Periodically forced ensemble of nonlinearly coupled oscillators: From partial to full synchrony

Yernur Baibolatov,1,2 Michael Rosenblum,1 Zeinulla Zh. Zhanabaev,2 Meyramgul Kyzgarina,2 and Arkady Pikovsky1

1Department of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Str. 24/25, D-14476 Potsdam-Golm, Germany2Department of Physics, Al-Farabi Kazakh National University, Tole bi Str. 96, 050012 Almaty, Kazakhstan

Received 29 July 2009; published 22 October 2009

We analyze the dynamics of a periodically forced oscillator ensemble with global nonlinear coupling.Without forcing, the system exhibits complicated collective dynamics, even for the simplest case of identicalphase oscillators: due to nonlinearity, the synchronous state becomes unstable for certain values of the couplingparameter, and the system settles at the border between synchrony and asynchrony, what can be denoted aspartial synchrony. We find that an external common forcing can result in two synchronous states: i a weakforcing entrains only the mean field, whereas the individual oscillators remain unlocked to the force and,correspondingly, to the mean field; ii a strong forcing fully synchronizes the system, making the phases of alloscillators identical. Analytical results are confirmed by numerics.

DOI: 10.1103/PhysRevE.80.046211 PACS numbers: 05.45.Xt

I. INTRODUCTION

Ensembles of globally, or all-to-all, coupled oscillatorsserve as models for many physical, chemical, biological, etc.,systems allowing one to describe diverse natural phenomenaas synchronous blinking of fireflies, pedestrian synchrony onthe millennium bridge, emergence of brain rhythms, just tomention a few 1–7. The main effect is the self-synchronization and the appearance of a collective mode inan ensemble of generally nonidentical elements—the prob-lem relevant for Josephson junction and laser arrays 8,9,electrochemical reactions 10, neuronal dynamics 11, hu-man social behavior 12–14, etc. This collective mode arisesdue to an adjustment of frequencies and phases of at leastsome group of the elements; the larger is this synchronizedgroup, the larger the amplitude of the collective mode.

An ensemble, exhibiting the collective mode, can be con-sidered as a macroscopic oscillator. A natural idea is to ana-lyze the response of this oscillator to an external perturbationor to a coupling to another macroscopic oscillator en-semble. This study is relevant, e.g., for modeling of neu-ronal rhythms in the case when an ensemble of interactingneurons is influenced by rhythms from other brain regions15–17. In particular, one can analyze a collective phaseresetting by pulsatile stimuli 18–20, entrainment of the col-lective mode by external forcing 21–23, mutual synchroni-zation of two ensembles 24,25, suppression or enhance-ment of the collective mode by a feedback 26,27, etc. Inthe first approximation, the macroscopic oscillator behaveslike a usual limit cycle oscillator and the description of thedynamics in this approximation can be obtained by means ofa model equation for the collective mode, incorporating theterms responsible for forcing, coupling 24, or feedback26,27, respectively. However, there are interesting effectswhich go beyond this simplistic description. For example,the coupling between populations of identical units can de-stroy synchrony in one of the populations and cause a par-tially synchronous chimera state 28, when the phases ofindividual units are not adjusted, but the collective modeexist, though its amplitude is reduced with respect to that inthe synchronous state. Another effect is the appearance of

synchronous subgroups with different frequencies in a har-monically forced ensemble of nonidentical elements 21.

In the present paper we study the effects of periodic forc-ing on an oscillator ensemble that by itself exhibits complexdynamics even in the simplest setup of identical units.Namely, we consider ensembles with global nonlinear cou-pling see 29,30 and a detailed discussion below. Due tononlinearity in the coupling, the completely synchronousstate loses its stability upon increase in the coupling strength,whereas the completely asynchronous state remains unstableas well. As a result, for a sufficiently large coupling, suchensembles demonstrate an intermediate state of partial syn-chrony, when the phases of oscillators are nonuniformly dis-tributed between 0 and 2. Especially interesting is the re-gime of self-organized quasiperiodicity, when the frequencyof the collective mode differs from the frequency of indi-vidual oscillators. Here we demonstrate that nonlinearlycoupled ensembles, subject to external forcing, exhibit non-trivial synchronization properties. We show that a harmonicexternal force can destroy partial and induce full synchrony;however, it is also possible that the forcing entrains only thecollective mode but not individual oscillators.

The paper is organized as follows. In the next section wedescribe the basic model of the ensemble of phase oscillatorswith global nonlinear coupling. In Sec. III we use theWatanabe-Strogatz theory to derive the equation for the col-lective dynamics of the driven system. In Secs. IV and V weanalyze regimes of full synchrony and of mean-field locking,respectively. In Sec. VI we summarize and discuss our re-sults.

II. BASIC PHASE MODEL

A. Kuramoto-Sakaguchi model

A paradigmatic model for the description of the dynamicsof large ensembles of weakly interacting units is the Kura-moto model 1,2 of each-to-each, or globally coupled phaseoscillators. The model reads

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k = k +

Nj=1

N

sin j − k + , 1

where k and k are the phase and the natural frequency ofthe k-th oscillator, N is the number of oscillators in the en-semble, is the strength of the interaction within each pair ofoscillators. Generally, the model can also include a constantphase shift ; this case is also called the Kuramoto-Sakaguchi model 31. It is convenient to rewrite Eq. 1 as

k = k +

Nj=1

N

Imeijei−k+ = k + ImeiZe−ik ,

2

where Z is the complex Kuramoto order parameter, or themean field

Z = rei = N−1k=1

N

eik.

