Content uploaded by Dane Taylor
Author content
All content in this area was uploaded by Dane Taylor on Dec 01, 2014
Content may be subject to copyright.
arXiv:0906.2124v2 [cond-mat.stat-mech] 1 Feb 2010
Spontaneous synchronization of coupled oscillator systems with frequency adaptation
Dane Taylor,1, ∗Edward Ott,2and Juan G. Restrepo3
1Department of Electrical, Computer, and Energy Engineering,
University of Colorado, Boulder, Colorado 80309, USA
2Institute for Research in Electronics and Applied Physics,
University of Maryland, College Park, Maryland 20742, USA
3Applied Mathematics Department, University of Colorado, Boulder, Colorado 80309, USA
(Dated: February 1, 2010)
We study the synchronization of Kuramoto oscillators with all-to-all coupling in the presence of slow, noisy
frequency adaptation. In this paper we develop a new model for oscillators which adapt both their phases and
frequencies. It is found that this model naturally reproduces some observed phenomena that are not qualitatively
produced by the standard Kuramoto model, such as long waiting times before the synchronization of clapping
audiences. By assuming a self-consistent steady state solution, we find three stability regimes for the coupling
constant k, separated by critical points k1and k2: (i) for k < k1only the stable incoherent state exists; (ii) for
k > k2, the incoherent state becomes unstable and only the synchronized state exists; (iii) for k1< k < k2
both the incoherent and synchronized states are stable. In the bistable regime spontaneous transitions between
the incoherent and synchronized states are observed for finite ensembles. These transitions are well described
as a stochastic process on the order parameter rundergoing fluctuations due to the system’s finite size, leading
to the following conclusions: (a) in the bistable regime, the average waiting time of an incoherent→coherent
transition can be predicted by using Kramer’s escape time formula and grows exponentially with the number of
oscillators; (b) when the incoherent state is unstable (k > k2), the average waiting time grows logarithmically
with the number of oscillators.
I. INTRODUCTION
Many natural and engineered systems can be described as
an ensemble of heterogeneous limit-cycle oscillators influenc-
ing each other. Examples include glycolytic oscillations in
yeast cell populations [1], pedestrians walking over a bridge
[2], arrays of Josephson junctions [3], the power grid [4],
lasers [5], and some species of fireflies [6]. A central issue
is that of understanding the mechanism of coherent behavior
that is often observed for these systems.
The Kuramoto model [7] (for a review, see [8]) addresses
this problem by considering the simplified case in which os-
cillators are all-to-all coupled, and each oscillator, labeled by
an index n, has an intrinsic frequency ωnand an oscillator
state that can be specified solely by its phase angle θn. The
evolution of the phase of each oscillator nis given by
˙
θn=ωn+k
N
N
X
m=1
sin(θm−θn),(1)
where Nis the number of oscillators, m= 1,2,...,N indi-
cates different oscillators, and kis a parameter that represents
the strength of the coupling between oscillators. Kuramoto
found that for the N→ ∞ coupling limit (approximating the
typical case of large Narising in most applications), the col-
lective behavior of the oscillator ensemble, quantified by the
order parameter r=|PN
m=1 exp(iθm)|, undergoes a transi-
tion from incoherence (r= 0) to synchronization (r∼1) as
the coupling strength is increased past a critical value kc. The
Kuramoto model provides a simple mathematical model cap-
∗dane.taylor@colorado.edu
turing the essential mechanisms for synchronization of limit-
cycle oscillators. Despite its long-standing status as a classical
model of synchronization, some advances in the theoretical
understanding of Kuramoto-type models have been achieved
only very recently (e.g., Refs. [9]).
Due to the ubiquity of synchronization phenomena in com-
plex systems, there is current interest in understanding the ef-
fect of network structure interactions and adaptation on the
synchronization of oscillators [10]. The new model presented
in this Article aims to investigate synchronization in coupled
oscillator systems where each oscillator’s natural frequency
ωnslowly adapts, while being subjected to random noise-like
fluctuations. One motivation for considering adaptive syn-
chronization is the observation that, in many biological sit-
uations, synchronization seems to serve a useful function. A
fairly clear example is the synchronization of pacemaker cells
in the heart. Another example is the observed evolving pat-
terns of neuronal synchrony in the brain, which have been
conjectured to play a key role in organizing brain function. In
some cases, adaption of frequencies has been experimentally
observed: fireflies of the species pteroptyx-malaccae slowly
adapt their flashing frequency in response to the flashes they
observe [6]. Assuming the utility of synchronization in such
biological cases, it is reasonable that there might be evolu-
tionary pressure for the development of adaptive mechanisms
that promote synchronization or maintain it in the presence
of disruptive influences (e.g., noise). In addition, one could
imagine technological and social situations where adaptation
to promote synchronization might be relevant. A familiar so-
cial example is that of an audience clapping their hands and
seeking to synchronize [11, 12]. Therefore, it is important to
consider the possibility that various mechanisms might oper-
ate independently to promote synchronization in a noisy envi-
ronment. In this paper we introduce and analyze a model of
2
all-to-all coupled phase oscillators with noisy frequency adap-
tation, where, as seems reasonable in the above-cited biolog-
ical examples, the coupling of the oscillators’ phases occurs
on a faster timescale than the frequency adaptation dynamics.
