Geometric signature of complex synchronisation scenarios

1.5 MB·English

by  Jan F. Feldhoff

epl draft
Geometric signature of complex synchronisation scenarios
J.H. Feldhoff1,2 (a), R.V. Donner1 , J.F. Donges1,2 , N. Marwan1 and J. Kurths1,2,3
1 Potsdam Institute for Climate Impact Research - P.O. Box 60 12 03, 14412 Potsdam, Germany
2 Department of Physics, Humboldt University - Newtonstr. 15, 12489 Berlin, Germany
3 Institute for Complex Systems and Mathematical Biology, University of Aberdeen, Aberdeen AB24 3FX, United
Kingdom
PACS 05.45.Tp–Time series analysis
3
1 PACS 05.45.Xt–Synchronization; coupled oscillators
0 PACS 89.75.Hc–Networks and genealogical trees
2
Abstract. - Synchronisation between coupled oscillatory systems is a common phenomenon

n in many natural as well as technical systems. Varying the strength of coupling often leads to
a
qualitative changes in the complex dynamics of the mutually coupled systems including different
J
types of synchronisation such as phase, lag, generalised, or even complete synchronisation. Here,
4
westudythegeometricsignaturesofcouplingalongwiththeonsetofgeneralisedsynchronisation

between two coupled chaotic oscillators by mapping the systems’ individual as well as joint
]
D recurrences in phase space to a complex network. For a paradigmatic continuous-time model
system, the transitivity properties of the resulting joint recurrence networks display distinct
C
variations associated with changes in the structural similarity between different parts of the
.
n consideredtrajectories. Theythereforeprovideausefulindicatorfortheemergenceofgeneralised
i synchronisation.
l
n
[ This paper is decidated to the 25th anniversary of the introduction of recurrence plots by

Eckmann et al. (EPL, 4 (1987), 973).
1
v
6
0
8
0
. Introduction. – The scientific interest in studying coupled systems evolve in an identical way with a fixed
1
and qualitatively understanding synchronisation between mutual time shift as y(t) = x(t−τ). Finally, phase syn-
0
coupledoscillatorysystemstracesbackatleasttoobserva- chronisation(PS)ischaracterisedbyalockingbetweenthe
3
1tionsofweaklycoupledpendulumclocksmadebyChristi- phases of two systems, mφy(t) = nφx(t) with m,n ∈ N,
:aanHuygensinthe17thcentury[1,2]. Inthelastdecades, whereas the amplitudes may remain uncorrelated. PS is
v
ia detailed quantitative knowledge of synchronisation phe- typical for non-identical chaotic systems at moderate or
X
nomenahasbeenestablishedforbothperiodicandchaotic even weak coupling strengths.
roscillators [3–7].
a Even though PS is typically considered a weaker form
In the study of synchronisation processes between cou-
of synchronisation than GS and CS, the sequence of syn-
pled chaotic oscillators, different phenomena have to be
chronisation phenomena arising for two coupled oscilla-
distinguished: First, complete synchronisation (CS) [6]
torswithincreasingcouplingstrengthdependscruciallyon
can occur for identical systems with a sufficiently strong
the structural differences between both systems as well as
coupling. Inthiscase,thetrajectoriesx(t)andy(t)ofthe
the specific coupling configuration. For several examples,
two coupled systems become identical, i.e. y(t) = x(t).
it has been demonstrated that as the coupling becomes
Second, generalised synchronisation (GS) [8–11] refers to
stronger, the dynamics of both systems becomes succes-
a general fixed and deterministic functional relationship
sivelysynchronisedonmoreandmorecharacteristictime-
betweenbothtrajectories,y(t)=f(x(t)),wheref isadif-
scales [13,14]. This observation implies the emergence of
feomorphism. This type occurs for non-identical chaotic
PS at relatively low coupling strengths if the characteris-
oscillators at rather high coupling strengths. A closely
tic oscillation frequencies of both systems are sufficiently
related form is lag synchronisation (LS) [12], where both
similar, whereas LS, GS, and CS typically occurat higher
(a)E-mail: [email protected] couplingstrengths. However,undercertainconditionsthe
p-1
J.H. Feldhoff et al.
