Stochastic phenomena of synchronization in ensembles of mean-field coupled limit cycle oscillators with two native frequencies

0.6 MB·English

by  K. Okumura

epl draft
Stochastic phenomena of synchronization in ensembles of mean-
field coupled limit cycle oscillators with two native frequencies
K. Okumura(a), A. Ichiki and M. Shiino
1
Department of Physics, Faculty of Science, Tokyo Institute of Technology, 2-12-1 Oh-okayama, Meguro-ku, Tokyo
1
152-8551, Japan
0
2

n
a PACS 05.45.Xt–Synchronization; coupled oscillators
J
PACS 05.10.Gg–Stochastic analysis methods (Fokker-Planck,Langevin, etc.)
1
PACS 05.70.Fh–Phase transitions: general studies

] Abstract. - We study effects of independent white noise on synchronization phenomena in
h
c ensemblesofcoupledlimitcycleoscillatorswithdifferentnativefrequencies. Weconsiderasimple
e model where the ensemble consists of two inter-connected clusters with own native frequencies
m and mean-field couplings are introduced between intra- and inter-clusters. Taking advantage
- of nonlinear mean-field coupling concept together with the law of large numbers valid in the
t thermodynamic limit, we employ a nonlinear Fokker-Planck equation approach that turns out
a
t to be noise level-free analysis, to analytically derive the time evolution of the order parameters.
s
Showing the occurrence of bifurcations from chaotic attractors in the deterministic limit to limit
.
t cycle ones with increasing noise intensity, we confirm the occurrence of nonequilibrium phase
a
m transitions including inter-cluster synchronization induced byexternal noise.
-
d
n
o
c
[ Introduction. – Synchronization phenomena are oscillatorshasbeenreported[4]. Incontrast,the opposite

ubiquitous ones observed in many fields of natural sci- phenomena of noise-induced synchronization are becom-
1
ences [1]. In particular, synchronization phenomena of ing an active field of the study of nonlinear dynamical
v
6limit cycle oscillatorswith different native frequencies are systems [5–13]. Synchronization phenomena induced by
4of paramountimportance in studies of nonlinear dynamic independent noiseincoupledexcitable systems(including
3systems and neurosciences. In neurosciences, neurons in active rotator models) have been investigated both ana-
0
the basal ganglia, which consists of several types of neu- lytically [5,6] and numerically [7]. A sparsely connected
.
1ronswithdifferentfiringfrequencies[2],exhibitmoresyn- networkofintegrate-and-fireneuronswithinhibitory cou-
0
chrony in Parkinson’s disease than in normal state, sug- plings has been found to exhibit noise induced synchro-
1
gesting that neural information coding associated with nization[8]. Among analyticalstudies onsynchronization
1
:action selections of motor controls is closely related to phenomena of ensembles of stochastic limit cycle oscilla-
v
the synchronization phenomena (Ref. [3] and references tors is the phase reduction analysis [9–13]. Such type of
i
Xtherein). For sucha reason,to study howthe noise exerts recent studies reveal the effects of common noise on syn-
rits influence onthe structure ofsynchronizationwill be of chronization of uncoupled oscillators [12] and uncoupled
agreatimportance fromthe viewpointofnonlineardynam- two populations of oscillators [13]. These studies, how-
ical controls involving changes in synchrony of oscillatory ever, are restricted to the case with weak noise. As far
systems. as we know, there are very few papers that concern an-
Relationships between effects of noise and synchroniza- alytical studies of the relationships between noise effects
tion phenomena have been studied using various types of and synchronization in the case of coupled general types
models. It can be intuitively supposed that the presence of limit cycle oscillators.
ofnoisemightdeterioratethedegreeofsynchronizationof We address the issue of synchronization phenomena of
oscillatorysystems. Breakdownofcoherenceandsynchro- coupled limit cycle oscillators, which is supposed to be
nizationduetoexternalnoiseinanensembleoflimitcycle subjected to independent noise. In stochastic systems of
a finite number of coupled limit cycle oscillators with ex-
(a)E-mail: [email protected] ternal noise, each oscillator behaves randomly under the
p-1
K.Okumura, A. Ichiki and M. Shiino
influenceofnoise. ThecorrespondingFokker-Planckequa-
tions involving all of the variables that describe the sys-
tems of many body oscillators are linear, as one of the
standard master equations for Markovian dynamics. In
this case, the probability densities of the systems as their
solutions,ingeneral,exhibitergodicpropertytosettleinto
equilibrium(i.e. fixedpointtype)probabilitydensitiesfor
sufficientlylargetimes. Thisimpliesthattheorderparam-
eters do not oscillate.
