Table Of ContentTransition from spatial coherence to incoherence in coupled chaotic systems
Iryna Omelchenko,1,2 Bruno Riemenschneider,1 Philipp H¨ovel,1,2 Yuri Maistrenko,3,4 and Eckehard Sch¨oll1,∗
1Institut fu¨r Theoretische Physik, TU Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
2Bernstein Center for Computational Neuroscience,
Humboldt-Universit¨at zu Berlin, Philippstraße 13, 10115 Berlin, Germany
3Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkivska str. 3, 01601 Kyiv, Ukraine
4National Center for Medical and Biotechnical Research,
National Academy of Sciences of Ukraine, Volodymyrska str.54, 01030 Kyiv, Ukraine
(Dated: January 19, 2012)
We investigate the spatio-temporal dynamics of coupled chaotic systems with nonlocal interac-
tions,whereeachelementiscoupledtoitsnearestneighborswithinafiniterange. Dependingupon
the coupling strength and coupling radius, we find characteristic spatial patterns such as wave-like
profilesandstudythetransitionfromcoherencetoincoherenceleadingtospatialchaos. Weanalyze
2 the origin of this transition based on numerical simulations and support the results by theoretical
1 derivations identifying a critical coupling strength and a scaling relation of the coherent profiles.
0 Todemonstratetheuniversalityofourfindingsweconsidertime-discreteaswellastime-continuous
2 chaotic models realized as logistic map and Ro¨ssler or Lorenz system, respectively. Thereby we
establish the coherence-incoherence transition in networks of coupled identical oscillators.
n
a
J PACSnumbers: 05.45.Xt,05.45.Ra,89.75.-k
Keywords: nonlinearsystems,dynamicalnetworks,coherence,spatialchaos
8
1
I. INTRODUCTION chaos [22, 23].
]
D This paper is structured as follows: Section II intro-
C Whileisolatednonlinearsystemsarefairlywellunder- duces the model equations. Sections III and IV describe
stood, much less is known about emergent dynamics of the general coherence-incoherence scenario and numer-
.
n coupled systems [1–6]. Complete synchronization, also ical results on the coherence-incoherence transition, re-
li known as zero-lag synchronization, where all nodes of spectively. Analytical results are presented in Sec. V. In
n a network behave identically, is just the simplest sce- Sec. VI we demonstrate that the observed scenario can
[
nario. Nevertheless, it has important applications, for also be found in the Lorenz model. Finally we conclude
2 instance, as chaos synchronization for secure communi- in Sec. VII.
v cations [7, 8]. Other types of synchronization have been
9
foundaswell,includingphase-lag,cluster,orgeneralized
6
synchronization [4]. Recently a new form of dynamics II. MODELS
8
called chimera state triggered a lot of research [9–19].
4
2. Thesechimerastatesariseinregularnetworksofidentical In order to investigate the universality of the
phaseoscillatorsandconsistofpartswithhighspatialco-
1 coherence-incoherence transition for N coupled oscilla-
herenceandregionswherespatialcoherenceislost. They
1 tors we consider different nonlinear models: As chaotic
1 can be realized with carefully prepared initial conditions systemsweemploythetime-discrete,logisticmapaswell
v: and a nonlocal network topology, which both are crucial as the R¨ossler system. The latter is paradigmatic for
ingredients for these states. Similar phenomena, local-
i chaos described by ordinary differential equations.
X ized patterns surrounded by a chaotic background, were
As a control parameter we use the coupling strength
r foundinpopulationsofthresholdelementswithnonlocal σ and the coupling radius r = P/N, where P denotes
a interaction, related to a continuity-discontinuity transi-
the number of nearest neighbors in each direction on a
tion [20].
ring. Varying the value of P from 1 to N/2 we cover the
In this paper we investigate in detail the transition
range of regular network topologies from local (nearest-
from spatial coherence to incoherence in both time-
neighbor) to global (all-to-all) coupling.
discrete and time-continuous chaotic systems, using the
well-known models of the logistic map and the R¨ossler
system, respectively. See also Ref. [21] for a first report
A. Logistic map
on this transition. We consider a ring topology and use
the coupling strength and coupling range as bifurcation
parameters. At the transition we observe partially in- Asanexamplefortime-discretesystems,westudycou-
coherent chimera-like states that we identify as spatial pled maps:
i+P
zt+1 =f(cid:0)zt(cid:1)+ σ (cid:88) (cid:2)f(cid:0)zt(cid:1)−f(cid:0)zt(cid:1)(cid:3), (1)
∗ correspondingauthor: [email protected] i i 2P j i
j=i−P
2
where z (i = 1,...,N) are real dynamic variables and 11
i
t denotes the discrete time. In the coupling term the
index i is periodic mod N. The function f(z) is a local 00..88
one-dimensional map that we choose as the logistic map aaa===444...000
f(z)=az(1−z). We fix the bifurcation parameter a at
00..66
thevaluea=3.8unlessstatedotherwise. Thisisavalue aaa===333...888
σσ
that yields chaotic behavior with a positive Lyapunov
00..44 aaa===333...666
exponent λ≈0.431.
