PHYSICAL REVIEW E 89, 062906 (2014)

Clustering in globally coupled oscillators near a Hopf bifurcation: Theory and experiments

Hiroshi Kori,1,2,* Yoshiki Kuramoto,3 Swati Jain,4 Istvan Z. Kiss,5 and John L. Hudson4

1Department of Information Sciences, Ochanomizu University, Tokyo 112-8610, Japan2CREST, Japan Science and Technology Agency, Kawaguchi 332-0012, Japan

3International Institute for Advanced Studies, Kyoto 619-0225, Japan4Department of Chemical Engineering, University of Virginia, Charlottesville, Virginia 22904, USA

5Department of Chemistry, Saint Louis University, St. Louis, Missouri 63103, USA(Received 17 January 2014; published 10 June 2014)

A theoretical analysis is presented to show the general occurrence of phase clusters in weakly, globally coupledoscillators close to a Hopf bifurcation. Through a reductive perturbation method, we derive the amplitude equationwith a higher-order correction term valid near a Hopf bifurcation point. This amplitude equation allows us tocalculate analytically the phase coupling function from given limit-cycle oscillator models. Moreover, usingthe phase coupling function, the stability of phase clusters can be analyzed. We demonstrate our theory withthe Brusselator model. Experiments are carried out to confirm the presence of phase clusters close to Hopfbifurcations with electrochemical oscillators.

DOI: 10.1103/PhysRevE.89.062906 PACS number(s): 05.45.Xt, 82.40.Bj

I. INTRODUCTION

The dynamics of weakly interacting oscillating units can bedescribed with phase models [1]. For example, for N identicaloscillators with global (i.e., mean field) coupling, we have

d!i

dt= " + #

N

N!

j=1

$(!i ! !j ), (1)

where !i (i = 1, . . . ,N) and " are the phase and the frequencyof oscillator i, respectively, # is the coupling strength, and $is the phase coupling function. Phase models have been for-mulated from ordinary differential equations describing, e.g.,chemical reactions [1]. Recently they have been formulatedfrom direct experiments as well [2–4]. A prominent featureof such phase models is the presence of higher harmonics inthe phase coupling function. As a result of higher harmonics,complex dynamics including chaos and multiphase clusterscan be observed [5–9]. Here we refer to phase clusters asclustering behavior purely attributed to phase dynamics. Clus-tering dynamics that cannot be described by phase models arereferred to amplitude clusters [10]. Phase clusters have beenexperimentally observed in a wide range of systems includingelectrochemical systems, light-sensitive BZ reactions, andcarbon monoxide oxidation on platinum [2,3,11–17]. Thenumber of clusters and their appearance with positive ornegative coupling or feedback is puzzling. Multiphase clustersare usually explained by the presence of higher harmonics inthe coupling functions in the phase model [5]; however, themechanism through which higher harmonics can develop isunclear.

In this paper we give a theoretical explanation for clustersclose to oscillations that develop through Hopf bifurcations.With analytical derivation, we show how the higher harmonicsoccur in the phase coupling function through two-step reduc-tions: an amplitude equation is derived from coupled limit-cycle oscillators through a reductive perturbation method, and

*kori.hiroshi@ocha.ac.jp

then the phase coupling function is derived from the amplitudeequation through the phase reduction method. Experiments arecarried out to confirm the presence of multiphase clusters closeto Hopf bifurcations with electrochemical oscillators.

II. THEORY

A. Overview

The Stuart-Landau (SL) oscillator, which is a local elementof the complex Ginzburg-Landau model [18,19], is consideredas a general skeleton model for the description of oscillatorsclose to Hopf bifurcations. It can be derived as a lowest-order amplitude equation through a reductive perturbationmethod [1], as summarized in Sec. II B. However, a populationof SL oscillators coupled globally and linearly cannot describemultiphase clusters because its corresponding phase couplingfunction $ in Eq. (1) does not contain second or higherharmonics; i.e., $(%!) " sin(%! + & ) + a0, where a0 and& are constant [1,10]. As shown in Table I(a), the stabilityanalysis of such a phase model predicts that the only stablebehavior is the one-cluster state (in-phase synchrony).

However, it should be noticed that infinitesimally smallperturbations given to the SL system may alter a neutral stateto a stable or unstable one; i.e., the SL system is structurallyunstable. Indeed, we will show that when we take into accounthigher-order correction to the amplitude equation, multiclusterstates may become asymptotically stable. The number ofclusters and whether they occur with negative or positivecoupling depends on the types of nonlinearity in the ordinarydifferential equations. The stability of the cluster state is weak;nonetheless it is expected that they can be observed in globallycoupled oscillators even very close to a Hopf bifurcation point.

B. Lowest-order amplitude equation

Consider a network of identical limit-cycle oscillators withlinear coupling

xi = f (xi ; µ) + µ#

N!

j=1

Aij D(xj ! xi), (2)

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KORI, KURAMOTO, JAIN, KISS, AND HUDSON PHYSICAL REVIEW E 89, 062906 (2014)

TABLE I. General properties of phase clusters in theoreticalmodels close to a Hopf bifurcation with weak, global coupling.(a) Properties in the lowest-order amplitude equation [Eq. (9)].(b) Properties in the higher-order amplitude equation [Eq. (21)].“Desync” refers to the desynchronized state, which is the state ofuniform phase distribution. This state is also called the splay state.Here we assumed that the one cluster state is asymptotically stablefor # > 0. When one cluster state is stable for # < 0, the stability ofthe desynchronized state is exchanged between # > 0 and # < 0.