To consider the synchronization phenomena in systems
of coupled stochastic oscillators by overcoming the prob-
Fig. 1: Thelimit cycleattractors exhibitedbythebasicunits.
lem mentioned above, one may introduce the concept of
The limit cycles show “loops” undersome parameter settings.
taking the thermodynamic limit based on a mean-field By adjusting the detuning parameter δ(y), we can change the
model. Taking advantage of these issues, it is useful to nativefrequencyand/orthenumberofloopsofthelimitcycle.
employthenonlinearFokker-Planckequation(NFPE)ap-
proach [14–17]. In general, probability densities of the
to transfer functions in analog neural networks [23–26].
limit cycle oscillator systems might be multimodal under
The oscillator can be of limit cycle if the functions are
the influence of weak external noise. In some cases, how-
appropriatelychosen. We here specify the nonlinearfunc-
ever,Gaussianapproximationhasbeenshowntobeuseful
tions as F(µ)(x) = sinx. Conducting a linear stabil-
toqualitativelyunderstandphasetransitionsinvolvingbi-
ity analysis of fixed points, we verify the occurrence of
furcations from non-oscillatory to oscillatory states with
Hopf bifurcations under certainconditions to bring about
changes in noise intensity [18].
limit cycle attractors. We set the parametervalues as fol-
Anotherwaytoavoidthedifficultyofdealingwithmul-
lows: a(x) = 0.5, a(y) = 1.0, b(x,x) = 0.5, b(x,y) = 1.5,
timodalprobabilitydensities is to restrictmodels to some
b(y,x) =−12.0,b(y,y) =−1.0, J(x) =18.0, J(y) =30.0. To
extent. Assuming that each element in the system is es-
describe limit cycle oscillators with different native fre-
sentiallygovernedbythe vectorfieldwithaquadraticpo-
quencies, we introduce detuning parameter δ(y) such that
tential except for nonlinear terms representing mean-field
a(y) = 1.0+δ(y). The limit cycle attractors appear for
couplings with other elements, one can take advantage of
δ(y) ∈(−0.4485,19.35)(Fig. 1).
using NFPEs to exactly derive the time evolution of the
We then consider a system of coupled limit cycle oscil-
order parameters in the thermodynamic limit for systems
lators consisting of two clusters. The oscillators are as-
ofnonlinearlymean-fieldcoupledoscillators. TheNFPEs,
sumedto be coupledvianonlinearglobalinteractionsand
whichmayexhibitbifurcations,haveturnedouttobeuse-
subjected to independent additive and multiplicative ex-
ful for studying the occurrence of chaos-nonchaos phase
ternal noise. The model dynamics we are concerned with
transitions [19–21].
isdescribedbyasetofLangevinequationsusingthebasic
In this letter, we study the influence of external noise
oscillator units mentioned above [19–22]:
on synchronization in globally nonlinear coupled oscilla-
for site i in cluster 1 (i=1,···,N ),
tors with two native frequencies. We focus our attention 1
on the inter-cluster synchronization phenomena. Using dz(x) N1
the NFPE of mean-field coupled oscillators,we derive the 1i =−a(x)z(x)+ J(1,x)F(x)(b(x,x)z(x)+b(x,y)z(y))
dt 1 1i ij 1 1 1j 1 1j
time evolution of the order parameters without any ap- j=1
X
proximations. Investigating so obtained ordinary differ- N2
ential equations for the order parameters instead of a set +ǫ K(2,x)F(x)(b(x,x)z(x)+b(x,y)z(y))+η(x)(t), (1)
ik 2 2 2k 2 2k 1i
of the Langevin equations, we confirm the occurrence of k=1
X