The mean-field amplitude, or the real order parameter r,varies from zero in the absence of synchrony to one, if allelements have identical phases.

B. Generalization to nonlinear coupling

Recently, the Kuramoto model has been generalized29,30 to cover the case of coupled ensembles with nonlin-ear interaction between oscillators. “Nonlinear” in this con-text means that the effect of the force on an oscillator quali-tatively depends on the amplitude of the forcing, e.g., a weakforce tends to induce in-phase synchrony, whereas a strongerforce tends to synchronize the oscillator in antiphase. In caseof a globally coupled ensemble, nonlinear interaction meansa dependence of the form of the coupling function on themean-field amplitude r. In other words, with a variation in abifurcation parameter, the interaction can change from anattractive one to a repulsive one. The generalized modelreads

k = kr, + ImAr,eir,Ze−ik , 3

where kr ,, Ar ,, and r , are some functions ofthe amplitude of the mean field r= Z. The model can betreated analytically for the case of identical oscillatorsk=, see 29,30. It is easy to show that the fully synchro-nous cluster state 1=2 . . . =N= , r=1 is stable ifAr ,cos r ,0. When either the amplitude function Aor the phase function cos changes its sign upon an increasein the coupling strength , the synchronous cluster dissolvesand a partial synchrony arises.

If the loss of synchrony occurs via zero crossing of the Afunction, whereas cos remains positive, a regime appearswhere the oscillator phases scatter on the circle, but theirdistribution is not uniform. It means that the mean-field am-plitude is 0r1. The frequency of the mean field equals

that of oscillators, = =osc= . The appearance ofthis partially synchronous state can be easily understoodqualitatively for a quantitative analysis see 30. Assume

for definiteness that A is a monotonically decreasing functionof and r and is positive for small and r=1, so that thecompletely asynchronous state r=0 is unstable, andA1,b=0 for some critical value b. Then the fully syn-chronous state is stable for b and unstable for b.Hence, for b the oscillators tend to desynchronize, butthey cannot desynchronize completely, because with the de-crease of r the function A becomes positive again, whatmeans that the interaction becomes attractive, so that theoscillators again tend to synchronize. As a result of thiscounterplay the system settles at the border of stability sothat Ar ,=0; this equation determines the mean-field am-plitude r as a function of . The described state is called theself-organized bunch state SOB 30.

The most interesting effect is observed if the functioncos becomes zero. In this regime the oscillators are notlocked to the mean field they produce, so that in this state theoscillators and the mean field have different frequencies,

osc. These frequencies are smooth functions of the cou-pling strength , what means that generally they are incom-mensurate. We denote this dynamics as self-organized quasi-periodicity SOQ. This state is also characterized by anonuniform scattering of phases, and, hence, by the mean-field amplitude in the range 0r1; it emerges from thefully synchronous state with r=1 via a desynchronizationtransition when the function cos crosses zero upon in-crease in the coupling strength . Again, the peculiar featureof the state is that with the variation in the coupling strengthbeyond a critical value q, the system always stays exactlyon the border of stability, when full synchrony is lost andpartial synchrony sets in. This border is determined by thecondition 1,q= /2, and for q the system is or-ganized in a way that it stays at this critical state, so thatr ,= /2; the latter expression provides the depen-dence r. Hence, the mean-field amplitude also smoothlydepends on . The quantitative analysis of SOQ dynamicscan be found in 29,30.

C. Forced ensembles

In this paper we analyze the dynamics of an ensemblewith global nonlinear coupling, described by Eq. 3, underan external periodic forcing. This study is relevant, e.g., formodeling of neuronal rhythms in case when an ensemble ofinteracting neurons is influenced by rhythms from otherbrain regions. On the other hand, this study is a first step toconsideration of an interesting problem of two interactingnonlinearly coupled ensembles. In our analysis we restrictourselves to the case of identical oscillators, when the model3 of the autonomous ensemble can be treated analytically29,30, and introduce a harmonic forcing eit, where and are the amplitude and the frequency of the external force.The model reads

k = r, + ImAr,eir,Ze−ik + eit−k . 4

Next, we take ,r==const and make a transforma-tion to a frame, rotating with the frequency . The phases ofoscillators and of the mean field are then transformed to slow

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phases as k→k+t, →+t and the equations in therotating frame are:

k = ImHe−ik , 5

H = Ar,eir,rei + eit, 6

where = − is the frequency mismatch. Here we have alsointroduced the complex effective force H, acting on indi-vidual oscillators.

In the following analysis we concentrate separately on theeffect of the amplitude function A ,r and of the phase shiftfunction ,r. In other words, we concentrate on two caseswhen the unforced system is partially synchronous either inthe SOB or in the SOQ state. We demonstrate and analyzetwo different synchronous regimes: i the external force en-trains only the mean field, but not individual oscillators andii the regime of full synchrony, when the force entrains alloscillators and, hence, destroys the SOQ/SOB regime.

