networks on the edge of chaos: global feedback control of turbulence in oscillator networks

Networks on the edge of chaos: Global feedback control of turbulence in oscillator networks

Santiago Gil* and Alexander S. MikhailovAbteilung Physikalishe Chemie, Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany

�Received 12 August 2008; revised manuscript received 28 November 2008; published 25 February 2009�

Random networks of coupled phase oscillators with phase shifts in the interaction functions are considered.In such systems, extensive chaos �turbulence� is observed in a wide range of parameters. We show that, byintroducing global feedback, the turbulence can be suppressed and a transition to synchronous oscillations canbe induced. Our attention is focused on the transition scenario and the properties of patterns, includingintermittent turbulence, which are found at the edge of chaos. The emerging coherent patterns represent variousself-organized active �sub�networks whose size and behavior can be controlled.

DOI: 10.1103/PhysRevE.79.026219 PACS number�s�: 05.45.Xt, 89.75.Fb, 05.45.Gg, 89.75.Hc

I. INTRODUCTION

Network-organized nonlinear systems are ubiquitous innature and in the human world �1�. They may well be evenmore important with respect to applications than continuousnonlinear media, such as reaction-diffusion systems, whichhave been extensively studied. Understanding principalmodes of self-organization in network-organized systemsand finding ways of controlling them is a major scientificchallenge. While studies on network dynamics are rapidlyprogressing �see Refs. �2,3��, only some of its aspects havebeen so far mainly considered. Studies are often focused onlinear stability of synchronous oscillations in such systemsand its dependence on the network architecture. Comparingthis to studies of nonequilibrium pattern formation in nonlin-ear continuous reaction-diffusion models �4–7�, one can no-tice that other kinds of coherent dynamical patterns, beyondof uniform oscillations, are possible too. What would be theanalogs of traveling or standing waves, Turing patterns, orhigh-dimensional chaos �turbulence� for the network-organized dynamical systems? What are the bifurcations cor-responding to transitions between different coherent patternsin the networks? How would the emergence and propertiesof self-organized coherent patterns or turbulence in dynami-cal network systems be controlled? Many of such questionsare awaiting their analysis.

Our study is motivated by previous investigations on con-tinuous nonequilibrium media. Diffusive coupling betweenperiodic oscillators on a lattice or in a continuous mediumcan destabilize uniform oscillations and lead to diffusion-induced extensive chaos, known as chemical turbulence�5,8�. In experimental and theoretical investigations �9–13�,it has been shown that this kind of turbulence can be con-trolled by introducing global feedback. Such feedback cannot only induce uniform oscillations, but also produce vari-ous coherent dynamical patterns, observed at the edge ofchaos within the transition region.

Here we consider the network analog of the problem stud-ied in Ref. �11�. We take a network system where individualelements, occupying the nodes of a network, are periodicphase oscillators. We introduce interactions between the os-

cillators, which depend only on the differences in the phasestates of the network neighbors and thus correspond to thediffusive coupling in continuous media. We show, by directcomputation of Lyapunov exponents and embedding dimen-sions, that such interactions are sufficient to break down uni-form oscillations and to establish high-dimensional chaos�network turbulence�. We introduce global feedback anddemonstrate that, through its action, not only synchronizationcan be restored, but also various, complexly organized dy-namical regimes can be achieved. Then, we systematicallystudy coherent structures and intermittent turbulence in thetransition from developed network turbulence to synchro-nous oscillations under the variation of the global feedbackintensity.

Our analysis reveals that the transition can be divided intotwo distinct stages. In the first stage, groups of elementsbecome active, that they repeatedly perform excursions fromthe common synchronous state. These groups are organizedinto �sub�networks whose collective dynamics depends onthe size of a �sub�network and its architecture. As the feed-back intensity is decreased, the fraction of active elementsgrows. The networks formed by these elements increase insize and become more numerous. When a certain feedbackintensity is reached, such networks merge in a percolationtransition, so that a large connected network of active ele-ments is formed. Even before this transition, network dy-namics becomes chaotic, as signalled by positive values ofthe maximal Lyapunov exponent.

In the next stage, changes in the internal organization ofthe dynamical system are best monitored by looking at theglobal Kuramoto order parameter which specifies the ob-served degree of synchronization. We find that, as the feed-back intensity is further decreased, this order parameter rap-idly decreases and becomes very small below a certain finitethreshold. Because this global parameter determines themagnitude of the global feedback signal in the system, itsvanishing indicates the failure of the global feedback to en-train the system. Our calculations show that the maximalLyapunov exponents and the Kaplan-Yorke embedding di-mension of the system strongly increase as the feedbackbreakdown transition is approached.

After this transition has taken place, all system elementsstill receive relatively strong signals from their networkneighbors and they do not become effectively independent.Analyzing the internal organization of the system in this tur-*[email protected]

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bulent state, we find that it is characterized by repeated syn-chronization bursts that involve only small groups of net-work elements. Remarkably, each next burst appears to beproduced by a different group of network elements.

The considered model is formulated in the next section.The chaos transition scenario is outlined in Sec. III. This isfollowed by a more detailed numerical analysis of the systembehavior at two characteristic stages in the subsequent twosections. The paper ends with conclusions and discussion ofobtained results.