The paper is organized as follows. In Sec. II we present and
analyze our model. Analytical results are derived and numer-
ically tested in Secs. II A and II B, respectively. In Sec. III we
study the statistics of spontaneous transitions to the synchro-
nized state due to finite size effects. In Sec. IV we discuss our
results and their relation with previous work. In Sec. V we
present our conclusions.
II. FREQUENCY ADAPTION MODEL
We consider the classical Kuramoto model supplemented
with a dynamical equation for the evolution of the oscillators’
natural frequencies ωn,
˙
θn=ωn+k
N
N
X
m=1
sin(θm−θn),(2)
˙ωn=τ−1k
N
N
X
m=1
sin(θm−θn) + ηn−dVn(ωn)/dωn,
where τis assumed to be much larger than the spread in the
oscillator period and ηnis a Gaussian uncorrelated noise term
such that hηn(t)ηm(t′)i= 2Dδnmδ(t−t′), where h·i is an en-
semble average and δnm is the Kronecker delta. The motiva-
tion for the different terms in this natural frequency adaption
term is the following:
•We assume that each oscillator nmay only have knowl-
edge of the aggregate input to it from the other oscilla-
tors, kPN
m=1 sin(θm−θn). This frequency-coupling
term was originally introduced in Ref. [6], who consid-
ered frequency adaptation without phase coupling.
•The form of the coupling guarantees that if the phase of
oscillator nis behind (ahead of) the average phase [so
that sin(θm−θn)is, on average, positive (negative)],
its frequency increases (decreases).
•Frequency adaptation occurs on a time scale τ, much
slower than the phase dynamics.
•The intrinsic frequencies ωnare subject to random
noise ηn. This is partly motivated by observations of
frequency drift in biological oscillators [13].
•The confining potential Vn(ω)represents physical
mechanisms that, depending on the application, con-
strain the natural frequencies to some reasonable range.
The dynamics we find for our system (2) is related to that
for the Kuramoto model with inertia [14, 15]; however, the
differences are significant and will be discussed in Sec. IV.
Also in Sec. IV, we discuss the relation between (2) and a
model for circadian rhythms that implements wandering, un-
coupled frequencies [16].
We note that we could have added a noise term to the θ
equation in (2). However, the effect of such a noise term has
been already studied in the Kuramoto model, and it has been
found that it shifts the transition to synchronization to larger
values of the coupling strength k, maintaining the same qual-
itative behavior. Therefore, for analytical simplicity, we will
not consider this term. As we will see in our numerical sim-
ulations, a role analogous to fluctuations in θwill be played
by the fluctuations resulting from having a finite number of
oscillators.
A. Model Analysis
We consider Nto be very large and adopt a continuum de-
scription. Thus, we assume that the ensemble of oscillator
intrinsic frequencies can be regarded as being drawn from a
continuous distribution function G(ω, t).
We analyze our proposed model (2) by using the assumed
separation of timescales between the oscillator dynamics and
the frequency adaptation. Rewritting Eqs. (2) in terms of the
mean field reiψ =1
NPN
m=1 eiθm, we obtain
˙
θn=ωn−kr sin(θn−ψ),(3)
˙ωn=−τ−1kr sin(θn−ψ) + ηn−dV (ωn)/dωn.(4)
Here we have dropped the subscript non the potential Vn, as
we will henceforth consider all oscillators to have the same
Vn. In addition, we will assume that V(ω) = V(−ω).
Since the frequencies vary on a timescale much longer than
the phases, on the fast time scale we can approximate ωnin
Eq. (3) as constant. As we shall soon see, it is relevant to
assume that G(ω, t)is symmetric in ωand monotonically de-
creasing away from its maximum. Furthermore, without loss
of generality, it suffices to take the maximum of Gto occur at
ω= 0 (if it does not, it can be shifted to 0by the change of
variablesω→ω+Ω,θ→θ−Ωt). As originally noted by Ku-
ramoto, in the saturated state (i.e., rconstant on the fast time
scale), the phase dynamics is of two types depending on the
value of ωn. For |ωn|< kr oscillator nis said to be “locked”
and its phase settles at a value given by
sin(θn−ψ) = ωn/(kr).(5)
For |ωn|> kr the phase is said to “drift” and θncontinually
increases (decreases) with time for ωn> kr (ωn<−kr). For
a given frequency |ω|> kr, the drifting oscillators have a dis-
tribution of phases ρ(θ, ω )determined from the conservation
of oscillator density by the condition ρdθ/dt =constant. This
yields
ρ(θ, ω) = pω2−(kr)2
2π|ω−kr sin(θ−ψ)|,(6)
where the factor pω2−(kr)2/(2π)normalizes ρ(θ, ω)so
that Rπ
−πρdθ = 1.
