and(D)
(D)
diameter
for varying
recurrence
Fig. 7. (A)
T , (B)path
average
path length
L, assortativity
(C) assortativityR,
R, and
diameter
D forD
varying
recurrence
window sizewindow
W for thesize W for
logistic
map
(Eq.drifting
(18)) withbifurcation
drifting bifurcation
parameter
r (see
text)
andinitial
initial condition
x1 x
=10.7.
W was
linearly linearly
in the interval
18)
with
parameter
r (see
text)
and
condition
= 0.7.
W varied
was varied
in the interval
the logistic map (Eq.
[100,600], the
recurrence
window
step size
wastofixed
∆W
10 time
steps.No
No embedding
embedding was
usedused
and the
set to ε =set
0.05σ
[100,600], the recurrence
window
step
size was
fixed
1Wto =
10=time
steps.
was
andthreshold
the threshold
to ε = 0.05σ
(Marwan et al., 2009), where σ denotes the standard deviation of the time series segment within the recurrence window. Vertical dashed lines
(Marwan et al., 2009),
where σ denotes the standard deviation of the time series segment within the recurrence window. Vertical dashed
indicate the critial values of r discussed in the main text.
lines indicate the critial values of r discussed in the main text.
danger
of confusing
statisticaletfluctuations
with
proper dysystems (Marwan
et al.,
2009; Donner
al., 2010a,
2011b),
namical
changes
substantially.
Time
series
from
T and R abruptly increase to their maximum value of 1geological
folarchives are typically characterised by a variety of different
lowing this transition,
whereas
at
the
same
time
L
and
types of nonstationarities, including (i) changes in theDlongsharply decrease
theiror minimum
value
of proxies,
1. Among
all
termtomean
variance of the
recorded
(ii) variations
in the Tamplitudes
almost
periodic
variability
the four measures,
and R of
most
clearly
detect
the components
termi(e.g., such attributed
to Milankovich-type
variations
caused
nation of the period-doubling
cascade
following the
period-3
by periodic changes in the Earth’s orbit), or (iii) even mulbehaviour at the
accumulation
point
r ≈ 3.849,
while
T ,and
L intimodal
behaviour (e.g.,
transitions
between
glacial
and D highlightterglacial
the merger
of the
three
periods).
All subsequently
these three typesformed
of nonstationarities
contained
our data
2 and Trauth
et al. (2009)).
chaotic bands atarethe
interiorincrisis
at r(Fig.
≈ 3.857
(Wackerbauer
While
these
different
phenomena
can
be
analysed
usinginmore
et al., 1994). The latter transition is only weakly visible
specific methodological approaches, we propose RN analyR. Additionally,
much fine-structure is resolved by the netsis as a general exploratory tool for detecting time intervals
work measures,containing
e.g. a narrow
window
. 3.89 bechanges period-4
in the dominating
typeat
of rdynamical
haviour.
In the following,
we will illustrate
the robustness
that is most clearly
indicated
by an increased
transitivity
T
across all W . Generally, the transitions appear more and
more blurred as W increases, which is due to the growing
number of samples from both periodic and chaotic dynamical
regimes contained in the recurrence windows when sliding
over the bifurcation point. In consequence, some of the narrow periodic windows appearing for r < 3.83 and r > 3.86
are only visible for small recurrence window sizes W . As a
rule of thumb, we can expect a periodic/chaotic window of
width wr embedded within a chaotic/periodic background to
be detectable if wr & W 1r.
www.nonlin-processes-geophys.net/18/545/2011/
of
this approach
for feature
the four is
marine
in a clear
Another
notable
that records
both Lintroduced
and D show
Sec.
2.1
and
briefly
discuss
the
possible
climatological
backtendency to increase with growing W in the chaotic paramground of the observed dynamical changes.
eter ranges (Fig. 7b and d). This is theoretically expected,
since
measures are
extensive,
i.e. they depend explic4.1 both
Time-dependence
of network
properties
itly and nonlinearly on the number of vertices W in the RN
consider the
four space
marine distribution
palaeoclimate records
forWe
a general
phase
of stateembedvectors as inded in a three-dimensional reconstructed phase space with a
duced by chaotic dynamics
∗ (Donner et al., 2010a). In contime delay of approximately τ = 10 kyr, resulting in the emtrast,
L and
D do not
changeinwith
theanperiodic
bedding
parameters
described
Sec. W
2.3.inFor
initial in-windows,
most
notably
in the
large period-3
window
the kyr
logistic map
spection,
we use
recurrence
windows of
size W ∗of
= 410
with7b
a mutual
offset
of can
subsequent
windows
∆W ∗ = 41 kyr.
(Fig.
and d).
We
explain
this behaviour
by recalling
Note that the latter two parameter choices correspond to
that
for discrete-time systems in a p-periodic regime, the RN
those used in previous work on the ODP site 659 dust flux
reduces
to a setetof
fully
connected
components
record (Marwan
al.,p
2009;
Donner
et al., 2011).
The selec- (Donner
et tion
al., of
2010a).
Following
the
definitions
in Sect.
2.5, this in
both parameters
results
from
a compromise
between
turn leads to L = D = 1 in any periodic regime and independent of W .
3.2
Lorenz system
To illustrate the performance of windowed RN analysis for
detecting transitions in continuous-time dynamical systems,
we consider the Lorenz system with a time-dependent bifurcation parameter r = r(t),
Nonlin. Processes Geophys., 18, 545–562, 2011
554
J. F. Donges et al.: Identification of dynamical transitions in marine palaeoclimate records by RN
analysis
11
J. F. Donges et al.: Identification of dynamical transitions in marine palaeoclimate records by RN analysis
Transitivity
Average path length
Assortativity
Diameter
Fig. 8. (A)
T , (B)
average
pathL,
length
(C) assortativity
and(D)
(D) diameter
for for
varying
recurrence
window window
size W forsize W for
Fig. 8. (A) Transitivity
T ,Transitivity
(B) average
path
length
(C)L,assortativity
R,R,and
diameterD D
varying
recurrence
the (Eq.
Lorenz
system
(19)) with
drifting bifurcation
parameter
initial condition
condition (x0(x
,y0,y
,z ) = (10,10,10). Because we are
the Lorenz system
19)
with(Eq.
drifting
bifurcation
parameter
r (seer (see
text)text)
andandinitial
0 00 ,z0 ) = (10,10,10). Because we are
interested in the performance of our method for scalar time series, we chose the x-component of the trajectory sampled with sampling time
interested in the performance
of our method for scalar time series, we chose the x-component of the trajectory sampled with sampling time
∆t = 0.05 and embed it with embedding dimension m = 3 and delay τ = 15. W was varied linearly in the interval [100,600], the recurrence
1t = 0.05 and embed
it
with
embedding
dimension
m = 3We
and
delay
τ = 15. W
was varied
linearly
in the
interval
[100,600],
window step size was fixed to ∆W
= 10 samples.
varied
the recurrence
threshold
ε to yield
a fixed edge
density
ρ = 0.05
(Donner etthe
al.,recurrence
window step size2010b).
was fixed
to dashed
1W =lines
10 indicate
samples.
varied
threshold
Vertical
the We
critial
valuesthe
of r recurrence
discussed in the
main text.ε to yield a fixed edge density ρ = 0.05 (Donner et al.,
2010b). Vertical dashed lines indicate the critial values of r discussed in the main text.
high temporal resolution of the finally produced RN mea-
which has not been considered in the aforementioned stud-
sures (small W ∗ , ∆W ∗ ) and larger statistical
ies. As we will show in the following, the main features reconfidence in
d
8
∗
detect
at r ≈ changed
163.5, rwhen
≈ 164.5 and
results
(large −
W z)
). −The
choice
critical others
covered
by ourweaker
analysistransitions
are not qualitatively
(x,y,z) = the
10(y
− x),x(r
y,xy
− ofz W. ∗ is more(19)
dt
3 directly influences the r ≈
than that of ∆W ∗ , because the former
applying
detrending.
