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Accepted Manuscript
Twisted states in nonlocally coupled phase oscillators with bimodal
frequency distribution
Yuan Xie, Shuangjian Guo, Lan Zhang, Qionglin Dai, Junzhong Yang
PII:
DOI:
Reference:
S1007-5704(18)30270-3
https://doi.org/10.1016/j.cnsns.2018.08.008
CNSNS 4622
To appear in:
Communications in Nonlinear Science and Numerical Simulation
Received date:
Revised date:
Accepted date:
15 April 2018
26 June 2018
20 August 2018
Please cite this article as: Yuan Xie, Shuangjian Guo, Lan Zhang, Qionglin Dai, Junzhong Yang,
Twisted states in nonlocally coupled phase oscillators with bimodal frequency distribution, Communications in Nonlinear Science and Numerical Simulation (2018), doi:
https://doi.org/10.1016/j.cnsns.2018.08.008
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ACCEPTED MANUSCRIPT
Highlights
• Twisted states in nonlocally coupled phase oscillators with bimodal frequency distribution are studied.
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• Two different types of twisted states, twisted standing waves and stationary twisted
states, appear successively with the increase of the coupling strength.
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• The twisted states and the stabilities are theoretically analyzed with the assistance
of Ott-Antonsen ansatz.
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Twisted states in nonlocally coupled phase oscillators with
bimodal frequency distribution
Yuan Xie, Shuangjian Guo, Lan Zhang, Qionglin Dai, Junzhong Yang∗
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School of Science, Beijing University of Posts and Telecommunications,
Beijing, 100876, People’s Republic of China
Abstract
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Twisted states, referred to traveling waves in an array of nonlocally coupled phase oscillators, have drawn some attention in recent years. In this work, we study a one-dimensional
ring of phase oscillators with nonlocal coupling and a bimodal natural frequency distribution. We show that twisted standing waves and stationary twisted states appear
successively with the increase of the coupling strength. In the continuum limit, we derive a low-dimensional reduced equation using the Ott-Antonsen ansatz, which verifies
the twisted states in the simulations of finite networks of oscillators. We also theoretically investigate the stationary twisted states and their stabilities by using the reduced
equation.
1. Introduction
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Network of coupled oscillators provides a platform for studying dynamical behaviors
and spatiotemporal patterns in different areas of science [1, 2, 3]. One well-known system
is the Kuramoto model [4, 5, 6] of phase oscillators. In an original Kuramoto model, N
phase oscillators are globally coupled to each other by a sinusoidal function of phase
difference with a same coupling strength. The natural frequencies of phase oscillators
are randomly drawn from a given probability distribution. The Kuramoto model together
with its variants have played a central role in the study of coupled oscillator systems in
physics and biology since it was first proposed by Kuramoto in 1975 [7].
Over the last decade, a ring of identical phase oscillators coupled nonlocally has attracted much attention [4, 8, 9, 10, 11, 12]. Several interesting phenomena have been
studied in this system. One of them is the chimera dynamics which shows the alternations
between the regions of synchronous oscillators and asynchronous oscillators [4, 10, 11].
Another simpler dynamics is the twisted state, a traveling wave state where the phase
difference between neighboring oscillators is fixed and a fixed profile of oscillators’ phases
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Keywords: Twisted standing waves, Nonlocal coupling, Twisted states, Ott-Antonsen
ansatz
15
∗ Corresponding
author
Email address: jzyang@bupt.edu.cn (Junzhong Yang)
Preprint submitted to Journal of LATEX Templates
August 21, 2018
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travels at a constant speed. Generally, a uniformly twisted state is characterized by an
integer winding number q in which the phase of oscillators varies linearly with spatial
position in the form of θj = ωt + 2πqj/N + C, j = 1, . . . , N (ω is the natural frequency
and C is any constant) [13]. Lately, partially coherent twisted states have been investigated in non-identical phase oscillators [14, 15, 16]. A partially coherent twisted state in
heterogeneous networks is described by a spatially-dependent complex order parameter
whose argument is a linear function of space position. The partially coherent twisted
states and their stabilities have been theoretically investigated in the continuum limit by
using the Ott-Antonsen (OA) ansatz [17, 18].