(a) One cluster Desync Multiclusters

# > 0 Stable Unstable Unstable# < 0 Unstable Neutral or neutral

(b) One cluster Desync Multiclusters# > 0 Stable Unstable Unstable,# < 0 Unstable Unstable or neutral neutral, or stable

where xi # RM is the state variable of the ith oscillator(i = 1, . . . ,N), D is a M $ M matrix, and µ# is the cou-pling strength. We assume f (x = 0; µ) = 0 without loss ofgenerality. We also assume that, in the absence of coupling(# = 0), the trivial solution xi = 0 is stable for µ < 0 andundergoes a supercritical Hopf bifurcation at µ = 0, such thateach unit becomes a limit-cycle oscillator with the amplitude|xi | = O(

%µ) for µ > 0. Hereafter we consider only µ ! 0

and put µ = '2 for convenience.To derive the amplitude equation for Eq. (2) near a Hopf bi-

furcation, it is sufficient to focus on the subsystem in which anoscillator is coupled to another. Let x and x& be the state vectorsof the oscillators. The dynamical equation for x is given as

x = f (x; '2) + '2#Dx&. (3)

We expand f (x; '2) around x = 0 as

f (x; '2) = n1(x; '2) + n2(x,x; '2) + n3(x,x,x; '2)

+O(|x|4), (4)

where nk (k = 1,2,3) is the kth-order term in the expansion(the precise definitions of n2 and n3 are given in Appendix A).We further expand nk with respect to '2 to obtain

f (x; '2) = L0x + '2L1x + n2(x,x) + n3(x,x,x) + O(|x|4),

(5)

where nk(·) (k = 1,2) denotes nk(·; '2 = 0). Note that O('2)terms in n2 and n3 are irrelevant to the calculations below andthus omitted in Eq. (5). Because of the assumption of Hopfbifurcation, L0 has a pair of purely imaginary eigenvalues±i"0. The right and left eigenvectors of L0 corresponding tothe eigenvalue i"0 are denoted by u (column vector) and v(raw vector), respectively; i.e., L0u = i"0u and vL0 = i"0v.They are normalized as vu = 1. The solution to the linearizedunperturbed system, x = L0x, is given by

x0(t) = wei&(t)u + we!i&(t)u, (6)

where w is an arbitrary complex number, which we referto as the complex amplitude; u and w denote the complexconjugate of u and w, respectively; and & (t) = "0t .

In Eq. (2), x(t) generally deviates from x0(t). By inter-preting w as a time-dependent variable w(t), it is possible to

describe the time-asymptotic behavior of x(t) in the followingform:

x = x0(w,w,& ) + !(w,w,w&,w&,& ), (7)w = g(w,w) + '2#h(w,w,w&,w&), (8)

where !,g, and h are the functions to be determined pertur-batively. Note that g and h are free from & (t). Equation (8)is called the amplitude equation. In Ref. [1] the amplitudeequation to the lowest-order is derived as

w = '2(w ! )|w|2w + '2#* w&, (9)

where (,), and * are the complex constants with the followingexpressions:

( = vL1u, (10)

) = !3vn3(u,u,u) + 4vn2"u,L!1

0 n2(u,u)#

+ 2vn2(u,(L0 ! 2i"0)!1n2(u,u)), (11)

* = vDu. (12)

C. Phase reduction for the lowest-order amplitude equation

Following the method in Ref. [1], we may further reducethe amplitude equation to a phase model. For # = 0, Eq. (9)has the stable limit-cycle solution given by

w0(t) = rei!(t), (13)

where r = '%

(R/)R (the subscripts R and I denote thereal and imaginary parts, respectively), !(t) = "t + !0, " ='2(R(c0 ! c2),c0 = (I/(R, c2 = )I/)R, and !0 is an arbitraryinitial phase. For sufficiently small # , the trajectory of w(t)deviates only slightly from that of the unperturbed limit cycle.In this case, w(t) is well approximated by rei!(t) with the phase!(t) obeying the following phase model:

! = " + '2#$(! ! !&). (14)

The phase coupling function $ is obtained through

$(! ! !&) = 'z(!) · h(w0,w&0)(, (15)

where z(!) is the phase sensitivity function, a · b = (ab +ab)/2 denotes the inner product in a complex form, and'f (!,!&)( = 1

2+

$ 2+

0 f (! + µ,!& + µ) dµ denotes averaging.The phase sensitivity function is determined by the nature oflimit-cycle oscillation. In the case of Eq. (9), we have [1]

z(!) = !c2 + ir

ei! . (16)

For convenience, we expand the phase coupling function $as

$(,) = a0 +)!

-=1

(a- cos -, + b- sin -,). (17)

Substituting Eq. (16) and h(w0,w&0) = *w&

0 = * rei!&into

Eq. (15), we find

a1 = *R(c1 ! c2), (18)b1 = !*R(1 + c1c2), (19)

where c1 * *I/*R. Importantly, all other coefficients vanish.

