PHYSICAL REVIEW E 80, 046215 共2009兲
Existence of hysteresis in the Kuramoto model with bimodal frequency distributions
Diego Pazó1 and Ernest Montbrió2,3
1
Instituto de Física de Cantabria (IFCA), CSIC–Universidad de Cantabria, E-39005 Santander, Spain
Computational Neuroscience Group, Department of Information and Communication Technologies, Universitat Pompeu Fabra,
08003 Barcelona, Spain
3
Center for Neural Science, New York University, New York, New York 10012, USA
共Received 13 April 2009; revised manuscript received 4 August 2009; published 23 October 2009兲
2
We investigate the transition to synchronization in the Kuramoto model with bimodal distributions of the
natural frequencies. Previous studies have concluded that the model exhibits a hysteretic phase transition if the
bimodal distribution is close to a unimodal one due to the shallowness of the central dip. Here we show that
proximity to the unimodal-bimodal border does not necessarily imply hysteresis when the width, but not the
depth, of the central dip tends to zero. We draw this conclusion from a detailed study of the Kuramoto model
with a suitable family of bimodal distributions.
DOI: 10.1103/PhysRevE.80.046215
PACS number共s兲: 05.45.Xt
I. INTRODUCTION
Understanding the dynamics of large populations of heterogeneous self-sustained oscillatory units is of great interest
because they occur in a wide range of natural phenomena
and technological applications 关1兴. Often a macroscopic system self-organizes into a synchronous state, in which a certain fraction of its units acquires a common frequency. This
occurs as a consequence of the mutual interactions among
the oscillators and despite the differences in their rhythms
关2兴. Examples of collective synchronization include pacemaker cells in the heart and nervous system 关3,4兴, synchronously flashing fireflies 关5兴, collective oscillations of pancreatic beta cells 关6兴, and pedestrian induced oscillations in
bridges 关7兴.
A fundamental contribution to the study of collective synchronization was the model proposed by Kuramoto 关8兴. This
model, and a large number of extensions of it, has been extensively studied because it is analytically tractable but still
captures the essential dynamics of collective synchronization
phenomena 共for reviews see 关1,9–11兴兲. The original Kuramoto model consists of a population of N oscillators interacting all-to-all. The state of an oscillator i is described by its
phase ␪i共t兲 that evolves in time according to
N
K
␪˙ i = ␻i − 兺 sin共␪i − ␪ j兲.
N j=1
共1兲
The parameter K determines the strength of the interaction
between one oscillator and another. The oscillators are considered to have different natural frequencies ␻i, which are
taken from a probability distribution g共␻兲. In his analysis
Kuramoto adopted the thermodynamic limit N → ⬁ and considered g共␻兲 to be symmetric. In this case and without loss
of generality, the distribution can always be centered at zero,
i.e., g共␻兲 = g共−␻兲, by going into a rotating framework ␪ j
→ ␪ j + ⍀t.
Kuramoto found useful to study the synchronization dynamics of system 共1兲 in terms of a complex order parameter
z = N−1兺Nj=1 exp共i␪ j兲. Note that z is a mean field that indicates
the onset of coherence due to synchronization in the population. System 共1兲 possesses an incoherent state with z = 0 共that
1539-3755/2009/80共4兲/046215共9兲
exists for all values of the coupling strength K兲 in which the
oscillators rotate independently as if they were uncoupled,
␪i共t兲 ⬃ ␻it. Using a self-consistency argument, Kuramoto
found that for a unimodal distribution g共␻兲, above the coupling’s critical value
Kc =
2
,
␲g共0兲
共2兲
a new solution with asymptotics
兩z兩 ⬇
4
K2c
冑
K − Kc
− ␲g⬙共0兲
共3兲
branches off the incoherent 共z = 0兲 solution. This emerging
solution is a partially synchronized 共PS兲 state, in which a
subset of the population S entrains to the central frequency
共␪i苸S = const兲.
Equation 共3兲 shows that the orientation of the PS bifurcating branch depends on whether the distribution is concave or
convex at its center. As a consequence of that, at K = Kc the
PS state is expected to bifurcate supercritically for unimodal
distributions 关g⬙共0兲 ⬍ 0兴 and subcritically for bimodal distributions 关g⬙共0兲 ⬎ 0兴. However, Kuramoto’s analysis did not
permit to study the stability of the solutions and thus one
cannot conclude whether bimodal distributions show bistability close to the transition point 关Eq. 共2兲兴 共see discussion in
p. 75 in 关8兴兲. In fact, Kuramoto discarded the possibility of
bistability. Instead he expected the incoherent state to become unstable earlier, i.e., at a certain critical value Kc⬘
⬍ Kc, via the formation of two symmetric clusters of synchronized oscillators near the distribution’s maxima 共later
Crawford called this state standing wave 共SW兲 关12兴兲. As the
coupling is increased further, he predicted that the interaction
between the clusters would tend to synchronize them forming a single synchronized group, i.e., a PS state.