Still invoking the time scale separation, and consequently
assuming that the deterministic term in Eq. (4) can be aver-
3
aged over time, we obtain an approximation to Eq. (4) for the
drifting oscillators
˙ωn≈ −τ−1kr Zπ
−π
ρ(θ, ωn) sin(θ−ψ)dθ +ηn−dV/dωn.
(7)
[Here the deterministic term has been replaced by its time av-
erage, while the fast-varying noise term has been retained.
This can be justified by noting that the difference between the
original equation, Eq. (4), and the equation where the deter-
ministic part was averaged, Eq. (7), is what would be obtained
in the noiseless case. Since in the noiseless case this differ-
ence can be argued to be small by averaging, we conclude our
procedure is justified.]
Integrating Eq. (7) and recalling that for entrained oscilla-
tors kr sin(θn−ψ) = ωn, Eq. (4) can be rewritten as
˙ωn=h(ωn) + ηn−dV/dωn,(8)
where
h(ω) = −ω/τ, |ω| ≤ kr,
−ω/τ +sign(ω)pω2−(kr)2/τ, |ω|> kr.
We now seek a steady state solution (in a statistical sense)
for Eqs. (2). More precisely, for a given value of k, we
seek a time-independent probability distribution of frequen-
cies Gand a value of rthat make Eqs. (2) consistent. Such a
steady-state frequency distribution can be obtained by solving
the time-independent Fokker-Planck equation corresponding
to Eq. (8)
d
dω h−dV
dω G=Dd2G
dω2.(9)
The solution of Eq. (9) with no-flux boundary conditions
(dG/dω = 0 at either ω=±∞ or ω=±L) is G∝
exp[Rh(ω)dω/D−V(ω)/D],from which we obtain Eq. (10),
where σ2=Dτ :
G(ω, kr)∝
exp[−ω2/(2σ2)−V(ω)/σ2],|ω| ≤ kr,
h|ω|
kr 1−p1−(kr/ω)2i(kr)2/(2σ2)exp h−ω2
2σ21−p1−(kr/ω)2−V(ω)/σ2i,|ω|> kr. (10)
In all our numerical plots we use D= 0.01 and τ= 50,
which yields σ2= 1/2.
This distribution depends on the value of the order parame-
ter r. In order to make this solution self-consistent, the value
of rhas to be determined from the classical Kuramoto results
corresponding to an ensemble of oscillators with frequency
distribution G(ω, kr). That is, ris equal to the average of
exp(iθ)over all the oscillators. As shown, e.g., in Refs. [8],
the average over drifting oscillators (|ω|> kr) is zero, and ris
thus determined entirely by the locked oscillators (|ω|< kr)
whose phase angles are given by Eq. (5). Thus we obtain
r=Zkr
−kr
G(ω, kr)p1−(ω/kr)2dω.
Besides the solution r= 0, other possible values of rare
given by the solutions of the nonlinear equation [8]
1 = kZ1
−1
G(zkr, kr)p1−z2dz. (11)
We now study numerically the solutions of Eqs. (10) and
(11). As we will later argue, noise in the θevolution (either
extrinsic or due to finite N) causes the dynamics to be insensi-
tive to the choice of confining potential V(ω). Therefore, for
simplicity, we will choose an infinite potential well to simplify
the analysis. The potential is defined as V(ω) = 0 if |ω|< L,
and V(ω) = ∞otherwise. This corresponds to frequencies
that evolve freely in a box of size 2Lwith hard walls. We
will later show that Lcan be chosen large enough so that the
dynamics is insensitive to its value. Except for Fig. 1, all our
plots use L= 5.
Solving Eq. (11) numerically, a bifurcating pair of solutions
rs(k)and ru(k)is found to appear at a finite value of the cou-
pling strength k=k1as shown in Fig. 1. The upper branch
(solid black line) in these figures was numerically found to
be stable, while the lower branch (colored lines in Fig. 1 and
dashed line in Figs. 2aand 3) was numerically found to be
unstable. The trivial solution r= 0 was numerically found to
be stable until the lower branch crosses r= 0 at a value of
the coupling constant k=k2(see Fig. 1). For larger coupling
strength, k > k2, the nonzero unstable solution disappears
and the solution r= 0 becomes unstable.