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Note that, if the coupling term in Eq. (3) is diffusive[i.e., '2#D(x& ! x) instead of '2#Dx&], we have h(w0,w

&0) =

* (w&0 ! w0). In this case we obtain a0 = !a1 in addition to

Eqs. (18) and (19).We have seen that, for the lowest-order amplitude equation

given by Eq. (9), the corresponding phase coupling functiondoes not contain the second and higher harmonics. Therefore,as briefly mentioned in Sec. II A, this amplitude equation doesnot admit multiphase clusters. Note that strong coupling mayresult in amplitude clusters [10].

D. Higher-order correction

Next, we derive higher-order correction terms to theamplitude equation and the corresponding phase couplingfunction $. We here consider only weak coupling; i.e. thecorrections of O(#2) are neglected. We also consider onlylinear coupling, as given in Eq. (2). Our result would changeif we consider nonlinear coupling.

At first, we discuss higher-order terms in g(w) in Eq. (8).We know that g(w) consists only of |w|nw (n = 0,1,2, . . .),called the resonant terms [20]. The dynamical equationw = g(w) is invariant under the transformation w + wei! ;i.e., the system has the rotational symmetry. This impliesthat w0(!) and z(!) have the following forms: w0(!) =w0(0)ei! and z(!) = z(0)ei! . Then, obviously, $(! ! !&) ='z(!) · *w&

0(!&)( contains only the first harmonics. Note that,however, the term |w|nw provides the corrections of O('3)and O('1) to w0(!) and z(!), respectively. These correctionsgive rise to the corrections of O('2) in a1 and b1.

Therefore, for $ to possess higher harmonics, we needto consider a higher-order correction to h(w,w&). In gen-eral, h(w,w&) is described as a polynomial of w,w,w&,w&.Let us suppose that h(w,w&) = w-1w-2w&-3w&-4 with integers-1,-2,-3,-4 ! 0. Because w0 = O(') and z = O('!1) [seeEqs. (13) and (16)], we have z · h = O('-1+-2+-3+-4!1). Wealso have z · h(w,w&) " ei(-1!-2!1)!ei(-3!-4)!&

. This term con-tributes to a- and b- (- > 0) when this term is a function ofonly ±-(! ! !&); i.e., -1 ! -2 ! 1 = ±- and -3 ! -4 = ,-.Obviously w-!1w&- = O('2-!1) gives a leading contributionto a- and b-. We thus find

a-,b- = O('2(-!1)). (20)

Other terms in h(w,w&) together with higher-order terms ing(w) provide minor corrections to the coefficients a- and b-.

Let us focus on the resonant terms of O('3) in h(w,w&).There are five resonant terms: ww&2, |w|2w&, w2w&, |w&|2w&,and w|w&|2. As already discussed, the first term yields thesecond harmonic of O('2), thus providing leading terms ofO('2) to a2 and b2. The other resonant terms yield minorcorrections. Namely, the next three resonant terms yield thefirst harmonic of O('2), and the final resonant term yieldsthe zeroth harmonic of O('2). Therefore, only the term ww&2

gives a major effect on phase dynamics. Thus, as a second-order amplitude equation for weakly coupled oscillators neara supercritical Hopf bifurcation point, it is appropriate toconsider the following equation:

w = '2(w ! )|w|2w + '2#(*w& + .ww&2), (21)

where . is a complex constant. One of the main results in thepresent paper is, as shown in Appendix A, the derivation ofthe expression for ., which has the following concise form:

. = 2vn2(u,(L0 ! 2i"0)!1D(L0 ! 2i"0)!1n2(u,u)). (22)

Moreover, calculation of 'z(!) · .w0(!)w&20(!&)( yields the

Fourier coefficients of $, given as

a2 = r2.R(c3 ! c2), (23)b2 = !r2.R(1 + c2c3), (24)

where c3 = .I/.R. As discussed in Sec. III A, the stability ofphase clusters crucially depends on these Fourier coefficients.

III. DEMONSTRATION

Our theory is applied to the prediction of cluster states inglobally coupled oscillators. First, we briefly summarize theexistence and stability of cluster states in the phase model withglobal coupling. We then numerically confirm our predictionabout clustering behavior in the Brusselator model.

A. Existence and stability of balanced cluster states

A globally coupled system with N oscillators is obtainedby replacing x& in Eq. (3) with X * 1

N

%Nj=1 xj . The corre-

sponding phase models given in Eq. (1) with # being replacedby '2# . By assuming # > 0 and rescaling time scale, we dropthis #'2. For # < 0, all the eigenvalues given below will havethe opposite sign.

The balanced n-cluster state (n ! 2) is the state in which thewhole population splits into equally populated n groups (herewe assume that N is a multiple of n), oscillators in group m(m = 0,1, . . . ,n) have an identical phase ,m, and the phases ofgroups are equally separated (,m = /t + 2m+/n). In Eq. (1)this solution always exists for any n.

The stability of the balanced cluster states was studiedby Okuda [5]. The n-cluster state with n ! 2 possesses twotypes of eigenvalues, which are associated with intraclusterand intercluster perturbations. The eigenvalue associatedwith intracluster perturbations is given by 0intra

n =%)

k=1 bkn.Therefore, in the absence of the -th harmonics with - ! n in $,the n-cluster state has a zero eigenvalue; that is, the n-clusterstate may not be asymptotically stable. This is the reasonwhy any multiphase clusters do not appear in the lowest-orderamplitude equation given by Eq. (9).