A. Sum of unimodal distributions with different means
After Kuramoto’s seminal work 关8兴, several articles have
further investigated the synchronization transition in model
共1兲 with symmetric bimodal distributions 关12–17兴. These
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©2009 The American Physical Society
PHYSICAL REVIEW E 80, 046215 共2009兲
DIEGO PAZÓ AND ERNEST MONTBRIÓ
0.3
ξ=γ
γ = 0.05
1
ξ = 0.5
0.3
ξ=γ
0.8
γ = 0.7
0.6
0.2
ξ
0.2 γ = 0.5
D
0.1
0.1
Bimodal
A B
Unimodal
0.4
0.2
γ = 0.59
0
0
-2
0
ω
2
0
-2
0
ω
2
FIG. 1. 共Color online兲 Examples of bimodal frequency distributions given by Eq. 共5兲 with ␦ = 1. Left panel: ␰ = ␥ 关what implies
g共0兲 = 0兴. Note that as ␥ decreases the maxima of the distribution
become closer. For all these distributions 共with ␰ = ␥兲 the route to
synchronization as K is increased from zero is I → SW→ PS 共cf.
Fig. 3兲. Right panel: two examples with ␰ ⬍ ␥. The distribution
depicted with a continuous line has well separated peaks and shows
a transition I → SW→ PS, whereas the other distribution is closer to
the unimodal limit 关Eq. 共7兲兴 and presents hysteresis in the route to
synchronization 共cf. Fig. 4兲.
studies assumed g共␻兲 to be the superposition of two identical
even unimodal distributions g̃共␻兲 centered at ⫾␻0: g共␻兲
= g̃共␻ + ␻0兲 + g̃共␻ − ␻0兲 关19兴. Parameter ␻0 controls the separation of the peaks. Decreasing ␻0 the distribution’s maxima
approach each other and, at the same time, the central distribution’s dip becomes shallower 关i.e., g共0兲 increases兴. Eventually, at a value ␻0 = ␻0B that satisfies
g⬙共␻ = 0兲兩␻0=␻0B = 0
共4兲
the peaks merge and the distribution becomes unimodal. The
dynamics of the Kuramoto model for distributions of this
type is as follows 关13兴: when the peaks are well separated
共␻0 larger than a certain value ␻0D兲 the transitions increasing
K are as Kuramoto foresaw: incoherence 共I兲 → SW→ PS.
However, if the peaks are near 共␻0D ⬎ ␻0 ⬎ ␻0B兲 there exists
a range of K below Kc where bistability between incoherence
and either a PS or a SW state is observed, as Eq. 共3兲 suggested 关20兴.
B. Difference of unimodal distributions with different widths
In this paper we are interested in understanding the synchronization transition in the Kuramoto model with bimodal
distributions in situations that cannot be achieved summing
even unimodal distributions. In particular summing even distributions implies that if the peaks are brought closer the
central dip becomes less deep 共unless the distributions are
Dirac deltas兲. Thus we cannot approach the peaks arbitrarily
near while keeping the central dip’s depth 共see, e.g., in the
left panel of Fig. 1 for a distribution family with constant
depth but arbitrary distance between the peaks兲.
We will use a family of bimodal distributions that are
constructed as the difference of two unimodal even functions
0.2
0.4
γ
0.6
0.8
0
1
FIG. 2. The parameter space of distribution 关Eq. 共5兲兴 关not defined above the bisectrix ␰ = ␥ neither at point 共1,1兲兴. Function 关Eq.
共5兲兴 is unimodal below line B and bimodal above it 共shaded regions兲. Three lines signal the loci of codimension-two bifurcations
共A, B, and D兲 projected on the 共␥ , ␰兲 plane. Between lines D and B
共dark gray region兲 the transition to synchronization involves
hysteresis.
with the same mean and different widths: g共␻兲 = g̃1共␻兲
− g̃2共␻兲. These distributions could be useful to model systems in which a fraction of the central natural frequencies of
a population g̃1 is missing due to, for example, some resonance, symmetry, or external disturbance.
We choose the functions g̃i to be Lorentzians because of
their mathematical tractability. Assuming ␦ ⬎ ␥ the normalized distribution reads
g共␻兲 =
冋
冉
␦2
␥
⌶
2
2 −␰
2
␲ ␻ +␦
␻ + ␥2
冊册
共5兲
with ␰ ⱕ ␥ to be well defined and ⌶ = 1 / 共␦ − ␰兲 is the normalization constant. Without loss of generality we assume ␦ = 1
hereafter because this can be always achieved rescaling ␻,
time, and the parameters ␻⬘ = ␻ / ␦, t⬘ = t␦, K⬘ = K / ␦, ␥⬘
= ␥ / ␦, and ␰⬘ = ␰ / ␦. We will also drop the primes to lighten
the notation. Figure 1 shows several examples of distributions 关Eq. 共5兲兴. Distribution family 关Eq. 共5兲兴 can exhibit an
arbitrarily deep minimum while keeping the maxima as near
as wished.
The left panel of Fig. 1 shows two examples for the case
␰ = ␥, which will be analyzed in detail below. This case implies g共0兲 = 0, which corresponds to the maximal value of the
ratio ␰ / ␥ = 1. As ␥ → 0, the central dip becomes infinitely
narrow and at ␥ = 0 the distribution becomes unimodal. This
unimodal transition is therefore discontinuous and satisfies
关21兴
lim g⬙共␻ = 0兲 = ⬁.
␥→0+
共6兲
In addition, distribution 关Eq. 共5兲兴 also presents the regular
unimodal-bimodal border via g⬙共0兲 = 0 at
␰B = ␥3
共7兲
with ␥ ⫽ 0 共line B in Fig. 2兲.
The outline of the paper is as follows. Section II summarizes recent theoretical results that permit to reduce the Kura-
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EXISTENCE OF HYSTERESIS IN THE KURAMOTO MODEL…
moto model to a system of ordinary differential equations
with complex variables. These results are then used to find
the two ordinary differential equations 共ODEs兲 that describe
the dynamics of the Kuramoto model with distribution 关Eq.