II. THE MODEL

We consider a dynamical system representing a networkof N identical phase oscillators with phase-shifted pairwiseinteractions. It is described by a set of differential equations

�̇i = � +1

�N�

j=1,j�i

N

Tij sin�� j − �i + 2�Aij� . �1�

Because all oscillators are identical, the common natural fre-quency can be assumed equal to zero ��=0�, which corre-sponds to the use of a corotating reference frame. T is theadjacency matrix of the interaction network and � is its av-erage connectivity �Ti,j�, so that �N is the mean degree of anode. The adjacency matrix is not symmetric and thereforedirected interactions are present in the system. Interactionsbetween the elements are characterized by random phaseshifts 2�Aij. Since directed interactions are considered, thephase shifts are generally different in two directions i→ j andj→ i. Note that in the globally coupled case �Tij =1 for any iand j� and for phase shifts taking only one of two valuesAij =0 or Aij =1 /2, the considered model becomes reduced tothat previously studied by Daido �14�. However, in contrastto the Daido model, our system is not variational and there-fore its dynamical properties are significantly different. Notealso that the model �1� can be derived in the weak interactionlimit from the model of coupled phase oscillators with timedelays �see Ref. �15��; however, we do not perform such areduction in this paper and take Eq. �1� as the starting pointof our analysis.

The matrix of phase shifts is chosen as Aij =aij�, where aijare random numbers independently drawn from the interval�0, 1� for any pair �i , j�. The parameter � specifies the char-acteristic magnitude of the phase shifts in the model. Ournumerical investigations show that, for sufficiently large val-ues of �, network dynamics is chaotic and has a large em-bedding dimension, so that high-dimensional chaos �turbu-lence� is observed in the system. In the numericalsimulations that follow, the value �=0.4 is chosen, a valuehigh enough to induce chaotic dynamics.

To control the behavior of the network, global feedback isused in our investigations. Generally, global feedback controlmeans that information about the states of all elements in asystem is gathered and used to generate a common �global�control signal which is applied back to all elements �13�. Forour study, the global signal shall be chosen as

Z =1

N�i=1

N

exp�ı�i� . �2�

Thus, it coincides with the Kuramoto order parameter �8�used to measure the degree of synchronization in a system ofoscillators. We assume that the global signal is applied as anexternal force to all oscillators in the network

�̇i =1

�N�

j=1,j�i

N

Tij sin�� j − �i + 2�Aij� +�

2i�Ze−i�i − c.c.� ,

�3�

where � is the parameter that specifies feedback intensity.Introducing the magnitude R and the phase � of the glo-

bal signal as Z=R exp�i��, we arrive at the final set of equa-tions that describe the considered model

�̇i =1

�N�

j=1,j�i

N

Tij sin�� j − �i + 2�Aij� + �R sin�� − �i� .

�4�

In our investigations, we will perform numerical simulationsof systems composed of N=1000 identical oscillators. As apattern of interactions T, standard networks of the Erdös-Renyi type will be used. In such networks, �directed� linksconnecting two elements �i→ j� are chosen independently atrandom with some probability �. Explicitly, the elements ofmatrix T are

Tij = �1 with probability � ,

0 with probability 1 − � , �5�

where the first condition excludes loop edges. Therefore, inthe limit of large networks the mean degree of a node is�k�=�N. For our simulations, we have set �=0.006, havingchecked that, despite the low connectivity, the considerednetworks are still fully connected, so that each node can bereached from any other node in the network.

III. LYAPUNOV EXPONENTS AND KAPLAN-YORKEDIMENSION

To characterize chaotic dynamics, two measures havebeen chosen in our investigations. The first of them is themaximal Lyapunov exponent that measures the rate of theexponential divergence of the trajectories. The second is theKaplan-Yorke embedding dimension, which tells what is theminimal number of independent variables generally neededto describe the chaotic attractor �16�.

Lyapunov exponents are a set of numbers �i that charac-terize the average rate of local divergence �or convergence�of close trajectories within an attractor. In particular, themaximal Lyapunov exponent �1 is the largest of these num-bers and it indicates whether neighboring trajectories sepa-rate or converge in time. For the calculation of the full spec-trum of Lyapunov exponents, we shall follow the methoddeveloped by Benettin et al. �17� and Shimada and Na-gashima �18� �see also Ref. �19��.

The maximal Lyapunov exponent can be computed bytracking the evolution of small perturbations to an orbit be-

SANTIAGO GIL AND ALEXANDER S. MIKHAILOV PHYSICAL REVIEW E 79, 026219 �2009�

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longing to the attractor. For this, it is convenient to use thelinearized form of the evolution equations. This amounts to

observing the evolution of a perturbation vector q1� that obeys

�

�tq1� �t� = J„�i�t�…q1

� �t� , �6�

where J is the Jacobian matrix of the system. Independentlyof the initial condition q�1�0�, this vector tends to adopt thedirection of maximal elongation �or minimal compression� inphase space. The average rate of growth of the module ofq�1�t� over the attractor yields the maximal Lyapunov expo-nent.

The remaining Lyapunov exponents can be calculated byrestricting the possible directions of perturbations. For ex-ample, �2 measures the rate of growth of a perturbation q�2 ina direction orthogonal to q�1. Generally, �i represents the av-erage growth rate of a perturbation vector q� i which belongsto the subspace orthogonal to vectors q�1 , . . . ,q� i−1. It is clearthen that the Lyapunov exponents calculated in this way sat-isfy the condition �1�2 ¯ �N−1�N.

In numerical implementations, several precautions need tobe taken �20�. In our case, in particular, we know that thesystem is invariant under rigid translations, that is, undertransformations of the variables �i�→�i+C for all i. Thismeans that perturbations along the direction q�1= �1, . . . ,1�will remain unchanged, resulting in an exponent equal to 0.These perturbations provide no information on the dynamicsof the system, and we therefore force q�1 to be orthogonal tothe vector �1,…,1� in every time step, letting it evolve freelyin all other directions. This means that the Lyapunov spec-trum of our system will be defined by N−1 exponents, ex-cluding the zero exponent that is always present.