However,
thisbe
step
appearswhen
relevant
166. Note
that one
has to
careful
comparing
number of vertices W in the RNs via Eq. (6). Since a formal these
in other
kinds
of
statistical analyses,
e.g., fordistinct
estimating
results
to
bifurcation
studies
where
realisations
While the system
is evolving,
r increases
linearly
from
r0 =
criterion
for determining
an optimal
choice of
W ∗ and
∆W ∗
spectrograms or time-dependent coefficients of autoregresthe processes,
Lorenz system
with fixed parameter r (not varying in
so far,
we study
the500,
robustness
since the data show considerable long-term
160 at time t0 =is0nottoavailable
rf = 170
at time
tf =
i.e. of our results of sive
∗ time) are studied (e.g. Donner et al., 2011a), since transients
with respect to variations in the more critical parameter W
trends in both mean and variance (Fig. 2).
rf −inrSec.
0 4.2.
influence
thethe
results
and(Fig.
cannot
excluded
by construcRegarding
transitivity
9), webefind
a synchronous
r(t) = r0 +
(t − t0 ).
(20)
behaviour
two geographically
distinct
ODP
when of
r the
is continuously
varied
inrecords
time. at However,
our
tf − tWe
0 additionally apply local detrending by removing the tion
∗
sites 659
721/722 during
Pliocene
Myret
BPal. (1996)
long-term average taken over windows of WD
= 500 kyr, results
areand
consistent
withthethe
work(∼5.3-2.6
of Trulla
(before present)) and Early Pleistocene (2.6-1.0 Myr BP)2 ,
For consistencywhere
with the analysis of scalar palaeoclimate
who observed that in transient scenarios bifurcations may
time series to be performed below, we use an embedding of
2
appear
for“Early
larger
bifurcation
than
in their nonHere
Pleistocene”
does notparameters
refer to any of the
archetypthe x-component
time series
for RN analysis without prior(21)
∗
WD = bWD
/h∆T ic,
ical stagesequivalents.
(Upper, Middle and
Lower
Pleistocene). of
Its timing
2.6transient
The
dependence
the results
on the
detrending (see caption of Fig. 8 for details). The RN mearecurrence window size W is more pronounced than that desures indicate two major transitions towards increasingly irscribed above for the logistic map. This is likely due to the
regular dynamics at r ≈ 161 and r ≈ 166.5 (Fig. 8). The forfact that transients play a larger role in continuous-time sysmer possibly reflects an initial transient due to the chosen
tems like the Lorenz model than in discrete-time systems.
initial condition. The latter agrees well with the major shift
from periodic (large T , large L and D for continuous-time
systems; Donner et al., 2010a, 2011b; Zou et al., 2010) to
4 Dynamical transitions in palaeoclimate records
chaotic (small T , small L and D) behaviour which is present
in the Lorenz system’s non-transient bifurcation scenario at
Our studies in the previous section demonstrated that RN
r ≈ 166 (Barrio and Serrano, 2007; Donner et al., 2011a). On
analysis can be meaningfully applied for detecting dynama shorter time scale, the path-based measures L and D among
ical transitions in non-stationary time series from different
Nonlin. Processes Geophys., 18, 545–562, 2011
www.nonlin-processes-geophys.net/18/545/2011/
12palaeoclimate records
J. F. by
Donges
et al.: Identification of555
dynamical tran
J. F. Donges et al.: Identification of dynamical transitions in marine
RN analysis
model systems by applying this kind of analysis to running
windows. This is a necessary, but not sufficient condition for
ensuring the feasibility of RN analysis for detecting regime
shifts in palaeoclimate records as well. However, the application of our simple significance test (Sect. 2.6) diminishes
the danger of confusing statistical fluctuations with proper
dynamical changes substantially. Time series from geological archives are typically characterised by a variety of different types of nonstationarities, including (i) changes in the
long-term mean or variance of the recorded proxies, (ii) variations in the amplitudes of almost periodic variability components (e.g. such attributed to Milankovich-type variations
caused by periodic changes in the Earth’s orbit), or (iii) even
multimodal behaviour (e.g. transitions between glacial and
interglacial periods). All these three types of nonstationarities are contained in our data (Fig. 2 and Trauth et al., 2009).
While these different phenomena can be analysed using more
specific methodological approaches, we propose RN analysis as a general exploratory tool for detecting time intervals
containing changes in the dominating type of dynamical behaviour. In the following, we will illustrate the robustness
of this approach for the four marine records introduced in
Sect. 2.1 and briefly discuss the possible climatological background of the observed dynamical changes.
4.1
Time-dependence of network properties
We consider the four marine palaeoclimate records embedded in a three-dimensional reconstructed phase space with
a time delay of approximately τ ∗ = 10 kyr, resulting in the
embedding parameters described in Sect. 2.3. For an initial
inspection, we use recurrence windows of size W ∗ = 410 kyr
with a mutual offset of subsequent windows of 1W ∗ =
41 kyr. Note that the latter two parameter choices correspond to those used in previous work on the ODP site 659
dust flux record (Marwan et al., 2009; Donner et al., 2011a).
The selection of both parameters results from a compromise
between high temporal resolution of the finally produced RN
measures (small W ∗ , 1W ∗ ) and larger statistical confidence
in the results (large W ∗ ). The choice of W ∗ is more critical
than that of 1W ∗ , because the former directly influences the
number of vertices W in the RNs via Eq. (6). Since a formal
criterion for determining an optimal choice of W ∗ and 1W ∗
is not available so far, we study the robustness of our results
with respect to variations in the more critical parameter W ∗
in Sect. 4.2.
We additionally apply local detrending by removing the
long-term average taken over windows of WD∗ = 500 kyr,
where
WD = bWD∗ /h1T ic,
(21)
which has not been considered in the aforementioned studies. As we will show in the following, the main features recovered by our analysis are not qualitatively changed when
applying detrending. However, this step appears relevant in
www.nonlin-processes-geophys.net/18/545/2011/
Fig. 9. Evolution of RN transitivity T for (A) the δ 18 O record from
ODP
659, andofthe
flux records
sites
(B) 659,
Fig. 9.site
Evolution
RNdust
transitivity
T forfrom
(A) ODP
the δ 18
O record
from
(C)
721,
(D)
967.
reveals
in theODP
regularity
ODP
siteand
659,
and
theTdust
flux changes
records from
sites of
(B)African
659, (C)
climate during the Plio-Pleistocene for the latter three records. Here
721, and (D) 967. T reveals changes in the regularity of African cliwe used a detrending window size WD∗ = 500 kyr, recurrence winmate during the Plio-Pleistocene. Here we used a detrending window size W ∗∗= 410 kyr and step size 1W ∗ = 41 kyr, embedding
window size W ∗ = 410 kyr
dow size W = 500 kyr, recurrence
dimension mD= 3 and
delay τ ∗ = 10 kyr. The recurrence thresh∗
and step size ∆W = 41 kyr, embedding dimension m = 3 and deold ε ∗was chosen adaptively to yield a fixed edge density ρ = 0.05.
lay τ = 10 kyr. The recurrence threshold ε was chosen adaptively
The grey bars represent the 5 % and 95 % quantiles with respect to
to yield
a fixed edgeobtained
density from
ρ = 0.05.
grey barsofrepresent
the
test distribution
10 000The
realisations
our null-the
5%
and
95%
quantiles
with
respect
to
the
test
distribution
obtained
model for each record separately. Vertical dashed lines indicate
the
from
10,000
realisations
of
our
null-model
for
each
record
sepadetected epochs of transitions discussed in the main text.
rately. Vertical dashed lines indicate the detected epochs of transitions discussed in the main text.
other kinds of statistical analyses, e.g. for estimating spectrograms
or time-dependent
of autoregressive
including
two periods of coefficients
extraordinarily
large values ofproT at
cesses,
since
the
data
show
considerable
long-term
trends in
about 3.45-3.05 and 2.2-2.1 Myr BP, related to pronounced
both
mean
variance
(Fig.
