kuramoto model

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The Kuramoto model: A simple paradigm for synchronization

phenomena

Juan A. Acebrón*

Departamento de Automática, Universidad de Alcalá, Crta. Madrid-Barcelona, km 31.600,

28871 Alcalá de Henares, Spain

L. L. Bonilla†

Grupo de Modelización y Simulación Numérica, Universidad Carlos III de Madrid, Avenida

de la Universidad 30, 28911 Leganés, Spain

Conrad J. Pérez Vicente‡ and Félix Ritort§

Department de Fisica Fonamental, Universitat de Barcelona, Diagonal 647, 08028

Barcelona, Spain

Renato Spigler

Dipartimento di Matematica, Università di Roma Tre, Largo S. Leonardo Murialdo 1, 00146

Roma, Italy

Published 7 April 2005

Synchronization phenomena in large populations of interacting elements are the subject of intense

research efforts in physical, biological, chemical, and social systems. A successful approach to the

problem of synchronization consists of modeling each member of the population as a phase oscillator.

In this review, synchronization is analyzed in one of the most representative models of coupled phase

oscillators, the Kuramoto model. A rigorous mathematical treatment, specific numerical methods, and

many variations and extensions of the original model that have appeared in the last few years are

presented. Relevant applications of the model in different contexts are also included.

CONTENTS

I. Introduction 138

II. The Kuramoto Model 139

A. Stationary synchronization for mean-field coupling 140B. Stability of solutions and open problems 141

1. Synchronization in the limit N = 141

2. Finite-size effects 143

III. The Mean-Field Model Including White-Noise Forces 144

A. The nonlinear Fokker-Planck equation 144

B. Linear stability analysis of incoherence 144

C. The role of g : Phase diagram of the Kuramoto

model 145

D. Synchronized phases as bifurcations from

incoherence, D0 146

1. Bifurcation of a synchronized stationary

phase 147

2. Bifurcation of synchronized oscillatoryphases 148

3. Bifurcation at the tricritical point 149

IV. Variations of the Kuramoto Model 151

A. Short-range models 151

B. Models with disorder 154

1. Disorder in the coupling: the oscillator glass

model 154

2. The oscillator gauge glass model 155

C. Time-delayed couplings 155

D. External fields 156

E. Multiplicative noise 157

V. Beyond the Kuramoto Model 157

A. More general periodic coupling functions 158

B. Tops models 159

C. Synchronization of amplitude oscillators 160

D. Kuramoto model with inertia 161

VI. Numerical Methods 163

A. Simulating finite-size oscillator populations 163

1. Numerical treatment of stochastic

differential equations 163

2. The Kuramoto model 164

B. Simulating infinitely many oscillators 165

1. Finite differences 1662. Spectral method 166

3. Tracking bifurcating solutions 168

C. The moments approach 168

VII. Applications 170

A. Neural networks 170

1. Biologically oriented models 170

2. Associative memory models 171

B. Josephson junctions and laser arrays 173

1. Josephson-junction arrays 173

2. Laser arrays 175

C. Charge-density waves 176

*Electronic address: [email protected]†Electronic address: [email protected]‡Electronic address: [email protected]§Electronic address: [email protected] address: [email protected]

REVIEWS OF MODERN PHYSICS, VOLUME 77, JANUARY 2005

0034-6861/2005/771/13749/$50.00 ©2005 The American Physical Society137


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D. Chemical oscillators 177

VIII. Conclusions and Future Work 177

Acknowledgments 178

Appendix A: Path-Integral Derivation of the Nonlinear

Fokker-Planck Equation 178

Appendix B: Calculating Bifurcations for the Nonlinear

Fokker-Planck Equation by the Method of Multiple Scales 180

Appendix C: Calculation of the Degenerate Bifurcation to

Stationary States Near 0 =D / 2 180Appendix D: Calculation of the Bifurcation at the Tricritical

Point 181

Appendix E: Stationary Solutions of the Kuramoto Model are

not Equilibrium States 181

Appendix F: Derivation of the Kuramoto Model for an Array

of Josephson Junctions 182

References 182

I. INTRODUCTION

Time plays a key role for all living beings. Their activ-ity is governed by cycles of different duration which de-termine their individual and social behavior. Some of these cycles are crucial for their survival. There are bio-logical processes and specific actions which require pre-cise timing. Some of these actions demand a level of expertise that can only be acquired after a long period of training, but others take place spontaneously. How dothese actions occur? Possibly through synchronization of individual actions in a population. A few examples fol-low. Suppose we attend a concert. Each member of theorchestra plays a sequence of notes that, properly com-bined according to a musical composition, elicit a deepfeeling in us as listeners. The effect can be astonishing ora fiasco apart from other technical details simply de-pending on the exact moment when the sound was emit-

ted. In the meantime, our hearts are beating rhythmi-cally because thousands of cells synchronize theiractivity. The emotional character of the music can accel-erate or decelerate our heartbeats. We are not aware of the process, but the cells themselves manage to changecoherently, almost in unison. How? We see the conduc-tor rhythmically moving his arms. Musicians know ex-actly how to interpret these movements and respondwith the appropriate action. Thousands of neurons inthe visual cortex, sensitive to specific space orientations,synchronize their activity almost immediately while thebaton describes a trajectory in space. This information istransmitted and processed through some remarkably

fast mechanisms. What more? Just a few seconds afterthe last bar, the crowd occupying the auditorium startsto applaud. At the beginning the rhythm may be inco-herent, but the wish to get an encore can transform in-coherent applause into perfectly synchronized applause,despite the different strength in beating or the locationof individuals inside the concert hall.

These examples illustrate synchronization, one of themost captivating cooperative phenomena in nature. Syn-chronization is observed in biological, chemical, physi-cal, and social systems, and it has attracted the interestof scientists for centuries. A paradigmatic example is the

synchronous flashing of fireflies observed in some SouthAsian forests. At night, myriad fireflies rest in the trees.Suddenly, several fireflies start emitting flashes of light.Initially they flash incoherently, but after a short periodof time the whole swarm is flashing in unison, creatingone of the most striking visual effects ever seen. Therelevance of synchronization has been stressed fre-quently although it has not always been fully under-stood. In the case of the fireflies, synchronous flashingmay facilitate the courtship between males and females.In other cases, the biological role of synchronization isstill under discussion. Thus perfect synchronizationcould lead to disaster and extinction, and therefore dif-ferent species in the same trophic chain may developdifferent circadian rhythms to enlarge their probabilityof survival. Details about these and many other systems,together with many references, can be found in the re-cent excellent book by Strogatz 2003.

Research on synchronization phenomena focuses in-evitably on ascertaining the main mechanisms respon-sible for collective synchronous behavior among mem-bers of a given population. To attain a global coherent

activity, interacting oscillatory elements are required.The rhythmical activity of each element may be due tointernal processes or to external sources externalstimuli or forcing. Even if the internal processes respon-sible for rhythmicity have different physical or bio-chemical origins and can be very complex, one mayhope to understand the essence of synchronization interms of a few basic principles. What might these prin-ciples be?

There are different ways to tackle this problem. Sup-pose that the rhythmical activity of each element is de-scribed in terms of a physical variable that evolves regu-larly in time. When such a variable reaches a certain

threshold, the element emits a pulse action potential forneurons, which is transmitted to the neighborhood.Later on, a resetting mechanism initializes the state of this element. Then, a new cycle starts. Essentially thebehavior of each element is similar to that of an oscilla-tor. Assuming that the rhythm has a certain period, it isconvenient to introduce the concept of phase, a periodicmeasure of the elapsed time. The effect of the emittedpulse is to alter the current state of the neighbors bymodifying their periods, lengthening or shortening them.This disturbance depends on the current state of the os-cillator receiving the external impulse, and it can also bestudied in terms of a phase shift. The analysis of the

collective behavior of the system can be carried out inthis way under two conditions: i the phase shift in-duced by an impulse is independent of the number of impulses arriving within an interspike interval, and iithe arrival of one impulse affects the period of the cur-rent time interval, but memory thereof is rapidly lostand the behavior in future intervals is not affected.

There is another scenario in which synchronization ef-fects have been studied extensively. Let us consider anensemble of nonlinear oscillators moving in a globallyattracting limit cycle of constant amplitude. These arephase- or limit-cycle oscillators. We now couple them

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weakly to ensure that no disturbance will take any of them away from the global limit cycle. Therefore onlyone degree of freedom is necessary to describe the dy-namic evolution of the system. Even at this simple levelof description it is not easy to propose specific models.The first scenario of pulse-coupled oscillators is perhapsmore intuitive, more direct, and easier to model. How-ever, the discrete and nonlinear nature of pulse couplinggives rise to important mathematical complications.While the treatment of just a few pulse-coupled ele-ments can be done within the framework of dynamicalsystems, the description becomes much more compli-cated for a large number of such elements. Proposing amodel within the second scenario of coupled limit-cycleoscillators leaves ample room for imagination. We areforced to consider models that are mathematically trac-table, with continuous time and specific nonlinear inter-actions between oscillators. Our experience tells us thatmodels with the latter property are exceptional. Never-theless some authors have been looking for a “solvable”model of this type for years. Winfree, for example, per-sistently sought a model with nonlinear interactions

Winfree, 1967, 1980. He realized that synchronizationcan be understood as a threshold process. When thecoupling among oscillators is strong enough, a macro-scopic fraction of them synchronizes to a common fre-quency. The model he proposed was hard to solve in itsfull generality, although a solvable version has been re-cently found Ariaratnam and Strogatz, 2001. Hence re-search on synchronization proceeded along other direc-tions.