In this way, the entire spectrum of Lyapunov exponentscan be calculated. An important quantity to derive from thespectrum is the dimensionality of the attractor. This is calcu-lated using the Kaplan-Yorke definition �21� which reads

D = k +�i=1

k �i

�k+1, �7�

where k is such that �i=1k �i0 but �i=1

k+1�i�0. The dimensionis in general a noninteger number. This means that the small-

est integer larger than or equal to D is the number of inde-pendent variables that would be necessary to describe thedynamics within the attractor.

The maximal Lyapunov exponent and the Kaplan-Yorkedimension are shown as functions of the control parameter �in Figs. 1 and 2, respectively. Additionally, four histogramsof the distributions of all N−1 Lyapunov exponents areshown in Fig. 3 for different values of �. In Fig. 1, we canimmediately see that the maximal Lyapunov exponent ispositive in the absence of global feedback, that is, at �=0.This implies that the behavior is chaotic, as anticipated in theprevious section. Furthermore, the embedding dimension ofthis chaotic attractor is high �D=137.2 at �=0� and compa-rable to the dimension �size� of the system N=1000. Thus,our system can be considered as exhibiting network turbu-lence in absence of the feedback.

In contrast, for large values of � we can see that �1 isnegative, indicating the presence of a stable fixed-point at-tractor. The dimension is accordingly D=0. This correspondsto a state of synchronization, induced by the global

0 0.2 0.4 0.6 0.8 1µ

-0.2

-0.1

0

0.1

0.2

Max

imal

Lya

puno

vE

xpon

ent

µ3µ1µ2

FIG. 1. �Color online� Dependence of the maximal Lyapunovexponent on the control parameter �.

0 0.2 0.4 0.6 0.8 1µ0

100

200

300

Attr

acto

rdi

men

sion

µ3 µ2 µ1

FIG. 2. �Color online� Dependence of the Kaplan-Yorke embed-ding dimension on the control parameter �.

0

20

40

60

80

-3 -2 -1 00

20

40

60

80

-3 -2 -1 0

Lyapunov exponents

µ=0.1µ=0.4

µ=1.0 µ=0.8

(a) (b)

(d)(c)

FIG. 3. �Color online� Histograms of Lyapunov exponents fordifferent values of the control parameter �.

NETWORKS ON THE EDGE OF CHAOS: GLOBAL… PHYSICAL REVIEW E 79, 026219 �2009�

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feedback.1 The distribution of Lyapunov exponents at�=1.0 is approximately symmetric and roughly centered at−1.3 �Fig. 3�a��.

When the intensity of the global feedback is decreased,the synchronous stationary state becomes unstable at �1�0.92 and we get �1=0 �Fig. 1�. The maximal Lyapunovexponent remains equal to zero in an interval of � below thistransition, so that limit-cycle oscillations take place there. Itis interesting that the Kaplan-Yorke dimension of the attrac-tor grows below �=�1, even though chaos is absent. Thisindicates that the system has several zero Lyapunov expo-nents, including �1 �indeed, if k zero Lyapunov exponentsare present, D=k according to Eq. �7��. The presence of de-generate zero Lyapunov exponents is directly demonstratedby Fig. 3�b�, where a clear peak is present at 0.

The attractor becomes chaotic at �2�0.77, where �1 firstbecomes positive. Both the value of the maximal Lyapunovexponent and the dimension of the attractor grow monoto-nously when decreasing the intensity of the global feedback.Thus, the behavior becomes more complex as chaos devel-ops. The dimension reaches high values, up to D=244.8,significantly higher than that in the absence of global feed-back. As seen in Fig. 3�c�, a large number of Lyapunov ex-ponents are then positive �approximately 10% of them� andthe distribution is now clearly asymmetric.

An abrupt drop occurs in both the maximal Lyapunovexponent and the Kaplan-Yorke dimension at about�3=0.26, indicating a drastic change in the properties of theattractor. The distribution of Lyapunov exponents becomesmore narrow and the fraction of positive exponents is ap-proximately 6%. Remarkably, further changes in � belowthis critical value seem to have no effect on the behavior ofthe system. This suggests that the sudden drop is associatedwith the breakdown of the global feedback.

The results displayed in these figures indicate that thereare several interesting aspects of the transition from synchro-nization to chaos. In the following we study this transitionusing different statistical tools to better describe and under-stand the complex behavior observed in networks at the edgeof chaos.

IV. VELOCITY DISTRIBUTIONS AND ACTIVEELEMENTS

The global feedback in the considered system tends toimpose synchronization. If only global interactions corre-sponding to this feedback are present, the system undergoescomplete synchronization, with all phases becoming identi-cal, �i=�=const. In the presence of network interactions,this state is no longer possible because of the phase shifts inthe interactions between individual oscillators, which bringthe system to a state of frustration �14�. Instead, a stateof frequency synchronization can be reached where�̇=�=const. If this state is stable, all deviations from it are

damped, except for a perturbation representing a uniformshift of all phases. Thus, it will be characterized by negativeLyapunov exponents �the trivial zero Lyapunov exponentcorresponding to uniform phase shifts is discarded�.

As follows from Figs. 1 and 3�a�, frequency synchroniza-tion takes place in the considered system if global feedbackis strong enough, i.e., if ��1. For weaker feedbacks, thisstate is destroyed and internal dynamics develops in the sys-tem. To monitor the gradual destruction of frequency syn-chronization, it is natural to observe the behavior of the ve-locities �̇i of network elements. Below we consider long-time average velocities i, defined as

i =1

�T�

0

�T

�̇i�t�dt =1

�T��i��T� − �i�0�� , �8�

where the time interval �T is large.The mean, maximum and minimum values of all veloci-

ties i at different feedback intensities � have been computed�Fig. 4�. We see that for ��1 the maximum and minimumvelocities coincide, so that all velocities are equal and thesystem is in the state of velocity synchronization. At��3, the mean velocity is close to zero � i��0 and dis-persion of velocities is small. Inside the transition region�1��3, the velocities of individual oscillators arespread broadly.

