PHYSICAL REVIEW E 77, 036107 共2008兲
Synchronization in networks of networks: The onset of coherent collective behavior in systems
of interacting populations of heterogeneous oscillators
Ernest Barreto,1,* Brian Hunt,2 Edward Ott,3 and Paul So1
1
Department of Physics & Astronomy, The Center for Neural Dynamics and The Krasnow Institute for Advanced Study,
George Mason University, Fairfax, Virginia 22030, USA
2
Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park,
Maryland 20742, USA
3
Institute for Research in Electronics and Applied Physics, Department of Physics, and Department of Electrical and Computer
Engineering, University of Maryland, College Park, Maryland 20742, USA
共Received 7 August 2007; revised manuscript received 10 December 2007; published 6 March 2008兲
The onset of synchronization in networks of networks is investigated. Specifically, we consider networks of
interacting phase oscillators in which the set of oscillators is composed of several distinct populations. The
oscillators in a given population are heterogeneous in that their natural frequencies are drawn from a given
distribution, and each population has its own such distribution. The coupling among the oscillators is global,
however, we permit the coupling strengths between the members of different populations to be separately
specified. We determine the critical condition for the onset of coherent collective behavior, and develop the
illustrative case in which the oscillator frequencies are drawn from a set of 共possibly different兲 Cauchy-Lorentz
distributions. One motivation is drawn from neurobiology, in which the collective dynamics of several interacting populations of oscillators 共such as excitatory and inhibitory neurons and glia兲 are of interest.
DOI: 10.1103/PhysRevE.77.036107
PACS number共s兲: 89.75.Fb, 05.45.Xt, 05.45.⫺a, 87.19.L⫺
In recent years, there has been considerable interest in
networks of interacting systems. Researchers have found that
an appropriate description of such systems involves an understanding of both the dynamics of the individual oscillators
and the connection topology of the network. Investigators
studying the latter have found that many complex networks
have a modular structure involving motifs 关1兴, communities
关2,3兴, layers 关4兴, or clusters 关5兴. For example, recent work
has shown that as many kinds of networks 共including isotropic homogeneous networks and a class of scale-free networks兲 transition to full synchronization, they pass through
epochs in which well-defined synchronized communities appear and interact with one another 关3兴. Knowledge of this
structure, and the dynamical behavior it supports, informs
our understanding of biological 关6兴, social 关7兴, and technological networks 关8兴.
Here we consider the onset of coherent collective behavior in similarly structured systems for which the dynamics of
the individual oscillators is simple. In seminal work, Kuramoto analyzed a mathematical model that illuminates the
mechanisms by which synchronization arises in a large set of
globally coupled phase oscillators 关9兴. An important feature
of Kuramoto’s model is that the oscillators are heterogeneous
in their frequencies. And, although these mathematical results assume global coupling, they have been fruitfully applied to further our understanding of systems of fireflies, arrays of Josephson junctions, electrochemical oscillators, and
many other cases 关10兴.
In this work, we study systems of several interacting
Kuramoto systems, i.e., networks of interacting populations
of phase oscillators. Our motivation is drawn not only from
the applications listed above 共e.g., an amusing application
*[email protected]
1539-3755/2008/77共3兲/036107共7兲
might be interacting populations of fireflies, where each
population inhabits a separate tree兲, but also from other biological systems. Rhythms arising from coupled cell populations are seen in many of the body’s organs 共including the
heart, the pancreas, and the kidneys, to name but a few兲, all
of which interact. For example, the circadian rhythm influences many of these systems. We draw additional motivation
from consideration of how neuronal tissue is organized. Although we do not consider neuronal systems specifically in
this paper, we note that heterogeneous ensembles of neurons
often exhibit a “network-of-networks” topology. At the cellular level, populations of particular kinds of neurons 共e.g.,
excitatory neurons兲 interact not only among themselves, but
also with populations of other distinct neuronal types 共e.g.,
inhibitory neurons兲. At a higher level of organization, various
anatomical regions of the brain interact with one another as
well 关6兴.