Up to now, twisted states have been investigated on phase oscillators whose natural
frequency distribution g(ω) follows a unimodal one. For globally coupled phase oscillators
with g(ω) symmetrical about its maximum at ω = 0, there exists a critical coupling
strength Kc = 2π/g(0) above which the coupled oscillators transit from the incoherent
states to partially synchronous states consisting of a unique synchronous cluster [19, 20,
21, 22]. On the other hand, if g(ω) follows a bimodal distribution, globally coupled
oscillators allow for standing wave states consisting of two synchronous clusters with
different average frequencies [23, 24, 25]. Increasing the coupling strength turns the
incoherent state to standing wave states and further to partially synchronous states
with a unique synchronous cluster. Now, it is natural to ask how bimodal frequency
distribution influences the twisted states in nonlocally coupled phase oscillators. In this
work, we investigate the effects of bimodal frequency distribution on twisted states in
an array of nonlocally coupled phase oscillators. We find twisted standing wave states
when incoherence states become unstable and stationary twisted states at strong coupling
strength where the profile of spatially-dependent complex order is frozen in space.
The paper is arranged as following. In Section 2, we describe the model. In Section 3,
we present numerical results on different types of twisted states. In the continuum limit,
we derive a low-dimensional reduced equation by using the OA ansatz and theoretically
study these twisted states. The paper concludes with a summary in Section 4.
We consider a one-dimensional ring of N nonlocally coupled phase oscillators in which
the individual unit is coupled to M neighbors on each side with the coupling strength
K,
θ̇j = ωj +
k=M
X
K
sin(θj+k − θj ).
2M + 1
(1)
k=−M
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2. Model
The subscript j denotes the unit index and the periodic boundary condition is imposed. M denotes the coupling range. To be convenient, we introduce the coupling
radius σ = M/N , which ranges from 1/N (nearest neighbour coupling) to 0.5 (global
coupling). The natural frequency ωj is randomly chosen from the frequency distribution
g(ω). We consider g(ω) to be a superposition of two Lorentzian distributions
g(ω) =
γ
1
1
[
+
],
2
2
2π (ω − ω0 ) + γ
(ω + ω0 )2 + γ 2
3
(2)
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Wj
=
k=M
X
1
eiθj+k ,
2M + 1
k=−M
we reformulate Eq. (1) as
θ̇j
= ωj +
K
[Wj e−iθj − W j eiθj ],
2i
3. Results
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where the overbar indicates complex conjugate.
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where γ is the half width of each Lorentzian and ±ω0 are the center frequencies of each
Lorentzian. The distribution g(ω) is a bimodal one if ωγ0 > √13 and becomes a unimodal
one otherwise [23]. In this work, we focus on the case with bimodal distribution g(ω)
and set γ = 0.5 and ω0 = 1.5.
By defining spatially-dependent complex order parameter Wj = Rj eiΘj as
(3)
(4)
In this section, we start with the numerical simulations of Eq. (1) by using fourthorder Runge-Kutta algorithm with a time step δt = 0.01. Throughout this work, we let
N = 2000. Traditionally, the coherence in coupled phase oscillators is characterized by
the global order parameter defined as
iΘ
M
Re
N
1 X iθj
=
e .
N j=1
Here R (0 ≤ R ≤ 1) measures the coherence of the oscillator population and Θ is the
average phase. However, as shown by Eq. (4), each oscillator is driven by a unique
spatially-dependent order parameter.
P To reflect this fact, we consider another global
quantity h|W |i, defined as h|W |i = j |Wj |, to characterize the coherence in the model.
We investigate the transition scenario in Eq. (1) from the incoherence to synchronous
behaviors with the coupling strength K. For each coupling strength, we consider dozens
of realizations with random initial conditions. For each realization, we calculate R and
h|W |i. Fig. 1(a) presents R and h|W |i against the coupling strength K at the coupling
radius σ = 0.035. The incoherence becomes unstable for the coupling strength above a
threshold around K = 2. For strong coupling strength such as K > 5, R and h|W |i,
acquired from different realizations, against K fall onto several curves, which suggests the
coexistence of several distinct states at a given coupling strength. Actually, these distinct
states represent partially synchronized twisted states with different winding numbers. To
verify it, we calculate R and h|W |i against K for prepared initial conditions θj = 2πqj/N
with q = 0, ±1, ±2, ±3, · · · . The corresponding results presented in Fig. 1(a2) reproduce
the characteristics in Fig. 1(a1). Figs. 1(a1) and (a2) suggest that twisted states with
the winding number |q| ≥ 4 do not exist when K < 10 and the critical coupling strengths
for twisted states with different winding numbers increase with |q|. In these figures, we
find that h|W |i is always larger than R except for the twisted state with q = 0. The
reason is that the contributions to R by two antiphased oscillators in a twisted state with
q 6= 0 are always offset by each other. As shown below, the partially coherent states in
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(5)
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1
1
(a1)
0
2
4
6
8
10
0
0
2
4
6
1
1
(b2)
(a2)
W
R,
W
R,
q= 2
q= 1
0
0
q= 3
2
4
6
8
10
0
0
2
4
6
K
q= 2
8
10
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K
10
q=0
q=0
q= 1
8
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R,
R,
W
W
(b1)
M
Figure 1:
The global order parameter R (in black) and the average of spatially-dependent order
parameter h|W |i (in red) are plotted against the coupling strength K for the coupling radius σ = 0.035 in
(a1,a2) and for σ = 0.1 in (b1,b2). The top row shows the results for dozens of random initial conditions
at each K while the bottom row shows the results for prepared initial conditions θj = 2πqj/N with
q = 0, ±1, ±2, ±3, · · · . The green arrows mark the different twisted states. Blue dashed arrows show
critical coupling strength for the partially synchronized twisted states with different winding number q.