However, as discussed in Sec. II D, the amplitude equationthat appropriately takes into account higher-order correctionsyields higher harmonics in the corresponding phase couplingfunction $. The order of higher harmonics is given byEq. (20), implying 0intra

n =%)

k=1 bkn = O('2(n!1)). Therefore,multiphase clusters can be asymptotically stable.

From here we focus on one- and two-cluster states. Byneglecting third and higher harmonics in $, eigenvalues forthe one-cluster (0intra

1 ) and the balanced two-cluster states(0intra

2 ,0inter2 ) are given by

0intra1 = $&(0) = b1 + 2b2, (25)

0intra2 = 1

2 ($&(0) + $&(+ )) = 2b2, (26)

0inter2 = $&(+ ) = !b1 + 2b2. (27)

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KORI, KURAMOTO, JAIN, KISS, AND HUDSON PHYSICAL REVIEW E 89, 062906 (2014)

By substituting Eqs. (19) and (24), we can obtain theexpressions for these eigenvalues. At this point, we shouldrecall that these expressions for b1 and b2 involve the errors ofO('2) and O('4), respectively. We should thus keep in mindthe following estimation when we substitute the expressionsgiven by Eqs. (19) and (24) into Eqs. (25)–(27):

0intra1 = b1 + O('2), (28)

0intra2 = 2b2 + O('4), (29)

0inter2 = !b1 + O('2). (30)

Equations (28)–(30) imply that the parameter region inwhich the balanced two-cluster state is asymptotically stablecoincides well with the region with b1 > 0 and b2 < 0.

There is another type of two-cluster states, in which thephase difference between the clusters is different from + . Thistype of two-cluster states is usually unstable. However, a pairof two-cluster states may form attracting heteroclinic cycles.In such a case, an interesting dynamical behavior called slowswitching appears [6–8]. Although the clusters are generallynot equally populated in this type of two-cluster states, weconsider only two equally populated clusters for simplicity inthe present paper. For convenience, we refer to the two-clusterstates with the phase difference + and other phase differencesas antiphase and out-of-phase cluster states, respectively.

For out-of-phase cluster states, one may show that thephase difference %! between the clusters is given by %! =arccos(!b1/2b2). The solution to %! exists only when b1 iscomparable to or even smaller than b2. As b1 is generally muchsmaller than b2, this situation typically occurs near the stabilityboundary of the one-cluster state at which b1 changes its sign.

There are three types of eigenvalues for out-of-phase clusterstate, given as

0(i)ss = 1

2($&(0) + $&(%!)) = !a1

2sin %! + O('2), (31)

0(ii)ss = 1

2($&(0) + $&(!%!)) = a1

2sin %! + O('2), (32)

0(iii)ss = 1

2($&(%!) + $&(!%!))

= (b1 + 2b2)(b1 ! 2b2)2b2

= !01(1 + cos %!). (33)

Local stability conditions necessary for the slow switchingdynamics are [6,7] 0(i)

ss 0(ii)ss < 0, 0(iii)

ss < 0, and

0(i)ss + 0(ii)

ss = b1(b1 + 2b2)2b2

= 01 cos %! < 0. (34)

As a1 is generally of O(1), the first condition always holdstrue. This means that any out-of-phase cluster state is saddle.The second condition holds only when 01 > 0 (i.e., the onecluster state is unstable). Then the last condition is satisfiedonly when cos %! < 0, i.e., b1 and b2 have the same sign. Then01 = b1 + 2b2 > 0 implies b1 > 0 and b2 > 0. Therefore, theslow switching dynamics may arise only when b1 > 0, b2 > 0and b1 is comparable to b2.

We summarize general properties of the cluster states. Herewe consider both positive and negative coupling strength #:

(1) #b1 < 0: one-cluster state is stable, antiphase clusterstate is unstable.

(2) #b1 > 0,#b2 < 0: one-cluster state is unstable, an-tiphase cluster state is stable.

(3) #b1 > 0,#b2 > 0: one-cluster state is unstable, bal-anced two-cluster state is unstable, slow switching may arisenear the stability boundary for one-cluster state.

B. Numerical verification with limit-cycle oscillators

To verify our theory, we consider a population of Brus-selator oscillators with global coupling, whose dynamicalequations are given by

dxi

dt= A ! (B + 1)xi + x2

i yi + #

N

N!

j=1

(xj ! xi), (35a)

dyi

dt= Bxi ! x2

i yi + #d

N

N!

j=1

(yj ! yi). (35b)

We treat B as a control parameter and fix other parameters Aand d. In the absence of coupling (i.e., # = 0), the Hopf bifur-cation occurs at B = Bc * 1 + A2. The bifurcation parameteris defined as '2 = B!Bc

Bcfor B > Bc. As shown in Appendix B,

using the expressions given in Sec. II, we obtained Fouriercoefficients a1,a2,b1, and b2 of $ as functions of A,B,d, and'. Figure 1 displays the phase diagram in the parameter space(A,d), where the lines show the predicted stability boundariesgiven by b1 = 0 and b2 = 0. Slow switching dynamics ispredicted to occur in the narrow region left of the line b1 = 0for # < 0.

We carried out direct numerical simulations of Eqs. (35a)and (35b). Starting from random initial conditions, the systemtypically converged to balanced cluster states. Two snapshotsare displayed in Fig. 2. The symbols in Fig. 1 display theparameter sets at which the indicated cluster state is obtained.