Thus, three regimes are found with our model. For k < k1,
the oscillator ensemble is incoherent, as only the r= 0 stable
solution exists. For k > k2, only the synchronized state is sta-
ble. These correspond to the traditional regimes of the origi-
nal Kuramoto model without adaptation [8]. The third regime
corresponds to intermediate coupling strengths on the finite
interval k1< k < k2, where both the synchronized and inco-
herent states are locally stable, and whose basins of attraction
4
0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
k
r
k2
k1
L=20
ru for L=5,...,20
rs for all L
L=5
FIG. 1. Stable (upper black line, rs(k)) and unstable (lower colored
lines, ru(k)) branches are shown for D= 0.01,τ= 50, and L=
5,10,15,20 and the curve kr =√2σ(dotted line), above which the
frequency distribution is normalizable. The values of k1and k2for
L= 5 are indicated by vertical arrows at k1≈1.8and k2≈6.37.
are separated by the unstable solution of Eq. (11). A similar
regime was also found in Ref. [14] for an inertial version of
the Kuramoto model without noise [in which ˙
θnis replaced
by m¨
θn+˙
θnin (1)]. In Sec. III we address the important
issue of noise-induced spontaneous transitions between stable
solutions which was not addressed in Ref. [14].
To study the behavior of our model close to the incoherent
state r= 0, we expand Gin Eq. (10) for (kr/ω)2≪1, find-
ing that the ωdependence of Gfor large |ω|is G(ω, kr)∼
(kr/ω)(kr)2/(2σ)2. Thus the frequency distribution G(ω)can
be normalized in (−∞,∞)if kr > √2σ. We conclude that,
as long as kr > √2σ,Gshould be insensitive to Lif the bulk
of Gis contained within (−L, L). In contrast, for kr < √2σ,
the frequency distribution is not normalizable in (−∞,∞)
and thus we expect Gto be broadly distributed in (−L, L),
and the dynamics to depend on the value of L. In particular,
the distribution of frequencies for the incoherent state r= 0 is
uniform with G(ω) = 1/(2L). More generally, Gshould be
insensitive to the specific form of the confining potential Vas
long as Vis negligible in the synchronized state, V(σ)≪σ2.
This can be interpreted as requiring that Vdoes not itself pro-
mote synchronization.
Figure 1 shows the stable and unstable branches for var-
ious values of Land the curve kr =√2σ(dotted line),
above which the frequency distribution is normalizable. Note
that the stable solution rs(ω)to Eq. (11) is above the curve
kr =√2σin Fig. 1 and is, as expected, insensitive on the
value of L. One interesting result from Fig. 1 is that the upper
critical coupling strength k2depends on the frequency bound
L. Recalling that G(ω) = 1/(2L)for the incoherent state, one
can integrate Eq. (11) to find k2= 4L/π. In reality, physical
limitations typically bound the natural frequency distribution.
However, it is also interesting to consider the unbound case
(V≡0, or L=∞) which leads to the following scenario: the
oscillators’ natural frequencies wander in (−∞,∞);k2=∞
and the incoherent state r= 0 remains stable for all coupling
strengths. However, it is important to note that, even though
the incoherent state for L=∞remains stable for arbitrarily
a
0 0.5 1 1.5 2 2.5 3 3.5 4
0
0.2
0.4
0.6
0.8
1
k
r
c
b
rs
ru
ak1
b
−5 0 5
10−6
10−4
10−2
100
ω
G(ω)
Ga
Gb
Gc
Expa
Expb
Expc
kr
−kr
a
b
c
FIG. 2. (a)Order parameter robtained from numerical simulation
of Eqs. (2) for decreasing values of kfor N= 104(triangles) with
D= 0.01,τ= 50, and L= 5. The solid and dashed lines indicate
stable rs(k)and unstable ru(k)solutions to Eq. (11), respectively.
The letters a, b and c indicate values of kat which the frequency
distribution is sampled for Fig. 2b.(b)Frequency distribution ob-
tained directly from Eqs. (2) (symbols) and from Eq. (10) (black
lines). Curves labeled a, b, and c correspond to k= 1,2, and 4,
indicated by arrows in Fig. 2a. The dashed vertical lines indicate
±kr for k= 2.
large k, its basin of attraction shifts to zero as k→ ∞ (the
unstable equilibrium ruapproaches zero as k→ ∞). The
situation is analogous to that applying in the study of the tran-
sition of stable laminar pipe flow to turbulence (e.g., Ref. [17]
and references therein). As in pipe flow, this situation points
to the possibly crucial role of noise which we address subse-
quently.
B. Model Simulation
In order to test our theoretical results, we compared them
with direct simulation of Eqs. (2) using τ= 50 and D= 0.01.
Due to the stability characteristics of the solutions, hystere-
sis phenomena and dependence on the initial conditions are
expected. To probe these characteristics, we let L= 5 and
initiate a simulation with strong coupling (k > k2) and with
the phases and frequencies of the oscillators clustered around
θn≈0and ωn≈0. The oscillators remain synchronized,
and their natural frequencies adopt a distribution given by
Eq. (10). For a given value of k, we simulate Eqs. (2) for
5
1000 seconds and then decrease the value of kby 0.1, keep-
ing the values of the phases and frequencies (this corresponds
to a coarse grained rate dk/dt ≈10−4). As this process is
repeated and the value of kdecreases below k1, the synchro-
nized solution disappears and the oscillators desynchronize.