The most successful attempt was due to Kuramoto1975, who analyzed a model of phase oscillators run-ning at arbitrary intrinsic frequencies and coupledthrough the sine of their phase differences. The Kura-

moto model is simple enough to be mathematically trac-table, yet sufficiently complex to be nontrivial. Themodel is rich enough to display a large variety of syn-chronization patterns and sufficiently flexible to beadapted to many different contexts. This “little wonder”is the object of this review. We have reviewed theprogress made in the analysis of the model and its ex-tensions during the last 28 years. We have also tried tocover the most significant areas where the model hasbeen applied, although we realize that this is not an easytask because of its ubiquity.

The review is organized as follows. The Kuramotomodel with mean-field coupling is presented in Sec. II.

In the limit of infinitely many oscillators, we discuss thecharacterization of incoherent, phase-locked, and par-tially synchronized phases. The stability of the partiallysynchronized state, finite-size effects, and open prob-lems are also considered. Section III concerns the noisymean-field Kuramoto model, resulting from adding ex-ternal white-noise sources to the original model. Thissection deals with the nonlinear Fokker-Planck equationdescribing the one-oscillator probability density in thelimit of infinitely many oscillators which is derived inAppendix A. We study synchronization by first analyz-ing the linear stability of the simple nonsynchronized

state called incoherence, in which every phase of theoscillators is equally probable. Depending on the distri-bution of natural frequencies, different synchronizationscenarios can occur in parameter regions where incoher-ence is unstable. We present a complete analysis of thesescenarios for a bimodal frequency distribution using bi-furcation theory.

Our original presentation of bifurcation calculationsexploits the Chapman-Enskog method to construct the

bifurcating solutions, which is an alternative to themethod of multiple scales for degenerate bifurcationsand is simpler than using center-manifold techniques.Section IV describes the known results for the Kura-moto model with couplings that are not of the mean-field type. They include short-range and hierarchicalcouplings, models with disorder, time-delayed couplings,and models containing external fields or multiplicativenoise. Extensions of the original model are discussed inSec. V. Section VI discusses numerical solutions of thenoisy Kuramoto model, both for the system of stochasticdifferential equations and for the nonlinear Fokker-Planck equation describing the one-oscillator probability

density in the limit of infinitely many oscillators. Appli-cations of the Kuramoto model are considered in Sec.VII. They include neural networks, Josephson junctionsand laser arrays, and chemical oscillators. These applica-tions are often directly inspired by the original model,share its philosophy, and represent an additional steptoward the development of new ideas. The last sectioncontains our conclusions and discusses some open prob-lems and hints for future work. Some technical detailsare collected in five appendixes.

II. THE KURAMOTO MODEL

The Kuramoto model consists of a population of N

coupled phase oscillators it having natural frequencies i distributed with a given probability density g , andwhose dynamics are governed by

˙ i = i + j =1

N

K ij sin j − i, i = 1, ... ,N . 1

Thus each oscillator tries to run independently at its ownfrequency, while the coupling tends to synchronize it toall the others. By making a suitable choice of rotating

frame, i→ i−t , in which is the first moment of g , we can transform Eq. 1 to an equivalent system

of phase oscillators whose natural frequencies have zeromean. When the coupling is sufficiently weak, the oscil-lators run incoherently, whereas beyond a certainthreshold collective synchronization emerges spontane-

ously. Many different models for the coupling matrix K ij have been considered such as nearest-neighbor coupling,hierarchical coupling, random long-range coupling, oreven state-dependent interactions. All of them will bediscussed in this review.

In this section, we introduce the Kuramoto modelwith mean-field coupling among phase oscillators. Forthis model, synchronization is conveniently measured by

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an order parameter. In the limit of infinitely many oscil-

lators, N =, the amplitude of the order parameter van-ishes when the oscillators are out of synchrony, and it ispositive in synchronized states. We first present Kura-moto’s calculations for partial synchronization of oscilla-tors and bifurcation from incoherence, a state in which

the oscillator phase takes values on the interval − , with equal probability. The stability of incoherence is

then analyzed in the limit N =. For coupling constantK K c, a critical value of the coupling, incoherence isneutrally stable because the spectrum of the operatorgoverning its linear stability lies on the imaginary axis.This means that disturbances from incoherence decaysimilarly to the Landau damping in plasmas Strogatz et al., 1992. When K K c and unimodal natural frequencydistributions are considered, one positive eigenvalueemerges from the spectrum. The partially synchronized

state bifurcates from incoherence at K = K c, but a rigor-ous proof of its stability is still missing. Finally, finite-size

effects N on oscillator synchronization are dis-cussed.

A. Stationary synchronization for mean-field coupling

The original analysis of synchronization was accom-plished by Kuramoto in the case of mean-field coupling,

that is, taking K ij = K / N 0 in Eq. 1 Kuramoto, 1975,1984. The model of Eq. 1 was then written in a moreconvenient form, defining the complex-valued orderparameter

rei =1

N j =1

N

ei j . 2

Here r t with 0r t 1 measures the coherence of theoscillator population, and t is the average phase. Withthis definition, Eq. 1 becomes

˙ i = i + Kr sin − i, i = 1,2, ... , N , 3

and it is clear that each oscillator is coupled to the com-

mon average phase t with coupling strength given byKr . The order parameter 2 can be rewritten as

rei = −

ei 1N

j =1

N

− j d . 4In the limit of infinitely many oscillators, they may beexpected to be distributed with a probability density , , t , so that the arithmetic mean in Eq. 2 nowbecomes an average over phase and frequency, namely,

rei = −

−

+

ei , ,t g d d . 5

This equation illustrates the use of the order parameter

to measure oscillator synchronization. In fact, when K

→0, Eq. 3 yields i it + i0, that is, the oscillatorsrotate at angular frequencies given by their own naturalfrequencies. Consequently, setting t in Eq. 5, bythe Riemann-Lebesgue lemma, we obtain that r →0 as

t → and the oscillators are not synchronized. On the

other hand, in the limit of strong coupling, K →, theoscillators become synchronized to their average phase, i , and Eq. 5 implies r →1. For intermediate cou-plings, K cK , part of the oscillators are phase

locked ˙ i = 0, and part are rotating out of synchronywith the locked oscillators. This state of partial synchro-

nization yields 0r 1 and will be further explained be-

low. Thus synchronization in the mean-field Kuramotomodel with N = is revealed by a nonzero value of theorder parameter. The concept of order parameter as ameasure of synchronization is less useful for models withshort-range coupling. In these systems, other conceptsare more appropriate to describe oscillator synchroniza-tion since more complex situations can happen Strogatzand Mirollo, 1988a, 1988b. For instance, it could hap-pen that a finite fraction of the oscillators have the same

average frequency ˜ i, defined by

˜ i = limt →

1

t

0

t

˙ idt , 6

while the other oscillators might be out of synchrony, orthat the phases of a fraction of the oscillators couldchange at the same speed and therefore partial synchro-nization would occur, while different oscillator groupshad different speeds and therefore their global orderparameter could be zero and incoherence would result.See Sec. IV for details.

A continuity equation for the oscillator density can befound by noting that each oscillator in Eq. 1 moveswith an angular or drift velocity vi = i + Kr sin − i.Therefore the one-oscillator density obeys the continu-ity equation

t

+

+ Kr sin

− = 0 , 7

to be solved along with Eq. 5, with the normalizationcondition

−

, ,t d = 1, 8

and an appropriate initial condition. The system of Eqs.5–8 has the trivial stationary solution = 1 /2 , r = 0,corresponding to an angular distribution of oscillatorshaving equal probability in the interval − , . Then,the oscillators run incoherently, and hence the trivial so-

lution is called the incoherent solution, or simply inco-herence. Let us now try to find a simple solution corre-sponding to oscillator synchronization. In the strong-coupling limit, we have global synchronization phaselocking, so that all oscillators have the same phase, i= = it + i0, which yields r =1. For a finite coupling,a lesser degree of synchronization with a stationary am-plitude, 0r 1, may occur. How can this smaller value

of r arise? A typical oscillator running with velocity v

= −Kr sin − will become stably locked at an anglesuch that Kr sin − = and − / 2 − /2. Allsuch oscillators are locked in the natural laboratory




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frame of reference. Oscillators with frequencies Kr cannot be locked. They run out of synchrony withthe locked oscillators, and their stationary density obeysv = C constant, according to Eq. 7. We have obtaineda stationary state of partial synchronization, in whichpart of the oscillators are locked at a fixed phase whileall others are rotating out of synchrony with them. Thecorresponding stationary density is therefore

= − − sin−1 Kr H cos , Kr

C

− Kr sin − elsewhere.