0

0.2

0.4

0.6

0.8

1

1 1.5 2 2.5 3

A

0

0.2

0.4

0.6

0.8

1

1 1.5 2 2.5 3

(a) (b)

b1=0b2=0

one cluster stateanti-phase cluster state

slow switching

A

d

FIG. 1. (Color online) Phase diagram of cluster states in theBrusselator system for (a) positive (# = 0.001) and (b) negativecoupling (# = !0.001). On the solid and dashed lines, b1 and b2

change their signs, respectively. The regions b1 > 0 and b2 < 0 areright of the lines. Numerical data are obtained by direct numericalsimulations of Eqs. (35a) and (35b) with N = 4, B = (1 + '2)Bc, ' =r&

2+A2

A2(1+A2) , r = 0.1.

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-0.4

-0.2

0

0.2

0.4

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

y

x

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

y

x

(a) (b)

FIG. 2. (Color online) Snapshots of cluster states in the Brusse-lator system. For a better presentation, displayed snapshots are thosebefore complete convergence. (a) The one-cluster state is obtained.(b) The balanced two-cluster state is obtained. Parameter values: N =12, # = !0.001,d = 0.4,B = (1 + '2)Bc, ' = r

&2+A2

A2(1+A2 , r = 0.1,(a) A = 2.8, (b) A = 1.1.

We also found the slow switching dynamics at the filledtriangle in Fig. 1(b), as theoretically expected. We expect thatthe slow switching dynamics would appear anywhere in thenarrow region left to the line b1 if the parameter A and dwere more finely varied. All together, we have an excellentagreement between the analytical and numerical results.

We next observed clustering behavior for various B valuesfar from the bifurcation point Bc with N = 24 oscillators.We fixed A = 1.0, so that Bc = 2.0. At each B value, weemployed 100 different random initial conditions. For eachinitial condition, we checked the number of phase clusters aftertransient time. Figure 3 shows the frequency of appearance ofthe n-cluster state for each B value. As the system is fartherfrom the bifurcation point, n-cluster states with larger n valueswere more likely to appear. In this particular case, the numberof clusters tends to increase as the system is farther from thebifurcation point. This result indicates that higher harmonicsin phase coupling function are developed as the bifurcationparameter increases, which is consistent with our estimationgiven by Eq. (20).

0

20

40

60

80

100

2 3 4 5 6 7

freq

uenc

y

2 clusters3 clusters4 clusters5 clusters6 clusters

B

FIG. 3. (Color online) Clustering behavior in the Brusselatorsystem farther from the Hopf bifurcation point. Each line indicateshow many times the n cluster state (not necessarily perfectly balanced)was obtained out of 100 different initial conditions. N = 24, A = 1.0,d = 0.7, # = !0.01. Initial values of xi and yi are random numberstaken from the uniform distribution in the range [!0.05,0.05].

IV. EXPERIMENTS

Experiments were conducted with a population of nearlyidentical N = 64 electrochemical oscillators. Each oscillatoris represented by a 1 mm diameter Ni wire embedded in epoxyand immersed in 3 mol/L sulfuric acid. The oscillators exhibitsmooth or relaxation waveforms for the current (the rate ofdissolution), depending on the applied potential (V) versus aHg/Hg2SO4/saturated K2SO4 reference electrode. The currentof the electrodes became oscillatory through a supercriticalHopf bifurcation point at V = 1.0 V; the oscillations aresmooth near the Hopf bifurcation point. As the potential isincreased, relaxation oscillations are seen that disappear intoa steady state through a homoclinic bifurcation at about V =1.31 V [21]. The parameters (applied potential) were chosensuch that the the oscillators exhibit smooth oscillations near theHopf bifurcation without any coupling. The electrodes werethen coupled with a combination of series (Rs) and parallel(Rp) resistors such that the total resistance Rtot = Rp + 64Rs

is kept constant at 652 ohms. The imposed coupling strengthcan be computed as K = NRs/Rp (more experimental detailsare given in Ref. [22]). Negative coupling was induced withthe application of negative series resistance supplied by a PAR273A potentiostat in the form of IR compensation. The clusterstates are obtained from nearly uniform initial conditions thatcorrespond to zero current (no metal dissolution).

Three-cluster states were observed near the Hopf bifurca-tion (V = 1.05 V) with negative coupling. Figure 4(a) showsthe current from one oscillator of each of the three clusters.The nearly balanced three-cluster state with configuration(25:20:19) is shown on a grid of 8 $ 8 circles [Fig. 4(b)]. Eachshade represents one cluster. In the previous work on the samesystem it had been shown that with positive coupling close tothe Hopf bifurcation only one-cluster state is present [2]. Phaseresponse curves [Fig. 4(c)] and coupling functions [Fig. 4(d)]for these oscillators were determined experimentally byintroducing slight perturbations to the oscillations [2]. Thestability of these cluster states was determined by computingthe eigenvalues of the phase model [5]. The maxima of realparts of these eigenvalues, for the same potential as in Fig. 4(a),are shown in Fig. 4(e). It is clear that with negative couplingmulticluster states should be observed, and the three-clusterstate is the most stable state.