Although our network is simple, it is novel in that we
include heterogeneity at several levels. Each population consists of a collection of phase oscillators whose natural frequencies are drawn at random from a given distribution. To
allow for heterogeneity at the population level, we let each
population have its own such frequency distribution. In addition, our system is heterogeneous at the coupling level: we
consider global coupling such that the coupling strengths between the members of different populations can be separately
specified.
The assumption of global 共but population-weighted兲 coupling permits an analytic determination of the critical condition for the onset of coherent collective behavior, as we will
show. While this assumption may not strictly apply in some
applications, our results provide insight into the behavior of
networks of similar topology even if the connectivity is less
than global.
We begin by specifying our model and deriving our main
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©2008 The American Physical Society
PHYSICAL REVIEW E 77, 036107 共2008兲
BARRETO et al.
results. We then discuss several illustrative examples. Consider first a two-species Kuramoto model. We label the separate species ␪ and ␾ and assume that there are N␪ and N␾
such oscillators in each population, respectively. Thus the
system equations are
G ␴共 ␻ ␴兲 =
+␩
and the order parameters are given by
r ␴e i␺␴ =
k␪␾
兺 sin共␾ j − ␪i − ␣␪␾兲,
N␾ j=1
冉
N
⬘
共3兲
Here, ␩ is an overall coupling strength, the ␣’s provide additional heterogeneity in the coupling functions, and the matrix
冊
共1兲
defines the connectivity among the populations 关11兴.
For arbitrarily many different species, let ␴ range over the
various population symbols ␪ , ␾ , . . . with the understanding
that depending on the context, ␴ may represent either a label
共when subscripted兲 or a variable. Thus we have
冋
册
k␴␴⬘ ␴⬘
d␴i
= ␻ ␴i + 兺 ␩
兺 sin共␴⬘j − ␴i − ␣␴␴⬘兲 .
N␴⬘ j=1
dt
␴⬘
The incoherent state has ␴ distributed uniformly over
关0 , 2␲兴, so that f ␴ = 1 / 2␲ and r␴ = 0. These satisfy Eq. 共3兲
trivially. We test the stability of this solution by perturbing it.
Note that a perturbation to f ␴ leads to a perturbation of r␴,
and we expect these perturbations to either grow or shrink
exponentially in time, depending on the overall coupling
strength ␩. The marginally stable case occurs at a critical
value ␩ⴱ at which coherent collective behavior emerges.
Thus we write f ␴ = 1 / 2␲ + 共␦ f ␴兲est and r␴ = 共␦r␴兲est, where
共␦ f ␴兲 and 共␦r␴兲 are small. Inserting the first of these into the
continuity equation 关Eq. 共3兲兴, and keeping only first-order
terms, we obtain
s共␦ f ␴兲 + ␻␴
The ␻␴i are the constant natural frequencies of the oscillators
when uncoupled, and are distributed according to a set of
time-independent distribution functions G␴共␻␴兲 关12兴.
We define the usual Kuramoto order parameter for each
population, i.e.,
r ␴e i␺␴ =
冊
⳵ f␴ ⳵
兵关␻␴ + 兺 ␩k␴␴⬘r␴⬘ sin共␺␴⬘ − ␴ − ␣␴␴⬘兲兴f ␴其 = 0.
+
⳵ t ⳵␴
␴
N
N
共2兲
and if we write F␴共␴ , ␻␴ , t兲 = f ␴共␴ , ␻␴ , t兲G␴共␻␴兲, we have
k␾␾ ␾
+␩
兺 sin共␾ j − ␾i − ␣␾␾兲.
N␾ j=1
k␪␪ k␪␾
k␾␪ k␾␾
F ␴e i␴d ␴ d ␻ ␴ .
⳵ F␴ ជ
d␴
+ ⵜ · F␴ ␴ˆ = 0,
dt
⳵t
k␾␪ ␪
d␾i
= ␻ ␾i + ␩
兺 sin共␪ j − ␾i − ␣␾␪兲
N␪ j=1
dt
冉
冕冕
In this context, the distribution functions satisfy the equations of continuity, i.e.,
N␾
K=
F␴共␴, ␻␴,t兲d␴ ,
0
N
k␪␪ ␪
d␪i
= ␻␪i + ␩ 兺 sin共␪ j − ␪i − ␣␪␪兲
N␪ j=1
dt
冕
2␲
⬘
⫻cos共␺␴⬘ − ␴ − ␣␴␴⬘兲.