nonequilibrium phase transitions involving changes from dz(y) N1
synchronousto asynchronousstates with changesin noise 1i =−a(y)z(y)+ J(1,y)F(y)(b(y,x)z(x)+b(y,y)z(y))
dt 1 1i ij 1 1 1j 1 1j
strength. Part of this work has been briefly reported in j=1
X
the conference proceedings [22]. N2
+ǫ K(2,y)F(y)(b(y,x)z(x)+b(y,y)z(y))+η(y)(t), (2)
ModelandnonlinearFokker-Planckequationap- ik 2 2 2k 2 2k 1i
k=1
proach. – To study globally coupled limit cycle oscil- X
lators with different native frequencies, first let us con- for site i in cluster 2 (i=1,···,N2),
sider basic limit cycle oscillator units. We define a ba-
sic two dimensional oscillator of z(x),z(y) with the form dz2(xi) =−a(x)z(x)+ N2 J(2,x)F(x)(b(x,x)z(x)+b(x,y)z(y))
dz(x)/dt = −a(x)z(x) + J(x)F(x)(b(x,x)z(x) + b(x,y)z(y)), dt 2 2i ij 2 2 2j 2 2j
j=1
dz(y)/dt = −a(y)z(y) + J(y)F(y)(b(y,x)z(x) + b(y,y)z(y)), X
N1
wherea(µ),b(µ,ν),J(µ)(µ=x,y)areconstantsandF(µ)(·) +ǫ K(1,x)F(x)(b(x,x)z(x)+b(x,y)z(y))+η(x)(t), (3)
are functions specifying the nonlinearity that correspond ik 1 1 1k 1 1k 2i
k=1
X
p-2
Stochastic inter-cluster synchronization
dz2(yi) =−a(y)z(y)+ N2 J(2,y)F(y)(b(y,x)z(x)+b(y,y)z(y)) hFα(µ)2i≡ dzαFα(µ)2(b(αµ,x)zα(x)+b(αµ,y)zα(y))Pα(t,zα). (7)
dt 2 2i ij 2 2 2j 2 2j Z
j=1
X The total number of dynamic variables of the system is
+ǫ N1 K(1,y)F(y)(b(y,x)z(x)+b(y,y)z(y))+η(y)(t), (4c)onsequently reduced from 2(N1 +N2) to 4 in the ther-
ik 1 1 1k 1 1k 2i modynamic limit. Thus one has a set of the Langevin
Xk=1 equations as dz(µ)/dt = −a(µ)z(µ) + J(α,µ)hF(µ)i +
α α α α
where z(µ)(µ = x,y)(α = 1,2) are the real valued- ǫK(α¯,µ)hF(µ)i + η′(µ)(t), where the original noise is
αi α¯ α
dynamical variables of the 2D-oscillators at site i of clus- transformed to effective white Gaussian noise η′(µ) as
α
aterrecαonusntdaenrtsthanednFatα(uµr)a(·l)baoruenndoanrlyinecaorndcoituiopnlisn,gaf(αuµn),ctbiα(oµn,νs). hDη(′µ(αµ))(=t)iD=(µ)0,+hηD˜′(α(µµ))((th)Fη′(β(µν))2(it′+)iǫ=2hF2D(µe)(f2µfαi))δ.µνTδαhβeδ(ctou−ptle′)d,
Themean-fieldintra-andinter-clustercouplingstrengths effα α α¯
J(α,µ)(t),K(α,µ)(t) may include noise as NFPEs for the empirical probability densities Pα(t,zα)
ij ik corresponding to above the set of the effective Langevin
equations read
J(α,µ)
(α,µ) (α,µ)
J (t)= +ξ (t),
ij Nα ij ∂ P (t,z )=− ∂ −a(µ)z(µ)
K(α,µ)(t)= K(α,µ) +ζ(α,µ)(t). (5) ∂t α α µX=x,y∂zα(µ)(cid:18) α α
ik Nα ik +J(α,µ)hF(µ)i+ǫK(α¯,µ)hF(µ)i−D(µ) ∂ P . (8)
In cluster 1 and 2, we postulate the different constraint α α¯ effα∂zα(µ)(cid:19) α
parametersasa2(µ) =a(1µ)+δ(µ),withδ(µ)beingessentially A Gaussian probability density satisfies Eqs. (8)
responsibleforthedifferenceofnativefrequenciesbetween as a special solution. Since the H theorem [19] en-
theoscillatorsinthetwoclusters. Weherenotethatunder sures that the probability density satisfying Eqs. (8)
the setting of Eqs. (5), the native frequency distribution converges to the Gaussian-form for sufficiently large
corresponds to p(ω)=(δ(ω−ω1(a1))+δ(ω−ω2(a2)))/2. times, we represent the Gaussian probability densities
The inter-cluster coupling strengths are controlled by ǫ. as P (t,z ) = exp −1sTC−1(t)s /(2π detC (t)),
Gα α 2 α α α α
Thisparametermaytakeanyrealconstantvalue,without where sT = (z(x)−hz(x)i , z(y)−hz(y)i ) ≡ (u(x), u(y)),
constraining to weak connections. The external noise α α α(cid:2) G α α(cid:3) G p α α
C (t)=hs s i andh·i denotesexpectationoverP .