Figure 2ashows the value of robtained by this process (trian-
gles). The solid and dashed lines indicate the stable rs(k)and
unstable ru(k)solutions, respectively obtained from Eq. (11).
The numerically obtained values of rfollow the stable branch
found theoretically.
In Fig. 2bwe show the steady-state frequency distribution
observed at values of kcorresponding to the arrows labeled a,
b, and c in Fig. 2a. The black solid, dashed, and dotted lines
indicate the theoretical expression given by Eq. (10) normal-
ized on ω∈[−5,5] for cases a, b, and c. The cross, circle, and
square symbols show the corresponding observed frequency
distributions which are in good agreement with the theory.
To observe hysteresis phenomena similar to that noted
in [14], the system was was brought to steady state with
a dispersed frequency distribution described by Eq. (10)
for small coupling strength (k < k1). The coupling
strength kwas slowly increased until the system underwent
an incoherent→synchronized state transition at the transition
coupling strength k∗, which is found on the interval k1<
k∗< k2. The precise value of k∗fluctuates slightly from run
to run, but its mean is observed to depend on the ensemble
size N. This is shown in Fig. 3, where k∗approaches k2as
Nincreases, as previously noted in [14] for the inertial Ku-
ramoto model. This is due to fluctuations in the order param-
eter ∆r∼N−1/2resulting from the system’s finite size: it
is hypothesized that fluctuations cause the system to cross the
barrier imposed by the unstable solution to Eq. (11) (dashed
line in Fig. 3). When the size of these fluctuations becomes
large enough to place rabove the unstable solution, the oscil-
lators begin to synchronize and the value of the order parame-
ter increases to the value corresponding to the stable solution
(upper solid line).
It should also be noted that, for this simulation with tempo-
rally increasing coupling strength, the k∗approach k1as the
simulation duration for each kis increased. In other words,
the hysteretic nature of this system depends not only on the
size of the network (as noted in [14]), but also on the rate at
which the coupling strength kis varied. We hypothesize this
phenomenology to also describe other Kuramoto-type models
with hysteretic behavior (e.g., [14, 18]). The fluctuations of
the order parameter rare stochastic, and thus the time required
for the transition to occur is a random variable. The longer a
simulation is run at constant coupling strength k1< k < k2,
the more likely an incoherent →synchronized transition has
occurred. In fact, oscillations between states, as hypothesized
in [14], were observed for our model in this bistable regime
(see Fig. 7). Describing such spontaneous state transitions is
the focus of the next section of our paper.
0 1 2 3 4 5 6 7
0
0.2
0.4
0.6
0.8
1
k
r
k1k2
N=105
N=104
N=103
N=200
FIG. 3. For increasing coupling strength, synchronization occurs for
each network when the order parameter fluctuations ∆rallow rto
surmount the barrier of the unstable solution ru(k)(dashed line).
Simulation used D= 0.01,τ= 50, and L= 5. Note that the transi-
tion coupling strengths k∗approach k2as network size Nincreases.
III. SPONTANEOUS STATE TRANSITIONS
Given the observed phenomenology of fluctuations driving
the system from one stable solution to another across an un-
stable solution, it is natural to conjecture that, for a fixed value
of k, the average time τsync (N)for a transition from the inco-
herent state r≈0to the coherent state r∼1can be obtained
by treating the problem as an escape over a potential barrier
under the influence of random noise (see Fig. 4). Conceptu-
ally, it is helpful to relate such transitions to a Brownian par-
ticle moving from one equilibrium to a second by traversing
an energy barrier under the influence of random noise. For the
case of oscillator system state transitions, fluctuations of the
order parameter ∆roccur due to a network’s finite size Nand
are akin to random noise. In addition, in some applications,
Eq. (2) may be subject to extrinsic noise [8].
For the traditional Kuramoto model, understanding finite
size fluctuations ∆rhas been a major area of interest [8, 19].
In general, fluctuations are typically O(N−1/2), although it
has been shown that these fluctuations increase in amplitude
near the critical coupling kc[19] for a traditional Kuramoto
oscillator system. Similarly, for our model, fluctuations in
rwere observed to be larger in the bistable regime than in
the traditional Kuramoto regimes. However, as with the tradi-
tional Kuramoto model, further study of these fluctuations for
our model remains open to future research.