As the potential was varied the number of cluster stateschanged. Four- and five-cluster states were observed athigher potentials. Examples of the oscillation waveformsand configurations for the four- and five-cluster states areshown in Figs. 5(a)–5(d). Further increase in the potentialresulted in complete desynchronization of the 64 oscillators. Athigher potentials, for moderately relaxational oscillators, onlya one-cluster state was observed. Figure 5(e) summarizes theeffect of changing the parameter (potential) on the existence ofdifferent cluster states. The presence of these clusters can beexplained by the most stable clusters from the experimentallydetermined phase model [Fig. 5(f)]. (We were not able to derivea phase model for the two-cluster state because the amplitudeof the oscillations was too small for response functions to bemeasured in experiments.)

The experiments thus confirm that varying number ofclusters (2–5) can be observed in the electrochemical system

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(a) (b)

(c) (d)

(e)

2.94

0.1

0

-0.131 2 3 4 5 6

No. of Clusters

! max

FIG. 4. (Color online) Experiments: Three-cluster state close to Hopf bifurcation with negative global coupling of 64 electrochemicaloscillators. K = !0.88. (a) Current time series and the three cluster configuration at V = 1.05 V (close to a Hopf bifurcation). Solid, dashed,and dotted curves represent currents from the three clusters. (b) Cluster configuration. (White, black, and gray circles represent the threeclusters.) (c) Response function and waveform (inset) of electrode potential E = V ! IRtot, where I is the current of the single oscillator.(d) Phase coupling function. (e) Stability analysis of the clusters with experiment-based phase models for K = !1: the maximum of the realparts of eigenvalues for each balanced cluster state.

close to Hopf bifurcation with negative global coupling.Note that these clusters are different than those reportedpreviously that had been obtained with relaxation oscillators

with positive coupling [22]. Similar to the results obtained withthe Brusselator model, when the system is shifted farther awayfrom the Hopf bifurcation, the number of clusters increased

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CLUSTERING IN GLOBALLY COUPLED OSCILLATORS . . . PHYSICAL REVIEW E 89, 062906 (2014)

(a) (b)

(c) (d)

(e) (f)

FIG. 5. Experiments: Multiple cluster states observed as parameters are moved away from Hopf bifurcation with negative global couplingfor 64 electrochemical oscillators. Top row: four clusters, V = 1.09 V, K = !0.88. Middle row: five clusters. V = 1.11 V, K = !0.88. (a)Current time series of the four clusters. (b) Cluster configuration (17:14:15:18) for four-cluster state. (c) Current time series. (d) Clusterconfiguration (15:13:14:7:15) of five-cluster state. (e) Experimentally observed cluster states as a function of applied potential. (f) The moststable cluster state as a function of potential predicted by the experimentally obtained phase model.

due to the emergence of stronger higher harmonics in thecoupling function. In agreement with theory, the clustersrequired relatively strong negative coupling (K - !0.88) incontrast with the one-cluster state with positive coupling thatrequired very weak coupling (K < 0.05) [23]. Therefore, wesee that weak higher harmonics can play important role indetermining the dynamical features of cluster formation when

the contribution of dominant harmonics does not induce astable structure.

V. CONCLUDING REMARKS

In summary, we have shown theoretically and confirmednumerically and experimentally the development of higherharmonics in the phase coupling function and the appearance

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KORI, KURAMOTO, JAIN, KISS, AND HUDSON PHYSICAL REVIEW E 89, 062906 (2014)

of phase clusters in globally coupled oscillatory systems. Wefound that the only relevant higher-order terms that shouldbe included in the amplitude equation for weakly coupledoscillators are w-!1w&- with - ! 2. In particular, we derivedthe expression for the coefficient of the term ww&2, whichhas a very concise form. The relevance of higher harmonicsin the phase coupling function has been well recognized. Ourstudy uncovered how higher harmonics are developed in limit-cycle oscillators near a Hopf bifurcation point. The derivedamplitude equation will serve as an analytically tractable limit-cycle oscillator model that produces higher harmonics in thephase coupling function.

ACKNOWLEDGMENTS

The authors are very thankful to Dr. Yoji Kawamura for hiscareful check of the manuscript. This work was supported byJSPS KAKENHI Grant No. 24740258.

APPENDIX A: DERIVATION OF THEAMPLITUDE EQUATION

Our aim is to reduce Eq. (3) to the amplitude equationgiven by Eq. (21). Because the expressions for (,), and * areobtained in Ref. [1], we focus on ..

For convenience, we rewrite Eqs. (3) and (8) as

x = L0x + '2L1x + n2(x,x) + n3(x,x,x) + '2#Dx&,(A1)

w = G(w,w,w&,w&), (A2)

respectively. Here n2 and n3 are defined as

n2(u,v) =M!

i,j=1

12!

'12 f

1xi1xj

(

x=0uivj , (A3)

n3(u,v,w) =M!

i,j,k=1

13!

'13 f

1xi1xj1xk

(

x=0uivjwk, (A4)

where x = (x1,x2, . . . ,xM )T and similar definitions areapplied to u,v, and w. Note that we consider only linearcoupling in Eq. (A1). In the presence of nonlinear coupling,the expression for . will be different from Eq. (22) while (, ),and * are unchanged.

By substituting Eq. (7) into Eq. (A1) and using Eq. (A2),we obtain

L0! = G exp(i& )u + G exp(!i& )u + b(w,w,w&,w&,& ),(A5)

where

L0 = L0 ! "01

1&, (A6)

b = !'2L1x ! n2(x,x) ! n3(x,x,x) ! '2#Dx&

+G1!