9

Here H x =1 if x0 and H x =0 otherwise, that is,H x is the Heaviside unit step function. Note that onecan write equivalently = K 2r 2− 2 −Kr sin − H cos for −Kr Kr . The normalization con-dition 8 for each frequency yields C = 2− Kr 2 /2 .We can now evaluate the order parameter in the state of partial synchronization by using Eqs. 5 and 9,

r = − /2

/2

−

+

ei − − − sin−1 Kr

g d d

+ −

Kr

ei − Cg

− Kr sin − d d . 10

Let us assume that g = g− . Then, the symmetry re-lation + ,− = , implies that the second termin this equation is zero. The first term is simply

r = Kr cossin−

1

Kr g d =

− /2

/2

cos gKr sin Kr cos d ,

that is,

r = Kr − /2

/2

cos2 gKr sin d . 11

This equation always has the trivial solution r =0 corre-

sponding to incoherence, =2 −1. However, it also has

a second branch of solutions corresponding to the par-tially synchronized phase 9, satisfying

1 = K − /2

/2

cos2 gKr sin d . 12

This branch bifurcates continuously from r =0 at the

value K = K c obtained by setting r =0 in Eq. 12, whichyields K c = 2/ g0. Such a formula and the argumentleading to it were first found by Kuramoto 1975. Con-sidering, as an example, the Lorentzian frequency distri-bution

g = /

2 + 2, 13

allows an explicit evaluation of the integrals above to beaccomplished, which was done by Kuramoto 1975. Us-ing Eq. 13, he found the exact result r = 1− K c / K forall K K c = 2 . For a general frequency distribution

g , an expansion of the right-hand side of Eq. 11 inpowers of Kr yields the scaling law

r − 16K − K c K c

4 g0 , 14

as K →K c. Throughout this review, we use the followingdefinitions of the symbol asymptotic which com-pares two functions or one function and an asymptoticseries in the limit as →0 Bender and Orszag, 1978:

f g ! lim →0

f

g = 1, 15

f

k=0

k f k!

f −

k=0

m

k f k

m, "m. 16

According to Eq. 14, the partially synchronized phasebifurcates supercritically for K K c if g00, and sub-critically for K K c if g00; see Figs. 1a and 1b.Notice that Kuramoto’s calculation of the partially syn-chronized phase does not indicate whether this phase isstable, either globally or locally.

B. Stability of solutions and open problems

1. Synchronization in the limit N =

Kuramoto’s original construction of incoherent andpartially synchronized phases concerns purely stationarystates. Moreover, he did not establish any of their stabil-ity properties. The linear stability theory of incoherencewas published by Strogatz et al. 1992 and interestingwork on the unsolved problems of nonlinear stabilitytheory was carried out by Balmforth and Sassi 2000. Toascertain the stability properties of the incoherent andpartially synchronized solutions, it is better to work with

the probability density , , t . Let us explain first whatis known in the limit of infinitely many oscillators de-scribed by Eqs. 5–7. The linearized stability problemfor this case is obtained by inserting = 1 /2 + ˜ , t ; with ˜ , t ; =expt , in Eqs. 5–8,and then ignoring terms nonlinear in :

−

+

K

2 Re e−i

−

−

+

ei ,

g d d = , 17

−

, d = 0. 18

If Re 0 for all possible , incoherence is linearly

stable, while it is unstable if some admissible has a




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positive real part. The periodicity condition implies

= n=− bn e

in , which, inserted in Eq. 17, yields

+ in bn =K

2 n,1 + n,−11, bn. 19

Here we have used b−n = bn a bar over a symbol denotes

taking its complex conjugate, and defined the scalarproduct

, =1

2

−

−

+

, , g d d . 20

Equation 19 shows that n =−in , n = ± 1 , ± 2 , . . . ,belong to the continuous spectrum describing the linear

stability problem, provided is in the support of g .When g is a unimodal natural frequency distributionbeing even and nonincreasing for 0, Strogatz et al.1992 have shown that the incoherent solution is neu-trally stable for K K c = 2 / g0. In fact, the afore-mentioned continuous spectrum lies on the imaginary

axis, in this case. For K K c, a positive eigenvalue ap-pears Strogatz et al., 1992. Although incoherence isneutrally stable for K K

c, the linearized order param-

eter Rt =e−i , ˜ , t ; decays with time. Due tophase mixing of the integral superposition of modes inthe continuous spectrum, such a decay is reminiscent of the Landau damping observed in plasmas Strogatz et al., 1992. This can be understood by solving the linear-ized problem with the initial condition ˜ , 0 ; = 2ei / 2 + 4 +c.c. for g = 1 + 2−1 and K = 1.The calculations can be carried out as indicated by Stro-gatz et al. 1992, and the result is

˜ ,t ; = 18 2 + 4

−

5

2i − 1+

1

2 − i ei − t

9

+5ei −t /2

9 2i − 1 +

ei −2t

9 2 − i + c.c., 21

Rt =10

9 e−t /2 −

4

9e−2t 22

Balmforth and Sassi, 2000. The function ˜ contains aterm proportional to e−i t , which is nondecaying andnonseparable, and does not correspond to a normalmode. As time elapses, this term becomes progressivelymore crenellated, and through increasing cancellations,

integral averages of ˜ decay. Besides this, Eq. 21 con-tain two exponentially decaying terms which contributeto the order parameter 22. If g has bounded sup-port, the order parameter may decay more slowly, alge-braically with time Strogatz et al., 1992.

Numerical calculations for K K c show that the order

parameter r t of the full Kuramoto model behaves simi-larly to that of the linearized equation Balmforth andSassi, 2000. However, the probability density may de-velop peaks in the , plane for intermediate timesbefore decaying to incoherence as t →. See Fig. 6 of

Balmforth and Sassi 2000. For K K c, Balmforth andSassi 2000 show that the probability density evolves to

a distribution that corresponds to Kuramoto’s partiallysynchronized phase given by Eq. 9. Balmforth andSassi 2000 obtained this result by numerically simulat-ing the full Kuramoto model with K K c. They also car-ried out different incomplete exact and perturbation cal-culations:

• Exact solution of the Kuramoto model for g = .

• Attempted approximation of the solution for otherfrequency distributions near K c assuming an unreal-

istic r that depends on .

FIG. 1. Bifurcation and stability of the Kuramoto model: asupercritical bifurcation in a diagram of r vs K ; b subcriticalbifurcation; c phase diagram of the noisy Kuramoto model, Dvs K , showing the regions of linear stability for the incoherent

solution 0 = 1 / 2 provided the frequency distribution is uni-modal and Lorentzian with width . The incoherent solution is

linearly stable if 0K 2D + , and unstable otherwise.




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• Regularizing the Kuramoto model by adding a diffu-sive term to Eq. 7 and constructing the stationaryprobability density in the limit of vanishing diffusiv-ity by means of boundary layer methods. This regu-larization corresponds to adding white-noise forcingterms to the Kuramoto model 1. The correspondingequation for the probability density is Eq. 7 with adiffusive term in its right-hand side, which is calledthe nonlinear Fokker-Planck equation. The nonlin-

ear Fokker-Planck equation will be studied in Sec.III.

• A mixture of multiple scales and boundary layerideas in the same limit of vanishing diffusivity of the

nonlinear Fokker-Planck equation, but for K nearvalues corresponding to the bifurcation of synchro-nized states from incoherence.

In the small-diffusivity limit of the nonlinear Fokker-Planck equation, D→0+, the calculations by Balmforthand Sassi 2000 indicate that the probability densitywith unimodal frequency distribution and K K c tendstoward a stationary phase that is concentrated around

the curve = Kr sin for 2K 2r 2. The peak of theprobability density is attained at Kr / 2 D. It wouldbe useful to have consistent perturbation results for theevolution of the probability density near bifurcation

points, in the low-noise limit D→0+ of the nonlinearFokker-Planck equation, and also for the Kuramoto

model with D =0. Both cases are clearly different, asshown by the fact that the synchronized phase is a gen-

eralized function a distribution when D =0, while it is asmooth function for D0. In particular, it is clear thatKuramoto’s partial synchronization solution 9 involv-ing a delta function cannot be obtained by small-amplitude bifurcation calculations about incoherence,

the laborious attempt by Crawford and Davies 1999notwithstanding.