The solution to this equation is
共 ␦ f ␴兲 =
N␴
1
兺 e i␴ j .
N␴ j=1
⳵
1
共 ␦ f ␴兲 =
兺 ␩k␴␴⬘共␦r␴⬘兲
⳵␴
2␲ ␴
冋
册
1
ei共␺␴⬘−␴−␣␴␴⬘兲 e−i共␺␴⬘−␴−␣␴␴⬘兲
␩
k
共
␦
r
兲
兺 ␴␴⬘ ␴⬘ s − i␻␴ + s + i␻␴ .
4␲ ␴
⬘
共4兲
Here, r␴ describes the degree of synchronization in each
population, and ranges from 0 to 1. Using this, the above
equations can be expressed as
Consistency demands that the perturbations 共␦ f ␴兲 and 共␦r␴兲
be related to each other via the order parameter equation, Eq.
共2兲. This yields our main result, as follows. Equation 共2兲
becomes
d␴i
= ␻␴i + 兺 ␩k␴␴⬘r␴⬘ sin共␺␴⬘ − ␴i − ␣␴␴⬘兲.
dt
␴
共␦r␴兲estei␺␴ =
⬘
冕
⬁
−⬁
Assuming that the N␴ are very large, we solve for the onset
of coherent collective behavior by using a distribution function approach. Thus instead of discrete indices, we imagine
continua of oscillators described by distribution functions
F␴共␴ , ␻␴ , t兲 such that F␴共␴ , ␻␴ , t兲d␴d␻␴ is the fraction of ␴
oscillators whose phases and natural frequencies lie in the
infinitesimal volume element d␴d␻␴ at time t. Note that in
the N␴ → ⬁ limit,
G ␴共 ␻ ␴兲
冕冉
2␲
0
冊
1
+ 共␦ f ␴兲est ei␴d␴d␻␴ .
2␲
The integral involving 1 / 2␲ is zero. Inserting the solution
for 共␦ f ␴兲 from Eq. 共4兲, one obtains 关13兴
共␦r␴兲ei␺␴ =
冋冕
1
2
⬁
−⬁
G ␴共 ␻ ␴兲
d␻␴
s − i␻␴
册兺
␴⬘
␩k␴␴⬘共␦r␴⬘兲ei共␺␴⬘−␣␴␴⬘兲 .
Define the bracketed expression to be g␴共s兲, and define z␴
= 共␦r␴兲ei␺␴. Then, we have
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SYNCHRONIZATION IN NETWORKS OF NETWORKS: THE ...
z␴ = g␴共s兲 兺 ␩k␴␴⬘e−i␣␴␴⬘z␴⬘ .
␴⬘
Now define the complex quantity k̄␴␴⬘ = ␩k␴␴⬘e−i␣␴␴⬘. Using
the Kronecker delta ␦␴␴⬘, the above equation can be written
兺
␴⬘
冉
k̄␴␴⬘ −
␦␴␴⬘
g␴共s兲
冊
tions; we set ⌬␪ = ⌬␾ = ⌬ and ⍀␪ = ⍀␾ = ⍀. 共A more generic
example will follow.兲 Denoting D = det共K兲 and T = tr共K兲, we
separate the real and imaginary parts of Eq. 共7兲 to obtain
D␩2 − 2⌬T␩ + 4⌬2 − 4共v + ⍀兲2 = 0,
共v + ⍀兲共4⌬ − ␩T兲 = 0.
z␴⬘ = 0.
One solution to these equations is
In matrix notation for the case of two populations labeled
␪ and ␾, this is
冢
k̄␪␪ −
1
g␪共s兲
k̄␪␾
1
k̄␾␾ −
g␾共s兲
k̄␾␪
冣
冉冊
z␪
= 0.
z␾
共5兲
This equation has a nontrivial solution if the determinant of
the matrix is zero. This condition determines the growth rate
s in terms of ␩, k̄␴␴⬘, and the parameters of the distributions
G␴共␻␴兲 关via g␴共s兲兴.