III. WATANABE-STROGATZ EQUATIONS FORTHE FORCED OSCILLATOR ENSEMBLE

A complete analysis of the model 5 and 6 can be per-formed by means of the Watanabe-Strogatz WS theory32,33. Their main result is that for any time-dependentfunction Ht the N-dimensional system of identical oscilla-tors 5 can be reduced to a three-dimensional system forglobal variables , , 40:

d

dt=

1 − 2

2ReHe−i , 7

d

dt=

1 + 2

2 ImHe−i , 8

d

dt=

1 − 2

2 ImHe−i . 9

Here 0 1 is the global amplitude and , are globalphases; their meaning is discussed below. The original phasevariables can be reconstructed from the WS variables bymeans of the time-dependent transformation

k = t + 2 arctan1 − t1 + t

tank − t2

. 10

Here k are N constants of motion, which should obey threeconstraints, otherwise the transformation is overdetermined;this is discussed in detail in 33. It is convenient to choosetwo constraints as follows:

k=1

N

cos k = 0, k=1

N

sin k = 0. 11

The third condition on k is somehow arbitrary, it only fixesthe common shift of k with respect to . It can be taken,e.g., as kk=0 this implies that constants are defined in the− , interval 33. Another convenient choice iscos2k=0. These conditions allow one to determine un-

ambiguously the variables 0, 0, 0, and constantsk from the initial conditions k0, except for the case wheninitial state contains too large clusters see 33 for details.

Our next goal is to relate the WS variables to the Kura-moto mean field Z=rei. This can be done by rewriting thetransformation Eq. 10 in the following form 34,35

eik = eit t + eik−t

1 + teik−t . 12

With the help of this equation the combination of two WSvariables, namely, ei, can be related to Z as follows 34:

rei = N−1k=1

N

eik = ei , , 13

where

, = N−1k=1

N1 + −1eik−

1 + eik− . 14

The obtained relation helps to understand the physical mean-ing of the WS variables. The amplitude variable is roughlyproportional to the mean-field amplitude r. Indeed, if =0,then from Eq. 12 with account of Eq. 11, we obtain r=0. From Eqs. 13 and 14 obviously follows that =1yields r=1. For intermediate values 0 1 the relationbetween and r generally depends also on . The phasevariable characterizes the position of the maximum in thedistribution of phases and is therefore close to the phase ofthe mean field . They coincide for =r=1; for 0 1, is shifted with respect to by an angle that depends on , .Finally, the second global phase variable determines theshift of individual oscillators with respect to . As it followsfrom Eq. 10, k grows by 2 when decreases by 2.Hence, for the time averaged phase growth we obtain k=osc= − . For the following it is useful to write, us-ing Eqs. 8 and 9, the expression for the oscillator fre-quency as:

osc = 2 2

1 + 2 . 15

We emphasize now an important case when WS Eqs.7–9 simplify. As it was shown in 34, for large N anduniform distribution of constants of motion k, factor inEq. 13 equals one, and, hence, =r, =. This also meansthat H depends on and and does not depend on . As aresult, from three Eqs. 7–9 for Watanabe-Strogatz vari-ables we obtain two equations for the amplitude and phase ofthe Kuramoto mean field r, :

dr

dt=

1 − r2

2ReHe−i , 16

d

dt=

1 + r2

2rImHe−i . 17

An analysis of these equations, performed in the followingsections, yields a description of synchronous states in theforced ensemble; there we also present numerical results for

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the case of nonuniformly distributed k. We note that thecase of uniformly distributed constants of motion, i.e., =1,corresponds to the recent Ott-Antonsen ansatz 23 this cor-respondence was shown in 34. Noteworthy, this is not justa solvable, but also the mostly interesting case, because for acontinuous distribution of oscillator frequencies the corre-sponding solutions are the only attractors 36. However, weemphasize that Eqs. 16 and 17 are also valid for an arbi-trary distribution of constants k in the particular case of fullsynchrony =1, since 1,=1.

For the further analysis of synchronous states it is conve-nient to present the phase shift =t− between the exter-nal force and the mean field and, with account of Eq. 6, torewrite Eqs. 16 and 17 as:

r =1 − r2

2Ar,r cos r, + cos , 18

= −1 + r2

2rAr,r sin r, + sin . 19

A solution of these equations with bounded describes syn-chronization of the ensemble by the external force. Belowwe consider different cases, starting with the full synchronyin Sec. IV and proceeding to the mean-field locking in Sec.V. Our starting point is the unforced system 18 and 19 ina state of partial synchrony with 0r1, resulting from thecondition Ar ,cos r ,=0. We analyze the effects of theforcing taking its amplitude and frequency as main pa-rameters.

IV. FULL SYNCHRONIZATION

In this section we analyze the case when the externalforce entrains all oscillators in the initially partially synchro-nized ensemble and imposes the regime of full synchronyFS, 1= . . . =N=, r=1. Then Eq. 19 yields

= − A1,sin 1, − sin . 20

This is a first-order equation for , it can have fixed pointsand running solutions. The former correspond to fully syn-chronous state. As follows from Eq. 18, such a solution isstable if

A1,cos 1, + cos 0, 21

or

cos A1,cos 1, . 22

We remind that the unforced ensemble is in the state ofpartial synchrony, what implies that A1,cos 1,0.This means that the fully synchronous regime exists only ifthe forcing is sufficiently strong, c, where

c = A1,cos 1, . 23

Thus, for the border of the domain of FS we have

sin = − A1,sin 1, , 24

cos = A1,cos 1, = c. 25

Excluding we obtain the equation for the border of thesynchronization region tongue

− A1,sin 1,2 + A1,cos 1,2 = 2.