Thus understanding synchronization in the hyperbolic

limit of the mean-field Kuramoto model with N →requires the following. First, a fully consistentasymptotic description of the synchronized phase andthe synchronization transition as D→0+ should befound. As indicated by Balmforth and Sassi 2000, thenecessary technical work involves boundary layers andmatching. These calculations could be easier if one

works directly with the equations for ln , as suggestedin the early paper Bonilla, 1987. Second, and mostlikely harder, the same problems should be tackled for

D =0, where the stable synchronized phase is expectedto be Kuramoto’s partially synchronized state which is adistribution, not a smooth function. Third, as pointedout by Strogatz 2000, the problem of proving stabilityof the partially synchronized state as a solution of theKuramoto model remains open.

2. Finite-size effects

Another way to regularize the hyperbolic equation 7is to study a large population of finitely many phase os-cillators, which are globally coupled. The analysis of this

large system may shed some light on the stability prop-erties of the partially synchronized state. The questioncan be posed as follows. What is the influence of finite-size effects on Kuramoto’s partially synchronized state

as N →?One issue with kinetic equations describing popula-

tions of infinitely many elements is always that of finite-size effects. This issue was already raised by Zermelo’sparadox, namely, that a system of finitely many particlesgoverned by reversible classical Hamiltonian mechanicswas bound to have recurrences according to Poincaŕ e’srecurrence theorem. Then, this system would come backarbitrarily close to its initial condition infinitely manytimes. Boltzmann’s answer to this paradox was that therecurrence times would become infinite as the numberof particles tend to infinite. Simple model calculationsillustrate the following fact. A nonrecurrent kinetic de-scription for a system of infinitely many particles ap-proximates the behavior of a system with a large butfinite number of particles during finite time intervals,after which recurrences set in Keller and Bonilla, 1986.

The same behavior, denoting the noncommutativity of

the limits N → and t →, is also present in the Kura-moto model. For instance, Hemmen and Wreszinski1993 used a Lyapunov-function argument to point outthat a population of finitely many Kuramoto oscillators

would reach a stationary state as t →. Our derivationof the nonlinear Fokker-Planck equation in Appendix A

suggests that fluctuations scale as N −1/2 as N →, a scal-ing that, for the order parameter, is confirmed by nu-merical simulations Kuramoto and Nishikawa, 1987;Daido, 1990.

More precise theoretical results are given by DaiPraand den Hollander 1996, for rather general mean-field

models that include Kuramoto’s and also spin systems.DaiPra and den Hollander 1996 obtained a centrallimit theorem, which characterizes fluctuations about the

one-oscillator probability density, for N =, as Gaussianfields having a certain covariance matrix and scaling as

N −1/2. Near bifurcation points, a different scaling is to beexpected, similarly to Dawson’s results for related mean-field models Dawson, 1983. Daido 1987b, 1989 ex-plored this issue by dividing the oscillator phase and theorder parameter in Eq. 2 into two parts: their limitsachieved when N → and their fluctuating parts whichwere regarded as small. In the equations for the phasefluctuations, only terms linear in the fluctuation of the

order parameter were retained. The result was then in-serted into Eq. 2, and a self-consistent equation for thefluctuation of the order parameter was found. For uni-modal frequency distributions, Daido found the scalingK c−K N

−1/2 for the rms fluctuation of the order pa-

rameter as K →K c− from below. For coupling strengths

larger than K c, he found that the fluctuation of the order

parameter was consistent with the scaling K −K c

−1/8N −1/2 as K →K c+ Daido, 1989. Balmforth and

Sassi 2000 carried out numerical simulations to inves-tigate finite-size effects, and discussed how sampling thenatural frequency distribution affects the one-oscillator




8/49

probability density. In particular, they found that sam-pling may give rise to unexpected effects, such as time-periodic synchronization, even for populations with uni-modal frequency distribution, g . Note that sucheffects do not appear in the limit N =. Further work onthis subject would be interesting, in particular, finding aformulation similar to Daido’s fluctuation theory for theorder parameter but for the one-oscillator probabilitydensity instead.

III. THE MEAN-FIELD MODEL INCLUDING WHITE-NOISE

FORCES

In this section, we analyze the mean-field Kuramotomodel with white-noise forcing terms. This generaliza-tion makes the model more physical, in that unavoidablerandom imperfections can be taken into account. At thesame time, the ensuing model turns out to be math-ematically more tractable. In fact, in the limit of infi-nitely many oscillators, the one-oscillator probabilitydensity obeys a parabolic equation the nonlinearFokker-Planck equation, instead of the hyperbolicequation 7, which is harder to analyze. The nonlinearFokker-Planck equation is derived in Appendix A. Its

simplest solution is also =2 −1, corresponding to in-coherence. First, we shall study its linear stability prop-

erties. When K K c the incoherence turns out to be lin-early stable, unlike what happens in the Kuramotomodel, for which incoherence is neutrally stable. Whenthe coupling strength is greater than the critical cou-pling, incoherence becomes linearly unstable, since thereal part of one of the eigenvalues in the discrete spec-trum of the linearized problem becomes positive. Then,different bifurcation scenarios and phase diagrams willoccur, depending on the distribution of natural frequen-

cies g . Synchronized phases branching off from inco-herence have been constructed by using different singu-lar perturbation techniques. In particular, a powerfultechnique, the Chapman-Enskog method, has been usedto study in detail these synchronization transitions for abimodal frequency distribution which has a very richphase diagram.

A. The nonlinear Fokker-Planck equation

The result of adding white-noise forcing terms to themean-field Kuramoto model is the system of stochasticdifferential equations

˙ i = i + it +K

N j =1

N

sin j − i, i = 1, . .. ,N . 23

Here the i’s are independent white-noise stochastic pro-cesses with expected values

it = 0, it j t = 2D t − t ij . 24

Introducing the order parameter 4, the model equa-tions 23 and 24 can be written as

˙ i = i + Kr sin − i + it , i = 1,2, . .. ,N . 25

The Fokker-Planck equation for the one-oscillator

probability density , , t corresponding to this sto-chastic equation is

t = D

2

2 −

v , 26

v , ,t = + Kr sin − , 27

provided the order parameter re i is a known function of

time and we ignore the subscript i. In the limit N →and provided all oscillators are initially independent, wecan derive Eq. 5:

rei = −

−

+

ei , ,t g d d , 28

which together with Eq. 26 constitute the nonlinearFokker-Planck equation see Appendix A for details.Notice that Eq. 26 becomes Eq. 7 if D =0. The system26–28 is to be solved with the normalization condi-

tion

−

, ,t d = 1, 29

the periodicity condition, + 2 , , t = , , t , andan appropriate initial condition for , , 0. In mostworks an exception is Acebrón et al., 1998, the naturalfrequency distribution g is a non-negative even func-tion, to be considered later.

B. Linear stability analysis of incoherence

The trivial solution of the nonlinear Fokker-Planck

equation, 0 = 1 / 2 , with order parameter r =0, repre-sents incoherent or nonsynchronized motion of all oscil-lators. A natural method for studying how synchronized

phases with r 0 may branch off from incoherence is toanalyze its linear stability as a function of the param-eters of the model, and then construct the possible solu-tions bifurcating from it. The first results were obtainedby Strogatz and Mirollo 1991. They studied the linearstability problem setting = 1/ 2 + ˜ , t ; with ˜ , t ; =expt , in Eqs. 26 and 8, and thenneglecting terms nonlinear in ,

D

2

2 −

+ K Ree−i e−i , = , 30

−

, d = 0. 31

Incoherence is linearly stable as long as Re 0, and itbecomes unstable if some admissible has positive real

part. The periodicity condition implies

=n=− bn e

in , which, inserted in Eq. 30, yields




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+ in + n2Dbn =K

2 n,1 + n,−11,bn . 32

Here we have used the relation b−n = bn, and the scalar

product defined in Eq. 20. As in the Kuramoto model,the numbers

n = − Dn2− in , n = ± 1, ± 2, .. . , 33

with belonging to the support of g , form the con-tinuous spectrum relevant to the linear stability prob-lem. Note that the continuous spectrum lies to the left

side of the imaginary axis when D0 b0 =0 because of the normalization condition 31. Then the eigenvalues33 have negative real parts and therefore correspondto stable modes.

The case n = ± 1 is special for two reasons. First, theright-hand side of Eq. 32 does not vanish, and thusb1 =K / 21 , b1 / + i + D. Then, provided 1 , b10,we find the following equation for Strogatz andMirollo, 1991:

K

2 −+ g

+ D + i d = 1. 34

The solutions of this equation are the eigenvalues forthe linear stability problem in Eq. 30. Clearly, they areindependent of . Since the continuous spectrum lies onthe left half plane, the discrete spectrum determines thelinear stability of the incoherence. Second, the nonlinearFokker-Planck equation and therefore the linear stabil-ity equation 30 are invariant under the reflection sym-metry, →− , →− , assuming g to be even. Thisimplies that two independent eigenfunctions exist, cor-

responding to each simple solution of Eq. 34,

1 , =

K

2 ei

D + + i , 2 , =

K

2 e−i

D + − i . 35

Note that these two linearly independent eigenfunctions

are related by the reflection symmetry. When is real,these eigenfunctions are complex conjugates of each

other. When is a multiple solution of Eq. 34, theeigenvalue is no longer semisimple Crawford, 1994.