To characterize the behavior of the system in the transi-tion region, distributions of velocities at three different val-ues of the feedback intensity � have been determined �Fig.5�. We see that, while the overall spread in the velocities islarge, the majority of the oscillators possess the same long-term average velocity, and only a small fraction of them havevelocities that are different.

In the reference frame that rotates with the velocity of thedistribution peak, one would see that most of the oscillatorsremain still or at most vibrate around a fixed point, so thattheir time-averaged velocities are all zero. Therefore, theseelements can be described as forming a synchronous conden-sate . In contrast, the oscillators outside of this group per-form repeated phase rotations around the condensate and canbe described as active elements.

1The synchronous state is invariant with respect to uniform shiftsof all phases. This is reflected in the presence of the additionaltrivial zero Lyapnov exponent, which is discarded.

0 0.2 0.4 0.6 0.8 1

µ

0

0.2

0.4

0.6

0.8

νi

<νι>max{νι}, min{νι}

µ1µ2µ3

FIG. 4. �Color online� Mean time-averaged velocity of the net-work �dots� as function of the feedback intensity �. Thin lines showthe maximum and minimum velocity values.


026219-4

This classification is further clarified if we look at theinstantaneous time-dependent velocities �̇i�t� of the elements�Fig. 6�. The elements belonging to the synchronous conden-sate exhibit only weak temporary deviations from the com-mon velocity. In contrast to this, active elements are charac-terized by repeated velocity pulses. Each of these pulsescorrespond to a full phase rotation, i.e. to a phase slip withrespect to the condensate. Their behavior is reminiscent ofspiking in neural models.

Note that the evolution equation �3� of the consideredmodel can also be written in the form

�̇i =1

2i��zi + �Z�e−i�i − c.c.� , �9�

where the local signal zi acting on an element i is introducedas

zi =1

�N�

j=1,j�i

N

Tijei��j+2�Aij�. �10�

In contrast to the global signal Z, local signals are generatedonly by network neighbors of a given element. Moreover,they depend on the phase shifts Aij that characterize connec-tions between the chosen element and its neighbors.

If we further introduce relative phases of elements as�i=�i−�, where the global phase is defined byZ=R exp�i��, the evolution equations become

�̇i = − � +1

2i��z̃i + �R�e−i�i − c.c.� , �11�

where

z̃i =1

�N�

j=1,j�i

N

Tijei��j+2�Aij� �12�

and �=� is the global velocity. Finally, we can write themin the form

�̇i = − � + �Ri sin��i − �i� �13�

with Ri exp�i�i�=R+�−1z̃i.In the synchronous state �i=0 and, therefore, relative

phases of all oscillators are determined by a set of algebraicequations

sin��i − �i� =�

�Ri. �14�

A solution to these equations exists only if �Ri�� for allnodes i. This is always the case if the global feedback isstrong ��→�� and contributions from the local signals arenegligible. As the global feedback gets weaker, the ratio� /�Ri increases and, eventually, becomes larger than unityfor some of the oscillators. These oscillators can no longer beentrained by the condensate and start to orbit around it, gen-erating phase slips. They form the subset of active elements.When orbiting elements have appeared, they act back on theelements in the condensate and force them to undergo smalltemporal deviations from the synchronous state. This behav-ior can be seen in Fig. 6.

The above analogy with spiking neurons is not farfetched. The transition to orbiting, described by Eq. �13�,represents the saddle-node bifurcation on the invariant cyclewhich is characteristic to type-I neuron models �22,23�. En-trained elements are effectively excitable and can make arotation �i.e., generate a spike� when a strong perturbationarrives. Individual orbiting elements are oscillatory �in thereference frame fixed by the condensate� and perform spon-taneous rotations.

Thus, the system can be understood to be composed ofoscillatory elements and excitable elements with differentthresholds: some active elements are in a suprathresholdstate, repeatedly generating spikes, whereas some elementsare in a subthreshold regime, susceptible to excitations.Threshold values for each element depend not only on thelocal topology of the network, but also on the phase shifts inthe interactions of this element with its network neighbors.

Spontaneous spiking is periodic for �2��1, as evi-denced by the fact that the maximal Lyapunov exponent iszero ��1=0� inside this interval. Actually, as seen in Fig.3�b�, several Lyapunov exponents can be zero inside thisinterval, indicating that many oscillators are orbiting inde-pendently. For the lower feedback intensities, the collectivedynamics of the system becomes chaotic.

ν0.0 0.2 0.4 0.6

020

040

060

080

010

00

µ=0.4

ν0.0 0.2 0.4 0.6

020

040

060

080

010

00

µ=0.6

ν0.0 0.2 0.4 0.6

020

040

060

080

010

00

µ=0.8

(c)(a) (b)

FIG. 5. Distributions of time-averaged velocities of all networkelements for three selected values of feedback intensity �.

100 120 140 160 180 200Time

-1

-0.5

0

0.5

1

-.φ

i

µ=0.87

FIG. 6. �Color online� Instantaneous velocities of several ran-domly chosen elements �=0.87.


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V. ACTIVE SUBNETWORKS

When velocity synchronization breaks down and nonen-trained active elements first appear, only a few of them arepresent. The number of active elements should however in-crease as the feedback intensity is further decreased. Some ofthem would have other active elements as their neighborsand, thus, active subnetworks would be formed. In this sec-tion we identify such subnetworks in the system and inves-tigate how their properties depend on the feedback intensity.