1w+ G

1!

1w+ G& 1!

1w& + G& 1!

1w& . (A7)

Regard Eq. (A5) formally as an inhomogeneous linear differ-ential equation for !(& ), where the right-hand side as a wholerepresents the inhomogeneous term. To solve Eq. (A5), !(& )

and b(& ) are expanded as

!(& ) =)!

-=!)!(-) exp(i-& ), (A8)

b(& ) =)!

-=!)b(-) exp(i-& ). (A9)

Note that the terms exp(i& )u and its complex conjugate inEq. (A5) are the zero eigenvectors of L0; i.e., L0(ei& u) =L0(e!i& u) = 0. Because the left-hand side in Eq. (A5) is freeof the zero-eigenvector components due to the operation ofL0, these components must be canceled in the right-hand sideas well. This condition is called the solvability condition.By substituting Eqs. (A8) and (A9) into Eq. (A5), andcomparing the component of exp(i& ) in both sides, we obtainthe solvability condition

G = !vb(1). (A10)

Further, by comparing other components, we obtain

!(-) = (L0 ! i-"0)!1b(-), (- .= ±1), (A11)

!(1) = (L0 ! i"0)!1(b(1) + Gu), (A12)

!(!1) = (L0 + i"0)!1(b(!1) + Gu). (A13)

Let b(-) and !(-) be further expanded in the powers of ':

b(-) =)!

2=2

'2 b(-)2 =

)!

2=2

b(-)2 , (A14)

!(-) =)!

2=2

'2 !(-)2 =

)!

2=2

!(-)2 . (A15)

Correspondingly, b and ! themselves are expanded as

b =)!

2=2

'2 b2 =)!

2=2

b2, (A16)

! =)!

2=2

'2 !2 =)!

2=2

!2 . (A17)

Let G be also expanded as

G =)!

2=1

'22+1G22+1 =)!

2=1

G22+1, (A18)

where we have anticipated the absence of even powers.To derive Eq. (21), we need to calculate G3 and G5. As

G3 is already obtained in Ref. [1], our main concern is G5,especially the higher-order coupling term in G5. To obtain G3and G5, we need the expressions for b2 (2 = 1, . . . ,5). Becausex0 = O('); b2,!2 = O('2) (2 ! 2); G2 = O('2) (2 ! 3), wefind

b2 = !n2(x0,x0), (A19)

b3 = !'2L1x0 ! 2n2(x0,!2) ! n3(x0,x0,x0) ! #'2Dx&0,

(A20)

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CLUSTERING IN GLOBALLY COUPLED OSCILLATORS . . . PHYSICAL REVIEW E 89, 062906 (2014)

b4 = !'2L1!2 ! 2n2(x0,!3) ! n2(!2,!2) ! 3n3(x0,x0,!2) ! #'2D! &2, (A21)

b5 = !'2L1!3 ! 2n2(x0,!4) ! 2n2(!2,!3) ! 3n3(x0,x0,!3) ! 3n3(x0,!2,!2) ! #'2D! &3

+G31!2

1w+ G3

1!2

1w+ G&

31!2

1w& + G&31!2

1w& . (A22)

We first calculate G3 = !vb(1)3 . Using

b(1)3 = !'2L1x(1)

0 ! 2n2(x0,!2)(1) ! n3(x0,x0,x0)(1) ! #'2Dx&(1)0

= !'2L1x(1)0 ! 2n2

"x(1)

0 ,!(0)2

#! 2n2

"x(!1)

0 ,!(2)2

#! 3n3

"x(1)

0 ,x(1)0 ,x(!1)

0

#! #'2Dx&(1)

0 , (A23)

!(0)2 = L!1

0 b(0)2 = !2L!1

0 n2"x(1)

0 ,x(!1)0

#, (A24)

!(2)2 = (L0 ! 2i"0)!1b(2)

2 = (L0 ! 2i"0)!1n2"x(1)

0 ,x(1)0

#, (A25)

we obtain Eq. (9) with Eqs. (10)–(12).Now we calculate G5 = !vb(1)

5 . We have

b(1)5 = !'2L1!

(1)3 ! 2n2(x0,!4)(1) ! 2n2(!2,!3)(1) ! 3n3(x0,x0,!3)(1) ! 3n3(x0,!2,!2)(1) ! #'2D!

&(1)3

+G31!

(1)2

1w+ G3

1!(1)2

1w+ G&

31!

(1)2

1w& + G&31!

(1)2

1w&

= !'2L1!(1)3 ! 2n2

"x(1)

0 ,!(0)4

#! 2n2

"x(!1)

0 ,!(2)4

#! 2n2

"!

(2)2 ,!

(!1)3

#! 2n2

"!

(1)2 ,!

(0)3

#! 2n2

"!

(0)2 ,!

(1)3

#! 2n2

"!

(!1)2 ,!

(2)3

#

! 2n2"!

(!2)2 ,!

(3)3

#! 3n3

"x(1)

0 ,x(1)0 ,!

(!1)3

#! 6n3

"x(1)

0 ,x(!1)0 ,!

(1)3

#! 3n3

"x(!1)

0 ,x(!1)0 ,!

(3)3

#! 6n3

"x(1)

0 ,!(2)2 ,!

(!2)2

#

! 6n3"x(1)

0 ,!(1)2 ,!

(!1)2

#! 3n3

"x(1)

0 ,!(0)2 ,!

