Content uploaded by Renato Spigler
Author content
All content in this area was uploaded by Renato Spigler on Jul 02, 2014
Content may be subject to copyright.
VOLUME
81, NUMBER 11 PHYSICAL REVIEW LETTERS 14S
EPTEMBER
1998
Adaptive Frequency Model for Phase-Frequency Synchronization in Large Populations
of Globally Coupled Nonlinear Oscillators
J.A. Acebrón
1,2
and R. Spigler
1,
*
1
Dipartimento di Matematica, Università di “Roma Tre,” Largo S. Leonardo Murialdo, 1, 00146 Roma, Italy
2
Escuela Politécnica Superior, Universidad Carlos III de Madrid, Butarque 15, 28911 Leganés, Spain
(
Received 17 November 1997; revised manuscript received 27 May 1998
)
Phase models describing self-synchronization phenomena in populations of globally coupled oscilla-
tors are generalized including “inertial” effects. This entails that the oscillator frequencies also vary
in time along with their phases. The model can be described by a large set of Langevin equations
when noise effects are also included. Also, a description of such systems can be given in the ther-
modynamic limit of infinitely many oscillators via a suitable Fokker-Planck-type equation. Numerical
simulations confirm that simultaneous synchronization of phases and frequencies is possible when the
coupling strength goes to infinity. [S0031-9007(98)07062-8]
PACS numbers: 05.45.+b, 05.20.–y, 05.40.+j, 64.60.Ht
A number of phase models have been proposed over
the recent years to describe the dynamic behavior of large
populations of nonlinear oscillators subject to a variety of
coupling mechanisms. A major phenomenon that can be
observed is the possibility of self-synchronization among
the members of the population. These can represent fire-
flies, pancreatic beta cells, heart pacemaker cells, and neu-
rons [1,2], as well as circuit arrays and other things (see
[3,4] for further references). Such models concern popu-
lations of N ¿ 1 as well as of infinitely many members,
and noise terms accounting for random imperfections
may also be included. However, doubt in the possi-
bility of effectively synchronizing an entire population
of oscillators in practice, both in phase and frequency,
has been cast by a recently found “uncertainty prin-
ciple,” in the mean-field coupling model (the Kuramoto-
Sakaguchi model [5,6]). Indeed, it was shown in [7] that
the Kuramoto-Sakaguchi model with noise terms does not
allow for simultaneous synchronization in both phase and
frequency.
In a more recent paper, Ermentrout [1] revisited the
special problem of self-synchronization in populations of
fireflies of a certain kind (the Pteroptyx malaccae). The
Kuramoto-Sakaguchi model yields a too fast approach to
the synchronized state (compared to the observed behav-
ior), and also requires an infinite value of the coupling
parameter to achieve full phase synchronization. There-
fore, Ermentrout proposed, rather, an adaptive frequency
model in terms of N ¿ 1 nonlinearly coupled second-
order differential equations for the phases, which can
handle both problems. Such a model differs from the
Kuramoto-Sakaguchi formulation in that the natural fre-
quency of each oscillator is allowed to vary in time, thus
leading to a new set of model equations.
It may be of some interest to stress that also certain
aftereffects in alterations of circadian cycles in mam-
malians may be explained by Ermentrout-type models;
cf. [1]. Other applications have also been pointed out,
for instance, to power systems described by the swing
equations [8], and also to extend the analysis of certain
Hamiltonian systems [9]. Moreover, several instances of
Josephson junctions arrays have been described in sim-
plified versions [10,11], where nonlinear first-order phase
equations govern the dynamics of zero temperature cir-
cuits. Nonzero temperature effects could be included,
however, adding suitable noise terms, and second-order
time derivatives might yield a physically more satisfac-
tory picture.