00..22
B. R¨ossler system
00
00 00..11 00..22 00..33 00..44 00..55
rr
In order to show that the coherence-incoherence tran-
sition scenario is universal, we will also discuss nonlo-
FIG. 1. (Color online) Blowout bifurcation curves in the
cally coupled networks of time-continuous systems. As
(r,σ)-parameter plane for N = 100 coupled logistic maps.
a prominent example for chaotic systems, we consider The system parameter is chosen as a = 4.0, 3.8, and 3.6 for
the R¨ossler model. In analogy to Eq. (1) the compound thesolid,dashed(red),anddotted(blue)curve,respectively.
system is given by
σ i(cid:80)+P III. STRUCTURE OF COHERENCE TONGUES
x˙ =−y −z + (x −x ),
i i i 2P j i
j=i−P
σ i+P Let us discuss first the coherence-incoherence transi-
(cid:80)
y˙ =x +ay + (y −y ), (2)
i i i 2P j i tion for coupled logistic maps described by Eq. (1). For
j=i−P
largevaluesofcouplingradius r andcouplingstrength σ
σ i+P
(cid:80)
z˙i =b+zi(xi−c)+ 2P (zj −zi), we observe complete chaotic synchronization. There the
j=i−P oscillators behave identically (z = z = ... = z ),
sync 1 N
but chaotically in time following the dynamics of f.
with i = 1,...,N. Again the index i is periodic mod
Using the fact that the coupling matrix of system (1)
N to realize the underlying ring topology. The system’s
is a circulant matrix, with cyclically permutated rows
parameters are chosen as a = 0.42, b = 2, and c = 4
(−2P,1,...,1,0,...,0,1,...,1), we obtain analytically the
showing chaotic behavior in the uncoupled case. (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125)
P P
The 3N coupled Eqs. (2) can also be written in com-
values for N −1 transversal Lyapunov exponents of the
pact form as follows
completely synchronized solution z :
sync
X˙ =F(X)+ σ (G⊗H)X (3) (cid:12)(cid:12) σ (cid:18)(cid:80)P (cid:18)2πmk(cid:19)
2P λ⊥,m =λ(cid:107)+ ln(cid:12)(cid:12)(1−σ)+ 2P cos N
with X = (x1,y1,z1,...,xN,yN,zN) and a 3 × 3 cou- + N(cid:80)−1 cos(cid:18)2πmk=k1(cid:19)(cid:33)(cid:12)(cid:12)(cid:12),
pling matrix H that determines, which components are N (cid:12)
k=N−P (cid:12)
transmitted. TheN×N matrixGdescribesthenetwork
topology. In the case of Eqs. (2), we have the identity m = 1,...,N − 1, where λ > 0 denotes the longi-
(cid:107)
matrix for H, and a circulant matrix with cyclically per- tudinal Lyapunov exponent corresponding to the one-
mutated rows (−2P,1,...,1,0,...,0,1,...,1) is chosen dimensionallocalmap,andthesecondtermontheright-
(cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125)
handsideisderivedfromtheeigenvaluesoftheJacobian
P P
for G such that gii =−2P. matrixofthesystem,usingthefactthatcouplingmatrix
This notation has been employed in the framework of is circulant.
themasterstabilityfunctioninordertoinvestigatecom- Withadecreaseofthecouplingradiusrorthecoupling
plete, potentially chaotic synchronisation [24]. Based on strength σ the completely synchronized solution loses its
a variational equations this formalism allows for calcula- stabilityinablowoutbifurcation,wherethelargesttrans-
tionoftheLyapunovexponenttransversetothesynchro- verseLyapunovexponentbecomespositive. Theposition
nisation manifold [25, 26]. The master stability function oftheblowoutbifurcationlineinthe(r,σ)-planedepends
techniquehasbeenusedinvariouscontextsrangingfrom on the local dynamics of the system elements. Figure 1
genericmodels[27,28]tocoupledlasers[29,30]andneu- shows examples for the blowout bifurcation curves for
roscience [31–33]. It can also be extended to cluster syn- a = 3.6, 3.8, and 4.0. For any of these values the local
chronization[34], butdoesnotgiveconclusiveresultsfor logistic maps operate in the chaotic regime. The region
the purpose of the investigation of spatially incoherent of complete chaotic synchronization is the region above
states. the blowout bifurcation curves.