共5兲兴. In Sec. III we study the special case ␰ = ␥, and we show
that there indeed exists a transition to synchronization in
absence of hysteresis independent of the separation between
the distribution’s maxima. Namely, in this case the route to
synchronization is always I → SW→ PS. In Sec. IV we study
the most general case g共0兲 ⬎ 0 and determine the disposition
of the different synchronization scenarios with respect to the
unimodal-bimodal border.
II. LOW DIMENSIONAL DESCRIPTION OF THE
KURAMOTO MODEL
We start considering the thermodynamic limit N → ⬁ of
model 共1兲. We drop hence the indices in Eq. 共1兲 and introduce the probability density for the phases f共␪ , ␻ , t兲 关8,22兴.
Then f共␪ , ␻ , t兲d␪d␻ represents the ratio of oscillators with
phases between ␪ and ␪ + d␪ and natural frequencies between
␻ and ␻ + d␻. The density function f obeys the continuity
equation
⳵f
⳵ 共f v兲
=−
,
⳵t
⳵␪
共8兲
where the angular velocity of the oscillators v is given by
v共␪, ␻,t兲 = ␻ − K
冕
2␲
f共␪⬘, ␻,t兲sin共␪ − ␪⬘兲d␪⬘ .
␣˙ = − i␻␣ +
冕冕
⬁
z共t兲 =
2␲
i␪
A. Main equations
In this section we use the OA ansatz considering frequency distribution 共5兲. This yields two ODEs governing the
dynamics inside the low-dimensional OA manifold. First of
all, it is convenient to express Eq. 共5兲 in partial fractions,
共9兲
e f共␪, ␻,t兲d␪d␻ .
共10兲
0
−⬁
Since the density function f共␪ , ␻ , t兲 is real and 2␲ periodic in
the ␪ variable, it admits the Fourier expansion
f共␪, ␻,t兲 =
ⴱ
f n = f −n
.
where
reduces to
冋
⬁
册
g共␻兲
1 + 兺 关f n共␻,t兲ein␪ + c.c.兴 ,
2␲
n=1
g共␻兲 =
zⴱ共t兲 =
冕
⬁
g共␻兲f 1共␻,t兲d␻ .
共12兲
−⬁
Substituting the Fourier series 关Eq. 共11兲兴 into the continuity
Eq. 共8兲 and using Eq. 共12兲 one gets an infinite set of integrodifferential equations for the Fourier modes,
ḟ n = − in␻ f n +
nK ⴱ
共z f n−1 − zf n+1兲.
2
共13兲
Recently Ott and Antonsen 共OA兲 found a very remarkable
result 关23兴: the ansatz
f n共␻,t兲 = ␣共␻,t兲n
共14兲
is a particular—and usually the asymptotic—solution of the
infinite set of Eq. 共13兲 if ␣ satisfies
冉
冊冉
冊
1
1
␰
␰
⌶
−
+
−
. 共16兲
2␲i ␻ − i ␻ + i
␻ − ␥i ␻ + ␥i
Then, according to Eq. 共12兲 the order parameter reads
zⴱ共t兲 = ⌶关␣1共t兲 − ␰␣2共t兲兴,
共17兲
with ␣1共t兲 = ␣共␻ = −i , t兲 and ␣2共t兲 = ␣共␻ = −i␥ , t兲. Using Eq.
共17兲 in Eq. 共15兲, we obtain the following two ODEs with
complex variables that govern the evolution of the order parameter 关Eq. 共17兲兴,
共11兲
Note that the order parameter 关Eq. 共10兲兴 now
共15兲
Equation 共15兲 reduces to a finite set of ODEs for distributions g共␻兲 with a finite set of simple poles out of the real
axis. Recalling f 1 = ␣ the order parameter can be calculated
by extending the integral in Eq. 共12兲 to a contour integration
in the complex plane. This is possible since ␣ has an analytic
continuation in the lower half ␻ plane 关23兴. In turn only that
the values of ␣ at the poles of g共␻兲 with negative imaginary
part are relevant.
Several recent studies show that ansatz 共14兲 yields predictions in agreement with numerical simulations 关13,23–28兴. In
addition Ott and Antonsen theoretically supported the validity of their ansatz for the case of a Lorentzian distribution
关29兴. So far, disagreement between the OA ansatz and numerical results has been shown for frequency distributions
with no spread and non-odd-symmetric coupling function.
This entails the freedom to select arbitrary values for some
constants of motion 关30兴.
0
In the continuous formalism, the complex order parameter
defined by Kuramoto becomes
K ⴱ
共z − z␣2兲.
2
␣˙ 1 = − ␣1 + k共␣1 − ␰␣2兲 − k共␣ⴱ1 − ␰␣ⴱ2兲␣21 ,
共18a兲
␣˙ 2 = − ␥␣2 + k共␣1 − ␰␣2兲 − k共␣ⴱ1 − ␰␣ⴱ2兲␣22 ,
共18b兲
with k = ⌶K / 2. The phase space of Eqs. 共18兲 is four dimensional, but due to the global phase shift invariance 共␣1 , ␣2兲
→ 共␣1ei␤ , ␣2ei␤兲 the dynamics is actually three dimensions
关see also Eqs. 共A1兲 in Appendix A兴.
B. Fixed points
According to Eq. 共17兲, the fixed points of Eqs. 共18兲 correspond to steady states of the order parameter z. The trivial
solution ␣1 = ␣2 = 0 yields z = 0, corresponding to the incoherent state.