Tanaka et al. [12], on the other hand, considered about
the same problem described by Ermentrout, but within the
mean-field coupling framework and with sinusoidal non-
linearities. In the light of a kind of uncertainty principle
[7], which governs phase-frequency synchronization pro-
cesses in the Kuramoto-Sakaguchi models, here we extend
the Ermentrout-Tanaka et al. analysis proposing a new
model. This consists of a system of N ¿ 1 (but N
,
`) second-order Langevin equations subject to a mean-
field interaction with a sinusoidal nonlinearity. Also, in
the thermodynamic limit N ! `, we propose a certain
nonlinear partial integro-differential (Fokker-Planck-type)
equation. The latter yields the time evolution of the one-
oscillator probability density of the system. As a justi-
fication of our assumptions, we stress that the sinusoidal
nonlinearity can indeed be representative of more general
types of nonlinearities as long as the natural frequencies
fall within the range of the adaptive frequency [1,12]. On
the other hand, a mean-field model can be adopted as a
reasonable one, as pointed out by Ermentrout [1], as long
as we are concerned with rather compact populations of
fireflies lying on nearby trees. Also, in case of Josephson
junctions arrays [11], the all-to-all coupling (correspond-
ing to the mean-field model) can indeed be justified by cir-
cuit analysis, rather than because of a merely simplifying
approximation. In addition, however, here we introduce
some noise terms, so as to account for unavoidable im-
perfections of various natures. Therefore, the model we
0031-9007y98y81(11)y2229(4)$15.00 © 1998 The American Physical Society 2229
VOLUME
81, NUMBER 11 PHYSICAL REVIEW LETTERS 14S
EPTEMBER
1998
propose is given by
m
¨
u
j
1
Ù
u
j
V
j
1 Kr
N
sinsc
N
2u
j
d1j
j
std,
j1,...,N, (1)
or by the system
Ù
u
j
v
j
,
Ù
v
j
1
m
f2v
j
1V
j
1Kr
N
sinsc
N
2u
j
dg
1
1
m
j
j
std, j 1,...,N, (2)
where u
j
, v
j
, V
j
denote phases, frequencies, and natural
frequencies, m . 0 is an “inertial term,” and K sizes the
nonlinearity. The complex order-parameter, defined by
r
N
e
i c
N
1
N
N
X
j1
e
i u
j
, (3)
measures the phase synchronization, and the j
j
’s are
Gaussian white noises, with kj
j
l 0, kj
i
stdj
j
ssdl
2Dd
ij
dst 2 sd.
Typically, N must be large, but we are also interested
in the limit of infinitely many oscillators. In this case
we obtain, for the one-oscillator probability density,
rsu, v, V, td, the evolution equation
≠r
≠t
D
m
2
≠
2
r
≠v
2
2
1
m
≠
≠v
fsss2v 1 V 1 Kr sinsc2uddddrg
2v
≠r
≠u
, (4)
which should be accompanied by initial and bound-
ary data (2p periodicity in u, and decay to zero as
v ! 6`, with sufficiently high rate), and normalization,
R
1`
2`
R
2p
0
rsu, v, V,0ddvdu 1.
In Eq. (4), r and c are given by
re
ic
Z
1`
2`
dv
Z
2p
0
du
3
Z
1`
2`
dV gsVde
iu
rsu, v, V, td , (5)
that is, the complex phase order-parameter, whose ampli-
tude measures the degree of the phase synchronization.
In (5), gsVd represents a given natural frequency distribu-
tion. In order to study, in cases of both finitely and infin-
itely many oscillators, simultaneous self-synchronization
in phase and frequency, it is convenient to introduce in ad-
dition, as in [7], the complex frequency order-parameter,
s
N
e
i f
N
1
N
N
X
j1
e
i v
j
sN ,`d, (6)
se
if
Z
1`
2`
dv
Z
2p
0
du
3
Z
1`
2`
dV gsVde
iv
rsu, v, V, td
sN `d . (7)
In the following, we take for simplicity identical oscil-
lators, gsVd dsVd. In order to analyze the spread in
phase and frequency, we solve the stationary equation as-
sociated with (4). To this purpose, we look for solutions
of the form rsu, vd xsudhsvd. Thus,
√
D
m
2
d
2
h
dv
2
1
1
m
v
dh
dv
1
1
m
h
!
x2
1
m
Kr sinsc2udx
dh
dv
2vh
dx
du
0. (8)
Numerical simulations show that the frequency distribu-
tion hsv, td
R
2p
0
rsu, v, tddu does not seem to depend
on the coupling strength K; cf. the time evolution of the
frequency order-parameter, jsstdj, Fig. 1.