observedsynchronisationscenariocanbemuchmorecom- proaches also to more general problems of data analysis,
plex. suchasforinferingso-calledfunctionalnetworksfrommul-
A formal description of GS has been given first only tivariate time series [24,25] or characterising structural
for driver-response systems. However, the typical signa- properties of time series. For the latter purpose, a mul-
tures of GS are not unique to systems with unidirectional tiplicity of complementary approaches (such as cycle net-
coupling, but can be observed in a similar way also for works,visibilitygraphs,andrecurrencenetworks,tomen-
symmetrically coupled oscillators [15] (in fact, GS has tion only a few) have been proposed [26–30], which trans-
beenfirstdescribedfortwosystemswithbidirectionalcou- form certain types of time series to the complex network
pling [5]). domain (see [31] for a corresponding review).
Fordriver-responserelationships,thedrivensystemun- Among the currently existing types of time series net-
dergoes systematic changes in its dynamical evolution as works, approaches characterising the mutual proximity of
the coupling strength is increased. These changes can state vectors in phase space have recently become partic-
be detected by a variety of methods such as the auxil- ularlypopular,sincetheresultingnetworks’structuresre-
iary system method [9], numerical estimates of the con- flectnontrivialgeometricpropertiesofthesupposed(low-
ditional Lyapunov exponent of the response system as a dimensional) attractor underlying the observed dynam-
measure for its asymptotic stability [10], or by quanti- ics. A particularly interesting representative of this class
fying the predictability of the driven system. The lat- arerecurrencenetworks[30–32], whichencodethemutual
ter class of approaches includes several methods based on proximity of state vectors in phase space arising from dy-
neighbourhood relationships in phase space (i.e., relying namical recurrences in the sense of Poincar´e. Specifically,
on geometric information), with the mutual false near- defining neighbourhoods of individual states by consider-
est neighbour method [8] and the synchronisation likeli- ing a spatial threshold distance ε around each observed
hood [16] as most prominent examples. However, there state vector offers a flexible way for capturing the geo-
are specific cases (such as systems exhibiting multistabil- metricbackboneofthesystemunderstudy. Formally,the
ity)whereatleastsomeofthesemethodsarenotapplica- system’s finite-time recurrence properties infered from a
ble. Even more, they are not always directly transferable time series {xi}Ni=1 are then represented by the binary re-
to the case of bidirectional coupling configurations. Con- currence matrix
sequently, recentlyseveralnewapproacheshavebeenpro-
R (ε)=Θ(ε−(cid:107)x −x (cid:107)), (1)
posedforstudyingGSforbothuni-andbidirectionalcou- ij i j
plings, including methods based on recurrence plots [17] where (cid:107)·(cid:107) is the maximum norm in phase space (how-
orageneralisedanglebetweenthetrajectoriesofbothsys- ever, other norms could be used here as well), and Θ(·)
tems [18]. denotes the Heaviside function. The visualisation of this
Since synchronisation has distinct effects on the spa- symmetricmatrixiscommonlyreferedtoastherecurrence
tial organisation of driven systems in phase space, geo- plot [33] and has become an important tool for nonlinear
metric methods are promising and widely applicable can- time series analysis in the last decades [34].
didates for detecting GS from time series data. In this Recurrence network (RN) analysis reinterprets matrix
workweproposeanewgeometricmethodbasedonacom- (1) as the adjacency matrix of an undirected simple
plex network approach. Specifically, our method is based graph [30] by setting
on a graph-theoretical interpretation of joint recurrence
A (ε)=R (ε)−δ (2)
plots describing the simultaneous occurrence of mutually ij ij ij
close pairs of state vectors in two or more coupled sys-
(where δ is Kronecker’s delta). The resulting networks’
ij
tems [19]. After presenting the details of our approach,
properties can be described analytically by interpreting
the paradigmatic example of two coupled R¨ossler oscilla-
RNs as random spatial graphs with the system’s invari-
tors in different dynamical regimes are discussed in order
antdensityuniquelydeterminingRNconnectivity[35,36].
to demonstrate the performance of the proposed method.