η(µ)(t), ξ(α,µ)(t), ζ(α,µ)(t) are ofthe white Gaussiantype, αij αi αj G G G
αi ij ik Now that Eqs. (6) and (7) are described in terms of the
hη(µ)(t)i = 0, hη(µ)(t)η(ν)(t′)i = 2D(µ)δ δ δ δ(t − first and second moments, one obtains a set of closed
αi αi βj ij µν αβ
t′), hξ(α,µ)(t)i = 0, hξ(α,µ)(t)ξ(β,ν)(t′)i = ordinary differential equations involving at most second
ij ij kl
moments. While Gaussian approximations ignore cumu-
2D˜(µ)δ δ δ δ δ(t − t′)/N , hζ(α,µ)(t)i = 0,
ik jl µν αβ α ik lantshigherthanthesecond[18],ourexpressionsbasedon
hζi(kα,µ)(t)ζl(mβ,ν)(t′)i = 2D˜(µ)δilδkmδµνδαβδ(t − t′)/Nα, the Gaussian probability densities are exact in the sense
where not common but independent noise is considered that a random variable representing the state of an indi-
and η(µ)(t), ξ(α,µ)(t), ζ(α,µ)(t) are also independent of vidual oscillatorof the nonlinearly coupled system under-
αi ij ik
eachother. Wenote thatinthe absenceofnoise,Eqs. (1) goesaquadraticpotential,thusconstitutingtheOrnstein-
- (4) with N1 = N2 = 1 describe the behavior of coupled Uhlenbeck process. The time evolutionofeachmomentis
limit cycle oscillators with two native frequencies, which calculated from Eqs. (8) and we have
can exhibit chaos due to the inter-cluster coupling.
(µ)
In the thermodynamic limit Nα → ∞, coupling dhzα iG =−a(µ)hz(µ)i
strengthsofEqs. (5)ensuretheconvergenceofeachmean- dt α α G
field coupling term that appears in the model equations +J(α,µ)hF(µ)i +ǫK(α¯,µ)hF(µ)i (9)
α G α¯ G
(1) - (4). This is attributed to the law of large numbers, (µ)2
giving rise to the validity of the self-averaging property. dhuα iG =−2a(µ)hu(µ)2i +2D(µ) (10)
dt α α G effα
Hence the NFPE [17] satisfied by the empirical proba-
bility density P(t,z) (zT = (z1(x), z1(y), z2(x), z2(y))) of the dhu(αµ)uβ(ν)iG =−(a(µ)+a(ν))hu(µ)u(ν)i , (11)
Langevin equations (1) - (4) can easily be written down dt α β α β G
in the limit N → ∞ [22]. We note that the empiri-
α where (α,β,µ,ν) = (1,1,x,y), (2,2,x,y), (1,2,x,x),
cal probability density can also be obtained for each clus- (1,2,y,y), (1,2,x,y), (1,2,y,x). Note that hu(µ)u(ν)i →
ter in our model setting by taking marginal distribution: α β G
0(t→∞), implying that the covariantcomponents of C
Pα(t,zα)= dzα¯P(t,z),where(zTα =(zα(x), zα(y)))andα¯ take zero in the stationary states. α
denotesthecounterpartofclusterα. Then,themean-field
R Forobservingqualitativeaswellasquantitativedynam-
coupling terms in Eqs. (1) - (4) are expressed in terms of
ical behaviors of the system, we proceed to solve numeri-
cally the aboveset ofequations. When the coupling func-
hFα(µ)i≡ dzαFα(µ)(b(αµ,x)zα(x)+b(αµ,y)zα(y))Pα(t,zα), (6) tions are specified as Fα(µ)(x) = sinx, then hFα(µ)iG and
Z
p-3
K.Okumura, A. Ichiki and M. Shiino
hF(µ)2i are calculated as
α G
E) 8 ε = 0.727
hFα(µ)iG =sin b(αµ,x)hzα(x)iG+b(αµ,y)hzα(y)iG nt (LL 6 δ(y) = 4.5
×exp −(cid:16)b(αµ2,x)2hu(αx)2iG− bα(µ2,y)2huα(y(cid:17))2iG!,(12) ov Expone 4
n
u
hF(µ)2i = 1 − 1cos 2b(µ,x)hz(x)i +2b(µ,y)hz(y)i yap 2
α ×eGxp 2−2b(2µ,x)2h(cid:16)u(x)α2i −α2b(µG,y)hu(yα)2i α. (1G3(cid:17)) argest L 0
α α G α α G L T LC
(cid:16) (cid:17) -2
0 0.002 0.004 0.006 0.008
Nonequilibrium phase transitions and stochastic
D(x)
inter-clustersynchronization.– NowthattheNFPE
asthe time evolutionofour systemhas been reducedto a
setofmomentequations(9)-(11)thatconstitutetheorder Fig. 2: Dependence of the largest Lyapunov exponents (LLE)
parameter equations, it will suffice to deal with them to on the Langevin noise intensity D(x), when D(y) = δ(x) = 0.