C. The role of g : Phase diagram of the Kuramotomodel

The mean-field Kuramoto model for infinitely manyoscillators may have different stable solutions alsocalled phases depending on the natural frequency dis-tribution g , the values of the coupling strength K , andthe diffusion constant D. Many phases appear as stablesolutions bifurcating from known particular solutions,which lose their stability at a critical value of some pa-rameter. The trivial solution of the nonlinear Fokker-Planck equation is incoherence, and therefore much ef-fort has been devoted to studying its stability properties,

as a function of K , D, and parameters characterizing g . As explained in Sec. II, we can always consider the

first moment of g to be equal to zero, shifting theoscillator phases if necessary. Most of the work reported

in the literature refers to an even function, g , g− = g . In addition, if g has a single maximum at =0, we call it a unimodal frequency distribution. For gen-eral even unimodal frequency distributions, Strogatz andMirollo 1991 proved that the eigenvalue equation 34has at most, one solution, which is necessarily real, and itsatisfies + D0. Explicit calculations can be carried

out for discrete g = and Lorentzian g = / / 2 + 2 frequency distributions. We find =−D− + K /2, with =0 for the discrete distribution.

Clearly, incoherence is linearly stable for points K , Dabove the critical line D =− + K /2, and unstable forpoints below this line; see Fig. 1c. In terms of the cou-pling strength, incoherence is linearly stable providedthat K K c 2D + 2 , and unstable otherwise. This con-clusion also holds for D = 0 in the general unimodal case,for which Strogatz and Mirollo 1991 recovered Kura-moto’s result K c = 2/ g0 K c = 2 in the Lorentziancase. When D =0, the stability analysis is complicated

by the fact that the continuous spectrum lies on theimaginary axis.For even or asymmetric multimodal frequency distri-

butions, the eigenvalues may be complex Bonilla et al.,1992; Acebrón et al., 1998. The simple discrete bimodaldistribution g = − 0+ + 0 /2 has been stud-ied extensively Bonilla et al., 1992; Crawford, 1994;Acebrón and Bonilla, 1998; Bonilla, Pérez-Vicente, andSpigler, 1998. In this case, Eq. 34 has two solutions,± =−D +K ± K 2−16 02 /4. The stability boundaries forthe incoherent solution can be calculated by equating tozero the greater of Re + and Re −. The resulting phase

diagram on the plane K , D is depicted in Fig. 2 Bon-

illa et al., 1992. When the coupling is small enough K 2D, incoherence is linearly stable for all 0, whereasit is always unstable when the coupling is sufficientlystrong, K 4D. For intermediate couplings, 2DK 4D, incoherence may become unstable in two differ-

ent ways. For 0D, ± are real and incoherence is

linearly stable provided that K K c = 2D1 + 0 / D2,

and unstable when K K c. For 0D, ± are complex

conjugate and have zero real parts at K c = 4D.What happens in regions of the phase diagram where

incoherence is unstable? Typically there appear stablesolutions with r 0, which correspond to synchronizedphases. As discussed below, their study has been based

on bifurcation theory, for parameter values close to criti-cal lines of the phase diagram where incoherence is neu-trally stable. These analytical results are supplementedby numerical simulations of the nonlinear Fokker-Planck equation far from critical lines, or by numericalcontinuation of synchronized solutions bifurcating fromincoherence Acebrón, Perales, and Spigler, 2001. Be-sides this, Acebrón and Bonilla 1998 have developed asingular perturbation description of synchronization for

multimodal g , arbitrarily far from critical lines. Theiridea is to consider a g with m maxima located at 0l , where 0→, g d l =1

m l −l d, and




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then to use a method of multiple scales. The main resultis that the solution of the nonlinear Fokker-Planck equa-tion splits at lowest order into l phases, each obeying anonlinear Fokker-Planck equation with a discrete uni-

modal distribution centered at l and a couplingstrength l K . Depending on the value of l K , the l thphase turns out to be either synchronized or incoherent.The overall order parameter is the weighted sum withweight l of the order parameters of the correspondingphases. If first-order terms are included, the method of multiple scales describes incoherence, as well as oscilla-tor synchronization for multimodal frequency distribu-tions, rather well. These results hold for general fre-quency distributions both with or without reflectionsymmetry and for relatively low values of 0 Acebrónand Bonilla, 1998. Related work on multimodal fre-quency distributions include that of Acebrón et al. 1998

and Acebrón, Perales, and Spigler 2001. An interestingopen problem is to generalize the method of Acebrónand Bonilla 1998 so that both the location and thewidth of the peaks in g are taken into account.

D. Synchronized phases as bifurcations from incoherence,

D0

At the parameter values for which incoherence ceasesto be linearly stable, synchronized phases stable solu-tions of the nonlinear Fokker-Planck equation, , , t ,with r 0 may bifurcate from it. In this rather technical

subsection, branches of these bifurcating solutions willbe constructed in the vicinity of the bifurcation point bymeans of the Chapman-Enskog method; see Bonilla2000. We shall study the Kuramoto model with the dis-crete bimodal natural frequency distribution, whosephase diagram is depicted in Fig. 2. The stability bound-aries in this rich phase diagram separate regions whereincoherence becomes unstable, undergoing a transition

to either a stationary state, when 0D and K K c= 2D1 + 0 / D2, or to an oscillatory state, when 0D and K K c = 4D. The bifurcating solutions are asfollows:

1 When 0D / 2, the synchronized phases bifurcat-ing from incoherence are stationary and stable. Thebifurcation is supercritical, hence the synchronized

phases exist for K K c.

2 When D / 2 0D, the bifurcation is subcritical.An unstable branch of synchronized stationary solu-

tions bifurcates for K K c, reaches a limit point at asmaller coupling constant, and there coalesces witha branch of stable stationary solutions having largerr .

3 When 0D, the synchronized phases bifurcatingfrom incoherence are oscillatory and the corre-sponding order parameter is time periodic. Two

branches of solutions bifurcate supercritically at K c= 4D, a branch of unstable rotating waves and abranch of stable standing waves.

4 At the special point 0 = D / 2 and K c = 3D, the bi-furcation to stationary solutions changes from super-critical to subcritical. Near this point, the bifurcationanalysis can be extended to describe analyticallyhow the subcritical branch of stationary solutionsturns into a branch of stable solutions at a limitpoint.

5 At the special point 0 = D and K c = 4D, which canbe called the tricritical point , a line of Hopf bifurca-tions coalesces with a line of stationary bifurcationsand a line of homoclinic orbits. The study of thecorresponding O2-symmetric Takens-Bogdanovbifurcation shows how the oscillatory branches dieat a homoclinic orbit of an unstable stationarysolution.

The Chapman-Enskog method is flexible enough to ana-lyze all these bifurcations and, at the same time, simpler

than alternatives such as constructing the center mani-fold Crawford, 1994. Other than at the two special bi-furcation points, the simpler method of multiple scalesexplained in Appendix B yields the same results Bonillaet al., 1992; Bonilla, Pérez-Vicente, and Spigler, 1998.The Chapman-Enskog method Chapman and Cowling,1970 was originally employed by Enskog 1917 in thestudy of the hydrodynamic limit of the Boltzmann equa-tion. It becomes the averaging method for nonlinear os-cillations Boltzmann and Mitropolsky, 1961 and isequivalent to assuming a center manifold in bifurcationcalculations Crawford, 1994.

FIG. 2. Linear stability diagram for the incoherent solution

0 = 1 / 2 and the discrete bimodal frequency distribution inthe parameter space K / D , 0 / D. 0 is linearly stable to theleft of the lines K = 4D, 0D where Hopf bifurcations take

place and K / 2D = 1 + 02

/ D2

, 0D where one solution of Eq. 34 becomes zero. To the right of these lines, the inco-herent solution is unstable. At the tricritical point, K = 4D, 0= D, two solutions of Eq. 34 become simultaneously zero.The dashed line separates the region where eigenvalues are

real below the line from that where they are complex conju-gate above the line. From Bonilla, Pérez-Vicente, andSpigler, 1998.




11/49

1. Bifurcation of a synchronized stationary phase

Let 0D and consider K close to its critical valueK c = 2D1+ 0 / D

2. The largest eigenvalue satisfies K −K c / 21−

2 / D2 as K →K c. As indicated in Eq.35, there are two eigenfunctions associated with thiseigenvalue, ei /D + i and its complex conjugate. Ex-cept for terms decaying exponentially fast in time, the

solution of the linearized stability problem at K = K c is

therefore Aei /D + i +c.c., where A is a constant andc.c. means complex conjugate of the preceding term. Let

us suppose now and justify later that K = K c +2K 2,

where is a small positive parameter. The probabilitydensity corresponding to initial conditions close to inco-herence will have the form 2 −1 + A ei /D+ i +c.c. The correction to incoherence will be close tothe solution of the linearized stability problem, but now

we can assume that the complex constant A variesslowly with time. How slowly? The linearized solution

depends on time through the factor et , and = OK −K c = O

2. Thus we assume =2t . The probabilitydensity can then be written as

, ,t ; 1

2 1 + A ;ei

D + i + c.c.