In order to calculate the nontrivial fixed points, note first
that invariance under the action of the global rotation ei␤
allows us to choose ␣1 = x1 + iy 1 real, i.e., ␣1 = x1. It follows
from Eq. 共18a兲 that the fixed points lie on the subspace
where ␣2 is real too. We can therefore take ␣1 and ␣2 as real
共keeping in mind that a continuous of fixed points is gener-
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DIEGO PAZÓ AND ERNEST MONTBRIÓ
ated under the action the neutral rotation ei␤兲. Hence, the
equations for the fixed points are
共19a兲
0 = − ␥x2 + k共x1 − ␰x2兲共1 − x22兲.
共19b兲
P共X兲 = k 共1 − ␥␰兲X − k关共2k − 1兲共1 − ␥␰兲 − 1 + k␰共␰ − ␥兲兴X
3
− k␰关␥ + k共␰ − ␥兲兴 = 0.
共20兲
Each of the solutions of this equation yields two twin solutions with coordinates
x 1 = ⫾ 冑X
冋
␰x2 = x1 1 −
册
1
.
k共1 − X兲
Standing
Wave
4
Hopf
3
Incoherence
2
0
2
+ 关共k2 − 2k兲共1 − ␥␰兲 + 1 + 2k2␰共␰ − ␥兲兴X
Partial Sync
5
Additionally, note that these equations are symmetric under
the reflection 共x1 , x2兲 → 共−x1 , −x2兲. This implies that the solutions 共with the exception of the solution at the origin兲 exist
always in pairs with opposite signs 共⫾x1 , ⫾ x2兲.
Subtracting Eq. 共19a兲 from Eq. 共19b兲 multiplied by ␥␰ , we
obtain x22 = ␥␰ 关x21 + 1k + ␥␰ − 1兴. This can be substituted back into
Eq. 共19a兲 to get a cubic equation in X ⬅ x21,
2
SNIC
K
0 = − x1 + k共x1 − ␰x2兲共1 − x21兲,
6
0.2
0.4
γ
0.6
0.8
1
FIG. 3. Phase diagram for ␰ = ␥. For this case the synchronization transition never involves hysteresis. The solid lines mark the
saddle-node 共SNIC兲 关from Eq. 共26兲兴 and the Hopf 关Eq. 共24兲兴 bifurcations. Symbols correspond to the numerical estimation of the bifurcation lines via numerical integration of the original Eq. 共1兲 with
N = 2000.
共21兲
␻ = ⫾ ␥.
共23兲
After some algebra we obtain the relation of the solutions
with order parameter,
2␰冑X
.
兩z兩 =
K共1 − X兲
A. Stability of the incoherent state
共22兲
A steady state 共x1 , x2兲 results in a time-independent value of
z and hence it should correspond to a partially synchronized
state. However, note that X can only take values within the
2
range X 苸 关0 , 1 − 2 K␰ 共 K␰ 2 + 1 − K␰ 兲兴 to have a z value consistent
with its definition, i.e., 兩z兩 苸 关0 , 1兴.
As the polynomial in Eq. 共20兲 is cubic, there is one real
solution, X共3兲, for all the parameter values. This solution lies
in the range 关0 , 1兴 共for k ⬎ 1 a better bound is 关1 − 1 / k , 1兴
since P共1 − 1 / k兲 = −␰2 ⬍ 0 and P共1兲 = 1 ⬎ 0兲. However, it turns
out that the fixed points associated to X共3兲 are “unphysical”
共even though in some parameter ranges 兩z兩 ⬍ 1兲. The reason is
that the x2 coordinate, corresponding to the solution X共3兲, is
always larger than 1 in absolute value. This implies 兩␣2兩 ⬎ 1,
and according to Eq. 共14兲 the Fourier series of the density
function f共␪ , ␻ , t兲 is divergent at ␻ = −i␥.
We will see below that for large enough values of K there
exist two more real solutions of P共X兲: X共1兲 ⱕ X共2兲 ⬍ 1 − 1 / k. In
this case 共except when X共1兲 becomes negative兲 such solutions
indeed correspond to PS states of the original Kuramoto
model 关Eq. 共1兲兴.
冑
In the incoherent state the oscillators are uniformly distributed in the interval 关0 , 2␲兲, and thus the order parameter
vanishes. This state corresponds to the fixed point at the
origin ␣1 = ␣2 = 0. A linear stability analysis of Eqs. 共18兲 reveals that this fixed point undergoes a degenerate Hopf bifurcation at kH = 共1 + ␥兲 / 共1 − ␥兲. In terms of the original coupling constant K, we find
KH = 2 + 2␥ .
ⴱ
= i冑␥
At this point the eigenvalues are imaginary ␭1,2 = ␭3,4
and twofold degenerate. Observe that as ␥ → 0, the critical
coupling for a 共unimodal兲 Lorentzian distribution of unit
width is recovered: KH共␥ → 0兲 = Kc = 2 / 关␲g共0兲兴 = 2. Figure 3
shows the boundary KH in the 共␥ , K兲 plane. As expected, we
find that as the central dip of the distribution broadens 共increasing ␥兲 the stability region of the incoherent state grows.