2). 6B,C. The first of these peclusters
of and
vertices
shown
in Figs.
we find asuppressed
synchronous
Regarding
the
transitivity
(Fig.
riods results from a time interval 9),
of strongly
and
behaviour
of
the
two
geographically
distinct
records
ODP2),
almost constant dust flux in the Mid Pliocene (seeatFig.
sites
721/722
during the
Pliocene
while659
the and
latter
one coincides
with
a period(∼5.3–2.6
of almostMyr
periBP; before present) and Early Pleistocene (2.6–1.0 Myr
odic2 Milankovich-type variations (Trauth et al., 2009). We
BP) , including two periods of extraordinarily large values
note that it is known (and empirically understood) that both
of T at about 3.45–3.05 and 2.2–2.1 Myr BP, related to protypes of dynamics typically lead to large values of T (Marnounced clusters of vertices shown in Figs. 6b and c. The
wan et al., 2009; Donner et al., 2010a, in press; Zou et al.,
first of these periods results from a time interval of strongly
2010), so that this result is consistent with theoretical expecsuppressed and almost constant dust flux in the Mid Pliocene
tations.
thethe
Early
Pleistocene,
the with
signatures
at both
(see
Fig.During
2), while
latter
one coincides
a period
of
sites
decouple
from
each
other,
which
could
be
the
result
almost periodic Milankovich-type variations (Trauth et al.,of
an enhancement
theitatmospheric
Walker
circulation
(Rav2009).
We note of
that
is known (and
empirically
underelo et al., 2004). For the last about 1.5 Myr, the variations of
transitivity
become
more similar
ODP
sites
721/722
2 Here “Early
Pleistocene”
does notbetween
refer to any
of the
archetypandstages
967, (Upper,
particularly
highlighting
the Mid Pleistocene
tranical
Middle
and Lower Pleistocene).
Its timing 2.6–
sition
and 0.7
Myr BP (Fig.but6A),
which corre1.0
Myrbetween
BP is not1.2
motivated
stratigraphically,
climatologically,
i.e. by the onset of the Mid-Pleistocene transition around 1.0 Myr
1.0 Myr BP is not motivated stratigraphically, but climatologically,
BP.
i.e., by the onset of the Mid-Pleistocene transition around 1.0 Myr
BP.
Nonlin. Processes Geophys., 18, 545–562, 2011
Fig.
recor
(B) 6
clima
icanc
spon
riodi
(Fig.
light
tocen
varia
at ab
mate
for a
Th
of b
ent c
nous
ity re
An i
pecta
syste
lar lo
to th
signi
lated
at ar
tent
sugg
clim
900
and i
J. F. Donges et al.: Identification of dynamical transitions in marine p
of dynamical556
transitions in marine
by RNofanalysis
J. F.palaeoclimate
Donges et al.:records
Identification
dynamical transitions in marine palaeoclimate records by RN analysis
rd from
59, (C)
can cling win410 kyr
and deaptively
sent the
btained
d sepatransi-
of T at
ounced
ese peed and
Fig. 2),
st peri9). We
at both
(Maret al.,
expecat both
sult of
n (Ravions of
21/722
e trancorre-
gically,
1.0 Myr
Tab
evol
and
(D)
unde
mar
Fig. 10. Evolution of RN average path length L for (A) the δ 18 O
record from ODP site 659, and the dust flux records from ODP sites
Fig. 10. Evolution of RN average path length L for (A) the δ 18 O
(B) 659, (C) 721, and (D) 967, indicating transitions in African
record from ODP site 659, and the dust flux records from ODP sites
climate dynamics during the Plio-Pleistocene. Parameters, signifi(B) 659, (C) 721, and (D) 967, indicating transitions in African
cance test, and vertical lines are the same as in Fig. 9.
climate dynamics during the Plio-Pleistocene. Parameters, significance test, and vertical lines are the same as in Fig. 9.
stood) that both types of dynamics typically lead to large
values of T (Marwan et al., 2009; Donner et al., 2010a,
sponds
a change
in the dominating
pe2011b; to
Zou
et al., 2010),
so that this Milankovich-type
result is consistent
riodicity.
The
results
obtained
for
the
average
path
length
with theoretical expectations. During the Early Pleistocene, L
(Fig.
10) are mostly
with these
findings,
also
highthe signatures
at bothconsistent
sites decouple
from each
other,
which
lighting
the
Mid
Pliocene,
Early
Pleistocene,
and
Mid
Pleiscould be the result of an enhancement of the atmospheric
tocene
periods (Ravelo
with changes
the long-term
flux
Walker as
circulation
et al., in
2004).
For the lastdust
about
variability.
Specifically,
L
tends
to
show
significant
peaks
1.5 Myr, the variations of transitivity become more similar
at
abruptODP
change
points
between
regular
and morehighlighterratic clibetween
sites
721/722
and 967,
particularly
mate
variability,
as
indicated
by
T
(see
Marwan
et
al. (2009)
ing the Mid Pleistocene transition between 1.2 and 0.7
Myr
for
theoretical
explanation
of thistobehaviour).
BP a(Fig.
6a), which
corresponds
a change in the dominating
Milankovich-type
periodicity.
The from
resultsthe
obtained
The oxygen
isotope anomaly
obtained
analysis
mostly consistent
for benthic
the average
path lengthcharacterises
L (Fig. 10) are
of
foraminifera
a distinctively
differwith
these
findings,
also
highlighting
the
Mid
Pliocene,
Early
ent climatic parameter (i.e., global ice volume) than terrigePleistocene,
and so
Mid
Pleistocene
periods with
changes
in
nous
dust flux,
that
it can beasexpected
that the
variabilthe
long-term
dust
flux
variability.
Specifically,
L
tends
to
ity recorded by this proxy differs from that of the dust flux.
show
significantofpeaks
at abrupt
change
points
betweenthis
regAn
inspection
the RN
properties
indeed
confirms
exular
and
more
erratic
climate
variability,
as
indicated
by
pectation. Specifically, the transitivity T does not show Tany
et al. at
(2009)
for a theoretical
explanation
of
(see Marwan
systematic
maxima
all (indicating
time intervals
with reguthis behaviour).
lar long-term dynamics) (Fig. 9A), which is in clear contrast
oxygen dust
isotope
anomaly
obtainedpath
from
the analysis
to The
the aeolian
flux.
The average
length
L shows
of
benthic
foraminifera
characterises
a
distinctively
differ-resignificant maxima around 2.9 Myr BP (possibly being
ent
climatic
parameter
(i.e.
global
ice
volume)
than
terrigelated to the intensification of Northern hemisphere glaciation
nous
dust flux,
so thatbetween
it can be1.8
expected
variabilat
around
this time),
and 1.3that
MyrtheBP
(consisity
recorded
by
this
proxy
differs
from
that
of
the
dust
flux.
tent with the corresponding results for the dust flux records,
An inspection of the RN properties indeed confirms this exsuggesting a high-latitude mechanism behind the large-scale
pectation. Specifically, the transitivity T does not show any
climatic changes during this time period), and after about
systematic maxima at all (Fig. 9a), which is in clear contrast
900 kyr BP (possibly resulting from the glacial terminations
and inceptions with a rather long – roughly 100 kyr – periodNonlin. Processes Geophys., 18, 545–562, 2011
Fig. 11. Evolution of RN assortativity R for (A) the δ 18 O record
from ODP site 659, and the dust flux records from ODP
Fig.
11. Evolution of RN assortativity R for (A) the δ 18 Osites
record
(B) 659, (C) 721, and (D) 967 during the Plio-Pleistocene. Paramfrom ODP site 659, and the dust flux records from ODP sites (B)
eters, significance test, and vertical lines are the same as in Fig. 9.