The critical coupling strength K = 4.6, 5.4, 5.8, 7.2 for q = 0, ±1, ±2, ±3 in (a2) and K = 4, 4.5, 9.1 for
q = 0, ±1, ±2 in (b2), respectively. Other parameters: ω0 = 1.5 and γ = 0.5.
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the intermediate coupling strength, 2 < K < 5, are also twisted states though they are
hardly identified in Figs. 1(a1) and (a2). The transition scenario described above holds
at larger coupling radius, for example σ = 0.1 in Figs. 1(b1) and (b2).
Then, we examine the twisted states for σ = 0.035 in details by using prepared initial
conditions θj = 2πqj/N (q = 1, 2). The top two panels in Fig. 2 show the results for K =
3, an intermediate coupling strength. The snapshots of θj in Figs. 2(a1,b1) show that
oscillators distribute their phases in the range of 2π non-uniformly and oscillators tend
to crowd themselves around the mean phase Θj of the spatially-dependent complex order
parameter Wj (the red curves). According to the definition of the partially synchronized
twisted states [15, 16], the states in Figs. 2(a1,b1) are the twisted ones with q = 1 and
q = 2. The partial coherence of the states is demonstrated by plotting the effective
frequency ωe , defined as ωe = h dθ
dt it with h·it the average over a long time interval after
transient, against the natural frequency ω. As illustrated in Figs. 2(a2,b2), the graph of
ωe (ω) for each twisted state exhibits two plateaus with opposite ωe . That is, there exist
two synchronous clusters with different average frequencies in the states, a typical feature
of standing wave states. The periodic variation of |Wj | in Figs. 2(a3,b3) and that Θj
oscillates periodically without drifting along space in Figs. 2(a4,b4) further demonstrate
the existence of twisted standing wave states in the model (1) at an intermediate coupling
strength.
The bottom two panels in Fig. 2 show the results at K = 10, a strong coupling
strength. The snapshots in Figs. 2(c1,d1) demonstrate the twisted states with q = 1 and
q = 2 where most oscillators condensate their phases to the mean phase Θj of spatially5
95
100
0
(a1)
0.5
(a3)
650
t
t
-5
-5
5
2000
(d3)
2000
0
6000
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j
2000
-20
t
20
0
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0
2000
-20
0
K=3,q=2
2000
(c4)
t
0
j
,
2000
6000
e
j
20
600
0
6000
K=10,q=1
2000
(d4)
t
-20
(d2)
20
M
(d1)
2000
t
,
2000
0
600
0
6000
j
-20
0
2
2000
(b4)
650
(c3)
(c2)
e
j
20
0
650
t
e
,
2000
(c1)
2
2000
(b3)
j
0
0
600
600
0
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(b2)
5
j
2
-5
-5
K=3,q=1
t
(b1)
j
650
,
2000
0
(t)
(a4)
j
j
0
2
0
1
|W |(t)
e
j
2
(a2)
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2000
j
2000
0
j
2000
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Figure 2: The partially synchronized twisted states in the phase oscillators model Eq. (1) with bimodal
natural frequency distribution Eq. (2). (a,b) The twisted standing wave states with q = 1 and q = 2 at
K = 3. (c,d) The stationary twisted state with q = 1 and q = 2 at K = 10. The first column shows the
snapshots of oscillators’ phases and red solid line shows the argument Θj of the spatially-dependent order
parameter Wj . The second column shows the graphs of the effective frequencies of oscillators ωe (ω).
The third and the fourth columns show the evolutions of the spatially-dependent order parameter |Wj |
and its argument Θj , respectively. Other parameters: σ = 0.035, ω0 = 1.5, and γ = 0.5.
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∂
∂f
+
(f v) = 0,
∂t
∂θ
v=ω+
115
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where
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dependent order parameter. Only one plateau with the frequency at around zero, the
mean natural frequency, in the graphs of ωe (ω) in (c2,d2) indicates that there is only
one synchronous cluster in the oscillators. The evolutions of |Wj | in (c3,d3) and Θj in
(c4,d4) suggest that the twisted states at K = 10 are roughly stationary.