To illustrate the resulting behavior, we note that g␴共s兲 can
be evaluated in closed form for a Cauchy-Lorentz distribution
G ␴共 ␻ ␴兲 =
1
⌬␴
,
␲ 共␻␴ − ⍀␴兲2 + ⌬␴2
共6兲
where ⍀␴, the mean frequency of population ␴, and the half
width at half maximum ⌬␴ are both real, and ⌬␴ is positive.
One obtains
␩ⴱ = ⌬
vⴱ = − ⍀,
Using this, the determinant condition for the two-population
case reduces to
关k̄␪␪ − 2共s + ⌬␪ + i⍀␪兲兴关k̄␾␾ − 2共s + ⌬␾ + i⍀␾兲兴 − k̄␪␾k̄␾␪ = 0.
For simplicity, we set the phase angles ␣␴␴⬘ to zero for the
remainder of this paper 关14兴. In this case, the matrix elements
k̄␴␴⬘ are purely real, so that k̄␴␴⬘ = ␩k␴␴⬘. The determinant
condition then becomes
关␩k␪␪ − 2共s + ⌬␪ + i⍀␪兲兴关␩k␾␾ − 2共s + ⌬␾ + i⍀␾兲兴
共7兲
Notice that if ␩ = 0, then s = −⌬␴ − i⍀␴, indicating that the
incoherent state is stable for zero coupling 共since −⌬␴ is
negative兲. We imagine increasing 共or decreasing兲 ␩ until s
crosses the imaginary axis at a critical value ␩ⴱ. At this point,
the incoherent state loses stability and coherent collective
behavior emerges in the ensemble. Thus the critical value ␩ⴱ
can be determined from the determinant condition by setting
s = iv, where v is real 共so that the perturbations are marginally stable兲, and equating the real and imaginary parts of the
left side of Eq. 共7兲 to zero. This results in two equations
which can be solved simultaneously for the two 共real兲 unknowns ␩ and v.
For our first example, we choose two identical popula-
冉
冊
T ⫾ 冑T2 − 4D
,
D
which is valid for T2 ⱖ 4D, since ␩ is assumed to be real.
Notice that the appropriate solution as D → 0 共using the
negative sign for T ⬎ 0 and the positive sign for T ⬍ 0兲 is
vⴱ = − ⍀,
␩ⴱ =
2⌬
,
T
as can be verified by setting D = 0 in Eq. 共8兲 directly. Another
solution is
vⴱ = − ⍀ ⫾
⌬
冑4D − T2,
T
␩ⴱ =
4⌬
.
T
Finally, setting T = 0 in Eq. 共8兲 yields vⴱ = −⍀ and ␩ⴱ
⌬
= ⫾ 冑2−D
for D ⬍ 0, and no solution for D ⱖ 0. These results
are summarized in Table I.
Thus the critical values ␩ⴱ are determined by T, D, and ⌬.
To illustrate this result, we begin by discussing a particular
example. Consider the matrix
K=
1
= 2共s + ⌬␴ + i⍀␴兲.
g ␴共 ␻ ␴兲
− ␩2k␪␾k␾␪ = 0.
共8兲
冉 冊
1 −1
1
0
,
which has trace T = 1 and determinant D = 1. This corresponds to case 2 in Table I, from which we find that ␩ⴱ
= 4⌬ / T. Figure 1 shows the results of a numerical simulation
of two populations of 10 000 oscillators each, using ⌬ = 1.
The order parameters r␴ vs ␩ are shown, and we can see that
the oscillator populations are incoherent for ␩ values below
the predicted critical value ␩ⴱ = 4, and that they grow increasingly synchronized as ␩ is increased beyond ␩ⴱ.
Next, we examined eight different connectivity matrices
K that were chosen to sample the various regions in T − D
space. Table II shows these matrices and the corresponding
value共s兲 of ␩ⴱ. These predictions were tested by numerically
calculating the order parameters r␴ for a range of coupling
TABLE I. Solutions to Eq. 共8兲 for two identical populations.