Since the basic oscillators show single loop limit cycles for
investigate the behavior of the system. The order param-
δ(y) = 4.5, the system consists of coupled 6 and 1 loop limit
eter equations (9)-(11) together with Eqs. (12) and (13)
cycles. Inthedeterministiclimit,theLLEhasapositivevalue,
are nonlinear, yielding various types of bifurcations with
implyingachaoticattractor. ForthenoiseintensityD(x)larger
changes in the parameters.
thanacertainvalue,theLLEalmosttakezero,andtheattrac-
For simplicity, we only treat the Langevin noise case, torchangesintothatoflimitcycletypeviatorustype(denoted
i.e., D˜x =D˜y =0, and do not study inter-cluster chaotic byLC and T, respectively).
synchronization in this paper [27]. From Eqs. (10), the
variances turn out to be constant for sufficiently large
times. Sinceweareconcernedwithinvestigatingnonequi- due to mean-field coupling (self-organization). This indi-
libriumstationarystates,particularlytheappearanceand catesthatalloscillatorsineachcluster showthe samebe-
disappearance of the inter-cluster synchronization phe- havior in the deterministic limit. Thus chaotic attractors
nomena in the thermodynamic limit, we confine ourselves inthedeterministiclimitimplyintra-clustersynchronized
to the appearance of limit cycle attractors with changes chaotic behavior.
in the noise intensity. To identify attractors of limit cycle
The property of worth noting of inter-cluster synchro-
type, we employ two criteria: (a) the numerically esti-
nization originates from effects of noise (Fig. 2). With
mated largest Lyapunov exponent (LLE) of the attractor increase of the Langevin noise intensity D(x), the chaotic
nearly equals zero and (b) all of the relevant order pa-
attractors with positive LLEs in weak noise limit change
rameters exhibit periodic motions with a common time
into those of limit cycle type via torus type, suggesting
period. Note that both of them exclude the case where
the appearance of the inter-cluster synchronization. Fur-
the attractor is of torus type.
therincreaseofthenoiseintensityleadstonegativevalues
Numericalcalculationswereperformedwiththefourth-
of the LLEs, implying fixed point type attractors. These
orderRunge-Kuttamethodunderthesettingofparameter
results arequalitatively consistentwith those reportedby
(x) (y) (x,x) (x,y)
values: a1 = 0.5, a1 = 1.0, bα = 0.5, bα = 1.5, Hakim and Rappel (Figs. 1 and 2 in Ref. [28]), where a
b(y,x) = −12.0, b(y,y) = −1.0, J(α,x) = K(α,x) = 18.0, set of Langevin equations were solved numerically.
α α
J(α,y) = K(α,y) = 30.0, a(x) = 0.5 + δ(x), a(y) = We show below the dynamics of the order parameters
2 2
1.0+δ(y). Nonequilibriumphase transitionsinvolvingthe in phase space. The changes of types of attractors ob-
inter-cluster synchronization of the limit cycle oscillators served in terms of the LLE in Fig. 2 can be more di-
are systematically investigated with changes in the inter- rectly seen in Figs. 3 and 4, where the corresponding
cluster coupling strength ǫ and the detuning parameters changes of trajectories are drawn. It is seen that as the
δ(µ) , and the Langevin noise intensity D(µ). For simplic- noise strength increases, the roughly estimated number
ity, we take δ(x) =D(y) =0 in what follows. of loops of chaotic trajectories decreases, while it gives
We begin with investigating behaviors of the system rise to bifurcations from chaotic to torus type attractor.