To construct active subnetworks, we fix the global feed-back intensity and let the system evolve. Analyzing its dy-namics after a transient, the set of elements which becomeactive is determined. The active elements occupy certain net-work nodes. Next we retain only these nodes and networkconnections between them, ignoring elements belonging tothe synchronous condensate and their connections. In thisway, the subnetworks of active elements are obtained.

Figure 7 shows the active subnetwork for several intensi-ties � of the global feedback. We see that initially the activesubnetworks consist of a small number of isolated elements��=0.9 and �=0.8�. As new active elements emerge at lowfeedback intensities, small network fragments with two,three, and four elements are formed ��=0.75 and 0.7�. The

connected network components increase in size ��=0.65 and0.6� and finally a large connected component develops ��=0.55�.

In Fig. 8, the total number of active elements and the sizeof the largest connected component are shown depending onthe control parameter �.2 Within a range of the control pa-rameter, the size of the largest connected component is wellbelow the number of active elements, implying that there aremany disconnected network fragments present. For��0.53, both curves become almost coincident, indicatingthat most of the active elements belong to a single giantconnected component. A subnetwork is viewed as a con-nected component if any of its elements is linked to at leastone other element in the subnetwork, neglecting interactiondirections.3

Decreasing the intensity of the global feedback increasesthe size and complexity of subnetworks of active elements.Recalling the curve for the maximal Lyapunov exponent inFig. 1, we notice that it becomes positive and chaos emergesalready for ��0.77, before active subnetworks merge into agiant component. Thus, the dynamics of relatively small sub-networks can already be chaotic.

An important structural property of any network is itsdiameter, defined as the longest of the shortest paths connect-ing pairs of elements. For networks with directed connec-tions, each link in a path should be traversed along its pre-scribed direction. If there are several disconnectedfragments, their diameters can be independently computed.In Fig. 9, the longest diameter is displayed as a function ofthe global feedback intensity. The diameters of active sub-networks can be compared with the the diameter of the fullnetwork, which is equal to 8 in the case considered.

2Since the distinction between active elements and element con-forming a synchronous condensate requires the clear existence ofthe latter, the classification might not be well defined when themajority of the elements are active. However, we have seen in Fig.5 that a pronounced peak for the velocity of the condensate persistseven when more than half of the elements have become active.Therefore, the values presented in Fig. 8 are valid for the very largeportion of the range.

3Note that this condition is weaker than the one used for matrix T.

0.2 0.4 0.6 0.8 1

µ0

200

400

600

800

1000

Size

ofla

rges

tclu

ster

FIG. 8. �Color online� Dependence of the number of active el-ements �line� and the size of the largest connected active subnet-work �dots� on the feedback intensity �.

0.2 0.4 0.6 0.8 1

µ0

5

10

15

20

25

30

Dia

met

er

µ3 µ1µ2

FIG. 9. The largest diameter �dots� of active subnetworks asfunction of the feedback intensity �.

µ=0.90µ=0.80 µ=0.75 µ=0.70

µ=0.55 µ=0.60 µ=0.65

FIG. 7. �Color online� The sequence of active subnetworks ob-served under gradual decrease of the feedback intensity �.


026219-6

As the feedback intensity is decreased, the largest diam-eter grows, starting from zero when only individual nodesare active. It reaches a maximum at��0.49 and then decreases. It is remarkable that, at its peak,the largest diameter of subnetworks is equal to 29, which ismuch higher than that of the underlying network structure.After passing the maximum, the diameter takes the valuesclose to 8, the same as in the underlying structure.

Generally, active subnetworks are characterized by thepresence of long paths and their diameters are significantlylarger than those of the comparable random networks. Forexample, the subnetwork of diameter 29 has 293 nodes andits mean degree is 3.37. Standard directed random networksof this size and the same mean degree would have the diam-eter of about 23.

Active subnetworks are dynamical structures. Their ele-ments perform repeated phase rotations with respect to themain condensate. Some of these elements are oscillatory andplay the role of pacemakes. Other elements in a subnetworkare effectively excitable and their phase rotations are trig-gered by perturbations arriving at them from their neighbors.Thus, wave propagation takes place inside active subnet-works.

This kind of wave progagation is illustrated in Fig. 10.Here, we have taken the feedback intensity �=0.49 yieldingthe subnetwork with the largest diameter of 29 and havechosen the chain of nodes in the path corresponding to thisdiameter. The raster plot in Fig. 10 shows the dynamicalactivity along this chain. The absolute value of the velocitiesof the elements in the chain with respect to the synchronouscondensate are displayed in gray scale, with the dark colorcorresponding to higher values. Repeated wave propagationalong chain fragments is apparent. It can be also noticed thatthe entire activity pattern is irregular.

The elements forming an active subnetwork are stronglyheterogeneous in terms of their intrinsic dynamics. The het-erogeneity is caused by the fact that each element receivessignals through its connections from a different number ofelements—some of them active and others in thecondensate—and these connections are characterized by dif-ferent phase shifts.

The dynamics of an active element is described by Eq.�11�. Separating in this equation the terms which correspondto the interactions of the chosen element with the condensatefrom those representing interactions with other active ele-ments in the subnetwork, we can write it as

�̇i = − � +1

2i��z̃i

c + z̃ia + �R�e−i�i − c.c.� , �15�

where z̃ic and z̃i

a are local signals received by the consideredelement from its neighbors belonging to the condensate andto the active subnetwork, respectively. Introducing Ri

c and �ic

as ric exp�i�i

c�=�R+ z̃ic and taking into account the definition

�12� of local signals, the evolution equation for elements inan active subnetwork is

�̇i = − � + ric sin��i

c − �i� +1

�N�j=1

N

� jTij sin�� j − �i + 2�Aij� .