B. Saddle-node bifurcation
The cubic equation 关Eq. 共20兲兴 for the nontrivial fixed
points becomes greatly simplified under the assumption ␰
= ␥,
Q共X兲 = k2共1 − ␥2兲X3 − k关共2k − 1兲共1 − ␥2兲 − 1兴X2
III. BIMODAL DISTRIBUTIONS VANISHING AT THEIR
CENTER (␰ = ␥)
In this section we consider ␰ = ␥ what implies that distribution 关Eq. 共5兲兴 vanishes at its center, g共0兲 = 0. In this case ␥
共or ␰兲 becomes the parameter controlling the width of the
central dip of g共␻兲, and the maxima of the distribution are
located at 共see Fig. 1, left panel兲
共24兲
+ 关共k2 − 2k兲共1 − ␥2兲 + 1兴X − ␥2k = 0.
共25兲
For ␥ = 0 the central dip vanishes, and we recover the solutions for a Lorentzian distribution X = 0 , 1 − 1 / k. When ␥
⬎ 0 there is a saddle-node bifurcation at k = kSN, i.e., there is
a transition from one 共for k ⬍ kSN兲 to three solutions 共for k
⬎ kSN兲. kSN and ␥ can be related imposing the condition that
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the discriminant of Q共X兲 vanishes. This gives the following
relation:
␥2 =
4
2 3/2
2
− 共1 + 8kSN
兲 + 20kSN
−1
8kSN
.
3
8kSN共kSN + 1兲
共26兲
There are two important asymptotic values for this bifurcation line, which expressed in terms of the original coupling
constant K are
冉冊
冑 冉
KSN共␥ → 0兲 = 2 + 6
␥
2
2/3
+ O共␥兲,
共27兲
冊
共28兲
KSN共␥ → 1兲 ⯝ 共3 + 8兲 1 −
1−␥
.
2
When K increases above KSN the born solutions depart from
each other X共2兲 − X共1兲 ⬃ 冑K − KSN+ higher order terms. One solution becomes progressively smaller 关dX共1兲共K兲 / dK ⬍ 0兴,
whereas the second one grows 关dX共2兲共K兲 / dK ⬎ 0兴. The latter
solution X共2兲 yields a monotonically growing value of 兩z兩 with
K. This is not surprising because in the Kuramoto model, at
large values of K, there exists always a stable PS solution
with d兩z兩 / dK ⬎ 0 共and limK→⬁兩z兩 = 1, i.e., full synchronization兲. We advance that the corresponding twin fixed points
from X共2兲 are stable, whereas the fixed points corresponding
to X共1兲 are saddle.
C. Numerical simulations and phase diagram
In this section we construct the phase diagram with the
loci of Hopf and saddle-node bifurcations that we have obtained above. Numerical simulations of the reduced Eqs. 共18兲
were carried out and compared with the full model 关Eq. 共1兲兴.
This permits to relate the dynamics of the variables ␣1,2 with
the actual dynamical states of the Kuramoto model.
As already mentioned, the four-dimensional system 关Eqs.
共18兲兴 is effectively three dimensional due to the existence of
a neutral global rotation. Interestingly the attractors of the
model are apparently embedded into a two-dimensional
plane. Numerical simulations of Eqs. 共18兲 using arbitrary
initial conditions show that the dynamics always collapses
into a plane which, by virtue of the neutral rotation ei␤, can
be made coincident with the 共x1 , x2兲 plane, hereafter referred
to as the “real plane.” The stability against perturbations
transversal to the real plane 共and not tangent to the global
rotation兲 is difficult to prove analytically. For the fixed point
X共2兲 born at the saddle-node bifurcation, the stability against
transversal perturbations is proven in Appendix A. Other attractors 共limit cycle兲 are transversally stable according to our
numerical simulations.
Numerical simulations of the reduced Eqs. 共18兲 with either real or complex variables, it is irrelevant, reveal the
following:
共i兲 The Hopf bifurcation at K = KH is supercritical and it
gives rise to a limit cycle around the origin. Due to the reflection symmetry of the equations z共t兲 vanishes twice per
period 关this occurs when ␣1 = ␥␣2; see Eq. 共17兲兴. It is therefore reasonable to assume that the limit cycle corresponds to
the SW state, for which the two counter-rotating clusters of
phase-locked oscillators are ␲ out of phase twice per period.
共ii兲 The oscillatory dynamics appearing at KH is destroyed
at K = KSN where twin saddle-node bifurcations give rise to
twin pairs of fixed points on the limit cycle. This bifurcation
is known as saddle-node on the invariant circle 共SNIC兲 or
saddle-node of infinite period 共SNIPER兲. As K approaches
KSN from below the period of 兩z共t兲兩 diverges due to the slowing down of the dynamics at the twin bottlenecks anticipating the cease of oscillations via the 共double兲 SNIC bifurcation.
Finally, numerical simulations of the full Kuramoto model
关Eq. 共1兲兴 confirm the scenario I → SW→ PS predicted by the
reduced Eqs. 共18兲. We have numerically determined the
boundaries of different behaviors: square symbols in Fig. 3
are points in which the incoherent state loses stability leading
to a SW state. Additionally, triangles indicate points where
the order parameter becomes stationary.
D. Concluding remarks
Distribution 关Eq. 共5兲兴 with ␰ = ␥ becomes unimodal only
for ␥ = 0. As ␥ → 0 the bimodal distribution tends to a unimodal, but the limit is non-regular, see Eq. 共6兲. The remarkable point is that bistability is not observed even if the central dip is extremely narrow 共␥ → 0兲. This is in sharp contrast
with the scenario found when the peaks are close to merge
with g⬙共0兲 → 0+ at the usual unimodal-bimodal transition
共see below兲.