3
The completely synchronized state represents the sim- (((aaa))) 111
2-
plestexampleofspatialcoherence. Tobeprecise,wewill period 4-
kkk===333 kkk===222 kkk===111 kkk===000 8-
call a network state zt, i=1,...,N, coherent on the ring
S1 asN →∞, ifforainypointx∈S1 thefollowinglimit 000...888 1
holds:
zi
lim lim sup (cid:12)(cid:12)zt−zt(cid:12)(cid:12)→0, for δ →0, (4) 000...666 00 i 100
i j
N→∞t→∞i,j∈UN(x) σσσ
δ CCCIIIBBB CCCIIIBBB
000...444 BBBBBB
whereUN(x)={j : 0≤j ≤N, |j/N −x|<δ}denotes
δ
a network neighborhood of the point x. Intuitively it 1 1 1
means that the profile becomes smooth in the thermo- 000...222
zi zi zi
dynamic limit N → ∞. If the limit (4) does not vanish
0.3 0.3 0.3
for δ → 0, at least for one point x, the network state is 0 i 100 0 i 100 0 i 100
000
considered incoherent. 000 000...111 000...222 000...333 000...444 000...555
rrr
Below the blowout bifurcation curve, several further
(((bbb))) 111
regions of coherence appear as the control parameters σ 2-
period 4-
or r decrease. Figure 2 depicts these regions as shaded kkk===111 kkk===000 ≥8-
(color) tongues for the logistic map and the R¨ossler sys- 000...888 kkk===222
tem in panels (a) and (b), respectively. The blowout
kkk===333
bifurcation line is added as dashed curve (BB), and the
regime of complete, chaotic synchronisation is indicated 000...666 00..22
by the light hatching. σσσ
Acoherentstatehasasmoothprofilecharacterizedby 00..11
000...444
the number of maxima, i.e., the wave number k. Only
00..22 00..33
regionsforwavenumbersk =1,2,and3areshown. The
corresponding coherent states zt are shown as snapshots 000...222
i
intheinsetsofFig.2(a)forthelogisticmap. Furtherde- BBBBBB
creaseofr yieldsadditionalthinhigher-orderregionsfol-
000
lowingaperiod-addingcascadek =4,5,... (notshown). 000 000...111 000...222 000...333 000...444 000...555
rrr
Inside the regions, the states are coherent in space and
periodic in time and undergo a period-doubling cascade
FIG.2. (Coloronline)Coherenceregionsinthe(r,σ)param-
in time as r or σ decrease. In the parameter space be-
eterplaneforN =100logisticmaps(a)andRo¨sslersystems
tween the coherence regions the network dynamics re-
(b),labeledbythewavenumberk. Grayscale(colorcode)in-
mains coherent but not periodic anymore. The states sidethecoherenceregionsdistinguishesdifferentperiodsand
alternate chaotically between the adjacent k-states and the coherence-incoherence bifurcation (CIB) curve separates
thus exhibit chaotic itineracy [2, 35]. The combination regimes with coherent and incoherent dynamics. The light
of period-adding in space and period-doubling in time hatchedregionboundedbytheblowoutbifurcationcurveBB
represents a remarkable feature of networks of coupled refers to completely synchronized chaotic states. The insets
chaotic oscillators with nonlocal coupling. inpanel(a)displaysnapshotsoftypicalcoherentstates. Sys-
temparameters: panel(a)a=3.8;panel(b)a=0.42,b=2,
Figure 2(b) shows stability diagram for the coupled
and c = 4.0. The vertical dashed lines refer to values of σ
R¨ossler systems again in the (r,σ)-plane. It is easy to
inFig.3. Trianglesdenoteparametervaluesusedtodescribe
see that the overall structure of the diagram is similar
the bifurcation scenario in Figures 4 and 5.