659, (C) 721, and (D) 967 during the Plio-Pleistocene. Parameters,
significance test, and vertical lines are the same as in Fig. 9.
to the aeolian dust flux. The average path length L shows
significant maxima around 2.9 Myr BP (possibly being related to the intensification of Northern hemisphere glaciation
at around this time), between 1.8 and 1.3 Myr BP (consistent with the corresponding results for the dust flux records,
suggesting a high-latitude mechanism behind the large-scale
climatic changes during this time period), and after about
900 kyr BP (possibly resulting from the glacial terminations
and inceptions with a rather long – roughly 100 kyr – periodicity) (Fig. 10a).
Figures 11 and 12 additionally show the time variability
of the two other RN properties assortativity R and diameter D. Since the latter one is closely related to the average path length L (Donner et al., 2010a), the variability
of both measures is very similar. Moreover, we also find
some much weaker similarities between the temporal variability patterns of transitivity T and assortativity R, which
are less pronounced, since both properties characterise less
obviously related aspects of the network geometry in phase
space. Specifically, the time interval of suppressed dust flux
in ODP 659 and 721/722 during the Mid Pliocene results not
only in an increased transitivity, but also a high assortativity.
The latter feature can be explained by the fact that a relatively
large12.
cluster
of state
vectors
representing
this
regime
Fig.
Evolution
of RN
diameter
D for (A)
thelaminar
δ 18 O record
from
emerges
theand
network,
is rather
densely
connected
ODP
site in
659,
the dustwhich
flux records
from
ODP sites
(B) 659,
6c).and (D) 967 during the Plio-Pleistocene. Parameters, sig(Fig.721,
(C)
nificance
test, andthat
vertical
lines
are the same
as in
Fig. 9.
We conclude
the RN
measures
are not
statistically
independent in their time evolution (Table 2). For the ODP
site 659 δ 18 O and dust flux records, the correlations between
icity) (Fig. 10A).
Figures www.nonlin-processes-geophys.net/18/545/2011/
11 and 12 additionally show the time variability
of t
eter
erag
of b
som
abil
are
obv
spac
in O
only
The
larg
eme
(Fig
W
inde
site
tran
age
man
retic
mor
the
721
Fig. 11. Evolution of RN assortativity R for (A) the δ 18 O record
from ODP site 659, and the dust flux records from ODP sites (B)
659, (C) 721, and (D) 967 during the Plio-Pleistocene. Parameters,
significance test, and vertical lines are the same as in Fig. 9.
D
0.37
0.74
0.35
1.00
T
L
R
D
T
1.00 0.65 0.61 0.23
J. F. Donges et al.: Identification of dynamical transitions in marine palaeoclimate
by RN 1.00
analysis
557
L 0.65
0.54 0.78
(D)records
R 0.61 0.54 1.00 0.16
0.23 0.78
0.16 correlations
1.00
Table 2. Spearman’sDρ measuring
rank-order
in the
time evolution of RN measures for (A) the ODP site 659 δ 18 O
record, and the dust flux records from ODP sites (B) 659, (C)
721/722, and (D) 967. Significant correlations having a p-value
of thethan
two
other
and
diamsmaller
0.05
underRN
the properties
assumption assortativity
of uncorrelatedR
data
of the
eter
D.
Since
the
latter
one
is
closely
related
to
the
avsame length are marked in bold.
Fig. 12. Evolution of RN diameter D for (A) the δ 18 O record from
ODP
site Evolution
659, and the
dustdiameter
flux records
ODP
(B) 659,
Fig. 12.
of RN
D forfrom
(A) the
δ 18sites
O record
from
(C)
721,
(D)and
967the
during
Parameters,
ODP
siteand
659,
dust the
fluxPlio-Pleistocene.
records from ODP
sites (B)sig659,
nificance
linesthe
arePlio-Pleistocene.
the same as in Fig.
9.
(C) 721, test,
and and
(D) vertical
967 during
Parameters,
significance test, and vertical lines are the same as in Fig. 9.
transitivity T and assortativity R as well as between average
path
length
icity)
(Fig.
10A).L and diameter D as measured by Spearman’s ρ are most pronounced, which is consistent with theFigures 11 and 12 additionally show the time variability
oretical expectations (Donner et al., 2010a). Correlations are
more clearly developed between all four measures in case of
the more highly sampled dust flux records from ODP sites
721/722 and 967 (Table 1). However, for all records the four
measures can be considered sufficiently independent to justify including all of them for a broad and thorough nonlinear
time series analysis of oxygen isotope and terrigenous dust
flux variability.
4.2
Robustness of the results
To assure the reliability and robustness of our results, we
systematically study their dependence on the relevant algorithmic parameters of our method, in particular, the widths
of the recurrence window (W ∗ ) and the detrending window
(WD∗ ) as well as the embedding delay (τ ∗ ). In Figs. 13–15,
the results of the significance test are presented as contours
at two prescribed significance levels obtained from the observed measure’s quantiles with respect to the corresponding test distribution. Green contours represent the lower
prescribed quantile (5 %), while black contours indicate the
upper one (95 %). This implies that values of the measure
under study enclosed by green contours can be considered
as exceptionally low, while those lying within black contours are exceptionally large, recalling the interpretation of
the applied null-model given in Sect. 2.6. It is, however,
www.nonlin-processes-geophys.net/18/545/2011/
erage path length L (Donner et al., 2010a), the variability
of both measures
T is very
L similar. RMoreover,
D we also find
some much weaker similarities between the temporal variT
1.00
−0.08
0.38
0.00
ability patterns
of transitivity
T and
assortativity
R, which
(A) L −0.08
1.00
−0.06
0.92
are less pronounced,
since
both
properties
characterise
not so
R 0.38
−0.06
1.00
0.03
obviouslyDrelated
of the network
geometry
in phase
0.00aspects0.92
0.03
1.00
space. Specifically, the time interval of suppressed dust flux
T
L
R
D
in ODP 659 and 721/722 during the Mid Pliocene results not
1.00 transitivity,
−0.05 but
0.12
0.03assortativity.
only in anTincreased
also a high
−0.05
1.00
−0.08
The(B)
latter L
feature
can be explained
by the fact0.77
that a relatively
R of0.12
−0.08
1.00 this0.23
large cluster
state vectors
representing
laminar regime
D the0.03
1.00 connected
emerges in
network,0.77
which is 0.23
rather densely
T
L
R
D
(Fig. 6C).
We conclude
that the0.50
RN measures
not statistically
T
1.00
0.40 are 0.37
(C) L in0.50
1.00
0.37
0.74For the ODP
independent
their time
evolution
(Tab. 2).
RO and
0.40dust flux
0.37records,1.00
0.35
site 659 δ 18
the correlations
between
D
0.37
0.74
0.35
transitivity T and assortativity R as well as1.00
between average path length
D
by SpearT L and diameter
L
R as measured
D
man’s ρ are
most
pronounced,
which
is
consistent
with
theoT
1.00
0.65
0.61
0.23
retical
expectations
(Donner
et
al.,
2010a).
Correlations
are
(D) L 0.65
1.00
0.54
0.78
more clearly
developed
between
all
four
measures
in
case
of
R 0.61
0.54
1.00
0.16
the more D
highly
sampled0.78
dust flux0.16
records 1.00
from ODP sites
0.23
721/722 and 967 (Tab. 1). However, for all records the four
important to recognise that the null-hypothesis of stationarity has been tested pointwise, while physical significance
requires the null-hypothesis to be rejected over a certain period of time, i.e. for several subsequent time points (Maraun
et al., 2007). Therefore, certain line-like structures, particularly those seen in Fig. 15, are likely to reflect statistical fluctuations rather than physically significant dynamical transitions. In the following, we will only present the results for
the ODP site 659 dust flux record.
i. Recurrence window size W ∗ : as for the model systems
in Sect. 3, we first discuss the sensitivity of our results to
the changing width of the recurrence window W ∗ . The
corresponding results for the four chosen RN measures
are shown in Fig. 13. We recognise that the most significant features persist under varying W ∗ , although the
relevant structures become broader and less significant
for larger windows. This is to be expected since more
and more data from time intervals not directly affected
by the origin of specific network properties (e.g. a laminar phase in the dynamics) contribute to the longer windows. As the window width is increased linearly, conelike structures emerge (which is especially well visible
for the Mid Pliocene transitivity maximum as the most
Nonlin. Processes Geophys., 18, 545–562, 2011
558
J. F. Donges et al.: Identification of dynamical transitions in marine palaeoclimate records by RN analysis
J. F. Donges et al.: Identification of dynamical transitions in marine palaeoclimate records by RN analysis
Transitivity
Average path length
Assortativity
Diameter
15
Fig. 13. Dependence
of (A) transitivity
T , (B)Taverage
path path
length
L, (C)
assortativity
onthe
therecurrence
recurrence
window
Fig. 13. Dependence
of (A) transitivity
, (B) average
length
L, (C)
assortativityR,
R,and
and(D)
(D) diameter
diameter DDon
window
∗ = 41 kyr, the detrending window
∗
∗
size W ∗ for
the
dust
flux
record
from
ODP
site
659.