Another interesting fact is that the counter-rotating clusters of the SW are born at the Hopf bifurcation 关Eq. 共24兲兴
with frequencies ⫾冑␥, although the maxima of the distribution are located at ⫾␥. This means that the relative shift
between distribution’s maxima and cluster frequencies at the
onset of the SW diverges as ␥ → 0. This is a consequence of
the extreme asymmetry of the peaks in this limit.
IV. BIMODAL DISTRIBUTIONS NONVANISHING AT
THEIR CENTER (␰ ⬍ ␥)
In this section we analyze the case ␰ ⬍ ␥, which is
complementary to the one studied in Sec. III 共␰ = ␥兲. Thus in
the present case we let ␰ and ␥ to be independent of each
other 共see Fig. 2兲. As we did in the Sec. III, we determine
first the local bifurcations of the fixed points, and then we
summarize our findings in the 共␥ , K兲 phase plane together
with the results obtained by numerical integration of the reduced Eqs. 共18兲 as well as of the full Kuramoto model 关Eq.
共1兲兴.
A. Fixed points
1. Incoherent state and its stability
The incoherent state becomes unstable in two possible
ways depending on the value of ␰ with respect to
␰A = ␥2
共29兲
共see line A in Fig. 2兲. For ␰ ⬍ ␰A, there is a degenerate Hopf
bifurcation at the critical value KH given by Eq. 共24兲 which is
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DIEGO PAZÓ AND ERNEST MONTBRIÓ
independent of ␰. For ␰ ⬎ ␰A, the instability of the incoherent
state occurs via a pitchfork bifurcation at
4
PS
D
共30兲
B. Numerical simulations and phase diagram
Our analytical results provide information about local bifurcations. In addition we have performed numerical simulations of the ODEs 关Eqs. 共18兲兴 in order to obtain the full
system’s picture. As we discussed in Sec. III C, we can assume that ␣ j are real variables. In addition, we have performed numerical simulations of the original system that indicate that this assumption yields to correct results.
Figure 4 shows the disposition of qualitatively different
dynamics in the parameters space spanned by ␥ and K for a
particular value of ␰. Like in 关13兴 we find that three
codimension-two points organize the parameter space:
Takens-Bogdanov 共A兲, degenerate pitchfork 共B兲, and saddlenode separatrix loop 共D兲 关32兴. The three codimension-two
points collapse at ␰ = ␥ = 0 关see Fig. 2 and expressions 共29兲
and 共7兲兴. Line D approaches the origin linearly: ␰D共␥ → 0兲
= a␥ with a ⯝ 0.493, suspiciously close to 21 .
One can better understand Fig. 4 by looking at the panels
of Fig. 5, in which phase portraits for qualitatively different
states are shown. In the rightmost part of Fig. 4, ␥ ⬎ ␥B
= ␰1/3, the distribution becomes unimodal, and thus the standard route to partial synchronization is found. In the leftmost
part, ␰ ⱕ ␥ ⬍ ␥D ⯝ 0.599 97 共KD ⯝ 3.7646兲, we have the same
route than in Sec. III, i.e., a SW state limited by Hopf and
SNIC bifurcations. In contrast, in the central part of the
phase diagram 共around point A兲, there exist two regions with
bistability where the observed asymptotic state depends on
the initial conditions. In one region 共SW/PS兲 standing waves
and partial synchronization coexist, and the SW state 共a limit
cycle兲 disappears via a heteroclinic collision with the saddle
points born at mirror saddle-node bifurcations. In the second
region 共I/PS兲 incoherence and partial synchronization coexist.
I/PS
SN
3
Pitchfork
Hopf
B
I
2
2. Nontrivial fixed points (partial synchronization)
A saddle-node bifurcation occurs when P共X兲 in Eq. 共20兲
has exactly two roots 共one of the twofold degenerate兲. And
this bifurcation point can be determined numerically finding
the value of k where the discriminant of P共X兲 vanishes. The
scenario is similar to the one observed for ␰ = ␥, but in this
case the saddle solution X共1兲 ⬎ 0 exists up to the pitchfork
bifurcation with the origin at K = K P. If the distribution is
unimodal X共1兲 ⬍ 0, which makes this solution not valid.
A
0.6
γ
Bimodal
Unimodal
The bifurcation is subcritical, and it switches to supercritical
when the distribution becomes unimodal at ␥ ⬎ ␰B1/3. The loci
of Hopf and pitchfork bifurcations collide at the
codimension-two point where KH = K P and ␰ = ␰A. This point
is of the double zero eigenvalue type 共Takens-Bogdanov兲
关31兴.
The boundaries 关Eqs. 共24兲 and 共30兲兴 for Hopf and pitchfork instabilities have also been obtained following a different approach in Appendix B.
SW/PS
Het
SW
K
2␥共1 − ␰兲
2
=
.
KP =
␲g共0兲
␥−␰
SNIC
0.8
FIG. 4. Phase diagram for ␰ = 0.5. Solid lines mark the bifurcations: saddle-node off the limit cycle 共SN兲, SNIC, Hopf bifurcation
关Eq. 共24兲兴, heteroclinic bifurcation 共found numerically using the
reduced equations兲, and pitchfork bifurcation 关Eq. 共30兲兴. Three big
circles signal the codimension-two points: 共a兲 Takens-Bogdanov, 共b兲
degenerate pitchfork, and 共d兲 saddle-node separatrix loop. The open
symbols correspond to different bifurcations found by numerical
integration of Eq. 共1兲 with N = 2000. Filled symbols inside each
region indicate parameter values for the phase portraits in Fig. 5.