�16�

Here, vector �i defines what elements belong to the consid-ered active subnetwork, i.e., � j =1 if the element i is in thesubnetwork and � j =0 otherwise. The second term in Eq.�16� describes interaction with the condensate and globalfeedback in the system. When the fraction of active elementsin a network is relatively small, the global feedback is domi-nated by condensate elements.

Therefore, the condensate acts as external forcing on ac-tive elements. However, the magnitude ri

c of such forcing isdifferent for different active elements, which explains theheterogeneity of emerging active networks. Those elementsi, which have ri

c��, are oscillatory and play the role ofpacemakers. Their intrinsic oscillation frequencies are differ-ent and determined by the difference �−ri

c. On the otherhand, the elements with ri

c� are excitable and their exci-tation thresholds are hi=ri

c−�. For excitable elements in anactive subnetwork, excitation thresholds are small enough, sothat the excitation coming either directly from pacemakers orfrom the neighboring excitable elements is sufficient to trig-ger their phase rotation. For the element at the end of a wavepropagation chain, all further network neighbors have thresh-olds so high that the excitation cannot be further transmitted.

Thus, inside an interval of feedback intensities precedingthe final feedback breakdown, the considered system cangenerate a variety of dynamical networks whose sizes anddynamical properties can be controlled by the feedback in-tensity. These networks are effectively built from oscillatoryand excitable elements. They are strongly heterogenous interms of the oscillation frequencies and excitation thresholdsof their elements.

As the feedback intensity � is decreased, the number ofactive, nonentrained elements grows and they begin to domi-nate the dynamics of the system. In the next section we con-sider the behavior corresponding to the final loss of coher-ence in the transition.

VI. BREAKDOWN OF GLOBAL FEEDBACK

Calculations of the maximal Lyapunov exponent and theembedding dimensions �Figs. 1 and 2� have indicated thepresence of a singularity at about �3=0.26. Below this point,both properties cease to depend on the feedback strength,thus suggesting feedback breakdown. Our investigations inSec. IV have shown �Fig. 4� that this breakdown is alsoaccompanied by a significant change in the distribution of

Time

Nod

esin

dire

cted

path

FIG. 10. Space-time diagram showing the dynamical activitypattern in the chain of 29 elements belonging to the activesubnetwork.


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average velocities. As the critical point �=�3 is approached,the mean time-averaged velocity of the network sharply de-creases. The dispersion of time-averaged velocities, i.e., thedifference between the largest and the smallest average ve-locities observed, becomes large in the region preceding thetransition. After the transition, both the mean velocity and itsdispersion remain small and do not depend on �. Below inthis section we provide further analysis of the feedbackbreakdown transition and dynamical properties of the net-work in its vicinity.

In Fig. 11, the computed dependence of the time-averagedglobal signal intensity W= �Z�t�� is shown. We see that itdrops down to �almost� zero at the critical point and remainssmall below it. This provides direct evidence of the feedbackbreakdown. The global signal Z= �1 /N��i=1

N exp�ı�i� coin-cides with the Kuramoto order parameter of the consideredoscillator population. Its vanishing indicates therefore theloss of synchronization at �=�3. However, the observedphenomenon is essentially different from the Kuramoto de-synchronization transition in globally coupled systems �5�.

In oscillator populations with global, all-to-all coupling,vanishing of the Kuramoto order parameter leads to the lossof interactions between elements. They become almost inde-pendent, with the rest interactions inversely proportional tothe size N of the population and vanishing in the limitN→�. This is not, however, the case in the considered net-work system.

The dashed line in Fig. 11 shows the computed depen-dence of the mean time-averaged local signal intensity w onthe feedback strength �, defined as

w =1

N�i=1

N

�zi�t�� , �17�

where zi�t� is given by Eq. �10�. We see that, while the meanlocal signal decreases together with the global signal as thecritical point �=�3 is approached, it does not vanish at thispoint. Below �=�3, the mean local signal remains strongand does not depend on the feedback intensity.

Thus, local interactions do not disappear after the feed-back breakdown transition in the considered network system.

Instead of full desynchronization—characteristic for globallycoupled populations—the feedback breakdown leads the sys-tem to the highly dimensional chaotic state of network tur-bulence. Previously, analogous global feedback transitionshave been investigated by Kawamura and Kuramoto �24� forcontinuous active media described by the complex Ginzburg-Landau equation. It is remarkable that, as seen in Figs. 1 and2, the maximal Lyapunov exponent and the embedding di-mension increase as the critical point �=�3 is approachedand abruptly drop below it.

To investigate this transition, we characterize the systemin terms of time-dependent power spectra, broadly employedin the statistical analysis of the brain EEG data �see, e.g.,Ref. �25��. First, the autocorrelation function S�� , t� of theglobal signal is calculated within a sliding time window ofwidth T,

S��,t� =1

T�

t

t+T

Z�t��Z*�t� + ��dt�, �18�

After that, the time-dependent spectral density S�� , t� of theglobal signal is determined by the Fourier transform

S��,t� = 1

T�

0

T

S��,t�e−i��d� .

In our investigations we use a sliding time window ofwidth T=110. In the plots showing the temporal evolution ofthe spectral density, a gray scale representation with the darkregions corresponding to the higher density is employed.