The
recurrence
window
step
size
is
fixed
to
1W
size W for the dust flux record from ODP site 659. The recurrence window step size is fixed to ∆W = 41 kyr, the detrending window
∗
size to WD∗size
= 500
kyr.
and
black
contours
correspond
to theto5the
% and
95 %
quantiles
thetest
testdistribution
distribution
obtained
from
500 kyr.
Green
and
black contours
correspond
5% and
95%
quantileswith
withrespect
respect to the
obtained
from
to W
D =Green
10,000
realisations
of our null-model.
Other parameters
embedding
dimension
anddelay
delay ττ ∗ = 10
waswas
chosen
10 000 realisations of our null-model.
Other parameters
were:were:
embedding
dimension
mm
==3 3and
10kyr,
kyr,the
thethreshold
threshold
chosen
to yield
a fixed
edgeρdensity
ρ =The
0.05.
The white
indicating
value”
at the
leftand
andright
rightmargins
margins of
because
we we
to yield a fixed
edge
density
= 0.05.
white
bandsbands
indicating
“no “no
value”
at the
left
ofeach
eachpanel
panelappear
appear
because
∗
µ,W ∗
µ
plot themeasures
network measures
at the mid-points
the windows
µ used
for
RN
analysis(Sect.
(Sec. 2.4).
2.4). As
the
mid-points
of of
thethe
firstfirst
(last)
As
the
mid-points
(last)
t µ of ttheofwindows
µ
used
for
RN
analysis
plot the network
g µ,W atg the mid-points
∗
window
move
further
into the
past (present)
for increasing
the white
bands
grow
linearlyfor
forlinearly
linearly increasing
increasing WW∗ ∗. .
window move
further
into
the past
(present)
for increasing
W ∗, W
the ,white
bands
grow
linearly
of time series analysis, the consideration of embedding with
relevant
feature).
general, iswenecessary
observeinthat
fortoour
properly
chosenInparameters
order
obtain
example
the
transitivity
is
most
robust
with
respect
to
feasible results.
changes of W ∗ , whereas the other network measures
In contrast
to other
techniques,
RNofanalysis
does not
may lose
significance
if this
parameter
our analysis
characterise
temporal
interrelationships
within
analysed
method
is varied.
We note,
however, that
the the
periods
records
(although
time
information
enters
indirectly
through
of interest identified in Sect. 4.1 are robust for a wide
embedding parameters, however, mostly on short time scales
range of recurrence∗ window sizes, presenting a trade-off
as typically mτ N h∆T i), but quantifies geometric propbetween
of identified
features
(small
ertiesgood
of thelocalisation
sampled dynamical
system in
its (reconstructed)
windows)
and
reasonable
statistical
confidence
phase space. The only implicit assumption is thatofthethe
availcalculated
network
properties
(large
windows).
able sample
of observed
state
vectors
{yiµ } represents the
spatial distribution of the true state vectors in the (properly
reconstructed)
phase
the underlying
dynamical sysii. Detrending
window
sizespace
WD∗ :ofregarding
the dependence
tem
sufficiently well.
of our
observations
on the choice of the detrending window, Fig. 14 shows that the general temporal variability pattern of the different network measures remains
unchanged as WD∗ is altered, whereas the actual significance levels are more strongly influenced. In general,
we can conclude, however, that the most significant time
Nonlin. Processes Geophys., 18, 545–562, 2011
In this respect, our approach is very generally
applicaundermoderate
variations
of WD∗ , which
bleperiods
and haspersist
comparably
requirements
in termsisofparexpressed
transitivity
theticularly
requestedwell
number
of datafor
(i.e.,thewindows
with during
O(100) the
Mid
Pliocene.
Together
the factanalysis
that RN
data
points
are sufficient
for awith
reasonable
of analysis
nonstationary
systems).
case
of palaeoclimate
records, thisproof the three
dust In
flux
records
without detrending
complementary
way results
for characterising
duces consistent
(Donges ettime
al., series
2011) avoids
this sugconceptual
of other
approaches
due to influence
uncertain on
gests thatproblems
trends do
not have
a significant
agethe
models
and non-uniform
sampling.
E.g.,
the∗ results
of i.
N h1T
outcomes
of RN analysis
as long
as W
RN analysis {g µ } are invariant to changes in the age model
However, this should be checked in any particular appli{ti }, only the associated windows’ mid-points {tµ } change
cation by comparing the results for the time series data
with variations in {ti } (Eq. (9)). However, the aforemenbefore
and after
detrending.
thenecessary
results for
tioned
problems
indirectly
persist inNote
termsthat
of the
undetrended
time
series
are
approximated
by
embedding of the data and have to be finally resolvedthose
in cor-dish1T i,work
played infuture
Fig. work.
14 forWhile
WD∗ ≈
sincefocussed
RN analysis
responding
theNpresent
on
is invariant to nearly uniform translations of the data.
iii. Embedding delay τ ∗ : our results are also seen to be
robust with respect to reasonable variations of the embedding delay τ ∗ around the previously chosen delay time τ ∗ = 10 kyr (Fig. 15). However, for the
www.nonlin-processes-geophys.net/18/545/2011/
J. F. Donges
of dynamical
transitionsofindynamical
marine palaeoclimate
records
by RN analysis
16 et al.: Identification
J. F. Donges
et al.: Identification
transitions in marine
palaeoclimate
records by RN analysis 559
Transitivity
Average path length
Assortativity
Diameter
Fig. 14. Dependence of (A) transitivity T , (B) average path length L, (C) assortativity R, and (D) diameter D on the detrending window
Fig. 14. Dependence of (A) transitivity T , (B) average path length L, (C) assortativity R, and (D)
D on the detrending window
∗ diameter
∗ size
∗
∗
size WD∗ for
the
dust
from
ODP
sitesite
659.
sizeisisfixed
fixedtoto
410
a step
of ∗1W
41 kyr.
for
the flux
dust record
flux record
from
ODP
659.The
Therecurrence
recurrence window
window size
WW
==
410
kyrkyr
withwith
a step
size size
of ∆W
= 41 =
kyr.
WD
Green andGreen
blackand
contours
correspond
to theto5 %
95 %95%
quantiles
the test
testdistribution
distribution
obtained
10 000
realisations
black contours
correspond
the and
5% and
quantileswith
withrespect
respect to
to the
obtained
fromfrom
10,000
realisations
∗ 10 kyr, the threshold was chosen to yield a fixed
of our null-model.
Other parameters
were: were:
embedding
dimension
mm
==
3 and
of our null-model.
Other parameters
embedding
dimension
3 anddelay
delayττ ∗ =
=
kyr, the threshold was chosen to yield a fixed
edge
= 0.05.
In regions
the black
dashed
results
areinfluenced
influenced by boundary
since
the the
effective
detrending
edge density
ρ density
= 0.05.ρ In
regions
outsideoutside
the black
dashed
lineslines
thethe
results
are
boundaryeffects,
effects,
since
effective
detrending
window
size
Wto
has to decrease
time series’
(Eq.
(3)).