Therefore, looking for solutions hsvd independent of
K, we obtain from Eq. (8)
D
m
2
d
2
h
dv
2
1
1
m
v
dh
dv
1
1
m
h 0,
2
1
m
Kr sinsc2udx
dh
dv
2vh
dx
du
0. (9)
0.0 4.0 8.0 12.0 16.0
t
0.0
0.2
0.4
0.6
0.8
1.0
|r(t)|
K=6
K=10
K=20
(a)
0.0 4.0 8.0 12.0 16.0
t
0.0
0.2
0.4
0.6
0.8
1.0
|s(t)|
K=6
K=10
K=20
(b)
FIG. 1. Time evolution of the order-parameter amplitudes,
jrstdj (a) and jsstdj (b). The parameter m is kept fixed to 1, the
coupling strength is K 6 (solid line), K 10 (dotted line),
and K 20 (dashed line), and D 1.
2230
VOLUME
81, NUMBER 11 PHYSICAL REVIEW LETTERS 14S
EPTEMBER
1998
Using the normalization conditions and the boundary
condition, hsvd ! 0 as v ! 6`, the solution to Eq. (9)
is promptly obtained:
hsvd
r
m
2pD
e
2smy2Ddv
2
,
xsud
e
sKyDd r cossc2ud
R
2p
0
e
sKyDdr cossc2ud
du
. (10)
Define the spread in phase and frequency as
sDud
2
ksu2cd
2
l2sku2cld
2
,
sDvd
2
kv
2
l 2 skvld
2
, (11)
brackets denoting average with respect to the density
distribution r. The symmetry properties of the stationary
solution to Eq. (4) can be exploited along with the
Laplace method to obtain, in the limit of large coupling
K ! `,
sDud
2
p
2 D
K
, sDvd
2
D
m
; (12)
0.0 4.0 8.0 12.0 16.0
t
0.0
0.2
0.4
0.6
0.8
1.0
|r(t)|
m=1
m=0.5
m=0.25
(a)
0.0 4.0 8.0 12.0 16.0
t
0.0
0.2
0.4
0.6
0.8
1.0
|s(t)|
m=1
m=0.5
m=0.25
(b)
FIG. 2. Time evolution of the order-parameter amplitudes,
jrstdj (a) and jsstdj (b), for three different values of m. The
coupling strength is kept fixed to K 6, and D 1.
cf. [7], and, from these, the “uncertainty relation”
DuDv
2
1y4
m
21y2
D
K
(13)
is obtained immediately.
Numerical simulations of the Monte Carlo type for a
large number of oscillators (N 30 000) were carried out
in the system of Langevin equations (2). The distribution
function and the amplitudes of both the order-parameters,
rstd and sstd, have been computed for different values of
the parameters m, D, K, when the natural frequency dis-
tribution is gsVd dsVd. Note, in particular, in Fig. 1
that (partial) synchronization in phase is achieved faster
and better for larger values of K, while synchronization
in frequency remains always constant, as we expected.
In Fig. 2, however, (partial) phase synchronization is ob-
served (for a fixed value of K), to be independent of
m, while the frequency synchronization decreases as m
gets smaller. This fact can also be observed in Fig. 3,
where the phase and frequency distributions are shown.
-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0
ω
0.00
0.10
0.20
0.30
0.40
0.50
η(ω)
m=1
m=0.5
(a)
0.0 1.0 2.0 3.0 4.0 5.0 6.0
θ
0.0
0.2
0.4
0.6
0.8
1.0
χ(θ)
3.0 3.5 4.0 4.5 5.0
θ
0.50
0.60
0.70
0.80
0.90
1.00
χ(θ)
Analytical
m=1
m=0.5
(b)
FIG. 3. Frequency (a) and phase (b) distributions for two
different values of m. Comparison between the analytical and
the numerical solutions is shown. The coupling strength is kept
fixed to K 6, and D 1. Details are shown in the inset.
2231
VOLUME
81, NUMBER 11 PHYSICAL REVIEW LETTERS 14S
EPTEMBER
1998
0.0 5.0 10.0 15.0 20.0
t
0.0
0.2
0.4
0.6
0.8
1.0
|r(t)|
D=1
D=0
(a)
0.0 5.0 10.0 15.0 20.0
t
0.2
0.4
0.6
0.8
1.0
|s(t)|
D=1
D=0
(b)
FIG. 4. Time evolution of the order-parameter amplitudes,
jrstdj (a) and jsstdj (b) for two different values of D. The
coupling strength is kept fixed to K 6, and m 1.