D = det共K兲 and T = tr共K兲, where K is the connectivity matrix 关Eq.
共1兲兴, and ⌬ is the width parameter in Eq. 共6兲.
Case
Condition
vⴱ
1
T2 ⬎ 4D
−⍀
2
T ⱕ 4D
3
D=0
036107-3
2
−⍀ ⫾ T 冑4D − T
−⍀
⌬
4
T = 0, D ⬍ 0
−⍀
5
T = 0, D ⱖ 0
no solution
␩ⴱ
⌬共
2
T⫾冑T2−4D
D
兲
4⌬
T
2⌬
T
2⌬
⫾ 冑−D
no solution
PHYSICAL REVIEW E 77, 036107 共2008兲
BARRETO et al.
TABLE II. Connectivity matrices K chosen to sample T-D
space.
Case
A
B
C
D
FIG. 1. Numerical calculation of the order parameters 共쎲 , △兲
vs ␩ for case A in Table II. The vertical line corresponds to the
predicted value ␩ⴱ = 4. The data point nearest ␩ⴱ is at ␩ = 4.15.
values ␩ 关15兴. Results are shown in Fig. 2. The various cases
are located by letter in the T − D plane according to the trace
and determinant of the matrices, and the corresponding inset
shows the numerically calculated order parameters plotted vs
␩, with the predicted critical coupling indicated by a vertical
line at ␩ = ␩ⴱ. For example, the inset to case A 关with 共T , D兲
= 共1 , 1兲兴 corresponds to Fig. 1, which was discussed above. It
can be seen that for all cases, the onset of synchronization
occurs as predicted.
Note that more than one prediction for ␩ⴱ may be specified by our analysis 共see Table I兲. The solutions closest to
␩ = 0 are the relevant ones, because we expect the incoherent
state to lose stability once the first ␩ⴱ solution is encountered. There are two possible cases depending on the sign of
D. First, if the two solutions have the same sign, then there is
only one critical value ␩ⴱ 共which may be positive or negative
depending on the sign of the trace兲 for the onset of synchronization. This occurs for D ⬎ 0 and T ⫽ 0, as can be seen in
E
F
G
H
T
D
␩ⴱ
1 −1
1 0
1
1
4
−2 −3
1 1
−1
1
−4
3
1
−3.5 −1
2
0.5
2共2 − 冑2兲 = 1.172
−3 1
−3.5 1
−2
0.5
2共−2 + 冑2兲 = −1.172
−1 −1
1 2
1
−1
−共1 ⫾ 冑5兲 = −3.236, 1.236
1 1
−1 −2
−1
−1
1 ⫾ 冑5 = −1.236, 3.236
2 1
−3 −2
0
−1
⫾2
1 −1
2 −1
0
1
none
Matrix
共 兲
共 兲
共 兲
共 兲
共 兲
共 兲
共 兲
共 兲
Fig. 2. 共Interestingly, for D ⬎ 0 and T = 0, synchronization
does not occur for any ␩.兲 The other case occurs for D ⬍ 0,
for which the two ␩ⴱ solutions have opposite signs. In this
case, there are two critical values ␩ⴱ for the onset of
synchronization—one on either side of ␩ = 0. This can also
be observed in Fig. 2.
In the more general case in which the various populations
have different natural frequency distributions, it is not typi-
FIG. 2. Numerical simulations using the matrices listed in Table II for identical populations.
The letters indicate the placement of each case in
the T-D plane, and the corresponding insets show
numerical calculations of the order parameters
共쎲 , △兲 vs ␩ for that case 共in all cases, ␩ = 0 is
in the center of the horizontal axis兲. Vertical lines
in the insets indicate the predicted value共s兲 ␩ⴱ
listed in Table II for the onset of coherent collective behavior. In all cases, we used ⌬ = 1. Note
that for D ⬎ 0, there is one value of ␩ⴱ, whose
sign corresponds to the sign of the trace T. If D
ⱖ 0 and T = 0, then synchronization is not possible for any coupling strength. For D ⬍ 0, there
are two values of ␩ⴱ: one positive, and one
negative.