26

The variation in the phase shift between the externalforce and the mean field across the tongue, i.e., for fixed ,is determined by Eq. 24. The range of this variation fol-lows from Eq. 25, so that −maxmax, where max=arccosc /; for large , max→ /2. For comparison,recall that synchronization regions for one driven oscillatoror for a driven ensemble of identical linearly coupled phaseoscillators are of a triangular shape, the synchronizationthreshold is zero, and the variation in the phase shift acrossthe region is − /2 /2.

Below we analyze the fully synchronous solution for twoparticular cases, when the partial synchrony in the unforcedsystem arises either due to the amplitude function Ar , ordue to the phase function r ,. These cases correspond tothe SOB and SOQ states in the unforced system, respec-tively.

A. SOB state in the unforced system

For definiteness, we consider the model 5 and 6 withAr , monotonically decreasing with the mean-field am-plitude r and the coupling . Suppose that Ar , vanishesfor some critical coupling b, determined from the conditionA1,b=0. Suppose also that r ,=const /2. Thenfor b the synchronous solution of the unforced system,=0, becomes unstable due to the amplitude function andwe observe an SOB state. For b we have A1,0 andEq. 23 yields

c = A1,cos 1, . 27

Using A1,sin 1,=−A1,2−c2, we obtain

from Eq. 26 the equation for the border of the tongue:

l,r = − A1,2 − c2 2 − c

2. 28

Here indices l and r denote the left and the right border ofthe tongue, respectively. The tip of the tongue has coordi-nates −A1,sin 1, ,A1,cos 1,, so thatfor a fixed it stays on a circle with radius A1,, cen-tered at the origin Fig. 1. Note, that the tongue is parabolicfor −cc. Note also that for 1,→ /2 the syn-chronization threshold vanishes and the tongue becomes tri-angular, l,r=−A1,.

We illustrate the theory by simulating an ensemble of1000 oscillators for = /4 and A=1−r2. For each point inthe parameter plane , we run the simulation with two setsof initial conditions: nearly uniform and nearly identical41. These numerical results are presented in Fig. 2; theydemonstrate that numerically obtained domain of the fullsynchrony has a good correspondence with the theory. Inaddition, we find that the system is multistable: in a certainregion the nearly identical initial conditions lead to full syn-

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chrony, whereas the nearly uniform initial conditions resultin an asynchronous state.

B. SOQ state in an unforced system

Now we consider the model 5 and 6 with the ampli-tude function A, which has no roots and is positive, and thephase-shift function r ,, that monotonically increaseswith the mean-field amplitude r and the coupling . We re-call that for q, where q is determined from the condi-

tion 1,q= /2, the system exhibits SOQ. Then, for q we have 1, /2 and Eq. 23 yields for the criti-cal forcing amplitude

c = A1,cos 1, .

Then from Eq. 26 we find the border of the tongue:

l,r = A1,sin 1, 2 − c2. 29

Again, like in the previous example, the region is not trian-gular but parabolic for −cc. The tip of the tongue hascoordinates A1,sin 1, ,A1,cos 1,. Vary-ing we see that the tip of the tongue moves along a spiral.For an illustration we take the model with A=1 and =0+2r2 and assume that the coupling is not too strong so thatcos 1, remains negative, 1,3 /2, i.e., the condi-tion for SOQ dynamics in an unforced system holds. Itmeans that the coupling satisfies q+q

2. The theoryis illustrated in Fig. 3 and the numerical results for 1000oscillators, =0.4, and 0=0.475 are shown in Fig. 4. Onecan see that the border of the full synchronization domaindetermined according to Eq. 29 corresponds to the numer-ics. Again, we find multistability with respect to initial con-ditions: for certain parameter values a fully synchronous so-lution coexists with an asynchronous one.

V. MEAN-FIELD LOCKING

The case of full entrainment with r=1 is not the onlypossible stable dynamical regime in our system. In the otherinteresting regime the external force entrains only the meanfield, while individual oscillators remain unsynchronized toforcing. We call this regime mean-field locking MFL. Firstwe illustrate it numerically and then provide a theoreticalanalysis.

A. Illustration of the effect

For the illustration of the effect of mean-field locking wesolve numerically Eqs. 18 and 19 for the case of the non-

-1.5 -1 -0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

ν

µ

β

FIG. 1. Regions tongues of full synchrony for the harmonically forced ensemble with global nonlinear coupling see Eqs. 5 and 6with =const and A=1−r2. With a variation in , the tip of the tongue moves along a circle with radius −1. The tongues are plotted for=0, =0.25, =0.4, and =0.5; the value of coupling is =1.4.