to the one obtained for nonlocally coupled logistic maps
(Fig.2(a)). Inanalogytothemaps,thereexistablowout
bifurcation boundary (BB curve) and a sequence of co-
herencetongueswithwavenumbersk indicatingtheuni- herence tongues Figs. 3(a) and (b) depict cuts through
versalityofthegeneraldynamicalscenario. Intheshaded the (r,σ)-parameter space for a fixed coupling radius
(color)regionsshowninFig.2(b)thecoherentstateswith r = 0.12 and 0.24, respectively (see dashed lines in
regular(periodicorquasiperiodic)dynamicsaretheonly Fig. 2(b)). The black and gray (red) values refer to the
stablesolution. Thebright(yellow)regionsinsidetheco- variable y at a Poincar´e section (x =0) for a very slow
1 1
herence tongues below the stable period-2 tongues refer decrease and increase of the coupling strength σ with
to multistability and more complex behavior including steps ∆σ = 0.001, respectively. For each value of σ 300
coexisting period-4 and torus solutions in between the iterations are shown, where each initial condition is cho-
period-2 and period-4 regions. The dark (blue) region sen as the final configuration of the previous value of σ.
is the region of stability for the solutions with periods 8 Panel (a) shows a sequence of chaotic, period-4, torus,
and larger. Since these regions appear to be very thin, period-2,coexistingperiod-2andperiod-4,period-4,and
we combine them for better visualization. again chaotic dynamics. In panel (b) there appears an
For further insight of the transitions within the co- additional regime of chaos in between the period-2 and
4
(a) 1σ=0.43 (d) 1σ=0.30
zi zi
α α
0.3 0.3 1 2
(b) 1σ=0.40 (e) 1σ=0.2
zi zi
x x
0.3 1 2 0.3
(c) 1σ=0.32 (f) 1σ=0.1
zi zi
α α
0.3 1 2 0.3
1 100 1 100
i i
FIG. 4. Snapshots of the dynamics of the logistic map for
different coupling strengths σ = 0.43,0.4,0.32,0.3,0.2, and
0.1. The gray regions labeled α and α refer to multistable
1 2
FIG. 3. (Color online) Bifurcation scenario at cuts through incoherent parts of the profile. The coupling radius is fixed
the k = 1 coherence tongue of Fig. 2(b) at (a) r = 0.12 and at r=0.32. Other parameters as in Fig. 2.
(b)r=0.24. ThePoincar´esection(x =0)ofy isshownfor
1 1
the Ro¨ssler systems in dependence on the coupling strength
σ. Theblackandgray(red)pointscorrespondtoaslowdown map: The smooth wave-like solution profile zt splits up
i
and up sweep of σ, respectively. Long transients up to 3000 into upper and lower branches. Furthermore, two nar-
oscillationperiodshavebeendiscarded. Otherparametersas
row boundary layers of incoherence appear around the
in Fig. 2.
points x and x in Fig. 4(b). They are indicated by
1 2
shadedstripesα andα inFigs.4(c)and(d). Thewidth
1 2
of the incoherence stripes increases for smaller coupling
period-4 region.
strength σ (see Fig. 4(d)) until the dynamics becomes
Thecoupling radiusappearstobe thecrucialparame-
completely incoherent as depicted in Figs. 4(e) and (f).
ter in the system with nonlocal couplings. With increas-
Below the bifurcation parameter value σ ≈ 0.40 nu-
ing the system size N and a proportional increase of the
merous hybrid, that is, partially coherent states arise.
number of coupled neighbors, we obtain similar coherent
Theyarecharacterizedbycoherenceonsomeintervalsof
solutions. If with increasing N the number of coupled
the ring S1 and incoherence on the complementary in-
neighbors P does not change, then the coupling radius
tervals. Figs.4(c)and(d)displayexemplaryrealizations
decreases, and the point of operation (r,σ) moves in the
of these partially coherent states. Within the incoher-
stability diagram (Fig. 2) to the left, obtaining coherent
ence intervals, i.e., α and α , any combinations of the
solutionswithlargerwavmenumbersoritineracyregions 1 2
upper and lower states are admissible for appropriately
between them.
prepared initial conditions.
ThebifurcationdiagramspresentedinFig.2havebeen
Thus the coherence-incoherence bifurcation leads to
obtained by means of numerical simulations. Note, how-
spatial chaos [22, 23] that appears first at narrow inco-
ever, that in the thermodynamic limit N →∞ it is also
herenceintervalsand,withdecreasingcouplingstrength,
possible to obtain some analytic results. Details on such
encompasses the whole ring. Therefore we identify a
derivationsaregiveninSec.V,e.g.,byidentifyingascal-
chimera-like state of coexisting coherent and incoher-
ing relation of the coherent profiles.