Bifurcation lines in Fig. 4 are calculated from analytical
results and from numerical integration of the ODEs 关Eqs.
共18兲兴. Empty symbols in the figure show the bifurcations
determined integrating the Kuramoto model with N = 2000.
The agreement is good and confirms the validity of the OA
ansatz.
Codimension-two point A
In this section we make a short digression about the
codimension-two point A and the importance of the symmetries in the model. Point A in Fig. 4 is a Takens-Bogdanov
point of system 关Eqs. 共18兲兴 that has O共2兲 symmetry. This
stems from the inherent O共2兲 symmetry of the Kuramoto
model 关with symmetric g共␻兲兴. Numerics show that the
asymptotic dynamics occurs in the real plane—i.e., Eqs. 共18兲
with real coordinates—where the symmetry group is only
Z2 傺 O共2兲. This symmetry imposes the global bifurcation
共Het兲 to be nontangent to the Hopf line 关31兴, in contrast with
a nonsymmetric Takens-Bogdanov point. Two scenarios are
possible around the odd-symmetric Takens-Bogdanov point
关31兴. Hence, one may wonder if the alternative scenario, involving a saddle-node bifurcation of limit cycles, might also
be found in the Kuramoto model.
The scenario that we have presented in this section 共see
also 关13兴兲 is apparently the same one Bonilla et al. 关15兴 uncovered in the neighborhood of the Takens-Bogdanov point
for the Kuramoto model with additive noise and a bidelta
frequency distribution. In that work the full O共2兲 symmetry
is taken into account. Crawford and Bonilla et al. 关12,15兴
found that, due to the O共2兲 symmetry, the degenerate Hopf
bifurcation gives rise to a branch of unstable traveling wave
solutions, in addition to the stable SW. According to 关15兴
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EXISTENCE OF HYSTERESIS IN THE KURAMOTO MODEL…
x1− x2
0.1
0.1
0
0
−0.1
−0.1
(a, ) PS
(b, ) PS
−1
0
1
−1
0
1
x1− x2
0.1
0.1
0
ACKNOWLEDGMENTS
0
−0.1 (c, ) SW/PS
−1
−0.1
(d, ) I/PS
0
1
−1
0
1
0.1
x1− x2
some region in the neighborhood of the unimodal limit
共␰ , ␥兲 = 共0 , 0兲 共see Fig. 2兲.
We expect a wide family of bimodal distributions to exhibit the same qualitative features as that of Fig. 2: the hysteretic region exists at the bimodal side of the unimodalbimodal border, and it shrinks as the nonregular unimodalbimodal transition 关g⬙共0兲 = ⬁兴 is approached. Moreover the
absence of hysteresis for g共0兲 = 0 should be found in any
bimodal distribution if the dependence is quadratic—as in
our distribution 关Eq. 共5兲兴—or has a larger power: g共␻兲
⬀ 兩␻兩␯ for small ␻, with ␯ ⱖ 2.
0.1
0
D.P. acknowledges supports by CSIC under the Junta de
Ampliación de Estudios Programme 共JAE-Doc兲 and by Ministerio de Educación y Ciencia 共Spain兲 under Project No.
FIS2006-12253-C06-04. E.M. acknowledges the financial
support provided by the Centre de Recerca Matemàtica
共CRM兲, 08193 Bellaterra, Barcelona, Spain.
APPENDIX A: PROOF OF THE TRANSVERSAL
STABILITY OF FIXED POINT X(2) IN SEC. III
0
−0.1
−0.1
(e, ) SW
−1
(f, ) I
0
x +x
1
1
−1
2
0
x +x
1
1
2
Global phase shift invariance, 共␣1 , ␣2兲 → 共␣1ei␤ , ␣2ei␤兲,
allows us to reduce Eqs. 共18兲 in one dimension by passing to
polar coordinates, ␣ j = ␳ jei␾ j, and defining the phase difference ␺ = ␾1 − ␾2. We obtain three ODEs,
FIG. 5. Phase portraits in 共rotated兲 x1 , x2 coordinates for qualitatively different cases. Each panel corresponds to a value of ␥ and
K at the position of a filled symbol in Fig. 4. 共a兲 and 共b兲 Partial
synchronization with K = 4 and 共a兲 ␥ = 0.6 and 共b兲 ␥ = 0.75; 共c兲 coexistence SW/PS—␥ = 0.67, K = 3.45; 共d兲 coexistence I/PS—␥ = 0.7,
K = 3.3; 共e兲 SW—␥ = 0.6, K = 3.5; and 共f兲 I—␥ = 0.6, K = 2.5.
these traveling wave solutions should disappear at a certain
K ⬍ K P in a local bifurcation with the saddle fixed points X共1兲
born at the SN bifurcations. This bifurcation reverses the
transversal stability of the saddle fixed points, which in turn
makes congruent the pitchfork bifurcation of these fixed
points with the completely unstable fixed point at origin. We
think these traveling wave solutions and their associated bifurcations are captured by the reduced Eqs. 共18兲 because the
OA ansatz has retained the O共2兲 symmetry of the model.
This means that although the relevant dynamics 共the attractors兲 are inside the real plane of 共␣1 , ␣2兲, physical unstable
objects 共traveling waves兲 “live” outside this plane.
V. CONCLUSIONS
We have investigated the routes to synchronization in the
Kuramoto model with a bimodal distribution constructed as
the difference of two unimodal distributions of different
widths. These distributions admit an arbitrarily deep and narrow central dip, which is not achievable in distribution types
considered in the past. This has allowed us to reinforce and
extend the results recently published in 关13兴.