This solid theoretical foundation allows using RNs as a
The obtained results indicate that the complex network-
widely applicable tool for detecting dynamical changes in
based approach has great potentials and may be applied
non-stationary time series from mathematical models as
to real-world problems in future work.
well as real-world applications [37–39]. One basic, yet im-
portant characteristic of RNs is the edge density
Methodology. – In the last decades, there has been
a rising interest in studying structural properties of net- 1 (cid:88)
ρ(ε)= A (ε), (3)
worked systems across disciplines [20,21]. For this pur- N(N −1) ij
i,j
pose, a variety of measures have been introduced that
quantitatively characterise different aspects of the ob- which is a monotonous function of ε [40]. In general, a
served networks’ complex connectivity properties [22,23]. suitableresolutionofsmall-scalenetworkfeaturesrequires
Besides investigating real-world systems with a physical thechoiceofalowedgedensity,typicallyρ(cid:46)0.05[36,40],
network substrate, recently there have been various suc- which still meets the condition ρ > ρ(ε ) where ε is the
c c
cessful attempts to applying complex network-based ap- percolation threshold of the RN [36].
p-2
Geometric signature of complex synchronisation scenarios
It has been shown that among others, the transitivity PS PS
y(t)
properties of RNs provide some particularly useful char- X Y
acteristics measuring the effective local as well as global
dimensionality of the underlying dynamical system [35].
Since in a given integer-dimensional manifold, the transi-
tivitypropertiescanbecomputedanalytically[41],gener- y
i
alising the corresponding relationship to non-integer val-
ues allows defining the so-called transitivity dimension x εy
i
logT(ε)
D (ε)= (4) y
T log(3/4) j
ε x
with the network transitivity [21,22] X j
x(t)
(cid:80)
A (ε)A (ε)A (ε)
T(ε)= i,j,k ij jk ki . (5)
(cid:80)
A (ε)A (ε)
i,j,k ij ki
Figure 1: Schematic illustration of a joint recurrence of two
OtherglobalRNcharacteristics(e.g., averagepathlength
dynamical systems X and Y in their respective phase spaces
L or global clustering coefficient C) have also proven their
PS and PS .
X Y
capabilities[30,31],butshallnotbefurtherdiscussedhere
for brevity. Specifically, the dimensionality interpretation
ofT willbethefoundationoftheapproachtostudyingge- Unlike IRNs, JRNs can be constructed for systems with
ometricsignaturesofsynchronisationthatwillbedetailed different phase spaces, but explicitly require simultaneous
below. observations of all systems and, thus, time series of the
MotivatedbythegreatpotentialsofRNanalysisofindi- same length. The system-specific thresholds ε should be
k
vidualdynamicalsystems, athoroughextensiontostudy- chosen such that the edge densities ρ (ε ) of the RNs of
k k
ingthecollectivedynamicsoftwoormorecoupledsystems the individual systems are fixed at the same value ρ. By
has been recently proposed in terms of inter-system re- varying ρ, a desired edge density ρ can be obtained for
J
currence networks (IRNs) [42]. IRNs link the RNs of two the resulting JRN.
different systems using the cross-recurrence matrix [43] The properties of JRNs can be widely interpreted in
a similar way as those of a classical RN in the higher-
CR (ε)=Θ(ε−(cid:107)x −y (cid:107)) (6)
ij i j
dimensionalphasespaceofthecomposedsystem,withthe
between the associated time series {x } and {y }. Asym- exceptionthatspatialproximityisevaluatedseparatelyin
i j
metries in the resulting cross-transitivity measures for in- thesubspacesbelongingtotheindividualsystems. Specifi-
terdependent networks [44] have unveiled geometric sig- cally,thetransitivityT(ε ,ε )(cf.Eq.(5))oftheJRN(or,
x y
natures of unidirectional couplings between structurally for short, joint transitivity T ) provides a measure for the
J
similarsystems[42]. NotethatIRNsdonotrequiresimul- “joint dimensionality” of the composed dynamical system
taneousobservationsoftheconsideredsystems,butimply (in a general sense). Due to the larger effective dimen-
that they share the same phase space – a condition which sionality of the joint phase space, for two systems X and
is often not met in real-world applications. Moreover, the Y in the absence of synchronisation we expect T ,T (cid:29)
X Y
application of IRNs is restricted to studying weakly uni- T (specifically, D = logT /log(3/4) ≈ D + D ,
J TJ J TX TY
directionally coupled systems before the onset of GS [42]. cf. Eq. (4)). In turn, when both systems become syn-
In order to study the geometric signatures accompany- chronised (in the sense of GS), the effective degrees of
ing the onset of GS in some more detail, we suggest util- freedom of both systems become mutually locked, result-
isinganothermultivariategeneralisationoftherecurrence ing in T → T ,T , i.e., both systems acting collectively
J X Y
plot concept, joint recurrence plots [19], capturing the si- as one with the same effective dimensionality as the indi-
multaneous occurrence of recurrences in two dynamical vidual systems. Consequently, studying the ratio between
systems (Fig. 1). The underlying joint recurrence matrix the joint transitivity and some mean value (e.g., arith-
metic or geometric mean) of the individual systems’ RN
JR (ε ,ε )=Θ(ε −(cid:107)x −x (cid:107))Θ(ε −(cid:107)y −y (cid:107)) (7)
ij x y x i j y i j transitivities can provide a useful geometric indicator for
the emergence of GS. Here, we use the arithmetic mean,
is defined as the point-wise product of the individual sys-
since it is never smaller than the geometric one, implying
tems’ recurrence matrices. This basic concept can be di-
a higher discriminatory power of the transitivity ratio
rectly generalised to the study of K >2 coupled systems.