-1 -0.5 0 0.50.2

0.4

0.6

0.8

1

ν

µ

FIG. 2. Color online Simulation of a periodically forced en-semble model Eq. 5 and 6 with =const= /4 and Ar ,=1−r2, for =1.4. For each point in the parameter plane , thesimulation was performed twice, with nearly uniform and nearlyidentical initial conditions for oscillator phases. Black dots denoteparameter values when both initial conditions lead to full syn-chrony; gray yellow in color dots denote the parameters whenonly nearly identical initial conditions lead to a fully synchronizedstate, while nearly uniform initial conditions lead to asynchrony.The border of the synchronization region nicely corresponds to thetheoretical curve black bold line, obtained according to Eq. 28.

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linearity in the amplitude function, i.e., we take A=1−r2,=1.4, and =0. Note that for the chosen coupling strength,the critical value for the full synchrony is c=0.4 see Eq.27. First, we fix the amplitude of the external force at =0.2c, vary the forcing frequency , and for each com-

pute r and =−, where · means time averaging.Then we repeat this numerical experiment for =0.6c.The results are shown in Fig. 5. We see that for c thereexists a parameter range, where the mean-field frequency islocked to the external force. This locking increases the co-herence of the ensemble; however, the force is not suffi-ciently strong in order to overcome the instability of the fullysynchronous state due to the nonlinearity in the coupling. Asa result, the amplitude of the mean field increases in thesynchronization region, but remains less than unity.

In this figure we also plot the frequency of individualoscillators osc, computed according to Eq. 15, and see thatin the MFL state the oscillators are not locked to the forceand, correspondingly, to the mean field. We remind that the

unforced system exhibits a bunch state, where the phases ofthe oscillators are scattered, but the frequencies of the oscil-lators and of the mean-field coincide. The forcing shifts bothfrequencies, but entrains only the frequency of the mean field. Thus, in this regime osc and = become generallyincommensurate, similarly to the SOQ state. For a strongerforce, c, the mean-field amplitude reaches unity in thecenter of the synchronization region, so that both regimes offull synchrony and MFL are present; the borders of full syn-chrony correspond to Eq. 28. Finally, we note that a directsimulation of an ensemble of oscillators according to Eqs.5 and 6 yields very similar results not shown.

B. Domain of mean-field locking

The next step is to determine the domain of MFL regimefrom Eqs. 18 and 19. This can be done semianalyticallyfor the case when in the MFL state r=const and =const,what corresponds to the fixed point solution of Eqs. 18 and19. Another possible type of solution is discussed below.Thus, we have:

0 = Ar,r cos r, + cos , 30

0 = −1 + r2

2rAr,r sin r, + sin . 31

Excluding , we obtain an equation for r:

Ar,r2 −4r2

1 + r2Ar,sin r, + 2r

1 + r22

− 2 = 0,

32

which we solve numerically for particular functions A and .Picking up real valued solutions 0r1 and computing theeigenvalues of the found fixed point solutions r , as de-scribed in Appendix A, we determine the existence domainof MFL regime for given , and the corresponding bifur-cations.

We illustrate the determination of MFL domain by thesame example as we used in Sec. IV A=1−1.4r2 and =0 in Fig. 6. The first result is that, contrary to the case of

-4 -3 -2 -1 0 1 2 30

0.5

1

1.5

2

ν

µ

β − π2

ε = 0.8

ε = 1.4

ε = 1.7

ε ≈ 1.79

FIG. 3. Regions tongues of full synchrony for the harmonically forced ensemble with global nonlinear coupling Eqs. 5 and 6 with=0+2r2 and A=1. With variation in the tip of the tongue moves along a spiral. The tongues are plotted for 0=0.475 and =0.8,=1.4, =1.7, and =+q

21.79.

0.3 0.35 0.4 0.45 0.5

0.04

0.06

0.08

0.1

ν

µ

FIG. 4. Color online Simulation of a periodically forcedensemble Eqs. 5 and 6 with Ar ,=1 and =0+2r2, for0=0.475 and =0.4. Black bold curve is determined by Eq.29. Gray yellow in color area corresponds to multistability cf.Fig. 2.

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FS, the MFL has no threshold. For a small force, the loss ofthe MFL occurs via a saddle-node bifurcation, when one ofthe two real eigenvalues becomes positive. The lines of thisbifurcation end in the vicinity of Takens-Bogdanov points,which can be found as discussed in Appendix B. Above thesepoints, a desynchronization occurs via a Hopf bifurcation,

and here we have to distinguish two cases. First, the limitcycle that appears as a result of a Hopf bifurcation is small sothat variation in is less than 2, see Fig. 7. Expressed inthe other way, the cycle in coordinates r cos , r sin doesnot revolve the origin. Physically, this means that the meanfield is modulated both in the phase and the amplitude, butits frequency in average remains locked to the forcing; wedenote such regimes as MFL with libration. The second caseappeares with the further variation in when the limit cyclegrows and the variation in becomes unbounded. The cyclein coordinates r cos , r sin now revolves the origin. As aresult, the mean-field locking is destroyed. Note that thesimilar dynamics is known in the analysis of a forced weaklynonlinear oscillator 4, in a system of globally coupled sto-chastic phase oscillators 37, and in a forced Kuramotomodel with the Lorentzian distribution 22. In the immedi-ate vicinity of the Takens-Bogdanov points complex bifurca-tions occur, see Appendix B.