ent regions as a transitional state in the coherence-
incoherence bifurcation scenario. Opposed to previ-
ouslyreportedchimerastatesintime-continuoussystems
IV. COHERENCE-INCOHERENCE [9, 10, 36–38], however, the temporal behavior in the
TRANSITION present case is periodic rather than chaotic. The com-
plexityarisesduetothehugevarietyofmultistableinco-
The described combination of period-adding in space herent states and corresponds to permutations of the se-
and period-doubling in time for coherent solutions is quence of upper and lower local states. With further de-
valid for sufficiently large coupling strength (σ (cid:38) 0.4). crease of the coupling strength, the chimera states cease
In Ref. [21] we have shown that with decreasing cou- to exist and we find completely incoherent behavior.
pling strength σ a coherence-incoherence bifurcation is Thefindingsabovearebasedonnumericalsimulations
observedinthetime-discretesystem(1). Thetypicalsce- ofcoupledchaoticlogisticmaps. Evenifthelocallogistic
nario of this transition is shown in Fig. 4 for the logistic maps operate in the periodic regime, one can observe a
5
sequence similar to Fig. 4 as reported in Ref. [21]. Note x (t), i = 1,...,N of the R¨ossler systems for the same
i
that, when the local dynamics of the system is periodic, parameter values as in Fig. 5. Again the time series of
the region of stability of the completely synchronized so- node i = 50 is highlighted as black (red). Note that
lutioncoversalmostallofthe(r,σ)parameterplane,and in Figs. 6(b) and (c) the time series of the points from
the system is characterized by multistability. theincoherentregionsaregroupedintotwoclustersthat
are formed by the two coherent parts of the solution. In
Fig. 6(d) one can visualize the chaotic dynamics already
intheshorttimeintervalshown. Inthetimedomainone
canobserveaperiod-doublingbifurcationinFigs.6(a)to
6(c) eventually leading to chaos in Fig. 6(d) (lower pan-
els). At the same time the spatial irregularity becomes
more pronounced as described above.
V. ANALYTICAL RESULTS
After the numerical exploration of the coherence-
incoherence transition, we provide some analytical
derivations in this Section. We will discuss an approxi-
mation for the value of the critical coupling strength at
the coherence-incoherence transition and present a scal-
ing of the profiles in different coherence tongues, which
holds in the limit N →∞.
A. Critical coupling strength
In the following, we derive an approximation of the
criticalcouplingstrengthatthetransitionfromcoherent
toincoherentstates,seethelineCIBinFig.2(a). Wewill
restrict the investigation to the logistic map introduced
inSec.IIA,butasimilarargumentholdsfortheR¨ossler
FIG. 5. (Color online) Snapshots of the variables x (upper
i system as well.
panels, x as black dot) and space-time plots (lower panels)
50
From a geometrical point of view and in the thermo-
for the Ro¨ssler system. The coupling strength is chosen as
dynamic limit N →∞, coherent solutions zt approach a
σ = 0.12,0.1,0.09, and 0.06 in (a) to (d), respectively. The i
coupling radius is fixed at r = 0.28. Other parameters as in smooth profile zt(x) of the spatially continuous version
Fig. 2. of Eq. (1) given by
Figures 5 and 6 display the typical scenario of zt+1(x)=f(cid:0)zt(x)(cid:1)+ σ (cid:90) x+r(cid:2)f(cid:0)zt(y)(cid:1)−f(cid:0)zt(x)(cid:1)(cid:3)dy.
2r
coherence-incoherence transition for nonlocally coupled x−r
(5)
R¨ossler systems, similar to the one described above for
A transition from coherence to incoherence occurs when
thelogisticmap. IntheupperpanelsofFig.5,snapshots the respective solution profile zt(x) becomes discontinu-
of the variables xi are shown. Here all nodes are shown ous in some points x of the ring S1.