A. State Transition Analysis
In order to study the statistics of spontaneous synchroniza-
tion transitions, we will assume that finite-size fluctuations
can be described approximately as produced by uncorrelated
Gaussian noise acting on the 1-dimensional dynamics of the
order parameter. Treating finite-size fluctuations as an un-
correlated Gaussian noise term has already proven sucessful
in studying synchronization of Kuramoto oscillators in net-
works [20]. Consequently, let us assume that the macroscopic
dynamics of the order parameter rcan be described by a
6
r=0
rs
h
U(r,k)
ru
FIG. 4. State transitions parameterized by rfor k1< k < k2are
schematically shown as Brownian motion in a 1-dimensional energy
landscape with two stable equilibria.
Langevin equation of the form
˙r=−U′(r, k) + L(t),(12)
where U(r, k)is an unknown pseudo-potential, U′(r, k) =
∂U/∂r, and L(t)is an uncorrelated Gaussian noise term such
that hL(t)i= 0 and hL(t)L(t′)i= 2Γδ(t−t′). Since the
noise represents finite-size fluctuations, the diffusion coeffi-
cient Γwill be assumed to be inversely proportional to N,
or Γ∝1/N. Note that this is consistent with ∆rbeing
O(N−1/2)for the dynamics of rmodeled as a linear Ornstein-
Uhlenbeck process for the incoherent state with k < k1.
In the bistable regime, k1< k < k2, we assume U(r, k)to
be of the form shown in Fig. 4. Potentials of this type have re-
ceived much attention in the literature for studying Brownian
motion in bistable potentials and for describing chemical reac-
tions. We will draw on this research and use Kramer’s escape
time equation [21], which describes the mean first-passage
time τesc for a particle subject to random noise with diffu-
sion coefficient Γto escape over a potential barrier of height
h, and is given by log(τesc )∝h/Γ. Recalling that Γ∝1/N,
we conclude that the mean first-passage time (i. e., wait time
before synchronization) for our bistable Kuramoto system de-
pends exponentially on N, yielding τsync ∝eKN for some
constant K.
A similar analysis can also be done on the regime where
the incoherent state is unstable, where we are interested in
the average time required for an incoherent system (r∼0)
to synchronize. To first order, the dynamics for small ris
described by ˙r=αr +L(t), with αbeing a positive constant.
Taking r(0) = 0 and setting hr(t∗)2i ≡ r∗2, we can estimate
for large Nthe time t∗it takes for the order parameter to reach
a given threshold r=r∗≫pΓ/α as t∗∼log Γ−1∼log N.
Thus, the waiting time τsync grows logarithmically with Nin
the strong coupling regime (k > kcin the Kuramoto model or
k > k2in our model).
Although this paper focuses on the model described by
Eqs. (2), the above estimates may apply to other Kuramoto-
type models [14, 15, 18].
B. State Transition Simulation
To test the previous findings, statistics were compiled for
our adaptive Kuramoto system by simulating 100 realizations
0.5 1 1.5 2 2.5
x 104
102
103
N
τsync [s]
FIG. 5. Synchronization time τsync averaged over 100 realizations
as a function of the number of oscillators Nfor k= 6, which is
within the bistable regime. (D= 0.01, τ = 50,and L= 5)
104
103105
1.5
2
2.5
3
3.5
4
N
τsync [s]
FIG. 6. Synchronization time τsync averaged over 100 realizations
as a function of the number of oscillators Nfor k= 7 > k2. (D=
0.01, τ = 50,and L= 5). Note that the scale is different than that
of Fig. 5.
of synchronization for an initially incoherent system. For each
realization, at a constant coupling strength kthe initial natural
frequencies and phases were chosen randomly (θnuniform in
[0,2π), and ωnuniform in [−5,5]). Once the order parameter
exceeded a given threshold r∗ensuring synchronization had
occurred, the time before synchronization was recorded and
simulation stopped.
Statistics of incoherence→synchronization transitions for
the bistable regime are shown in Fig. 5, where log(τsync (N))
vs. Nis plotted for k= 6.τsync is defined as the average
time required for the order parameter to first reach r∗= 0.7.
In principle any coupling strength kwithin the bistable regime
could be used; however, to decrease simulation time kwas
chosen to be close to k2≈6.37. Error bars are included to
show statistical uncertainty. As the plot shows, log(τsync (N))
is well described by a straight line, which is consistent with
the supposition that the transition times can be described by
Kramer’s escape time formula.
For comparison, τsync is shown in Fig. 6 for synchroniza-
tion with the incoherent state being unstable (k > k2). From
this figure we confirm that τsync ∝log N, which is consis-
tent with unstable exponential growth of perturbations from
the r= 0 incoherent state.