In Fig. 4, it is shown that the noise reduces the synchro-
nization in both phase and frequency, as we expected
from analytical considerations. Setting (formally) m 0
exactly, in Eq. (1), we recover the Kuramoto-Sakaguchi
model with noise, described in the limit of infinitely many
oscillators by a Fokker-Planck-type equation for the dis-
tribution rsu, V, td. In this case, the frequency v, called
“drift velocity,” arises naturally in the problem as a de-
pendent variable, and is given by
v V1Kr sinsc2ud. (14)
The frequency distribution can be obtained from the
phase distribution [7], and the spread of both, phase and
frequency distributions, becomes
sDud
2
p
2 D
K
, sDvd
2
p
2 DK , (15)
and, consequently, the uncertainty relation is
DuDv
p
2 D . (16)
It seemed natural to add a noise term in Eq. (1), which
corresponds to add a noise independent of the inertial
parameter m. One may consider, however, the possibility
to introduce such a noise term directly in Eq. (2), thus
scaling its effects in a rather different way. It may be
interesting to see that the ensuing results are as follows:
Eqs. (12) and (13) become now sDud
2
p
2 m
2
DyK,
sDvd
2
mD, and DuDv 2
1y4
m
3y2
DyK, since D has
to be replaced by m
2
D. The main difference is that
Du ! 0, Dv ! 0,asm!0, and hence the spread of
both phase and frequency, and thus the uncertainty, vanish
for vanishing m’s. All of this is in (qualitative) agreement
with what happens in the Kuramoto-Sakaguchi model for
vanishing noise [cf. Eq. (15), and Eq. (1) with mj
j
std
replacing j
j
std].
In summary, we stress that a new model to better
explain synchronization phenomena in populations of
fireflies has been formulated. It has been emphasized
in [1] that the same type of models should also yield
an improved picture for the interaction among neurons,
merely changing the time and space scales with respect
to the fireflies problem. The main feature of the present
model seems to be, however, that no uncertainty occurs in
synchronizing both phases and frequencies in the limit of
infinite coupling strength; cf. Eq. (13) with Eq. (16).
This work was supported, in part, by the GNFM of
the Italian CNR and by Italian MURST funds. J.A.A. is
grateful to the University of “Roma Tre,” Rome, Italy, for
hospitality while he was a visiting researcher, supported
by the GNFM-CNR.
*Author to whom all correspondence should be
addressed.
Email address: spigler@dmsa.unipd.it
[1] B. Ermentrout, J. Math Biol. 29, 571 (1991).
[2] C.J. Pérez Vicente, A. Arenas, and L. L. Bonilla, J. Phys.
A 29, L9 (1996).
[3] A.T. Winfree, The Geometry of Biological Time
(Springer-Verlag, New York, 1980).
[4] S.H. Strogatz, in Frontiers in Mathematical Biology,
Lecture Notes in Biomathematics Vol. 100 (Springer-
Verlag, Berlin, 1994).
[5] Y. Kuramoto, in International Symposium on Mathemati-
cal Problems in Theoretical Physics, edited by H. Araki,
Lecture Notes in Physics Vol. 39 (Springer-Verlag, Berlin,
1975); Chemical Oscillations, Waves, and Turbulence
(Springer-Verlag, Berlin, 1984).
[6] H. Sakaguchi, Prog. Theor. Phys. 79, 39 (1988).
[7] J.A. Acebrón and R. Spigler (to be published).
[8] F.M.A. Salam, J.E. Marsden, and P. P. Varaiya, IEEE
Trans. CAS 32, 784 (1983).
[9] T. Konishi and K. Kaneko, J. Phys. A 25, 6283 (1992).
[10] K.Y. Tsang, R.E. Mirollo, S. H. Strogatz, and K. Wiesen-
feld, Physica (Amsterdam) 48D, 102 (1991).
[11] K. Wiesenfeld, P. Colet, and S.H. Strogatz, Phys. Rev.
Lett. 76, 404 (1996).
[12] H. Tanaka, A.J. Lichtenberg, and S. Oishi, Physica
(Amsterdam) 100D, 279 (1997).
2232