ingray,andanexemplarynode(i=50)ishighlightedas
Let us now discuss in more detail this coherence-
ablackpoint. Similarlytothecoupledmaps,thesmooth
incoherence bifurcation. Consider a solution of sys-
profileofthecoherent(k =1)solutionofFig.5(a)breaks
tem (5) with wave number k =1 and period-2 dynamics
up into two clusters in Fig. 5(b). With further decrease
in time. Hence we can reduce the dynamics by even and
of the coupling strength a partially coherent solution
odd time steps z (x) and z (x), respectively. This leads
0 1
appears in the system as shown in Fig. 5(c). Finally
to
we observe a chimera-like state around the upper clus-
x+r
ter state in Fig. 5(d), which combines spatially coherent σ (cid:90)
and spatially chaotic parts. The lower panels of Fig. 5 z1−j(x)=(1−σ)f(zj(x))+ 2r f(zj(y))dy (6)
showspace-timeplots. Herethespatialstructureandthe x−r
breaking up of the profile is clearly visible. Furthermore
with j =0,1. Taking the spatial derivative yields
one can already infer the temporal behavior that will be
discussed in the following. z1(cid:48)−j(x)=(1−σ)f(cid:48)(zj(x))zj(cid:48)(x)
Figure 6 shows three-dimensional phase portraits of σ
+ [f(z (x+r))−f(z (x−r))]. (7)
the x variables and the superimposed time series of all 2r j j
i
6
FIG. 6. (Color online) Phase space trajectories (upper panels) and time series x (t), i = 1,2,...,N, (lower panels) for the
i
Ro¨sslersystem. Thecouplingstrengthischosenasσ=0.12,0.1,0.09,and0.06inpanels(a)to(d),respectively,corresponding
to Fig. 5. The exemplary time series x (t) is highlighted in black (red). The coupling radius is fixed at r = 0.28. Other
50
parameters as in Fig. 2.
At the point x where the smooth profile breaks up, as a deviation from condition (9).
the spatial derivative becomes infinite. Considering that
For fixed system parameters r and σ that correspond
z(cid:48)(x),z(cid:48)(x) diverge to infinity, we can neglect the cou-
0 1 to the region of coherent solutions with periodic behav-
pling term on the right-hand side of Eq. (7). The main
ior of period 2 in time, and thus fixing z (x) and z (x),
0 1
contribution comes from the first term.
we can calculate G(x) and determine values x, where
Multiplying the equations for even and odd time steps
G(x) = 0. The obtained values x are the locations at
we obtain:
which the smooth profile break up into two parts. Fig-
z(cid:48)(x)z(cid:48)(x)=(cid:2)(1−σ)2f(cid:48)(z (x))f(cid:48)(z (x))(cid:3)z(cid:48)(x)z(cid:48)(x), ure7illustratesG(x)forafixedcouplingradiusr =0.32
0 1 0 1 0 1 for the finite case of 100 elements. Numerical evaluation
(8)
shows that for a coupling strength σ ≈0.41 the function
which yields the following condition
Gapproacheszerointwopoints. Figure7(b)displaysthe
1=(1−σ)2f(cid:48)(z (x))f(cid:48)(z (x)). (9) correspondingsnapshotofthesolution. Onecanseethat
0 1
the profile breaks up at positions at which G vanishes as
Keeping in mind that the local dynamics is governed by
indicated by the vertical dashed lines.
the logistic map f(z)=az(1−z) and that its derivative
is equal to f(cid:48)(z)=a(1−2z), we introduce the function To derive an approximation for this coupling strength
let z∗ be the fixed point of the local logistic map: z∗ =
G(x)=(1−σ)2a2(1−2z (x))(1−2z (x))−1 (10) f(z∗) = az∗(1 − z∗), hence z∗ = 1 − 1/a. Here, for
0 1
7
convenience,weconsiderthecaseoftime-continuoussys-
(a) 0 tems, but our findings are also valid for maps as we will
show later (Fig. 8).
-0.4
The network model of a ring of N linearly coupled
-0.8 oscillators, each coupled to its P nearest neighbors,
G
-1.2 i+P
σ (cid:88)
x˙ =f(x )+ (Hx −Hx ) (13)
-1.6 i i 2P j i
j=i−P
(b) 1
z becomes, in the limit N →∞,
0
0.8 σ (cid:90) u+r
z* x˙(u)=f(x(u))+ [Hx(v)−Hx(u)]dv (14)
2r
u−r
0.6
z
1 with coupling range r and coupling strength σ as used
in the previous sections. u ∈ [0,1] denotes the spatial
0.4
position on the ring. Compare Eq. (5) for the spatially
1 x1 i x2 100 continuous version of time-discrete maps. The coupling
schemeisdeterminedbythematrixH. Inthecaseofthe
FIG. 7. (a) Deviation from the incoherence condition (9) for R¨ossler system introduced by Eqs. (2), H is the identity
coupled logistic maps for σ = 0.42 (dashed), 0.40919 (solid matrix and the vector describing each node is given by
curve),σ=0.40(dotted). (b)Snapshotsforeven(opendots) x =(x ,y ,z ). ConsideringthelogisticmapsofEq.(1),
i i i i
and odd (filled dots) solutions for σ =0.40919 and r =0.32. the coupling term Hx changes to f(zt), but the follow-
Other parameters as in Fig. 2. j j
ing argument still holds (Fig. 8(a),(b)).