+ n=2

n n , t , ; A, Ā

1

2 exp A ;ei

D + i + c.c.

+ n=2

n n , t , ; A, Ā . 36

The corrections to 1/ 2 can be telescoped into an ex-ponential with a small argument, which ensures that theprobability density is always positive. Typically, by theexponential ansatz, the parameter region, where theasymptotic expansion is a good approximation to the

probability density, is widened. The functions n and nare linked by the relations

1 = 1 = A ;ei

D + i + c.c., 2 = 2 +

12

2 ,

3 = 3 + 1 2 +

13

3! ,

4 = 4 + 1 3 + 2

2

2 +

12 2

2 +

14

4! , 37

and so on. They depend on a fast scale t correspondingto stable exponentially decaying modes, and on a slow

time scale through their dependence on A. All terms inEq. 36 that decrease exponentially in time will be omit-ted. In Eq. 36, the slowly varying amplitude A obeysthe equation

dA

d =

n=0

nF n A, Ā . 38

The functions F n A , Ā are determined from the condi-tions that n or n be bounded as t → on the fast timescale, for fixed A, and periodic in . Moreover, theycannot contain terms proportional to the solution of thelinearized homogeneous problem, e ±i /D ± i , because

all such terms can be absorbed in the amplitudes A or Ā

Bonilla, 2000. These two conditions imply that

e−i , n = 0, n 1. 39

The normalization condition together with Eq. 36yields

−

n ,t , ; A, Ā d = 0, n 2. 40

To find n, we substitute Eqs. 36 and 38 into Eq.26 and use Eq. 39 to simplify the result. This yields

the following hierarchy of linear nonhomogeneousequations:

L 2 t − D 2 + 2 + K c Im e

−i e−i , 2

= − K c 1 Im e−i e−i , 1 + c.c., 41

L 3 = − K c 2 Im e−i ei , 1

− K 2 Im e−i e−i , 1 − F

0 A 1 + c.c., 42

and so on. Clearly, 1 = At ei /D + i + c.c. obeys the

linearized stability problem 30 with = 0, L 1 =0 up toterms of order . Thus it is not obvious that each linear

nonhomogeneous equation of the hierarchy has abounded periodic solution. What is the necessary solv-ability condition to ensure that the linear nonhomoge-neous equation,

L n = h ; 1, . .. , n−1 = Qei + ¯ , 43

has a solution of the required features? To answer thisquestion, we assume that, in fact, Eq. 43 has a boundedperiodic solution of the form n = Pe

i +¯. Then P isgiven by

P =K c1,P

2D + i +

Q

D + i , 44

from which we obtain the nonresonance condition

1, QD + i

= 0, 45first found by Bonilla et al. 1992. Note that we obtainEq. 45 even when 1 , P 0. e i times the term propor-tional to 1 , P in Eq. 44 is a solution of L =0 andshould therefore be absorbed in the definitions of A and

Ā . This implies Eq. 39.Inserting 1 into the right side of Eq. 41, we find




12/49

L 2 =2 A2

D + i e2i + c.c., 46

whose solution is

2 = A2

D + i 2D + i e2i + c.c. 47

We see that 1 contains odd harmonics and 2 contains

even harmonics a possible -independent term is omit-ted in 2 because of the normalization condition. This isactually true in general: 2n contains harmonics e

i2 j , j

= 0 , ± 1 , . . . , ± n, and 2n+1 contains harmonics ei2 j +1 , j

=−n + 1 , . . . ,n. The nonlinearity of the Fokker-Planckequation is responsible for the appearance of resonant

terms in the equations for 2n+1, which should be elimi-

nated through the terms containing F 2n. Then, we can

set F 2n+1 =0 and we only need the scaling K −K c= O2. The ultimate reason for these cancellations is, of course, the O2 symmetry of our problem, i.e., reflec-

tion symmetry and invariance under constant rotations, → + Crawford, 1994.Similarly, the nonresonance condition for Eq. 42

yields

F 0 = 1, 1D + i 2

−12K 2 AK c

2

− 1, 1D + i 22D + i

A A2

=K 2 A

21 − 0

2

D2 −

21 − 2 02D2

A A2

1 − 0

2

D24 + 0

2

D2D. 48

Keeping this term in Eq. 38, we obtain dA / d F 0, which is a reduced equation with the stationarysolution A = K 2D / 44 + 0 / D2 / 1−2 0 / D2. Thecorresponding order parameter is

r K − K cD4 + 0

2

D2

K c21 − 2 02

D2

1/2

, 49

which was obtained by Bonilla et al. 1992 using a dif-ferent procedure. The solution 49 exists for K K c su-percritical bifurcation provided that 0D / 2,whereas it exists for K K c subcritical bifurcationwhen 0D / 2. The amplitude equation 38 impliesthat the supercritical bifurcating solution is stable andthat the subcritical solution is unstable.

We can describe the transition from supercritical to

subcritical bifurcation at 0 = D / 2, K c = 3D, by evaluat-ing F 4 and adding it to the right-hand side of Eq. 38.The result is

dA

d = K 21 − 2 K 2 − 2 2 2

D A

−

47K 2 − 4 2 22

9D2 A A2 −

2722

171D3 A A4; 50

see Appendix C. The stationary solutions of this equa-tion are the stationary synchronized phases. We see that

stable phases bifurcate supercritically for K 20 if K 22 2 2, whereas a branch of unstable stationary solu-tions bifurcates subcritically for K 20 if K 22 2 2.This branch of unstable solutions coalesces with abranch of stable stationary synchronized phases at the

limit point K 2 −1927K 2−4 2 22 /612.

2. Bifurcation of synchronized oscillatory phases

The bifurcation in the case of complex eigenvaluescan be easily described by the same method. The maindifference is that the solution to the linearized problemis now

1 = A+t

D + i + eit + + c.c. +

A−t

D + i − eit −

+ c.c., 51

where 2 = 02−D2, K c = 4D, the eigenvalues with zero

real part being K c = ± i. For the two slowly varyingamplitudes, A+ , A−, we assume that equations of theform

dA±

d

n=0

2nF ±

2n A+, A−, A+, A− 52

hold. Following the previous method, the nonresonanceconditions for Eq. 42, with 1 given by Eq. 51, yieldF

+0

and F −

0. The corresponding amplitude equations are

Bonilla, Pérez-Vicente, and Spigler, 1998

Ȧ+ = A+ − A−2 + A+

2 A+,

Ȧ−

= A−− A+

2 + A−

2 A−

, 53

where the overdot denotes differentiation with respect

to , and

=

1

4 −

iD

4 , =

D + iD2 + 0

2

K 24D2 + 0

2 ,

=

23D2 + 4 02 + iD

3D2 + 2 02

DK 29D2 + 16 0

2 . 54

To analyze the amplitude equations 53, we define thenew variables

u = A+2 + A

−2, v = A+

2− A

−2. 55

By using Eq. 53, we obtain for u and v the system




13/49

u̇ = 2 Re u − Re + u2 − Re − v2,

v̇ = 2 Re v − 2 Re uv . 56

Clearly, u = v or u =−v correspond to traveling-wave so-

lutions with only one of the amplitudes A± being non-

zero. The case v 0 corresponds to standing-wave solu-tions, which are a combination of rotating and counter-rotating traveling waves with the same amplitude. Wecan easily find the phase portrait of Eqs. 56 corre-sponding to , , and given by Eqs. 54 see Fig. 3.Up to, possibly, a constant phase shift, the explicit solu-tions

A+ = Re Re

ei , A−

0,

= Im − Im

Re

Re 57

or A + 0 and A− as A+ above are obtained inthe case of the traveling-wave solutions, while

A+ = A− = 2 Re Re +

ei ,

= Im − Im +

Re + Re 58

are obtained in the case of the standing-wave solutions.Note that both standing wave and traveling wave bifur-

cate supercritically with r SW / r TW 1. Re + andRe are both positive when K 2 =1, whereas the squareroots in Eqs. 57 and 58 become pure imaginary whenK 2 =−1. This indicates that the bifurcating branches can-not be subcritical. An analysis of the phase portrait cor-

responding to Eq. 56 shows that the standing-wave so-lutions are always globally stable, while the traveling-wave solutions are unstable. This result was first pointedout by Crawford 1994.