Next, we briefly discuss what changes if =const0;this case is illustrated in Fig. 8. The main effect here is anappearance of bistability: there are domains where the asyn-chronous solution coexists with the MFL solution. On theright border this happens due to fact that the Hopf bifurca-tion here is subcritical; it means that the fixed point whichcorresponds to the MFL coexists with a stable limit cycle.The amplitude of the latter is large enough so that the MFL isimmediately lost no librations. In the bistability domain onthe left border the fixed point solution MFL also coexistswith a limit cycle; the latter, however, cannot be found fromthe stability analysis.

As the last example we consider the model with phasenonlinearity A=1, =0+2r2; here we have taken 0=0.475 and =0.4. Qualitatively, the picture remains quitesimilar to the previous case of the model with amplitudenonlinearity see Fig. 9 and compare with Figs. 6 and 8, justthe synchronization regions becomes more asymmetric andthe domains of multistability increase. Finally, we note thatfor all examples the results of a direct numerical simulation

0.84

0.86

0.88

0.9

0.92

-0.2 -0.1 0 0.1 0.2 0.3

-0.2

-0.1

0

0.1

0.2

0.8

0.85

0.9

0.95

1

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

-0.4

-0.2

0

0.2

0.4

νν

rr

Ω−

ν,ω

osc−

ν

Ω−

ν,ω

osc−

ν

FIG. 5. Color online Illustration of the mean-field locking in a periodically driven oscillator ensemble with nonlinearity in the amplitudefunction see text for details. Left column: subcritical driving =0.2c. Right column: supercritical driving =0.6c. In the lowerplots red dashed and black solid lines show the frequencies of the individual oscillators and of the mean field, respectively. For =0.2 theforce is not strong enough to induce full synchronization; it entrains the mean field, but not individual oscillators, so that r1. For =0.6 there are both domains of full synchrony and of MFL. Nonlinearity in the amplitude function A ,r=1−r2 and =0.

-0.8 -0.4 0 0.4 0.80

0.2

0.4

0.6

0.8

1

ν

µ

FS

MFL ASAS

FIG. 6. Color online Bifurcation diagram for the forced en-semble with the amplitude nonlinearity see text for details. Do-mains of FS, MFL, and asynchrony AS are seen. The black dottedcurve is the border of the FS area, obtained via Eq. 28. Black solidand red dashed curves at the borders of the MFL domain correspondto the loss of synchrony via saddle-node and Hopf bifurcations,respectively. The black filled circles are codimension-2 Takens-Bogdanov points; the bifurcation structure in the vicinity of thesepoints is shown in Fig. 11. Also, near Hopf bifurcation lines thereexist narrow areas of MFL regimes with modulation in amplitudeand phase libration which are not shown here see text for details.The mean-field amplitude and frequencies of mean field and indi-vidual oscillators along the horizontal dashed lines are shown inFig. 5. Nonlinearity in the amplitude function A ,r=1−r2 and=0.

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of an ensemble of 1000 oscillators are in a good correspon-dence with the analysis of Eqs. 18 and 19.

C. Beyond the Ott-Antonsen manifold

To conclude this section, we remind that Eqs. 18 and19 have been derived under the assumption of uniformlydistributed constants of motion k. This assumption meansthat the dynamics of the ensemble is confined to the reducedOtt-Antonsen manifold; as has been shown in 36, this

manifold is the only attractor if the oscillators are not iden-tical but have a frequency distribution and that is why thisparticular solution is especially important. The dynamics onthis manifold is described by a simplified system of twoWatanabe-Strogatz equations. However, for identical oscilla-tors, generally one has to consider the full WS equation sys-tem, which generally possesses other solutions as well see30. Moreover, the transients generally lie outside of theOtt-Antonsen manifold.

To illustrate this issue, we numerically analyze an en-semble of 1000 oscillators with nonlinearity in the amplitudefunction, and with a nonuniform distribution of constants ofmotion k. For this goal, we uniformly distribute one half ofthe constants k along the arc 1−q

2 , 1+q 2 , while the

-1 -0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

r

δ/π

ν = −0.9

ν = −0.86

ν = −0.85

ν = −0.841ν = −0.8405

ν = −0.835

ν = −0.83

ν = −0.825

ν = −0.82

FIG. 7. Illustration of the desynchronization transition at large forcing here =0.8. MFL corresponds to the fixed point shown for=−0.82. With the decrease of forcing frequency the limit cycle appears; however, first it is small and the deviation of phase difference is bounded libration. It means that on average the frequency of the mean field is still locked to the force. With further decrease in , phasedifference becomes unbounded this takes place at −0.841 and the synchrony gets lost. Nonlinearity in the amplitude functionA ,r=1−r2 and =0.

-1.2 -0.8 -0.4 0 0.4 0.8 1.20

0.2

0.4

0.6

0.8

1

ν

µ

FS

ASAS

AS/MFLAS/MFL

FIG. 8. Color online The same as in Fig. 6, but for =0.25.In addition to domains of FS, mean-field locking gray yellowarea, and AS, there are two domains where bistability MFL/AS isobserved. Solid black and dashed red curves are lines of saddlenode and Hopf bifurcations, respectively. Black circle shows theTakens-Bogdanov point the second Takens-Bogdanov point laysoutside of the shown range of and .

0.2 0.3 0.4 0.50

0.02

0.04

0.06

0.08

0.1

ν

µ

FS

AS

AS

AS/MFL

AS/MFL

FIG. 9. Color online The same as in Figs. 6 and 8 but forthe model with phase nonlinearity: A=1, =0+2r2, and 0

=0.475.