In short, we will show that for fixed σ the spatially
k -periodic dynamics of an oscillator at the position u
a = 3.8 we have z∗ ≈ 0.737. Under the assumption 1
with coupling range r is identical to that at a rescaled
z (x)=z (x)=z∗ ifG=0,weobtainanapproximation 1
0 1 positionuk /k ofthecorrespondingk -periodicstatein
for σ: 1 2 2
a network with coupling range r = r k /k . From this
2 1 1 2
G(x)=[a(1−σ)(1−2z∗)]2−1=0 (11) we can conjecture to find a k2-periodic state at r2, if a
stable k -periodic state is found at r , such that k r =
1 1 i i
1
⇒σ ≈1− . (12) const.
a−2
For a=3.8 we get σ ≈0.44. i’
1 200
From the numerical experiments we obtain that the 1
transition from coherence to incoherence in system (1) 0.8
occurs for a coupling strength σ close to 0.4, which zi 0.6
roughlyagreeswithourapproximation. Thedeviationis
0.4
duetothefinitenumberofnodesinthesimulations. The (a) (b)
0.2
derivation above was presented for wavenumber k = 1. -2
Note that coherent solutions with higher wave numbers
-3
undergo the same bifurcation scenario, with the smooth
xi
profile breaking up into 2k parts. -4
(c) (d)
-5 r r
1 1 1001 2 100
i i
B. Scaling of coherent profiles
FIG. 8. (Color online) Scaling of coherent profiles: Black
Asasecondanalyticresultwewillderiveinthefollow- pointsaresnapshotsofnumericalsimulations(a),(b)forcou-
ing a scaling relation that allows a mapping of coherent pled logistic maps with N = 100, σ = 0.7, r = 0.22 and
r = 0.11, respectively, and (c),(d) for Ro¨ssler systems with
profiles in different regimes of coherence. These regimes
N =100, σ =0.6, r =0.12 and r =0.06, respectively. Gray
were depicted in Fig. 2 by shaded (color coded) tongues
(red) points in panels (b),(d) show rescaled k = 1 solutions
labeledk =1,2,.... Ourconsiderationsapplytoringsof
from panels (a) and (c). Other parameters as in Fig. 2.
oscillators in the thermodynamic limit N →∞, N being
thenumberofcoupledoscillators. Acomparisonwithnu-
merical simulation shows that for sufficiently large finite Assumethatastatex (u)withspatialperiodk exists.
s 1
N theresultholdsingoodapproximation. Fornotational For simplicity and notational convenience consider k =
1
8
1. This state evolves according to
(a) 8σ=8 (b) 8σ=6
σ (cid:90) u+r
x˙ (u)=f(x (u))+ [Hx (v)−Hx (u)]dv.
s s 2r u−r s s xi xi
(15)
-8 -8
Inordertocompareanotherk-periodicstatex withthe
k 1 i 100 1 i 100
state x , in the way described above, we assume 50 50
s
(cid:18) (cid:19)
u+m z z
x (u)=x (16)
s k k
5 5
-20 x 20 -20 x 20
withm=0,1,2,...,k−1. Viceversa,scalingthespatial
coordinate of x (u) with u ∈ [0,1], as well as r, by a
k
factor k−1 leads to (c) 8σ=5.5 (d) σ=1
8
x˙ (k−1u)=f(cid:0)x (k−1u)(cid:1) xi xi
k k
+ k2σr (cid:90) k−1(u+r)(cid:2)Hxk(v)−Hxk(k−1u)(cid:3)dv. 5-081 i 100 5-801 i 100
k−1(u−r)
(17)
z z
Substituting v =k−1w gives
5 5
x˙k(k−1u)=f(cid:0)xk(k−1u)(cid:1) -20 x 20 -20 x 20
+ σ (cid:90) u+r(cid:2)Hx (k−1w)−Hx (k−1u)(cid:3) dw
2r k k FIG.9. SnapshotsofnonlocallycoupledLorenzsystems(20)
u−r
(18) fordifferentvaluesofcouplingstrengthσ=8,6,5.5,and1in
panels(a)to(d),respectively. Thecouplingradiusisfixedat
and, using Eq. (16), results in r =0.3 and the number of nodes is N =100. Parameters of
the Lorenz system: ς =10, ρ=28, β =8/3.