thatexhibits a chaoticattractorinthe deterministic limit Eventuallythe attractorbecomes thatof limit cycle type,
(Fig. 2). The chaos originates from one pair system of which corresponds to the occurrence of inter-cluster syn-
different native frequencies. We note that “determinis- chronization in the system. Within the context of rough
tic limit (D(x) → 0)” and “essentially deterministic case argument, Fig. 4 (c) implies the occurrence of the so-
(D(x) = 0)” may be different dynamics. Our model is called “in-phase” synchronization. Note that when the
proposed so that once external noise is introduced to the attractor is of torus type, the clusters are asynchronous
system, the model constrains the form of the probability becausethere are no commontime period. Phenomena of
densities to the Gaussian ones for sufficiently large times noise induced chaos can be also reproduced (though not
p-4
Stochastic inter-cluster synchronization
Fig.4: Otherviewsinphasespace. (a),(b)and(c)correspond
to the attractors shown in Figs. 3 (a) and 3 (b), Figs. 3 (c)
and 3 (d) and Figs. 3 (e) and 3 (f), respectively. It is clearly
seen thatthesmall orlarge mean valueof hz1(x)iG follows that
of hz2(x)iG in thesynchronizedstate, which roughly implies so-
called in-phasesynchronization.
Frankowicz[29],areobservedforlargervaluesofδ(y)inthe
deterministiclimit. Theincreaseofnoiseintensityinduces
breakdown of synchronized states. We note that complex
behaviors can be observed in this model, depending on
Fig. 3: The behavior of theorder parameters of thesystem in initial conditions (the results are not shown here).
phase space. The left ((a), (c), (e)) and right ((b), (d), (f))
Summary. – We have shown the effects of indepen-
columns show the trajectories projected to the hz1(x)iG-hz1(y)iG dent Langevin noise on the behavior of synchronization
and hz2(x)iG-hz2(y)iG planes, respectively. (a), (b) In the deter- of mean-field coupled limit cycle oscillators in the ther-
ministic limit D(x) = 0, it exhibits chaotic behavior. (c), (d)
Certain levels of noise intensity (e.g. D(x) = 0.0018) give rise modynamic limit. The ensemble is assumed to consist of
two clusters of oscillators with different native frequen-
to torus type attractors. (e), (f) For larger noise intensities
(e.g. D(x) = 0.0022), limit cycle attractors appear, confirm- cies. To investigate stochastic inter-cluster synchroniza-
ing the occurrence of stochastic inter-cluster synchronization tionphenomena,wehaveusedtheNFPEapproach. Since
in macroscopic physicalquantities. the NFPE for our model yields a Gaussian type prob-
ability density as its solution for large times, the time
evolution equations of the relevant order parameters of
shown here), which are qualitatively in good agreement the system become to be analytically derived. It should
with those observed in Refs. [19,20]. be noted that the method turns out to be a noise level-
We also show the effects of the frequency differences free analysis. Solving the order parameter equations nu-
(essentially equivalent to the detuning parameter δ(y)) as merically, we have systematically investigated the occur-
well as the inter-cluster couping strength ǫ on synchro- rence of nonequilibrium phase transitions with changes in
nization phenomena with and without noise (Fig. 5). For the inter-cluster coupling strength ǫ and the detuning pa-
the weak inter-cluster coupling strengths (Fig. 5 (a)), it rameters δ(µ) responsible for native frequency difference,
is shownthat certainlevelsof noise intensity can giverise and the Langevin noise intensity D(µ). The results have
to synchronized states in a broad region of δ(y) even if shown various interesting bifurcation phenomena includ-
the clusters do not synchronize in the deterministic limit. ing the inter-cluster synchronization and chaotic attrac-
In the case of the strong inter-cluster coupling strengths torsinduced bynoise. Toourknowledge,papersstudying
(Fig. 5(b)), itisseenthatthe quenchedoscillatorystates analyticallysystematicmodelsofgeneraltypesoflimitcy-
(oscillatordeath),whichwerefirstreportedbyShiinoand clesfromourviewpointandalsoavailableforcomparisons
p-5
K.Okumura, A. Ichiki and M. Shiino
throughthe“NanoscienceandQuantumPhysics”Project
of the Tokyo Institute of Technology.
E) 10 (a) DeteDrm(xi)n =is 0tic.0 l0im40it
LL ε = 0.029
nt ( 8 REFERENCES
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Fig. 5: Dependence of the largest Lyapunov exponents (LLE)
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both figures, all the points where the LLE take nearly zero
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show limit cycle attractors.
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cluster synchronization will be reported elsewhere. Fur-
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models will be also reported elsewhere. 637.
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K.O. acknowledges the financial support from the
Global Center of Excellence Program by MEXT, Japan
p-6
This figure "NISR_sin_BasicUnits.png" is available in "png"(cid:10) format from:
http://arxiv.org/ps/1101.0346v1