Figure 12 shows time dependent spectral densities forseveral selected values of the feedback intensity, both aboveand below the feedback breakdown. Above the transition

0 0.2 0.4 0.6 0.8 1µ0

0.2

0.4

0.6

0.8

1W

,waverage local signalglobal signal

µ3 µ2 µ1

FIG. 11. �Color online� Time-averaged magnitude of the globalsignal �solid line� and mean time-averaged magnitude of local sig-nals �dash line� as functions of the feedback intensity �.

0

0.05

0.1

0

0.05

0.1

0

0.05

0.1

0

0.05

0.1

Time1000 1500 2000 2500 3000 3500 40000

0.05

0.1

Freq

uenc

y

µ=0.266

µ=0.265

µ=0.24

µ=0.20

µ=0

FIG. 12. Time-dependent spectral densities of the global signalat different feedback intensities �. The absolute magnitude of thesignal is much larger at �=0.266, than in other plots shown.


026219-8

��=0.266�, a peak is persistently present in the spectral den-sity of the global signal. The position of this peak corre-sponds to the frequency of coherent collective oscillations.At the critical point ��=0.265�, such persistent peak is ab-sent. Instead, the system still exhibits irregular synchroniza-tion bursts seen as temporary maxima in the spectral densityat the frequency of collective oscillations. This intermittentbehavior is retained below the transition ��=0.24,0.2�.However, as the feedback intensity is gradually decreased,synchronization bursts become more rare and the character-istic duration of a burst decreases. Finally, in absence of thefeedback ��=0�, spontaneous synchronization bursts havenot been found.

It is interesting to compare spectral properties of the glo-bal signal with the respective dynamical properties of asingle oscillator. The time-dependent power spectrum of anoscillator i is defined as

si��,t� = 1

T�

0

T

si��,t�e−i��d� , �19�

where

si��,t� =1

T�

t

t+T

ei��i�t��−�i�t�+��dt�. �20�

Figure 13 shows the typical time-dependent power spec-trum of an arbitrarily chosen element above the breakdowntransition ��=0.265�. The spectral peak, observed in thespectral density of the global signal, is also visible in thespectrum of a single oscillator above the transition. Thisshould indeed be expected if we take into account that alloscillators effectively experience external forcing generatedby the global feedback, and their spectra therefore reflect thepresence of such forcing.

Below the transition, the global signal becomes weak anddoes not exhibit strong influence on a typical individual os-cillator. A detailed examination reveals however that someoscillators still �temporarily� possess a small maximum intheir spectral density at the level of collective oscillations.Such oscillator groups may generate synchronization burstsvisible in the spectral density of the global signal.

To test the conjecture that only some oscillator groups areresponsible for synchronization bursts in the intermittent re-gime and that these groups are different in each next burst, aspecial investigation has been performed. We have chosentwo different bursts at time positions and and calculatedspectral densities si�� , t1� and si�� , t2� of all network ele-ments individually. Then, we have compared the computed

spectral densities and selected groups of 50 oscillators at twotimes, whose spectra were closest to that of the global signal.To quantify the similarity between two spectral densities, weuse the overlap

qi�t� =�si��,t�S��,t�d�

�si��,t�d��S��,t�d�. �21�

After each oscillator group � of 50 elements has been se-lected, we have generated its summary signal

Z��t� =1

50 �i��

ei�i�t�

and determined its time-dependent spectral density S�� , t�.Figure 14 shows time-dependent spectral densities of the

summary signals generated by two groups �1 and �2, whichhave been chosen as those mostly contributing to the spectraldensity of the global signal at t1=822 and t2=1384, respec-tively. For comparison, we also show in this figure the spec-tral density of the global signal within the considered timeinterval. Examining the figure, we see that, although tracesof the spectrum of the global signal are visible in both partialsummary spectra, the group of elements with the closest in-dividual similarity to the global signal at a certain burst isalready well collectively reproducing the global signal in therespective time interval. Thus, it seems plausible that only asubset of all network elements is strongly involved in thegeneration of a synchronization burst. Note that in the aboveanalysis we have not attempted to identify all the elements inthe respective groups and only demonstrated choosing a sub-set of 50 nearest elements is sufficient.

Analyzing composition of the two groups, correspondingto synchronization bursts at t1=822 and t2=1384, we findthat they have only two common elements. Thus, the subsetof partially synchronized elements is indeed different in dif-ferent bursts in the intermittent turbulence regime. The ele-ments of a group represent nodes in the considered networkand there are links connecting them. Essentially, each groupdefines a certain subnetwork which temporally undergoespartial synchronization. Therefore, chaotic intermittency inthe considered system can be understood as generating an

Time

Freq

uenc

y

400 600 800 1000 1200 1400 1600 18000

0.05

0.1

FIG. 13. Time-dependent spectral density of a single oscillatorabove the feedback breakdown transition ��=0.266�.

Time

Freq

uenc

y

400 600 800 1000 1200 1400 1600 18000.02

0.04

0.06

0.08

0.02

0.04

0.06

0.08

Time

Freq

uenc

y

400 600 800 1000 1200 1400 1600 18000.02

0.04

0.06

0.08

Γ1

Γ2

FIG. 14. �Color online� Time-dependent spectrum plot of�above� the global signal and �below� collective signals of two spe-cially selected groups of oscillators. �=0.265.


026219-9

irregular sequence of synchronization events imposing par-tial synchronization on different subnetworks. A synchro-nized subnetwork emerges, persists for some time, dissolvesand, after a pause, is replaced by a different synchronizedsubnetwork.

VII. DISCUSSION AND CONCLUSIONS

We have chosen a network model of coupled phase oscil-lators which exhibits highly dimensional chaos �network tur-bulence� and have shown how this chaos can be controlledand synchronization induced by introducing a global feed-back. Our attention has been focused on the properties ofdynamical patterns and intermittent turbulence characteristicfor the transition region. An important property of the con-sidered model was that individual elements were not chaoticand chaos emerged as a result of coupling between them.