σ (cid:90) u+r
x˙ (k−1u)=f(x (u))+ [Hx (w)−Hx (u)]dw.
k s 2r s s time-continuous system that exhibits the coherence-
u−r
(19) incoherence bifurcation scenario described above. Fig-
Comparing the right-hand side of Eq (19) to Eq. (15), ure 9 displays the coherence-incoherence transition for
we find x˙ (k−1u) = x˙ (u), if the coupling radius r is nonlocally coupled Lorenz systems:
k s
appropriately rescaled by k−1.
σ i+P
This derivation shows that one can recover the cou- x˙ =ς(y −x )+ (cid:80) (x −x ),
i i i 2P j i
plingparametersleadingtoastatewithwavenumberk j=i−P
2
from the parameters of a k1 state. This finding is illus- y˙ =x (ρ−z )−y + σ i(cid:80)+P (y −y ), (20)
trated in Fig. 8 for k = 1 and k = 2 by corresponding i i i i 2P j i
1 2 j=i−P
snapshots of 100 logistic maps and 100 R¨ossler system σ i+P
(cid:80)
z˙ =x y −βz + (z −z ),
in panels (a),(b) and (c),(d), respectively. Already for i i i i 2P j i
j=i−P
N =100 there is strong agreement between the numeri-
cally obtained and scaled profiles. In the case of logistic wherethelocalparametersarefixedatthevaluesς =10,
maps(R¨osslersystems), wechoosethecouplingstrength ρ = 28, β = 8/3. Fig. 9 reveals a striking similar-
as σ = 0.7 (0.6) and the coupling radius as r = 0.22 ity with the sequences of snapshots for nonlocally cou-
(0.12) in Fig. 8(a) (Fig. 8(c)) and, according to the scal- pled chaotic logistic maps (Fig. 4) and R¨ossler systems
ing relation, as r = 0.11 (0.06) for the k2 = 2 profile in (Fig. 5). For fixed coupling radius and with decreas-
Fig. 8(b) (Fig. 8(d)). The gray (red) dots in Figs. 8(b) ing coupling strength the smooth profile breaks up into
and (d) depict the scaled profile of Fig. 8(a) and (b), two parts (Fig. 9(b)). Further decrease of the coupling
respectively. strength leads to the appearance and growth of inco-
herent parts, repeating again the typical scenario of the
coherence-incoherence bifurcation described above.
VI. LORENZ MODEL Additional numerical evidence shows that system (20)
ischaracterizedbyhighmultistabilityinvolvingdifferent
To investigate how universal the results obtained so typesofsolutions. Therearenoclearlyseparatedtongues
far are, we provide one more example of a chaotic in the (r,σ)-parameter plane present as in logistic maps
9
and R¨ossler systems, but there is strong overlap of the strengthcompletelyincoherentstatesarefoundasasign
regimesofdifferentwavenumbersk =1,2,3,... andthe offullydevelopedspatialchaos. Asimilartransitionfrom
regionofspatiallycoherentsynchronizedchaoticdynam- coherence to incoherence is found for nonlocally coupled
ics (k =0). This is due to the completely different local Lorenzsystems. Ourfindingsholdforbothtime-discrete
dynamics, which is characterized by a double scroll and and time-continuous systems indicating the universality
two symmetric saddle-foci. The coupling induces stable of the transition.
spatially coherent steady states in the form of wavelike Inconclusion,wehaveanalyzedindetailthetransition
profiles with wave numbers k =1,2,3,... (Fig. 9(a)). between spatial coherence and incoherence that appears
in different systems with nonlocal couplings, and we find
similarbifurcationscenariofornonlocallycoupledlogistic
VII. CONCLUSION maps,R¨osslerandLorenzsystems. Beyondnumericalin-
vestigations,wehavedeterminedanalyticallythecritical
By both numerical and analytical means we have in- coupling strength for this bifurcation, and established a
vestigated the transition from coherence to incoherence scaling relation for profiles with different wave numbers.
in coupled chaotic systems. In dependence upon the
coupling strength and coupling range we have observed
a combination of period-adding in space and period- ACKNOWLEDGMENTS
doublingintimefornonlocallycoupledchaoticmapsand
R¨ossler systems. An inspection of the spatial profiles IO and PH acknowledge support by the BMBF un-
reveals a coherence-incoherence bifurcation, where the der the grant no. 01GQ1001B (F¨orderkennzeichen). ES
smooth coherent profile of the solution breaks up into and YM acknowledge support by Deutsche Forschungs-
parts and gives rise to partially coherent solutions and gemeinschaft in the framework of SFB910. We thank
chimera states. Upon further decrease of the coupling Edward Ott for stimulating discussions.
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