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second half of the constants of motion is distributed alongthe arc −1+q

2 ,−1−q 2 . Here 0q1 is a parameter;

its largest value q=1 corresponds to the uniform distributionof k, and, correspondingly, to the simplified theory pre-sented above. Note that this choice of k satisfies constraintsEq. 11 and k=0. The initial values for the phases k areobtained with the help of Eq. 10, using the following initialvalues for the Watanabe-Strogatz variables: ==0, =0.05. The parameters of the system are =1.4, =0.4, and=0.25 cf. Fig. 8, and the frequency of the forcing wasvaried. The result is as follows: in the middle of the synchro-nization region, e.g., for =0.2, the value of parameter qdoes not influence the dynamics and we observe the MFLwith a periodic mean field. Inside the synchronization regionbut close to its border, e.g., for =0.4, the synchronizedmean field becomes modulated both in the amplitude and thephase; in other words, we observe librations, see Fig. 10a.Outside of the synchrony domain, e.g., for =0.6, the dy-namics of the mean field also contains a second frequency:we see that the torus is born from the limit cycle, see Fig.10b. We conclude that nonuniform distribution of constantsof motion k generally leads to an appearance of an addi-tional modulation, so that a static solution becomes periodicand a periodic solution becomes quasiperiodic.

VI. DISCUSSION

We have analyzed the dynamics of periodically forcedensembles with global nonlinear coupling, for the case ofidentical phase oscillators. We have considered parametervalues where the unforced ensemble exhibits partial syn-chrony, with the mean-field amplitude 0r1. Dependingon the type of the coupling nonlinearity, the oscillators in theautonomous ensemble either have the same frequency as themean field, but different phases bunch state, or the fre-quency of oscillators differs from that of the mean field andtherefore the oscillators exhibit quasiperiodic dynamics.

It is natural to expect that a common external forcingincreases the coherence in the system and tends to synchro-

nize all elements of the ensemble. However, the forcing hasto overcome the instability of the synchronous solution dueto the coupling nonlinearity. As a result, the state of fullsynchrony appears if the forcing amplitude exceeds somethreshold value, even for identical oscillators. Next, thereexists another synchronous state, which appears already for avery small forcing. In this state only the mean field of thepopulation is locked to the external force, whereas the indi-vidual oscillators have a different, generally incommensuratefrequency. The distribution of the phases has a hump whichrotates with the same frequency as the force, and individualoscillators pass through this hump; the amplitude of themean field is, respectively, between zero and one. If the pa-rameters of the forcing are varied so that the system ap-proaches the border of full synchrony, the distribution be-comes more and more narrow, unless it collapses to a function at the border of full synchrony. Thus, comparingsynchronization properties of ensembles with linear and non-linear coupling we conclude that nonlinearity results in afinite threshold of full synchrony. Furthermore, it leads to asynchronous state when the mean field of the oscillator isentrained, but individual oscillators are in a quasiperiodicstate.

To conclude, there exist two levels of description of aforced ensemble. From the macroscopic viewpoint, theforced nonlinearly coupled ensemble behaves like a complexoscillator, which exhibits multistability at the borders of thesynchronization region. The limit cycle of the collectivemode is not very stable in the sense that a relatively weakforce not only entrains the mode, but can also change itsamplitude. The macroscopic description, based on the as-sumption of a uniform distribution of the constants of mo-tion, leads to a low-dimensional system of equations and canbe therefore fully analyzed. However, this macroscopic con-sideration is not sufficient for the detailed description of themean-field dynamics: in dependence on the microscopic stateof the ensemble, i.e., on the distribution of the constants ofmotion k, the collective mode dynamics may become qua-siperiodic. Nevertheless, the macroscopic description yields

-0.4 -0.2 0 0.2 0.40.5

0.6

0.7

0.8

0.9

-0.5 0 0.5

-0.5

0

0.5

1

r cos δr cos δ

rsi

nδ

rsi

nδ

(a) (b)

FIG. 10. Color online Simulation of the ensemble of N=1000 oscillators with nonlinearity in the amplitude function and with differentdistribution of constants of motion; the distribution is parameterized by q see text. Amplitude of the force is =0.4. a: =0.4. Blackcircle, solid red, and orange dashed lines correspond to q=1, q=0.9, and q=0.7, respectively. b: =0.6. Solid black and orange dashed linescorrespond to q=1 and q=0.7, respectively.

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a correct although “coarse grained” picture of the mainsynchronization effects. Moreover, according to Ott and An-tonsen 36, the solutions obtained from a macroscopic de-scription are the only asymptotically stable solutions if oscil-lators have a continuous frequency distribution.

ACKNOWLEDGMENTS

We gratefully acknowledge helpful discussions with G.Bordyugov and M. Zaks and financial support from DFGSFB 555 and FOR 868.

APPENDIX A: STABILITY ANALYSIS

For numerical analysis of Eqs. 18 and 19 it is conve-nient to present coordinates x=r cos , y=r sin and rewritethe system as:

x = − y +1 − x2 − y2

2Ax,yx cos x,y +

1 + x2 + y2

2

Ax,yy sin x,y +

21 − x2 + y2 = fx,y ,

y = x −1 + x2 + y2

2Ax,yx sin x,y +

1 − x2 − y2

2

Ax,yy cos x,y − xy = gx,y .