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6
6
4
Im(η)
Im(η)
4
2
0
−2
−4
−4
−3
v
−2
−1
−5
−4
−6
−5
−4
v
−3
−2
−1
−3
−2
−1
6
0
4
−1
Re(η)
Re(η)
−6
(a)
1
−2
2
0
−3
−4
0
−2
(a)
(b)
2
−4
−3
v
−2
−2
−1
(b)
FIG. 3. Case E, different populations. The upper panel shows
Im共␩共1,2兲兲, with the vertical lines identifying roots at v1 = −4.024
and v2 = −1.722. The lower panel shows Re共␩共1,2兲兲; values at the
roots found above are indicated by horizontal lines, yielding ␩共1兲
ⴱ
= 0.515 and ␩共2兲
ⴱ = −2.809. Thus we expect synchronization to occur
at these values as ␩ is either increased or decreased away from zero.
cally possible to describe the onset of synchronization in
terms of the determinant and trace of the coupling matrix
K = k␴␴⬘ alone. We now consider this situation, but retain the
Cauchy-Lorentz form of the natural frequency distributions
for convenience. We manipulate Eq. 共7兲 as follows. Let s
= iv 共i.e., purely imaginary, to consider the marginally stable
case兲 and define a = ⌬␪ + i共v + ⍀␪兲 and b = ⌬␾ + i共v + ⍀␾兲. We
obtain
v
FIG. 5. Case A, different populations. Panels as in Fig. 3. Note
that the standard branch cut leads to discontinuities and the occurrence of two roots for Im共␩共1兲兲共v兲 共solid lines, upper panel兲, and
none for Im共␩共2兲兲共v兲 共dotted lines, upper panel兲. From the lower
panel we find ␩ⴱ = 2.189, 4.501, taking care to obtain these from
Re共␩共1兲兲 共solid line, lower panel兲. We expect to find synchronization
onset at the smaller of these values.
There are two unknowns in this equation: ␩ and v. Equation
共9兲 is a quadratic equation in ␩ with complex coefficients,
and we can easily obtain two complex solutions ␩共1,2兲 as
functions of v. Since the critical values ␩共1,2兲
must be real,
ⴱ
we solve for the roots of Im共␩共1,2兲兲, and evaluate Re共␩共1,2兲兲 at
.
these roots. This yields the possible critical values ␩共1,2兲
ⴱ
Typically, these steps must be performed symbolically and/or
numerically; we used MATLAB® 关16兴. As before, the values
of ␩共1,2兲
that are closest to zero 共on either side兲 are the relⴱ
evant values.
To illustrate, we choose two populations with CauchyLorentz natural frequency distributions 关Eq. 共6兲兴 with parameters ⌬␪ = 1, ⍀␪ = 2, ⌬␾ = 0.5, and ⍀␾ = 4. We consider the
FIG. 4. Case E, different populations. Calculations of the order
parameters vs ␩ confirm that the onset of synchronization occurs at
␩ⴱ = −2.809 and 0.515 共vertical lines兲, as predicted in Fig. 3.
FIG. 6. Case A, different populations: The onset of synchronization occurs at ␩ⴱ = 2.189 共vertical line兲, as predicted in Fig. 5. No
synchronization is observed for smaller values of ␩ 共not shown兲.
D␩2 − 2␩共bk␪␪ + ak␾␾兲 + 4ab = 0.
共9兲
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6
3
4
2
Im(η)
Im(η)
BARRETO et al.
2
0
−6
(a)
1
0
−1
−5
−4
−3
v
−2
−1
0
−2
1
(a)
−4
−3
−4
−3
−2
−1
0
−2
−1
0
v
6
3
4
2
2
Re(η)
Re(η)
8
0
−2
−4
(b)
−6
FIG. 7. Case
= −1.429, 5.000.
−5
H,
−4
−3
v
different
−2
−1
0
populations.