Thebands
white indicating
bands indicating
“no value”
at left
the and
left and
right
D (t)
The
white
“no
value”
at
the
right
margins
window size
WD (t)
has
decrease
towards towards
the timethe
series’
limits limits
(Eq. 3).
∗
µ,W ∗ at the
µ of the
margins
of each
panel we
appear
we plot measures
the networkg measures
g µ,Wmid-points
at the mid-points
tµ of
the windows
µ used
for
RN
analysis(Sect.
(Sec. 2.4).
of each panel
appear
because
plotbecause
the network
t
windows
µ
used
for
RN
analysis
In contrast to Fig. 13 their width does not change
W ∗ is fixed here.
In contrast2.4).
to Fig.
13 their width does not change as W ∗ is as
fixed
here.
the technical
of applying
RNkyr
analysis
to palaeocliembedding
delayaspects
exceeding
τ ∗ = 20
the results
and
mate
time
series,
an
in-depth
discussion
of
the
significance levels change considerably. Thisresults
is ex-obtained for the three dust flux records in the light of addipected as for delays larger than 20 kyr, autocorrelational proxy records and palaeontological evidence is given
tionsininDonges
the time
series
do not decrease significantly anyet al.
(subm.).
more. In the case of the δ 18 O record from ODP site 659
they even increase again due to pronounced (obliquityAcknowledgements. This work has been financially supported by
driven)
Milankovich
cycles
with
a period
around
41Ministry
kyr
the Leibniz
association
(project
ECONS)
and the
Federal
(Fig.for3),
so
that
the
autocorrelation
criterion
for
the
Education and Research (BMBF) via the Potsdam Research
∗ does
Cluster
Georisk
Environmental
Change and Sustainchoice
of τfor
notAnalysis,
apply here
anymore.
5
ability (PROGRESS). JFD thanks the German National Academic
Foundation for financial support. We thank Roger Grzondziel and
Ciaron Linstead for help with the IBM iDataPlex Cluster at the Potsdam Institute for Climate Impact Research.
Conclusions
We have demonstrated that RN analysis allows detecting
dynamical transitions in non-stationary model systems as
well as real-world palaeoclimate data. Transitivity and average path length have been previously discussed as appropriate network properties indicating qualitative changes in the
www.nonlin-processes-geophys.net/18/545/2011/
References
dynamics
of the underlying system. Here we have provided
examples that also other global network measures such as asAbarbanel, H.D.I. [1996] Analysis of Observed Chaotic Data
sortativity
and
network
(Springer,
New
York). diameter trace qualitative changes in
dynamical
however,
do not
have a similarly
Babu, P. &systems,
Stoica, P. which,
[2010] “Spectral
analysis
of nonuniformly
sampled data –interpretation
a review”, Dig. Sign.
Proc. 20,
straighforward
in terms
of 359-378.
basic system propBarrio,
Serrano,
S. aforementioned
[2007] “A three-parametric
study of the
erties
asR.the& two
other
quantities.
Lorenz model”, Physica D 229, 43-51.
Our results
show V.,
that
the outcomes
analysis
Boccaletti,
S., Latora,
Moreno,
Y., Chavez, of
M. RN
& Hwang,
D. are
quite[2006]
robust
if the fundamental
parameters
of the method
“Complex
networks: Structure
and dynamics”,
Phys. (deRep. 424,
trending
and175-308.
recurrence window sizes, embedding delay) are
Brockwell,
P.J. a&reasonable
Davis, R.A. [1991]
to Time
varied
within
range.Introduction
Unlike for
otherSeries
methods
and Forecasting (Springer, New York, 2nd ed).
ofBrockwell,
time series
analysis,
the
consideration
of
embedding
P.J. & Davis, R.A. [2002] Time Series: Theory and with
properly
chosen
parameters
is necessary
in order to obtain
Methods
(Springer,
New York, 2nd
ed).
feasible results.
In contrast to other techniques, RN analysis does not
characterise temporal interrelationships within the analysed
records (although time information enters indirectly through
embedding parameters, however, mostly on short time
scales as typically mτ ∗ N h1T i), but quantifies geometric
Nonlin. Processes Geophys., 18, 545–562, 2011
560
Donges
et al.: Identification
of dynamical
transitions
in marine
J. F. DongesJ.etF.al.:
Identification
of dynamical transitions
in marine
palaeoclimate
recordspalaeoclimate
by RN analysisrecords by RN analysis
17
Transitivity
Average path length
Assortativity
Diameter
Fig. 15. Dependence
of (A) transitivity
T , (B)Taverage
path path
length
L, (C)
assortativity
(D)diameter
diameter
embedding
Fig. 15. Dependence
of (A) transitivity
, (B) average
length
L, (C)
assortativityR,
R, and
and (D)
DD
on on
thethe
embedding
delaydelay
time time
∗=
τ ∗ for the dust
ODP
659.
window
size
W∗∗==410
410kyr
kyrwith
with
a step
of ∗1W
41 kyr,
τ ∗ forflux
the record
dust fluxfrom
record
fromsite
ODP
site The
659. recurrence
The recurrence
window
sizeisisfixed
fixed to W
a step
sizesize
of ∆W
= 41
kyr,
∗ to W ∗ kyr.
kyr. Green
and black
contours
correspondtoto the
the 55%
quantiles
withwith
respect
to theto the
the detrending
window
size
the detrending
window size
to W
and black
contours
correspond
% and
and95%
95 %
quantiles
respect
D = 500Green
D = 500
test distribution
realisations
our null-model.
Verticalline-shaped
line-shaped contours
to correspond
to statistical
test distribution
obtained obtained
from 10from
000 10,000
realisations
of ourof null-model.
Vertical
contoursare
arelikely
likely
to correspond
to statistical
than physically
significant
time intervals
text).
Otherparameters
parameters were:
were: embedding
dimension
m =m3,=
the3,threshold
fluctuationsfluctuations
rather thanrather
physically
significant
time intervals
(see(see
text).
Other
embedding
dimension
the threshold
was chosen to yield a fixed edge density ρ = 0.05.
was chosen to yield a fixed edge density ρ = 0.05.
K.E. & dynamical
Sprott, J.C. [2005]
“A comparison
of correlaproperties Chlouverakis,
of the sampled
system
in its (recontion and Lyapunov dimensions”, Physica D 200, 156-164.
structed) phase
The only
implicit
assumption
is that
Csárdi, space.
G. & Nepusz,
T. [2006]
“The igraph
software
package
µ
the availableforsample
observed
stateInterJournal
vectors {y
}
complexof
network
research”,
Complex
Systems
i repreCX.18,
1695.
sents the spatial
distribution
of the true state vectors in the
Costa, L.F., Rodrigues,
F.A., Travieso
& Villas Boas,
(properly da
reconstructed)
phase space
of theG.underlying
dy-P.R.
[2007] “Characterization of complex networks: a survey of meanamical system
sufficiently well.
surements”, Adv. Phys. 56, 167242.
Analysis
in the Geosciences
(Springer,
Heidelberg).
technical
aspects
of applying
RN analysis
to palaeoclimate
Donner, R.V., Zou, Y., Donges, J.F., Marwan, N. & Kurths, J.
time[2010a]
series,“Recurrence
an in-depth
discussion
of
the
results
obtained for
networks – a novel paradigm for nonlinear
the three
dust New
fluxJ.records
the light of additional proxy
time series”,
Phys. 12, in
033025.
Donner,and
R.V.,palaeontological
Zou, Y., Donges, J.F.,
Marwan,isN.given
& Kurths,
J.
records
evidence
in Donges
“Ambiguities in recurrence-based complex network rep(2011).
et al.[2010b]
resentations of time series”, Phys. Rev. E 81, 015101(R).
Donner, R.V., Small, M., Donges, J.F., Marwan, N., Zou, Y., Xiang,
Acknowledgements. This work has been financially supported
R. & Kurths, J. [2011] “Recurrence-based time series analysis
by the
Leibniz
association
(project
ECONS)
and the
Federal
by means
of complex
network
methods”,
Int. J. Bifurc.