In order to determine the stability of a fixed point solution,we compute the eigenvalues of the corresponding Jacobianmatrix 1,2= SS2−4J /2, where

S = f

x+

g

y, J =

f

x

g

y−

f

y

g

x.

In the bunch state we have Ax ,y=1−x2+y2, x ,y=const /2. Presenting notation z=x2+y2, we write theterms of the Jacobian as:

f

x= − Ax2 +

z − 1

2A + 1 − zx2cos

+ Axy − 1 + zxysin − x ,

f

y= − − xyA + 1 − zcos

+ Ay2 +1 + z

2A − 1 + zy2sin + y ,

g

x= − Ax2 +

1 + z

2A − 1 + zx2sin

− xyA + 1 − zcos − y ,

g

y= − xyA − 1 + zsin

− Ay2 −1 − z

2A + 1 − zy2cos − x .

Next, we express x and y via z using Eqs. 30 and 31.Hence, all the terms of the Jacobian can be expressed via z.Substituting xz and yz in the above derivatives we obtain,after tedious but straightforward manipulations

S = 2a − bz , A1

J = a2 + d2 − ab + dcz , A2

where

a =1

21 + z1 − zcos ,

b = 1 + − 2zcos ,

c = 1 − − 2zsin ,

d =1 − z

1 + z −

1

21 − z1 − zsin .

Thus, for given and we solve numerically Eq. 32,which for a given nonlinearity becomes an equation for z=r21, and, substituting the solution in Eqs. A1 and A2,check its stability.

In the SOQ state we have Ax ,y=1, x ,y=0+2x2

+y2=0+2z. Proceeding in the similar way as for thebunch state, we obtain the eigenvalues via z, using

S = 2a − cz ,

J = a2 + d2 − ac + bdz ,

where the functions a, b, c, and d now take the form

a =1 + z

2 cos ,z ,

b = sin ,z + 1 + z2 cos ,z ,

c = cos ,z + 1 − z2 sin ,z ,

d =1 − z

1 + z −

21 − zsin ,z .

APPENDIX B: DYNAMICS IN THE VICINITYOF TAKENS-BOGDANOV POINTS

In the Takens-Bogdanov TB points both eigenvalues arezero, and, hence, in terms of variables introduced in Appen-dix A, S=0 and J=0. Two latter equations can be solved toobtain these critical points. Let us begin with the bunch statecase. Equation S=0 becomes a quadratic equation, whichroot is:

z = zc = 1 −1 −1

. B1

Here we took into account that 0z1 and 1, so wedropped the second root. For given zc the second equation,

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J=0, is a quadratic equation for d, its solution is

d1,2 =zc

2c b2 + c2 ,

where a, b, and c are now computed for z=zc. The expres-sion for d can be now solved for c; we obtain:

c =1 + zc

1 − zcd1,2 +

1 − zc1 − zc2

sin .

Finally, using Eqs. 30 and 31, we obtain

c =1 − zc2zc +4c

2zc

1 + zc2 −4czc

1 + zc1 − zcsin ,

what completes determination of Takens-Bogdanov points.The same scheme can be used in the case of the SOQ

state. However, instead of the quadratic Eq. B1 we have atranscendental equation

cot ,z = 21 − zz ,

which should be solved numerically to yield zc. Next stepsare exactly the same as in the previous case and the finalexpressions are:

c =1 + zc

1 − zcd1,2 +

1 − zc

2 sin ,zc ,

c =2zc +4c

2zc

1 + zc2 −4czc

1 + zc sin ,zc ,

where

d1,2 =zc

2b b2 + c2 .

We briefly illustrate the dynamics in the vicinity of TBpoints in Fig. 11, where we schematically present the bifur-cation diagram near the left TB point; the bifurcation dia-gram for the right TB can be obtained by reflection with

respect to the vertical axis. For computation of the bifurca-tion lines near TB points we used the AUTO 2000 package38,39. It is important to note that the diagram is qualita-tively similar to diagrams given in Ref. 37, where Zaks etal. analyze dynamics of an ensemble of linearly coupled sto-chastic phase oscillators; therefore for details of bifurcationsaround TB points we refer to Ref. 37. A similar bifurcationdiagram also appeared in a recent publication 22, whereChilds and Strogatz analyzed the dynamics of periodicallyforced Kuramoto model with Lorentzian distribution of natu-ral frequencies.

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ν

µ

I

I

I

II

III

IV

S

H

hL

sn1

sn1sn2

TB

FIG. 11. Sketch of the bifurcation lines around the left TB pointin Fig. 6. Here sn1 and sn2 are lines of the saddle-node bifurcation;H and h are lines of Hopf and homoclinic bifurcation, respectively;the line labeled by S marks the transition from librations to rota-tions. Domains I, II, and III correspond to fixed point solutions,librations, and rotations, respectively. In a very small domain IV weobserve coexistence of two fixed point solutions, corresponding totwo MFL states with different amplitudes of the mean field. Theline sn1 not shown in Fig. 6 goes from one TB point to the otherand separates domains with one stable fixed point beyond this lineand one stable and two unstable fixed points below this line.

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