(0)2

#! 6n3

"x(!1)

0 ,!(2)2 ,!

(0)2

#! 3n3

"x(!1)

0 ,!(1)2 ,!

(1)2

#! #'2D!

&(1)3

+G31!

(1)2

1w+ G3

1!(1)2

1w+ G&

31!

(1)2

1w& + G&31!

(1)2

1w& . (A26)

Out of the above terms, we select those which produce#'2Dw&2w. Checking term by term, we find that the followingterms may safely be excluded:

(1) Those which include !2

(2) Those which include x(1)0

(3) Those which include x(!1)0 twice.

The remaining terms are

!'2L1!(1)3 ! 2n2

"x(!1)

0 ,!(2)4

#! #'2D!

&(1)3 . (A27)

The first of the above three terms is further dropped becausethe coupling term included there is linear. The last term is alsodropped because the cubic term n3(x&

0,x&0,x

&0) yields neither w

nor w. Thus, the only relevant term in b(1)5 is the #-dependent

term in

!2n2"x(!1)

0 ,!(2)4

#. (A28)

The #-dependent term in !(2)4 is

(L0 ! 2i"0)!1"!#'2D!&(2)2

#. (A29)

Because

!&(2)2 = (L0 ! 2i"0)!1!!n2

"x&(1)

0 ,x&(1)0

#"

= !w&2(L0 ! 2i"0)!1n2(u,u), (A30)

Eq. (A29) becomes

#'2w&2(L0 ! 2i"0)!1D(L0 ! 2i"0)!1n2(u,u). (A31)

Thus, the relevant term in Eq. (A28) is

! 2#'2w&2wn2(u,(L0 ! 2i"0)!1D(L0 ! 2i"0)!1n2(u,u)),

(A32)

which yields . shown in Eq. (22).

APPENDIX B: AMPLITUDE EQUATION FOR THEBRUSSELATOR MODEL

We derive the expression for (,),* , and . for the Brussela-tor model given by Eq. (35). There are three parameters, A, B,and d, in Eq. (35). We consider B as a bifurcation parameterwhile A and d are fixed, so that the expression for (,),* , and. will be functions of A and d. Note that such expressionsexcept for . were already derived in Ref. citekuramoto84.

The steady solution to Eq. (35) is (x0,y0) = (a,b/a).Introducing 3 = x ! x0 and 4 = y ! y0 and substituting theminto Eq. (35), we obtain

d3i

dt= (B ! 1)3i + A24i + f (3i ,4i) + #

N

N!

j=1

(3j ! 3i),

(B1a)

d4i

dt= !B3i ! A24i ! f (3i ,4i) + #d

N

N!

j=1

(4j ! 4i), (B1b)

where

f (3,4) = B

A3 2 + 2A34 + 3 24. (B2)

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KORI, KURAMOTO, JAIN, KISS, AND HUDSON PHYSICAL REVIEW E 89, 062906 (2014)

In the absence of coupling (i.e., # = 0), the trivial solution(3i ,4i) = (0,0) undergoes a supercritical Hopf bifurcation atB = Bc * 1 + A2. We define the bifurcation parameter as'2 = B!Bc

Bc. We then obtain

L0 ='

A2 A2

!(1 + A2) !A2

(, (B3)

L1 = (1 + A2)'

1 0!1 0

(, (B4)

D ='

1 00 d

(, (B5)

u ='

1

!1 + iA!1

(, (B6)

v = 12

(1 ! iA ! iA) , (B7)

"0 = A, (B8)

L!10 = 1

A2

' !A2 !A2

A2 + 1 A2

(, (B9)

(L0 ! 2i"0)!1 = 13A2

'A2 + 2iA 3A2

!A2 ! 1 !A2 ! 2iA

(. (B10)

We introduce ui = (5i ,µi)T (i = 1,2,3) and write

n2(u1,u2) =)

1 + A2

A5152 + A(51µ2 + µ151)

*'1

!1

(,

(B11)

n3(u1,u2,u3) = 5152µ3 + µ15253 + 51µ253

3

'1

!1

(. (B12)

Substituting these expressions to Eqs. (10)–(12) and (22),we obtain

( = 12

+ A2

2, (B13)

) = 1A2

+ 12

+ i2

'4

3A3! 7

3A+ 4A

3

(, (B14)

* = 12

+ d

2+ i

2(!A + Ad), (B15)

. = !83

+ 49A6

+ 83A4

! 1A3

+ 289A2

+ 1A

+ 2A ! 32d

3+ 4d

9A6! 68d

9A2

+ i'

2 + 49A5

+ 4A3

+ 2A2

+ 889A

+ 16A

3

+ 14d

9A5+ 6d

A3+ 20d

9A! 16Ad

3

(. (B16)

We further obtain

c1 = *I

*R= !A(1 ! d)

1 + d, (B17)

c2 = )I

)R= 4 ! 7A2 + 4A4

3A(2 + A2), (B18)

c3 = .I

.R= A[4 + 5d + (2 ! 11d)A2 + (d ! 1)A4]

4 + d + (2 ! 10d)A2 + (7d ! 2)A4, (B19)

r = '

+(R

)R= '

,A2(1 + A2)

2 + A2. (B20)

Using these coefficients, it is straightforward to obtain theexpression for the Fourier coefficients a1,a2,b1, and b2 of $.

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