Next, as for the traditional RN we reinterpret the joint
T
recurrence matrix as the adjacency matrix of a so-called Q = J (9)
T (T +T )/2
joint recurrence network (JRN) by setting X Y
A (ε ,ε )=JR (ε ,ε )−δ . (8) which we will use as an indicator for GS in the following.
ij x y ij x y ij
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J.H. Feldhoff et al.
Results. – As a well-studied paradigmatic model ex- The results are qualitatively similar when studying the
hibiting different types of synchronisation phenomena, same coupling configuration with both systems being in
we study the performance of JRNs for two non-identical the non-coherent funnel regime (Fig. 2B). Here, λ be-
4
R¨ossler oscillators [45] comes negative already at rather low coupling strengths,
whereas λ < 0 for µ ≈ 0.3. The corresponding
3 XY
x˙1 =−(1+ν)x2−x3 emergence of GS is accompanied by a continuous tran-
x˙2 =(1+ν)x1+ax2+µYX(y2−x2) sition of JPR and QT to values of about 1, where the
JRN-based index again shows better convergence. More-
x˙ =b+x (x −c)
3 3 1
(10) over,forsymmetricbidirectionallycoupledfunnelsystems
y˙ =−(1−ν)y −y
1 2 3 (µ = µ = µ, Fig. 3B), we observe qualitatively the
XY YX
y˙2 =(1−ν)y1+ay2+µXY(x2−y2) samebehaviourasintheunidirectionalcase,withthecor-
y˙ =b+y (y −c) respondingtransitionstakingplaceatsmallervaluesofthe
3 3 1
coupling strength (i.e., λ < 0 for µ (cid:38) 0.02 and λ < 0
4 3
that are diffusively coupled via their second component for µ (cid:38) 0.17 [17]). Notably, the transition to GS as un-
as in [17]. The detuning parameter ν accounts for a fre- veiledbybothJPRandQ isevenmorecontinuousthan
T
quency mismatch of the nonlinear oscillations exhibited for unidirectional coupling, and the residual difference of
by both systems, making them non-identical and, thus, JRN totheidealvalueof1closetothetransitionpointis
prone to GS. In the following, we will keep it fixed at even larger, whereas Q is at the same time already very
T
ν = 0.02. In turn, for the characteristic parameters a, b close to 1.