3. Bifurcation at the tricritical point

At the tricritical point, K = 4D, 0 = D, a branch of os-cillatory bifurcating phases coalesces with a branch of stationary bifurcating phases and a branch of homoclinicorbits, in an O2-symmetric Takens-Bogdanov bifurca-tion point. Studying the bifurcations in the vicinity of such a point shows how the stable and unstable branches

of oscillatory phases, standing wave and traveling wave,respectively, end as the coupling is changed. Analyzingtransitions at the tricritical point is a little more compli-cated because it requires changing the assumptions onthe amplitude equation Bonilla, Pérez-Vicente, and Spi-gler, 1998; Bonilla, 2000. First of all, at the tri-critical point, 1 , D + i −2= ReD + iD−2 = 0. Thisinnocent-looking fact implies that the term −F 0 A 1 onthe right-hand side of Eq. 42 dissapears in the nonreso-nance condition, and therefore using the same ansatz asin Eqs. 36 and 38 will not deliver any amplitude equa-tion. Second, the oscillatory ansatz 51 breaks down

FIG. 3. Phase planes a u , v and b A+ , A− showing the critical points corresponding to traveling wave TW and standingwave SW solutions. From Bonilla, Pérez-Vicente, and Spigler, 1998.




14/49

too, because =0 at the tricritical point, and the factormultiplying A

− is simply the complex conjugate of the

factor multiplying A+. Therefore only one independentcomplex amplitude exists, and we are brought back toEq. 36. How should one proceed?

In order to succeed, one should recognize that a basic

slow time scale different from does exist near the tri-critical point. The eigenvalues with largest real part are

= − D +K

4+ i 02 −

K

4 2

= i 2D 2 − K 24 + K 22

4 + O3

along with their complex conjugates, provided that K −4D = K 2

2, 0−D = 22 with 2K 2 /4. Therefore the

time-dependent factors et appearing in the solution of the linearized problem indicate that the perturbations

about incoherence vary on a slow time scale T =t nearthe tricritical point. This leads to the Chapman-Enskogansatz

, ,t ; =1

2 1 + AT ;D + i ei + c.c.+

j =2

4

j j ,t ,T ; A, Ā + O

5 , 59d2 A

dT 2 = F 0 A, Ā + F 1 A, Ā + O2 . 60

The equation for A is second order rather than firstorder as in Eq. 38 because resonant terms appear atO3 for the first time. These are proportional to ATT = d2 A / dT 2. The quantities F 0 and F 1 are evaluated inAppendix D. The resulting amplitude equation is

ATT − D

2K 2 − 42 A −

2

5 A2 A

= K 22

AT − 23

25D A2 AT −

1

5D A2 AT + O2.

61

Equation 61 is in the scaled normal form studied byDangelmayr and Knobloch 1987, cf. their equations3.3, p. 2480. Following these authors, we substitute

AT ; = RT ;ei T ; 62

in Eq. 61, separate real and imaginary parts, and ob-tain the perturbed Hamiltonian system

RTT + V

R= K 2

2 −

38

25DR2RT ,

LT = K 22 −

28

25DR2L. 63

Here L = R2 T is the angular momentum, and

V V R =L2

2R2 −

D

4K 2 − 4 2R

2−

R4

10 64

is the potential. This system has the following specialsolutions:

i The trivial solution, L = 0, R =0, which corre-sponds to the incoherent probability density,

= 1/ 2 . This solution is stable for K 20 if 20and for K 2−4 20 if 20.

ii The steady-state solution, L = 0, R = R0= 5D 2− K 2 / 40, which exists provided that 2K 2 /4. This solution is always unstable.

iii The traveling-wave solutions, L = L0= R0

2 2D 2− 1956K 20, R = R0 = 5

2 DK 2 / 140,

which exist provided that K 20 and 219K 2 / 56. These solutions bifurcate from the

trivial solution at K 2 = 2 =0. When 2 = 19K 2 /56,the branch of traveling waves merges with thesteady-state solution branch. This solution is al-ways unstable.

iv The standing-wave solutions, L = 0, R = RT peri-odic. Such solutions have been found explicitlysee Sec. V.A of Dangelmayr and Knobloch,1987. The standing waves branch off from thetrivial solution at K 2 = 2 = 0, exist for 211K 2 / 190, and terminate by merging with ahomoclinic orbit of the steady state ii on the line 2 = 11K 2 / 19 see Eq. 5.8 of Dangelmayr andKnobloch, 1987. This solution is always stable.

All these results are depicted in Fig. 4, which corre-sponds to Fig. 4, IV, in the general classification of sta-bility diagrams reported by Dangelmayr and Knobloch1987, p. 267. For fixed 0D, the bifurcation diagramnear the tricritical point is depicted in Fig. 5. Note thatEq. 59 yields, to leading order,

, ,t ; 1

2 1 + Rei +

D + i + c.c. , 65

and hence rei Re−i / 2D. It follows that r R / 2D and − , which shows that, essentially, the

FIG. 4. Stability diagrams K 2 , 2 near the tricritical point.From Bonilla, Pérez-Vicente, and Spigler, 1998.




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solution AT ; to Eq. 61 coincides with the conjugateof the complex order parameter. For this reason, in Fig.

5 the ordinate can be either R or r . In Fig. 6 we havedepicted the global bifurcation diagram which completesthat shown in Bonilla et al. 1992, Fig. 5.

In closing, the Chapman-Enskog method can be usedto calculate any bifurcations appearing for other fre-quency distributions and related nonlinear Fokker-Planck equations. As discussed in Sec. II.B, nontrivialextensions are needed in the case of the hyperbolic limit,D→0+.

IV. VARIATIONS OF THE KURAMOTO MODEL

We have seen in the preceding section how the long-

range character of the coupling interaction in the Kura-moto model allows us to obtain many analytical results.Yet one might ask how far these results extend beyondthe mean-field limit in finite dimensions. Also, onemight ask how synchronization effects in the Kuramotomodel are modified by keeping long-range interactionsbut including additional sources of quenched disorder,

multiplicative noise, or time-delayed couplings. Unfortu-nately, many of the analytical techniques developed inthe preceding section hardly cover such new topics. Inparticular, the treatment of short-range couplings oscil-lators embedded in a lattice with nearest-neighbor inter-actions presents formidable difficulties at both the ana-lytical and the numerical level. This challenges ourcurrent understanding of the mechanisms lying behindthe appearance of synchronization. The next sections

are devoted to discussing a number of these cases. Thepresent knowledge of such cases is still quite modest,and major work remains to be done.

A. Short-range models

A natural extension of the Kuramoto model discussedin Sec. III includes short-range interaction effects Sak-aguchi et al., 1987; Daido, 1988; Strogatz and Mirollo,1988a, 1988b. Kuramoto and co-workers Sakaguchi et al., 1987 have considered the case in which oscillatorsoccupy the sites of a d-dimensional cubic lattice and in-teractions occur between nearest neighbors,

˙ i = i + K i, j

sin j − i, 66

where the pair i , j stands for nearest-neighbor oscilla-tors and the i’s are independent random variables cho-sen according to the distribution g . Compared to theKuramoto model 23, the coupling strength K does notneed to be scaled by the total number of oscillators.However, convergence of the model 66 in the limit of large d requires that K scale as 1/ d. Although this modelcan be extended so as to include stochastic noise, i.e.,finite temperature T , most of the work on this type of

model has been done at T =0. Solving the short-rangeversion of the Kuramoto model is a hopeless task ex-cept for special cases such as one-dimensional models—see below—or Cayley-tree structures due to the diffi-culty of incorporating the randomness in any sort of renormalization-group analysis. In short-range systems,one usually distinguishes among different synchroniza-tion regimes. Global synchronization, which implies thatall the oscillators are in phase, is rarely seen except forK N →. Phase locking or partial synchronization isobserved more frequently. This is the case in which a

local ensemble of oscillators verify the condition ˙ i=const, for every i. A weaker situation concerns cluster-

ing or entrainment. Although this term sometimes hasbeen used in the sense of phase locking, usually it refersto the case in which a finite fraction of the oscillators

have the same average frequency ˜ i defined by

˜ i = limt →

it

t . 67

There is no proof that such a limit exists. However, if itdoes not exist, no synchronization whatsoever is pos-sible. The condition of clustering is less stringent thanphase locking, and therefore its absence is expected topreclude the existence of phase locking. Note that the

FIG. 5. Bifurcation diagram K , R near the tricritical point for 0D fixed. K

* is the coupling at which a subcritical branch of

stationary solutions bifurcates from incoherence. From Bon-

illa, Pérez-Vicente, and Spigler, 1998.

FIG. 6. Global bifurcation diagram including all stationary-

solution branches. From Bonilla, Pérez-Vicente, and Spigler,

1998.




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previous definitions do not exhaust all possible types of synchronized stationary solutions, such as, for instance,the existence of moving traveling-wave structures. Theconcept of synchronization either global or partial isdifferent from the concept of phase coherence intro-duced in the context of the Kuramoto model in Sec. II;see Eq. 2. Phase coherence is a stronger condition thansynchronization as it assumes that all phases i are clus-tered around a given unique value, as are their velocities

˙ i. The contrary is not necessarily true, as phases canchange at the same speed synchronization takes placewhile having completely different values incoherence.Coherence seems less general than synchronization asthe former bears connection to the type of ferromag-netic ordering present in the Kuramoto model. Al-though this type of ordering is expected to prevail infinite dimensions, synchronization seems more appropri-ate for discussing oscillator models with structural disor-der built in.