We have found that bimodal distributions 关Eq. 共5兲兴 near
unimodality produce hysteretic phase transitions except in
␳˙ 1 = − ␳1 + k共␳1 − ␰␳2 cos ␺兲共1 − ␳21兲,
共A1a兲
␳˙ 2 = − ␥␳2 + k共␳1 cos ␺ − ␰␳2兲共1 − ␳22兲,
共A1b兲
␳1␳2␺˙ = − k关共1 − ␰兲␳21␳22 + ␳21 − ␰␳22兴sin ␺ .
共A1c兲
In Sec. III we took ␰ = ␥ and found that twin saddle-node
bifurcations 共namely, SNICs兲 give rise to two pairs of fixed
points. Here we prove 共we rather sketch the proof兲 the transversal stability of the mirror fixed points associated to X共2兲
via Eq. 共21兲.
First of all note that X共2兲 yields a fixed point 共x1 , x2兲 and
its mirror image, with x1 and x2 having the same sign, ␺ = 0.
This is a consequence of Eq. 共21兲 because X共2兲 ⬍ 1 − 1 / k. The
latter inequality stems from the fact that Q共1 − 1 / k兲 = −␥2
⬍ 0 and by continuation of the solutions from k = ⬁:
limk→⬁ X共1兲共k兲 = 0, limk→⬁ X共2,3兲共k兲 = 1.
Therefore we have to prove that factor
F = 共1 − ␥兲␳21␳22 + ␳21 − ␥␳22
共A2兲
in Eq. 共A1c兲 for ␺˙ is positive. Replacing ␳21 = X共2兲 and ␳22
= X共2兲 + 1 / k, we have
2
+ X共2兲共1 + 1/k兲兴 − ␥/k.
F = 共1 − ␥兲关X共2兲
共A3兲
As X共2兲 exists only above the saddle-node bifurcation 共k
ⱖ kSN兲 and kSN ⬎ kH = 共1 + ␥兲 / 共1 − ␥兲,
F ⬎ 共1 − ␥兲h
with
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PHYSICAL REVIEW E 80, 046215 共2009兲
DIEGO PAZÓ AND ERNEST MONTBRIÓ
2
h = X共2兲
+ X共2兲 − ␥/共1 + ␥兲.
共A5兲
Then h ⬎ 0 is a sufficient condition for the transversal stability of the fixed point.
It suffices to prove that h is positive at the locus of the
saddle-node bifurcation because X共2兲共k , ␥兲 exhibits its minimal value over k precisely at the bifurcation: X共2兲共k
⬎ kSN , ␥兲 ⬎ X共2兲共kSN , ␥兲. For our aim it is better to parametrize the SNIC line by k instead of ␥. Hence we introduce in
Eq. 共A5兲 the expressions
共i兲 ␥ as a function of kSN via Eq. 共26兲 and
共ii兲 X共2兲共kSN兲, determined from Eq. 共25兲 in the twofold
root case.
The calculation of terms 共i兲 and 共ii兲 can be readily done
with symbolic software such as MATHEMATICA. As a result
we obtain a function h共kSN兲 that is positive in all the domain
of kSN 苸 共1 , ⬁兲.
Moreover using expressions 共27兲 and 共28兲 we can get approximate expression for h 共as a function of ␥兲,
h共␥ → 0兲 =
冉冊
␥
2
2/3
+ O共␥兲,
h共␥ → 1兲 ⯝ 0.0858.
the incoherent state when the model is perturbed with additive white noises. In this case, the right-hand side of Eq. 共1兲
has to be supplemented with uncorrelated fluctuating terms
␩i satisfying 具␩i共t兲␩ j共t⬘兲典 = 2␴␦ij␦共t − t⬘兲. So far a counterpart
of the Ott-Antonsen ansatz for the stochastic problem has not
been found. It is nonetheless possible to obtain the stability
boundary of incoherence resorting to the Strogatz-Mirollo
relation for the discrete spectrum of eigenvalues ␭ 关22兴,
K
2
APPENDIX B: STABILITY OF THE INCOHERENT STATE
IN THE KURAMOTO MODEL WITH NOISE
KP =
−⬁
g共␻兲
d␻ = 1.
␭ + ␴ + i␻
共B1兲
KH = 2 + 2␥ + 4␴ ,
共B2兲
2共␥ + ␴兲共1 − ␰兲共1 + ␴兲
,
共␥ − ␰兲 + ␴共1 − ␰兲
共B3兲
which indeed reduce to Eqs. 共24兲 and 共30兲 for ␴ = 0. The
location of the Takens-Bogdanov point 关cf. Eq. 共29兲兴 also
varies and now pitchfork and Hopf bifurcations collide 共KH
= K P兲 at
For the sake of completeness and as a double check of
some of the results obtained, we study here the stability of
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⬁
Considering the distribution of frequencies 关Eq. 共5兲兴, this
equation can be solved for the eigenvalues ␭.
Noise increases the domain of the incoherent state. Hopf
and pitchfork bifurcations continue to occur, but the values
of K are shifted to larger values. We obtain
共A6兲
共A7兲
冕
␰A =
冉 冊
␥+␴
1+␴
2
.
共B4兲
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noise 关14,15兴.
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关18,33,34兴. In this context the bimodal distribution arises naturally as the superposition of the two unimodal distributions.
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EXISTENCE OF HYSTERESIS IN THE KURAMOTO MODEL…
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