and c, we will mainly keep the same parameters for both Studying two symmetrically coupled phase-coherent
systems throughout all further considerations. Specifi- R¨ossler systems (Fig. 3A), the synchronisation scenario
cally, we study the geometric signatures of the complex gets more complex due to the emergence of periodic win-
synchronisation scenarios arising for two different dynam- dows as µ is increased. Specifically, we find a first very
ical regimes with chaotic oscillations: the phase-coherent small periodic window (λ = 0,λ < 0) close to the
1 2
regime (a = 0.16,b = 0.1,c = 8.5) and the non-phase- onset of PS at about µ = 0.039, where CPR shows a
coherent funnel regime (a = 0.2925,b = 0.1,c = 8.5). sharp rise to values close to 1 [17]. Note that in this
In all simulations, we integrate the coupled system with as well as other supposedly periodic windows, we observe
step size h = 0.001 until T = 6,000, i.e., for 6,000,000 T ,T ,T ≈ 0.75 consistent with the expected attrac-
X Y J
time steps. The first 5,000,000 time steps are discarded tor dimension of 1. At µ (cid:38) 0.04, also JPR and Q in-
T
to avoid even very long transients, and the remaining crease considerably, reaching plateau values in the phase-
data are downsampled by a factor of 200 to obtain RNs synchronised chaotic regime [17]. For µ∈[0.08,0.092], we
and JRNs with N = 5,000 vertices. For comparative findanothermoreextendedwindowofapparentlyperiodic
purposes, we also consider the recurrence plot-based in- dynamics λ < 0 that probably corresponds to a regime
2
dices CPR and JPR [17]. CPR, an indicator for PS, of intermittent lag synchronisation [17], whereas for even
is based on the cross-correlation between the two sys- largercouplingstrength,weinferthepresenceofGSsince
tems’ recurrence-based generalised auto-correlation func- λ <0, JPR≈1 and Q ≈1.
3 T
tions, whereas JPR attempts to identify synchronisation Inordertoillustratethelimitationsoftheproposedap-
by comparing the systems’ RN edge densities ρX,Y to ρJ, proach as well as the established JPR index, we finally
which are conjectured to become very similar in case of studythesignaturesofsynchronisationforthemixedcase
GS. of a phase-coherent R¨ossler system X coupled to a funnel
Letusfirstexamineaunidirectionalcouplingconfigura- oscillator Y, which can be taken as an example for syn-
tion X → Y (µYX = 0), where the slightly faster system chronisation between two strongly non-identical systems.
X drivesthesloweroneY. Fortwophase-coherentR¨ossler In the case of unidirectional coupling (Fig. 2C), we find
systems (Fig. 2A), we find that when increasing the cou- λ < 0 for µ (cid:38) 0.1 accompanied by an increase of
4 XY
pling strength µXY beyond about 0.06, the fourth-largest CPR indicating the presence of PS. At µXY ≈ 0.3, also
Lyapunov exponent λ4 gets negative. At about the same λ3 approaches negative values, whereas both JPR and
value,CPRstartsincreasingindicatingthepossibleonset Q increase gradually, but do not approach values close
T
of PS. At µXY ≈ 0.17, the third-largest Lyapunov expo- to1evenforrelativelyhighcouplingstrengths. Thus, the
nentλ3becomessignificantlynegative,whichiscommonly threepossibleGSindicatorsλ3, JPR andQT donotgive
interpreted as an indicator for the onset of GS. At the consistent results in this case, which implies that we can-
sametime,weobserveasharpriseofJPRtowardsvalues not unambigously conclude the possible presence of GS
ofabout0.8,whereasQT immediatelyrisestovaluesclose here. A similar statement holds for the bidirectional case
to 1 (in fact, there is some weak overshooting because the (Fig. 3C), which displays again a more complicated se-
edgedensitiesofRNsandJRNdiffernecessarily,resulting quence of transitions.
inslightlydifferentestimatesofthecorrespondingnetwork
transitivities). This finding suggests that Q has better Conclusions. – 25 years after the introduction of re-
T
capabilities for detecting GS than JPR. currence plots by Eckmann et al. [33], the development
p-4
Geometric signature of complex synchronisation scenarios
A B C
Figure 2: Results of synchronisation analysis for two unidirectionally (X →Y) coupled Ro¨ssler systems being (A) both in the
phase-coherent regime, (B) both in the funnel regime, and (C) in phase-coherent (X) and funnel regime (Y): the four largest
Lyapunov exponents λ ,...,λ estimated using the Wolf algorithm [46] (using N = 9,000,000 data points from simulations
1 4
with step size h = 0.001 starting at T = 1,000); recurrence-based synchronisation indices CPR (black) and JPR (grey) [17];
transitivities of individual and joint recurrence networks T (dark grey), T (light grey) and T (black); transitivity ratio Q
X Y J T
(from top to bottom).
of recurrence-based techniques still continues. With this studying the Lyapunov spectrum.