Chaos
Ministry
for Education and Research (BMBF) via the Potsdam
21, 1019-1046.
Research
for Georisk
Analysis,J.F.,
Environmental
Change and
Donner, Cluster
R.V., Heitzig,
J., Donges,
Zou, Y., Marwan,
N. & Kurths,
J. [in press] JFD
“The thanks
geometry
chaotic dySustainability
(PROGRESS).
theofGerman
National
namics Foundation
– A complexfor
network
perspective”,
Phys.
J. B,Roger
Academic
financial
support.Eur.We
thank
doi:10.1140/epjb/e2011-10899-1
(online
Grzondziel
and Ciaron Linstead for
helpfirst).
with the IBM iDataPlex
Eckmann, J.-P., Kamphorst, S.O. & Ruelle, D. [1987] “Recurrence
Cluster at the Potsdam Institute for Climate Impact Research.
plots of dynamical systems”, Europhys. Lett. 4, 973-977.
P.B. approach
[1995] “Plio-Pleistocene
African Climate”,
In this deMenocal,
respect, our
is very generally
applica-Sci270, 53-59.moderate requirements in terms of
ble and has ence
comparably
deMenocal, P.B. [2004] “African climate change and faunal evothe requested
number
of Pliocene-Pleistocene”,
data (i.e. windows
with
O(100)
lution
during the
Earth
Planet.
Sci. Lett.
data points are
sufficient
for
a
reasonable
analysis
of non220, 3-24.
J.F., Donner,
N., Trauth,records,
M.H., Schellnhustationary Donges,
systems).
In caseR.V.,
of Marwan,
palaeoclimate
this
ber, H.J.
& Kurths,
J. [subm.] “Nonlinear
large-scale
complementary
way
for characterising
timedetection
seriesofavoids
transitions in Plio-Pleistocene African climate”.
conceptualDonner,
problems
other S.A.
approaches
due
to uncertain
R.V. &of
Barbosa,
(eds.) [2008]
Nonlinear
Time Series
age models and non-uniform sampling. E.g. the results of RN
Edited by: S. Barbosa
analysis {g µ } are invariant to changes in the age model {ti },
Reviewed by: two anonymous referees
only the associated windows’ mid-points {t µ } change with
variations in {ti } (Eq. 9). However, the aforementioned problems indirectly persist in terms of the necessary embedding
of the data and have to be finally resolved in corresponding future work. While the present work focussed on the
Nonlin. Processes Geophys., 18, 545–562, 2011
www.nonlin-processes-geophys.net/18/545/2011/
J. F. Donges et al.: Identification of dynamical transitions in marine palaeoclimate records by RN analysis
References
Abarbanel, H. D. I.: Analysis of Observed Chaotic Data (Springer,
New York), 1996.
Babu, P. and Stoica, P.: Spectral analysis of nonuniformly sampled
data – a review, Dig. Sign. Proc. 20, 359–378, 2010.
Barrio, R. and Serrano, S.: A three-parametric study of the Lorenz
model, Physica D, 229, 43–51, 2007.
Boccaletti, S., Latora, V., Moreno, Y., Chavez, M., and Hwang, D.:
Complex networks: Structure and dynamics, Phys. Rep., 424,
175–308, 2006.
Brockwell, P. J. and Davis, R. A.: Introduction to Time Series and
Forecasting, Springer, New York, 2nd Edn., 1991.
Brockwell, P. J. and Davis, R. A.: Time Series: Theory and Methods, Springer, New York, 2nd Edn., 2002.
Chlouverakis, K. E. and Sprott, J. C.: A comparison of correlation
and Lyapunov dimensions, Physica D, 200, 156–164, 2005.
Csárdi, G. and Nepusz, T.: The igraph software package for complex network research, InterJournal Complex Systems, CX.18,
1695, 2006.
Costa, L. da F., Rodrigues, F. A., Travieso, G. and Villas Boas, P.
R.: Characterization of complex networks: a survey of measurements, Adv. Phys., 56, 167–242, 2007.
deMenocal, P. B.: Plio-Pleistocene African Climate, Science, 270,
53–59, 1995.
deMenocal, P. B.: African climate change and faunal evolution during the Pliocene-Pleistocene, Earth Planet. Sci. Lett., 220, 3–24,
2004.
Donges, J. F., Donner, R. V., Marwan, N., Trauth, M. H., Schellnhuber, H. J., and Kurths, J.: Nonlinear detection of large-scale transitions in Plio-Pleistocene African climate, in review, 2011.
Donner, R. V. and Barbosa, S. A. (Eds.): Nonlinear Time Series
Analysis in the Geosciences, Springer, Heidelberg, 2008.
Donner, R. V., Zou, Y., Donges, J. F., Marwan, N., and Kurths,
J.: Recurrence networks – a novel paradigm for nonlinear
time series, New J. Phys., 12, 033025, doi:10.1088/13672630/12/3/033025, 2010a.
Donner, R. V., Zou, Y., Donges, J. F., Marwan, N., and Kurths,
J.: Ambiguities in recurrence-based complex network representations of time series, Phys. Rev. E, 81, 015101(R),
doi:10.1103/PhysRevE.81.015101, 2010b.
Donner, R. V., Small, M., Donges, J. F., Marwan, N., Zou, Y., Xiang, R., and Kurths, J.: Recurrence-based time series analysis by
means of complex network methods, Int. J. Bifurc. Chaos, 21,
1019–1046, 2011a.
Donner, R. V., Heitzig, J., Donges, J. F., Zou, Y., Marwan, N.,
and Kurths, J.: The geometry of chaotic dynamics – A complex
network perspective, Eur. Phys. J. B, doi:10.1140/epjb/e201110899-1 (online first), in press, 2011b.
Eckmann, J.-P. and Ruelle, D.: Fundamental limitations for estimating dimensions and Lyapunov exponents in dynamical systems,
Physica D, 56, 185–187, 1992.
Eckmann, J.-P., Kamphorst, S. O. and Ruelle, D.: Recurrence plots
of dynamical systems, Europhys. Lett., 4, 973–977, 1987.
Elsner, J. B., Jagger, T. H., and Fogarty, E. A.: Visibility network
of United States hurricanes, Geophys. Res. Lett., 36, L16702,
doi:10.1029/2009GL039129, 2009.
Foster, G.: Time Series Analysis by Projection, I. Statistical Properties of Fourier Analysis, Astron. J., 111, 541–554, 1996a.
Foster, G.: Time Series Analysis by Projection, II. Tensor Methods
www.nonlin-processes-geophys.net/18/545/2011/
561
for Time Series Analysis, Astron. J., 111, 555–566, 1996b.
Fraser, A. M. and Swinney, H. L.: Independent coordinates for
strange attractors from mutual information, Phys. Rev. A, 33,
1134–1140, 1986.
Gámez, A. J., Zhou, C. S., Timmermann, A., and Kurths, J.: Nonlinear dimensionality reduction in climate data, Nonlin. Processes
Geophys., 11, 393–398, doi:10.5194/npg-11-393-2004, 2004.
Gibson, J. F., Farmer, J. D., Casdagli, M., and Eubank, S.: An analytic approach to practical state space reconstruction, Physica D,
57, 1–30, 1992.
Hamilton, J. D.: Time Series Analysis Princeton University Press,
Princeton, 1994.
Hirata, Y., Shimo, Y., Tanaka, H. L., and Aihara, K.: Chaotic properties of the Arctic Oscillation Index, Scientific Online Letters
on the Atmosphere, 7, 033–036, 2011.
Jones, E., Oliphant, T. E., Petersen, P. and others: SciPy: Open
source scientific tools for Python, available at: http://www.scipy.
org/ (last access: 31 August 2011), 2011.
Kantz, H. and Schreiber, T.: Nonlinear Time Series Analysis, Cambridge University Press, Cambridge, 1997.
Kennel, M. B., Brown, R., and Abarbanel, H. D. I.: Determining
embedding dimension for phase-space reconstruction using a geometrical construction, Phys. Rev. A, 45, 3403–3411, 1992.