Figure 7 shows fluctuations between the synchronized and
incoherent states for a case where the coupling strength is
within the bistable range. Note that since transitions between
7
0 0.5 1 1.5 2 2.5 3
x 107
0
0.2
0.4
0.6
0.8
1
[s]
r
FIG. 7. Spontaneous bidirectional transitions between the synchro-
nized (dashed) and incoherent (dotted) states are observed for N=
10,k= 1.9,D= 0.01, τ = 50,and L= 5. Note that because
of the small system size, the incoherent state has an average order
parameter of hri ∼ 0.4.
states are related to the height hof the pseudo-potential bar-
rier relative to each respective equilibrium (see Fig. 4), fluc-
tuations between states can only be observed when the barrier
heights are roughly equal and when the system is observed
for a duration in which transition-events should occur. For
example, if the barrier height is large and the finite system
is large (large N), the order parameter rwill undergo small
fluctuations and state transitions would be rare. At the same
time, if the barrier height is much larger for a particular state,
then the system will remain in that state for the majority of
time and transitioning out of that state would also be rare. For
the model parameters chosen in our simulation, we found that
spontaneous bidirectional transitions could only be observed
for small numbers of oscillators (N= 10 in Fig. 7) and for
coupling strengths in the bistable regime just above k1(be-
low which the coherent solution disappears). In general, for
k1< k < k2, we find that synchronized→incoherent tran-
sitions are very rare, implying that the barrier height for the
synchronized state is generally larger than the barrier height
for the coherent state (as shown schematically in Fig 4).
IV. DISCUSSION
Our results discussed above are in striking agreement with
observations of rhythmically clapping audiences [11, 12]. In
particular, as opposed to the behavior of the classical Ku-
ramoto model without adaption, the transition to synchronized
clapping occurs after a relatively long waiting time, and once
it starts the order parameter quickly achieves its steady state.
Previous models of this phenomenon have artificially altered
the frequency distribution [11] or introduced additional dy-
namics such as a time-dependent tendency of the oscillators
to synchronize [12]. In contrast, the long waiting times arise
in our model as a natural consequence of the dynamics. Al-
though we have found that all-to-all coupling leads to waiting
times that depend exponentially on the number of oscillators,
shorter waiting times are expected for local coupling such as
that describing clapping synchronization in a large venue.
Another possible application of our model is circadian
rhythms [13], which have been modeled by ensembles of Ku-
ramoto oscillators with drifting, nonadaptive frequencies [16].
Because of the importance of synchronization in this system,
evolutionary pressures might have led to frequency adapta-
tion. Our model generalizes previous models [16] by allowing
for frequency adaptation. By removing frequency coupling
(i.e., τ→ ∞) and assuming a quadratic form for the poten-
tial V(ω), our model [Eqs. (2)] recovers the model of coupled
circadian oscillators presented in [16].
Our results are somewhat related to the Kuramoto model
with inertia (Eq. (1) with ˙
θnreplaced by m¨
θn+˙
θn[6, 14, 15]),
which is equivalent to
˙
θn=ωn,(13)
˙ωn=τ−1"−dVn(ωn)/dωn+k
N
N
X
m=1
sin(θm−θn) + ηn#,
where Vn(ωn) = 1
2(ωn−Ωn)2, with Ωnconstant for each os-
cillator. However, the differences between this model and our
model are significant. First, in contrast to (13), our model cou-
ples both phases and frequencies. Second, as a consequence of
our two types of coupling, we are able to introduce two time
scales, with the frequency adaptive time scale being slower
than that of the phase dynamics. We believe that this two time
scale dynamics will be crucial to the modeling of the various
potential applications mentioned in Sec. I (e.g., clapping audi-
ences). [Note that simulations were conducted to investigate
the effect of closing the timescale gap. By keeping σconstant
and reducing τ(i.e., by also increasing D), it was found that
no qualitative differences were observed as long as τ > 5.]
The analysis presented here to describe fluctuation-induced
spontaneous transitions from incoherence to synchronization
for our adaptive model could also be applicable to other
Kuramoto-type systems with hysteretic behavior. Such sys-
tems include Kuramoto models with an added inertial term
[14, 15] and situations where there is a heterogeneous dis-
tribution of interaction time delays [18]. Various questions
remain to fully understand the dynamics of the observed
transitions. While order parameter fluctuations are typically
O(N−1/2), this is not always the case and a better understand-
ing of these fluctuations is needed. Similarly, the existence of
a pseudo-potential U(r, k)was assumed [Eq. 12], but its shape
and dependence on kremain to be investigated.
V. CONCLUSIONS
We have presented a new model to study the synchroniza-
tion of Kuramoto oscillators that are able to slowly adapt their
natural frequencies to promote synchronization, but are in-
hibited from doing so completely by the influence of noise.
We found that the interplay of noise and adaptation results in
bistability and hysteresis. In the bistable regime, finite size
effects induce incoherent→synchronized state transitions (or,
when Nis small, vice versa), which are well described as a
1-dimensional Kramer escape process on the order parameter
r. For an oscillator ensemble governed by our adaptive model
8
with all-to-all coupling, it was shown that the time τsy nc re-
quired for the system’s state to transition from incoherent to
synchronized depended exponentially on Nin the bistable
regime (k1< k < k2) and logarithmically for strong cou-
pling (k > k2).