For short-range systems, one would like to understandseveral issues:

• The existence of a lower critical dimension abovewhich any kind of entrainment is possible. In particu-lar, it is relevant to prove the existence of phase lock-ing and clustering for large enough dimensionalityand the differences between both types of synchroni-zation.

• The topological properties of the entrained clustersand the possibility of defining a dynamical correla-tion length describing the typical length scale of these clusters.

• The existence of an upper critical dimension abovewhich the synchronization transition is of the mean-

field type.• The resulting phase diagram in the presence of ther-

mal noise.

Sakaguchi et al. 1987 have proposed some heuristicarguments showing that any type of entrainment globalor local can occur only for d2. This conjecture is sup-ported by the absence of entrainment in one dimension.Strogatz and Mirollo 1988a, 1988b, however, haveshown that no phase locking can occur at any finite di-mension. As phase locking occurs in mean-field theory,this result suggests that the upper critical dimension inthe model is infinite. Particularly interesting results were

obtained in one dimension chains of oscillators. In thiscase, and for the case of a normal distribution of naturalfrequencies i, it can be proven that the probability of

phase locking vanishes as N →, while it is finite when-

ever K N . The same result can be obtained for anydistribution not necessarily Gaussian of independentlydistributed natural frequencies. The proof consists of showing that the probability of phase locking is relatedto the probability that the height of a certain Brownianbridge is not larger than some given value which de-

pends on K and the mean of g . Recall that by Brown-ian bridge we mean a random walk described by n

moves of length xi extracted from a given probability

distribution, where the end-to-end distance l n =i=1n xi

is constrained to have a fixed value for a given number nof steps. However, one of the most interesting results inthese studies is the use of block renormalization-grouptechniques to show whether clustering can occur in finitedimensions. Nearly at the same time, Strogatz andMirollo 1988a, 1988b and Daido 1988 presented simi-lar arguments but leading to slightly different yet com-patible conclusions. For Strogatz and Mirollo 1988a,1988b, the goal was to calculate the probability P N , K that a cube S containing a finite fraction N 1 of the oscillators could be entrained to have a single com-mon frequency. Following Strogatz and Mirollo 1988a,1988b, let us assume the macroscopic cluster S to bedivided into cubic subclusters Sk of side l , the total num-

ber of subclusters being N s = N / l d which is of order N .

For each subcluster k, the average frequency k andphase k are defined as

k =1

l d iS

k

i, k =1

l d iS

k

i. 68

Summing Eq. 66 over all oscillators contained in eachsubcluster Sk we get

̇k =k +K

l d i, j Sk

sin j − i, 69

where the sum in the right-hand side runs over all linksi , j crossing the surface Sk delimiting the region Sk.Considering that there are 2dl d−1 terms in the surfaceand hence in the sum in Eq. 69, this implies that

̇k −k2dK

l . 70

If S is a region of clustered oscillators around the fre-

quency ˜ , then, after time averaging, the limit 67 gives ˜ −k2dK / l for all 1kN s. Since the k are un-correlated random variables, the probability that such a

condition is simultaneously satisfied for all N s oscillators

is pN s , p being the typical probability value such that

˜ −k2dK / l is satisfied for a given oscillator. Thisprobability is therefore exponentially small when the

number of subclusters N s is of order ON ,

P N ,K N exp− cN . 71

Here c is a constant, and the factor N in front of the

exponential is due to the number of all possible ways thecluster S can be embedded in the lattice. Therefore theprobability vanishes in any dimension in the limit of large population. This proof assumes that entrainedclusters have a compact structure such as a cubicalshape. However, this need not necessarily be true. Hadthe clusters a noncompact shape such as space-fillingsponges or lattice-animal treelike structures, the proof would not hold anymore, since the number of subclus-

ters N s does not have to scale necessarily as 1/ l d. There-

fore such a result does not prevent the existence of amacroscopic entrainment in noncompact clusters.




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This finding does not appear to contradict that re-ported by Daido 1988, who has shown the existence of a lower critical dimension d l , depending on the tails of a

class of frequency distributions g . When

g − −1

,

1, 72then normalization requires 0. Moreover, Daido

considers that 2 for distributions with an infinitevariance, the limiting case =2 corresponding to thecase of a distribution with finite variance such as theGaussian distribution. The argument put forth byDaido resorts to a similar block decimation procedure asthat outlined in Eqs. 68 and 69. However he reportsdifferent conclusions from those by Strogatz and Mirollo1988a, 1988b. Having defined the subcluster or blockfrequencies k and phases k in Eq. 68, Daido showsthat only for d / −1, and in the limit when l →,does Eq. 70 yield the fixed-point dynamical equation

̇k =k 73

for every k, which shows that no clustering can occur for

0 1 in any dimension. However, for 1 2, mac-roscopic entrainment should be observed for dimensions

above dl = / −1. For the Gaussian case, =2, en-trainment occurs above dl =2 as was also suggested bySakaguchi et al. 1987. Numerical evidence in favor of asynchronization transition in dimension d = 3 Daido,1988 is not very convincing. The correct value of theupper critical dimension remains to be solved.

The Kuramoto model in an ultrametric tree has also

been studied by Lumer and Huberman 1991, 1992.They considered a general version of the model in Eq.66, where N oscillators sit in the leaves of a hierarchi-cal tree of branching ratio b and L levels; see Fig. 7. The

coupling among the oscillators K ij is not uniform but

depends on their ultrametric distance l ij , i.e., the numberof levels in the tree separating the leaves from its com-mon ancestor,

K ij = Kdl ij , 74

where d x is a monotonically decreasing function of thedistance. The existence of a proper thermodynamic limit

requires, for d x in the limit of large L , N ,

i=1

N

K ij = K , 75

for all j . For a given value of K , as the ultrametric dis-

tance l ij increases, entrainment fades away. However, asK increases, more and more levels tend to synchronize.

Therefore this model introduces in a simple way theclusterization of synchronization, thought to be relevantin the perception problem at the neural level see Sec.VII.A. The simplest function d x that incorporatessuch effects has an exponential decay d x 1 / a x, wherethe coupling strength decreases by a factor a at consecu-tive levels. It can be shown Lumer and Huberman,1992 that a cascade of synchronization events occurswhenever b / a −1, where is defined by Eq. 72.In such a regime, the model displays a nice devil’s stair-case behavior when plotting the synchronization param-

eter as a function of K . This is characteristic of the emer-gent differentiation observed in the response of the

system to external perturbations. Other topologies be-yond the simple cubic lattice structure have also beenconsidered. Niebur, Schuster, et al. 1991 analyze spatialcorrelation functions in a square lattice of oscillatorswith nearest-neighbor, Gaussian the intensity of the in-teraction between two sites decays with their distanceaccording to a Gaussian law, and sparse connectionseach oscillator is coupled to a small and randomly se-lected subset of neighbors. Overall, they find that en-trainment is greatly enhanced with sparse connections.Whether this result is linked to the supposed noncom-pact nature of the clusters is yet to be checked.

An intermediate case between the long-range Kura-

moto model and its short-range version 66 occurs whenthe coupling among oscillators decays as a power law,1/ r , r denoting their mutual distance. The intensity of the coupling is then properly normalized in such a waythat the interaction term in Eq. 66 remains finite in thelimit of large population. For the normalized case, inone dimension, it has been shown Rogers and Wille,1996 that a synchronization transition occurs when cK with K = 0 = 2 / g = 0, corresponding tothe Kuramoto model see the paragraph just after Eq.11. It is found that 2 is required for a synchroniz-ing transition to occur at finite K . Note that the same

condition is found for one-dimensional Ising and XY

models in order to have a finite-temperature transition.For the non-normalized case results are more interestingMarodi et al., 2002, as they show a transition in thepopulation size rather than in the coupling constant K for d. In such a case, synchronization occurs pro-vided that the population is allowed to grow above somecritical value N cK , d. The relevance of such a resultstems from the fact that sufficiently large three-dimensional populations, interacting through a signal

whose intensity decays as 1/ r 2, such as sound or light,

can synchronize whatever the value of the coupling K might be.

FIG. 7. A hierarchical tree with N =9 two-level L = 2 oscilla-tors having branching ratio b =3. The distance l ij between two

oscillators is given by the number of levels between them and

their closest common ancestor. In this example, l 12= l 46 = l 78= 1

and l 15 = l 58 = 2.




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Before ending the present overview of short-rangemodels, it is worth mentioning that the complexity of thesynchronization phenomenon in two dimensions hasbeen emphasized in another investigation conducted byKuramoto and co-workers Sakaguchi et al., 1988. Herethey studied a model where th

kuramoto model

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