paper, we would like to honour the seminal work by these In general, the proposed method has the advantage of
authors and show how the paradigms of recurrence plots being applicable to relatively short time series of length
and complex networks have meanwhile been combined to N (cid:46) O(103), whereas the computational requirements
pave the way for new concepts in complex systems analy- for numerically estimating the Lyapunov spectrum are
sis. far higher and usually not met in case of real-world ap-
We have introduced a new fully geometric approach plications. In this spirit, we conclude that the proposed
for detecting GS between two coupled chaotic oscillators method has great potentials for future applications to ob-
based on time series data. Specifically, we have used a servational data from different fields of research.
complex network interpretation of the recurrence as well
as joint recurrence matrices, the transitivity properties of ∗∗∗
whichallowtracingchangesintheeffectivedimensionality
of the individual system’s attractors as well as the attrac- This work has been financially supported by the IRTG
tor of the combined higher-dimensional system. We have 1740/TRP2011/50151-0(fundedbytheDFG/FAPESP),
demonstrated that in the presence of GS, both systems the Leibniz Society (project ECONS), the German Na-
effectively behave as one, with the dimensionality of the tional Academic Foundation, and the Potsdam Research
composedsystemapproachingthatoftheindividualones. Cluster for Georisk Analysis, Environmental Change and
For the case of structurally similar systems such as Sustainability (PROGRESS, support code 03IS2191B).
slightly detuned chaotic oscillators with otherwise equal For calculations of complex network measures, the soft-
characteristic parameters, we have shown that the ratio ware package pyunicorn has been used on the IBM iDat-
Q betweentheJRNtransitivityandthearithmeticmean aPlexClusterofthePotsdamInstituteforClimateImpact
T
of the individual systems’ RNs transitivities quickly ap- Research.
proaches a value of 1 as GS takes place. In comparison
with the conceptually related (i.e., recurrence plot-based)
References
measure JPR, our geometric characteristic Q displays
T
a better convergence towards the value of 1 expected for [1] Pikovsky A., Rosenblum M.andKurths J.,Synchro-
GS.Inturn,forstructurallydifferentsystems,bothindices nization – A Universal Concept in Nonlinear Sciences
fail to clearly indicate the transition to GS as unveiled by (Cambridge University Press) 2001.
p-5
J.H. Feldhoff et al.
A B C
Figure 3: As in Fig. 2 for bidirectional coupling.
[2] Boccaletti S., Kurths J., Osipov G., Valladares [20] Albert R. and Barabasi A.-L., Rev. Mod. Phys. , 74
D. L. and Zhou C. S., Phys. Rep. , 366 (2002) 1. (2002) 47.
[3] Fujisaka H. and Yamada T., Prog. Theor. Phys. , 69 [21] Newman M. E. J., SIAM Rev. , 45 (2003) 167.
(1983) 32. [22] Boccaletti S., Latora V., Moreno Y., Chavez M.
[4] Kuramoto Y., Prog. Theor. Phys. Suppl. , 79 (1984) and Hwang D.-U., Phys. Rep. , 424 (2006) 175.
223. [23] da F. Costa L., Rodrigues F. A., Travieso G. and
[5] Afraimovich V. S., Verichev N. N. and Rabinovich Boas P. R. V., Adv. Phys. , 56 (2007) 167.
M. I., Radiophys. Quant. Electron. , 29 (1986) 795. [24] Zhou C., Zemanova L., Zamora G., Hilgetag C. C.
[6] Pecora L. M. and Carroll T. L., Phys. Rev. Lett. , and Kurths J., Phys. Rev. Lett. 97, 2006 (238103) .
64 (1990) 821. [25] Donges J. F., Zou Y., Marwan N. and Kurths J.,
[7] Rosenblum M. G., Pikovsky A. S. and Kurths J., Europhys. Lett. 87, 2009 (48007) .
Phys. Rev. Lett. , 76 (1996) 1804. [26] Zhang J. and Small M., Phys. Rev. Lett. 96, 2006
[8] Rulkov N. F., Sushchik M. M., Tsimring L. S. and (238701) .
Abarbanel H. D. I., Phys. Rev. E , 51 (1995) 980. [27] Xu X., Zhang J. and Small M., Proc. Natl. Acad. Sci.