To our knowledge, this work is the first to analyze spon-
taneous synchronization at constant coupling strength as a
1-dimensional stochastic escape process. It is expected that
the analysis presented in this paper is also valid for other
Kuramoto-type models with hysteretic behavior [14, 15, 18].
The work of D. Taylor and J. G. Restrepo was supported
by NSF (Applied Mathematics) and the work of E. Ott was
supported by the NSF (Physics) and by the ONR (N00014-
07-0734).
[1] S. Dano, M. F. Madsen, and P. G. Sorensen, Proc. Natl. Acad.
Sci. USA 104, 12732 (2007).
[2] S. H. Strogatz, D. M. Abrams, A. McRobie, B. Eckhardt, and
E. Ott, Nature 438, 43 (2005).
[3] K. Wiesenfeld, P. Colet, and S. H. Strogatz, Phys. Rev. Lett. 76,
404 (1996).
[4] G. Fillatrella, A. H. Nielsen, and N. F. Pedersen, Euro. Phys. J.
B61, 485-491 (2008).
[5] M. C. Cross, A. Zumdieck, R. Lifshitz, and J. L. Rogers, Phys.
Rev. Lett. 93, 224101 (2004).
[6] B. Ermentrout, J. Math. Bio. 29, 571 (1991).
[7] Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence,
Springer (1984).
[8] S. H. Strogatz, Physica D 143, 1-20 (2000); E. Ott, Chaos
in Dynamical Systems, Cambridge University Press (2002),
Sec. 6.5; J. A. Acebr´on, L. L. Bonilla, C. J. P. Vicente, F. Ritort,
and R. Spigler, Reviews of Modern Physics, 77, 137 (2005).
[9] E. Ott and T. M. Antonsen, Chaos 18, 037113 (2008); 19,
023117 (2009).
[10] P. Seliger, S. C. Young, and L. S. Tsimring, Phys. Rev. E, 65,
041906 (2002); P. M. Gleiser and D. H. Zanette, Euro. Phys. J.
53 2233-238 (2006); C. Zhou and J. Kurths, Phys. Rev. Lett.
96, 164102 (2006); Y. L. Maistrenko, B. Lysyansky, C. Haupt-
mann, O. Burylko, and P. A. Tass, Phys. Rev. E, 75, 066207
(2007); Q. S. Ren and J. Y. Zhao, Phys. Rev. E 76, 016207
(2007). G. He and J. Yang, Chaos 35 5, 1254-1259 (2007); T.
Aoki and T. Aoyagi, Phys. Rev. Lett. 102, 034101 (2009).
[11] Z. N´eda, E. Ravasz, Y. Brechet, T. Vicsek, and A. L. Barab´asi,
Nature 403, 849 (2000); Z. N´eda, E. Ravasz, T. Vicsek, Y.
Brechet, and A. L. Barab´asi, Phys. Rev. E 61, 6987 (2000).
[12] D. Xenides, D. S. Vlachos, and T. E. Simos, J. Stat. Mech.
P07017 (2008).
[13] E. Nagoshi, C. Saini, C. Bauer, F. Naef, and U. Schibler, Cell
119, 693 (2004); D. K. Welsh, S. H. Yoo, A. C. Liu, J. S. Taka-
hashi, and S.A. Kay, Curr. Biol. 14, 2289 (2004); A. J. Carr and
D. Whitmore, Nat. Cell Biol. 7, 319 (2005).
[14] H. A. Tanaka, A. J. Lichtenberg, and S. Oishi, Physica D 100,
279 (1997).
[15] J. A. Acebr´on and R. Spigler, Phys. Rev. Lett. 81, 2229 (1998);
J. A. Acebr´on, L. L. Bonilla, and R. Spigler, Phys. Rev. E 62,
3437 (2000).
[16] J. Rougemont, Phys. Rev. E 73, 011104 (2006); J. Rougemont
and F. Naef, Mol. Sys. Biol. 3, 93 (2007).
[17] B. Eckhardt, Phil. Trans. Roy. Soc. A 367, 449 (2009).
[18] W. S. Lee, E. Ott, and T. M. Antonsen, Phys. Rev. Lett. 103,
044101 (2009).
[19] H. Daido, J. Phys. A: Math. Gen. 20, L629-L636 (1987), Prog.
Theor. Phys. 81, 727-31 (1989), Prog. Theor. Phys. Suppl. 99,
288-294 (1989), J. Stat. Phys. 60, 753 (1989); M. A. Buice and
C. C. Chow, Phys. Rev. E 76, 031118 (2007).
[20] J. G. Restrepo, B. R. Hunt, and E. Ott, Phys. Rev. E 71, 036151
(2005).
[21] N. G. Van Kampen, Stochastic Processes in Physics and Chem-
istry, Elsevier (2007).