Lacasa, L., Luque, B., Ballesteros, F., Luque, J., and Nuño, J. C.:
From time series to complex networks: The visibility graph,
Proc. Natl. Acad. Sci. USA, 105, 4972–4975, 2008.
Larrasoaña, J. C., Roberts, A. P., Rohling, E. J., Winklhofer, A., and
Wehausen, R.: Three million years of monsoon variability over
the northern Sahara, Clim. Dynam., 21, 689–698, 2003.
Lomb, N. R.: Least-squares frequency analysis of unequally spaced
data, Astrophys. Space Sci., 39, 447–462, 1976.
Maraun, D., Kurths, J., and Holschneider, M.: Nonstationary Gaussian processes in wavelet domain: Synthesis, estimation, and significance testing, Phys. Rev. E, 75, 016707,
doi:10.1103/PhysRevE.75.016707, 2007.
Marwan, N., Thiel, M., and Nowaczyk, N. R.: Cross recurrence plot
based synchronization of time series, Nonlin. Processes Geophys., 9, 325–331, doi:10.5194/npg-9-325-2002, 2002.
Marwan, N., Trauth, M. H., Vuille, M., and Kurths, J.: Comparing
modern and Pleistocene ENSO-like influences in NW Argentina
using nonlinear time series analysis methods, Clim. Dynam., 21,
317–326, 2003.
Marwan, N., Romano, M. C., Thiel, M., and Kurths, J.: Recurrence
plots for the analysis of complex systems, Phys. Rep., 438, 237–
329, 2007.
Marwan, N., Donges, J. F., Zou, Y., Donner, R. V., and Kurths,
J.: Complex network approach for recurrence analysis of time
series, Phys. Lett. A, 373, 4246–4254, 2009.
Mudelsee, M., Scholz, D., Röthlisberger, R., Fleitmann, D.,
Mangini, A., and Wolff, E. W.: Climate spectrum estimation in
the presence of timescale errors, Nonlin. Processes Geophys., 16,
43–56, doi:10.5194/npg-16-43-2009, 2009.
Newman, M. E. J.: Assortative mixing in networks, Phys. Rev. Lett.,
89, 208701, doi:10.1103/PhysRevLett.89.208701, 2002.
Newman, M. E. J: The structure and function of complex networks,
SIAM Rev., 45(2), 167–256, 2003.
Nicolis, G., Garcı́a Cantú, A., and Nicolis, C.: Dynamical aspects
of interaction networks, Int. J. Bifurc. Chaos, 15, 3467–3480,
2005.
Nonlin. Processes Geophys., 18, 545–562, 2011
562
J. F. Donges et al.: Identification of dynamical transitions in marine palaeoclimate records by RN analysis
Oliphant, T. E.: Guide to NumPy, Brigham Young University,
Provo, UT, USA, available at: http://www.tramy.us/, 2006.
Packard, N. H., Crutchfield, J. P., Farmer, J. D., and Shaw, R. S.:
Geometry from a time series, Phys. Rev. Lett., 45, 712–716,
1980.
Poincaré, H.: Sur la problème des trois corps et les équations de la
dynamique, Acta Math., 13, A3–A270, 1890.
Ravelo, A., Andreasen, D., Lyle, M., Lyle, A., and Wara, M.: Regional climate shifts caused by gradual global cooling in the
Pliocene epoch, Nature, 429, 263–267, 2004.
Rehfeld, K., Marwan, N., Heitzig, J., and Kurths, J.: Comparison of correlation analysis techniques for irregularly sampled time series, Nonlin. Processes Geophys., 18, 389–404,
doi:10.5194/npg-18-389-2011, 2011.
Robinson, G. and Thiel, M.: Recurrences determine the dynamics,
Chaos, 19, 023104, doi:10.1063/1.3117151, 2009.
Scargle, J. D.: Studies in Astronomical Time Series Analysis II. Statistical Aspects of Spectral Analysis of Unevenly Spaced Data,
Astron. J., 15, 835–853, 1982.
Schinkel, S., Marwan, N., Dimigen, O., and Kurths, J.: Confidence
bounds of recurrence-based complexity measures, Phys. Lett. A,
373, 2245–2250, 2009.
Schulz, M. and Mudelsee, M.: REDFIT: Estimating red-noise
spectra directly from unevenly spaced paleoclimatic time series,
Comput. Geosci., 28, 421–426, 2002.
Schulz, M. and Stattegger, K.: SPECTRUM: Spectral analysis of
unevenly spaced paleoclimatic time series, Comput. Geosci., 23,
929–945, 1997.
Scott, D. W.: Multivariate Density Estimation: Theory, Practice,
and Visualization, Wiley, New York, 1982.
Sprott, J. C.: Chaos and Time Series Analysis, Oxford University
Press, Oxford, 2003.
Takens, F.: Detecting strange attractors in turbulence, in: Dynamical Systems and Turbulence, Warwick 1980, edited by: Rand, D.
and Young, L.-S., Springer, New York, 366–381, 1981.
Telford, R. J., Heegaard, E., and Birks, H. J. B.: All age-depth models are wrong: but how badly?, Quat. Sci. Rev., 23, 1–5, 2004.
Thiel, M., Romano, M. C., Read, P. L., and Kurths, J.: Estimation
of dynamical invariants without embedding by recurrence plots,
Chaos, 14, 234–243, 2004a.
Nonlin. Processes Geophys., 18, 545–562, 2011
Thiel, M., Romano, M. C., and Kurths, J.: How much information
is contained in a recurrence plot?, Phys. Lett. A, 330, 343–349,
2004b.
Thiel, M., Romano, M. C., and Kurths, J.: Spurious structures in
recurrence plots induced by embedding, Nonlin. Dyn., 44, 299–
305, 2006.
Tiedemann, R., Sarnthein, M., and Shackleton, N. J.: Astronomic
timescale for the Pliocene Atlantic δ 18 0 and dust flux records of
Ocean Drilling Program site 659, Paleoceanography, 9, 619–638,
1994.
Trauth, M. H., Bookhagen, B., Marwan, N., and Strecker, M. R.:
Multiple landslide clusters record Quaternary climate changes in
the northwestern Argentine Andes, Palaeogeogr. Palaeocl., 194,
109–121, 2003.
Trauth, M. H., Larrasoaña, J. C., and Mudelsee, M.: Trends,
rhythms and events in Plio-Pleistocene African climate, Quat.
Sci. Rev., 28, 399–411, 2009.
Trulla, L. L., Giuliani, A., Zbilut, J. P., and Webber Jr., C. L.: Recurrence quantification analysis of the logistic equation with transients, Phys. Lett. A, 223, 255–260, 1996.
van Rossum, G. and Drake, F. L.: Python reference manual (Python
Software Foundation), available at: http://docs.python.org/ref/
ref.html, 2006.
Wackerbauer, R., Witt, A., Atmanspacher, H., Kurths, J., and
Scheingraber, H.: A comparative classification of complexity
measures, Chaos Soliton. Fract., 4, 133–173, 1994.
Watts, D. J. and Strogatz, S. H.: Collective dynamics of “smallworld” networks, Nature, 393, 440–442, 1998.
Xu, X., Zhang, J., and Small, M.: Superfamily phenomena and motifs of networks induced from time series, Proc. Natl. Acad. Sci.
USA, 105, 19601–19605, 2008.
Yang, Y. and Yang, H.: Complex network-based time series analysis, Physica A, 387, 1381–1386, 2008.
Zhang, J. and Small, M.: Complex network from pseudoperiodic
time series: topology versus dynamics, Phys. Rev. Lett., 96,
238701, doi:10.1103/PhysRevLett.96.238701, 2006.
Zou, Y., Donner, R. V., Donges, J. F., Marwan, N., and Kurths,
J.: Identifying complex periodic windows in continuous-time
dynamical systems using recurrence-based methods, Chaos, 20,
043130, doi:10.1063/1.3523304, 2010.
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