[9] Abarbanel H. D. I., Rulkov N. F. and Sushchik USA , 105 (2008) 19601.
M. M., Phys. Rev. E , 53 (1996) 4528. [28] Lacasa L., Luque B., Ballesteros F., Luque J. and
[10] KocarevL.andParlitzU.,Phys.Rev.Lett.,76(1996) Nuno J. C., Proc. Natl. Acad. Sci. USA , 105 (2008)
1816. 4972.
[11] Boccaletti S., Valladares D. L., Kurths J., Maza [29] Yang Y. and Yang H., Physica A , 387 (2008) 1381.
D. and Mancini H., Phys. Rev. E , 61 (2000) 3712. [30] MarwanN.,DongesJ.F.,ZouY.,DonnerR.V.and
[12] Rosenblum M. G., Pikovsky A. S. and Kurths J., Kurths J., Phys. Lett. A , 373 (2009) 4246.
Phys. Rev. Lett. , 78 (1997) 4193. [31] Donner R. V., Small M., Donges J. F., Marwan N.,
[13] Hramov A. E. and Koronovskii A. A., Chaos , 14 Zou Y., Xiang R. and Kurths J., Int. J. Bifurcation
(2004) 603. Chaos , 21 (2011) 1019.
[14] HramovA.E.andKoronovskiiA.A.,PhysicaD,206 [32] DonnerR.V.,ZouY.,DongesJ.F.,MarwanN.and
(2005) 252 . Kurths J., New J. Phys. , 12 (2010) 033025.
[15] Zheng Z., Wang X. and Cross M. C., Phys. Rev. E , [33] EckmannJ.-P.,KamphorstS.O.andRuelleD.,Eu-
65 (2002) 056211. rophys. Lett. , 4 (1987) 973.
[16] Stam C.andvan Dijk B.,Physica D ,163(2002)236. [34] MarwanN.,RomanoM.C.,ThielM.andKurthsJ.,
[17] Romano M. C., Thiel M., Kurths J., Kiss I. Z. and Phys. Rep. , 438 (2007) 237.
Hudson J. L., Europhys. Lett. , 71 (2005) 466. [35] Donner R. V., Heitzig J., Donges J. F., Zou Y.,
[18] Liu Z., Zhou J. andMunakata T., Europhy. Lett. , 87 Marwan N.andKurths J.,Eur.Phys.J.B,84(2011)
(2009) 50002. 653.
[19] Romano M. C., Thiel M., Kurths J. and von Bloh [36] Donges J. F., Heitzig J., Donner R. V. and Kurths
W., Phys. Lett. A , 330 (2004) 214 . J., Phys. Rev. E , 85 (2012) 046105.
p-6
Geometric signature of complex synchronisation scenarios
[37] Donges J. F., Donner R. V., Rehfeld K., Marwan
N., Trauth M. H. and Kurths J., Nonlin. Proc. Geo-
phys. , 18 (2011) 545.
[38] DongesJ.F.,DonnerR.V.,TrauthM.H.,Marwan
N., Schellnhuber H. J. and Kurths J., Proc. Natl.
Acad. Sci. USA , 108 (2011) 20422.
[39] Zou Y., Donner R. V., Wickramasinghe M., Kiss
I. Z., Small M. and Kurths J., Chaos , 22 (2012)
033130.
[40] DonnerR.V.,ZouY.,DongesJ.F.,MarwanN.and
Kurths J., Phys. Rev. E , 81 (2010) 015101(R).
[41] Dall J.andChristensen M.,Phys. Rev. E ,66(2002)
016121.
[42] Feldhoff J. H., Donner R. V., Donges J. F., Mar-
wan N.andKurths J.,Phys. Lett. A ,376(2012)3504
.
[43] Marwan N. and Kurths J., Physics Letters A , 302
(2002) 299.
[44] Donges J. F., Schultz H. C. H., Marwan N., Zou Y.
and Kurths J., Eur. Phys. J. B , 84 (2011) 635.
[45] Ro¨ssler O. E., Phys. Lett. A , 57 (1976) 397.
[46] Wolf A., Swift J. B., Swinney H. L. and Vastano
J. A., Physica D , 16 (1985) 285.
p-7