0
−1
1
We
1
find
␩ⴱ
same K matrices as before, i.e., those listed in the second
column of Table II. Case E is straightforward; the analysis is
illustrated in Fig. 3 and the numerical verification is in Fig.
4. Note that since Eq. 共9兲 has complex coefficients, obtaining
␩ typically involves taking the square root of a complex
number. Therefore one must be mindful of branch cuts when
obtaining symbolic and/or numerical solutions. This is important in the analysis for case A, shown in Figs. 5 and 6.
Finally, case H, which exhibits no synchronization for identical populations for any value of ␩, does show synchronization in the present case. The analysis is shown in Figs. 7 and
8.
We close by giving an example with three different populations. We choose the same Cauchy-Lorentz distributions as
above, and add a third with ⌬␳ = 1 / 3 and ⍀␳ = 1. We use the
following K matrix:
FIG. 8. Case H, different populations. Synchronization occurs at
␩ⴱ = −1.429 and 5.000 共vertical lines兲, as predicted in Fig. 7.
(b)
v
FIG. 9. 共Color online兲 Three populations. The imaginary and
real parts of ␩共1,2,3兲 are plotted to illustrate the discontinuities due to
branch cuts. The analysis proceeds as in the previous cases. Because of branch cuts, two roots occur on ␩共1兲, one on ␩共2兲, and none
on ␩共3兲. Evaluating the real parts appropriately, we find ␩ⴱ
= −0.891, −0.564, 2.303.
冢
−1
1
1
1
−1
1
1
1
−1
冣
.
The procedure for deriving ␩ⴱ proceeds as above, except that
Eq. 共7兲 is replaced by a third-degree polynomial in ␩. Figure
9 shows the imaginary and real parts of the three ␩ solutions.
共Note that the branch cuts are more complicated.兲 The predicted onset of synchronization was verified, as shown in
Fig. 10.
FIG. 10. Three populations. Synchronization occurs at ␩ⴱ
= −0.564 and 2.303, as predicted in Fig. 9.
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SYNCHRONIZATION IN NETWORKS OF NETWORKS: THE ...
ACKNOWLEDGMENTS
In conclusion, we have described how to determine the
onset of coherent collective behavior in systems of interacting Kuramoto systems, i.e., systems of interacting populations of phase oscillators with both node and coupling heterogeneity.
E.B. was supported by NIH Grant No. R01-MH79502;
E.O. was supported by ONR 共Physics兲 and by NSF Grant
No. PHY045624.
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99, 7821 共2002兲.
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关11兴 A similar system of two asymmetrically interacting populations with a particular form of coupling matrix K was consid-
ered in E. Montbrio, J. Kurths, and B. Blasius, Phys. Rev. E
70, 056125 共2004兲. The formulation in the current paper is
more general in two important aspects: K is arbitrary, and we
allow for any number of interacting populations.
Our system is similar to that studied in J. G. Restrepo, E. Ott,
and B. R. Hunt, Chaos 16, 015107 共2006兲. However, our formulation permits different natural frequency distributions for
each population.
The bracketed expression is valid for Re共s兲 ⬎ 0 and may be
analytically continued into the region where Re共s兲 ⱕ 0. See E.
Ott, P. So, E. Barreto, and T. Antonsen, Physica D 173, 29
共2002兲; E. Ott, Chaos in Dynamical Systems, 2nd ed. 共Cambridge University Press, Cambridge, England, 2002兲, p. 240.
Many interesting states require ␣␴␴⬘ ⫽ 0, such as the chimera
state observed in D. M. Abrams and S.H. Strogatz, Int. J. Bifurcation Chaos Appl. Sci. Eng. 16, 21 共2006兲.
Simulations used fourth-order Runge-Kutta with a time step of
0.01 s, N = 10 000 or 50 000, and parameters as noted. The
system was initialized in the incoherent state and an initial
transient was discarded. The order parameters were then averaged over the subsequent 10 s. Because the standard deviations
over this interval were small, no error bars were plotted.
MATLAB®, The MathWorks, Inc., Natick, MA, http://
www.mathworks.com/
关12兴
关13兴
关14兴
关15兴
关16兴
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