Previously, investigations on global-feedback control ofturbulence have been performed for continuous oscillatorymedia, in the general model of the complex Ginzburg-Landau equation �9,10� and in experimental and theoreticalstudies of the catalytic CO oxidation reaction �11,12�. Actionof global feedbacks on chaotic oscillatory systems with non-local coupling between elements has also been considered�26�. In all these systems, the individual oscillators were notchaotic and highly dimensional chaos �chemical turbulence�emerged as a result of diffusive coupling between them.These investigations have revealed that the behavior of os-cillatory reaction-diffusion media on the edge of chaos ischaracterized by spontaneous appearance of various coherentactivity patterns �see also the review �13��.

Specifically, propagating phase defects �kinks� spontane-ously develop as the feedback intensity is decreased and cas-cades of reproducing kinks are observed. In a traveling kink,the phase undergoes a full 2� rotation, or a slip �9,13�. Re-peatedly appearing kink cascades destroy synchronous oscil-lations in the system and establish the state of intermittentturbulence.

Our work shows that in networks, the role of emergingcoherent structures is played by active subnetworks. Some ofthe elements, forming such networks, spontaneously undergorepeated phase slips. Other elements produce phase slips inresponse to the excitation coming from their network neigh-bors. As result, traveling waves of phase slips repeatedlydevelop and propagate over a subnetwork, spreading fromseveral emergent pacemaker centers. These traveling wavescorrespond to the cascades of kinks in continuous media.

While each next cascade in the continuous case is typi-cally originating at a different spatial location and involves adifferent set of the elements of a medium, active networksare permanent in the considered network model. Althoughthe patterns of wave activity in a subnetwork may be quitecomplex and chaotic, the subnetwork itself remains fixed.

This difference is due to the fact that the considered os-cillator networks are strongly heterogeneous, with the het-erogeneity imposed both by the structure of the network ofconnections and the phase shifts in interactions between in-dividual elements. As a result of such heterogeneity, someelements become easily excitable and others turn into pace-

makers. Together, such connected elements form permanentactive subnetworks.

When wave patterns in active subnetworks are observed,the resting elements, representing the majority of a network,are in a synchronous state and form the condensate. As thefeedback intensity is decreased, the number of active subnet-works increases and each network typically grows in size. Ina percolation transition, individual subnetworks merge toform a giant active component, already comprising a sub-stantial fraction of all network elements.

Starting from this point, distinguishing between active el-ements and the condensate becomes difficult. The systemshows developed chaos and its embedding dimension iscomparable to the total network size. Nonetheless, some in-trinsic coherence is still present in its dynamics, as revealedby the persistence of the relatively strong Kuramoto orderparameter characterizing synchronization.

As the feedback intensity is further decreased and ap-proaches the critical point, the Kuramoto order parameter�almost� vanishes, indicating the loss of persistent synchro-nization. Since this order parameter yields at the same timethe magnitude of the global signal, we conclude that break-down of the feedback control takes place. Analogousfeedback-breakdown transitions have previously been stud-ied for continuous oscillatory media under the global feed-back control �24�. In contrast to the global signal, local sig-nals corresponding to network interactions do not vanishafter the transition. Therefore, it is different from the desyn-chronization transition in globally coupled systems where,after the transition, interactions between elements almostvanish and the oscillators become effectively independent.

The time-dependent spectral analysis of the global signalhas shown that, after its breakdown, the feedback is able toinduce sporadic bursts of synchronization in the system. Thisbehavior is reminiscent of what has previously been seen inthe oscillatory systems comprising global and nonlocal cou-pling �26�. Our analysis suggest that different groups of ele-ments are responsible for each next synchronization burst.Taking into account their connections, this means that differ-ent subnetworks of the entire network spontaneously exhibit�partial� synchronization. Each synchronization episode in-volves a different subnetwork and such episodes are alternat-ing with complex asynchronous states.

This looks similar to the behavior characteristic for devel-oped hydrodynamic turbulence, where different coherentstructures are built and replace one another in an irregularsequence. In networks, spatial ordering is absent and there-fore spatiotemporal patterns cannot obviously develop. In-stead, coherent structures represent various dynamical sub-networks which get accentuated. The simplest form ofcoherence is partial synchronization, but other, more compli-cated kinds of coherent dynamics in the emerging networksare also possible.

The emphasis in this study has been on control of high-dimensional network chaos, rather than on detailed investi-gations of network systems showing such kind of turbulence.We wanted to demonstrate that, by applying global feedbackand adjusting its intensity, one can readily control the net-work dynamics. Therefore, we have chosen a particularmodel network, fixed its properties and then varied only the


026219-10

feedback parameters in our investigations. We have, how-ever, checked that analogous results hold as well for othernetwork realizations with various sizes.

While our investigations have been performed for a par-ticular model, we conjecture that their results should be gen-eral and similar behavior will be found in other network-organized systems of various origins. Indeed, it can beshown �27� that the considered model is closely related to thegeneric phase network model obtained by phase reduction ofamplitude oscillator dynamics on networks in the vicinity of

the supercritical Hopf bifurcation. The instability leading tothe network turbulence in the model is the analog of theBenjamin-Feir instability of uniform oscillations in the com-plex Ginzburg-Landau equation.

ACKNOWLEDGMENTS

Stimulating discussions with Y. Kuramoto are gratefullyacknowledged. Finacial support from the Volkswagen Foun-dation �Germany� is acknowleged.

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networks on the edge of chaos: global feedback control of turbulence in oscillator networks

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