The twisted states in Eq. (1) can be studied with the assistance of the Ott-Antonsen
ansatz (OA ansatz) [17, 18]. Supposing the length of the ring to be 2π and the position
of oscillator j to be xj = 2πj/N , we consider the continuum limit (N → ∞) with the
coupling radius σ unchanged. We consider the probability density function f (x, ω, θ, t)
defined as the fraction of oscillators with phases between θ and θ + dθ and natural
frequencies between ω and ω + dω at time t and position x. The density f (x, ω, θ, t)
satisfies the continuity equation
K
[W e−iθ − W eiθ ].
2i
(6)
(7)
The spatially-dependent complex order parameter W (x, t) can be reformulated as
Z 2π
Z ∞ Z 2π
W (x, t) =
G(x − y)
f eiθ dθdωdx
(8)
−∞
0
1
4πσ ,
M
with
G(x) =
0,
0
|x| < 2πσ
otherwise
(9)
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Using OA ansatz, we write the probability density function f (x, ω, θ, t) as
(10)
PT
∞
X
g(ω)
[a(ω, x, t)]n einθ + c.c.]
f (θ, ω, x, t) =
[1 +
2π
n=1
where c.c. is the complex conjugate of the previous term. By substituting Eqs. (7,10)
into Eq. (6), we have
For the bimodal frequency distribution Eq. (2), we have
Z ∞ Z 2π
1
f (θ, ω, x, t)eiθ dθdω = [a(ω0 − iγ, x, t) + a(−ω0 − iγ, x, t)].
2
−∞ 0
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da(x, t)
K
= iωa(ω, x, t) + [W (x, t) − a2 (ω, x, t)W (x, t)].
dt
2
Defining u1 (x, t) = a(ω0 − iγ, x, t) = b1 eiϕ1 and u2 (x, t) = a(−ω0 − iγ, x, t) = b2 eiϕ2 , we
have
∂u1 (x, t)
= (−γ + iω0 )u1 (x, t) +
∂t
∂u2 (x, t)
= (−γ − iω0 )u2 (x, t) +
∂t
K
[W (x, t) − W (x, t)u21 (x, t)]
2
K
[W (x, t) − W (x, t)u22 (x, t)]
2
7
(11)
0.0
(a1)
0
(b2)
650
600
2
650
2
(c2)
0
2
(d2)
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600
0
x
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0
600
600
0
2
(d3)
700
t
1,2
700
2
(c3)
t
(d1)
600
0
t
M
t
600
2
(b3)
700
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600
0
arg[W(x,t) ]
t
0
700
1,2
0
600
2
(a3)
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t
0
600
(c1)
2
600
650
1,2
2
650
(b1)
0
650
t
0
600
2
1.0
W(x,t)
t
650
1,2
2
0.50
(a2)
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600
0
x
2
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Figure 3: The partially synchronized twisted states in the reduced OA model Eq. (11). (a,b) The twisted
standing wave states with q = 1 and q = 2 at K = 3. (c,d) The stationary twisted states with q = 1
and q = 2 at K = 10. The first column shows the time evolutions of ϕ1 (in black) and ϕ2 (in red) for
one of OA oscillators. The second and the third columns show the evolutions of the spatially-dependent
order parameter |W (x, t)| and its argument arg[W (x, t)], respectively. Other parameters: σ = 0.035,
ω0 = 1.5, and γ = 0.5.
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125
R 2π
with W (x, t) = 12 0 G(x − y)[u1 (y, t) + u2 (y, t)]dy.
The dynamical behaviors in Fig. 2 can be reproduced by simulating Eq. (11) (the
OA oscillators model). We take the coupling radius σ = 0.035 as an example. To build
twisted states with the winding number q, we prepare initial conditions taking the form
qx iqx
)e . The top two panels in Fig. 3 show the twisted standing wave states
u1,2 (x, 0) ∼ ( 2π
with q = 1 and q = 2 at the intermediate coupling strength K = 3, respectively. The
time sequences of one of OA oscillators in Figs. 3(a1) and (b1) show that ϕ1 (x, t) and
ϕ2 (x, t) oscillate at the same frequency but in opposite directions. Based on the simulation results, we may express u1,2 as u1 (x, t) = b(t)eiΩt+qx and u2 (x, t) = b(t)e−iΩt+qx .
That is, u1 and u2 support their own traveling waves and the two waves propagate in
opposite directions. Then we have W (x, t) = b(t) sin(2πσ) cos(Ωt)eiqx /2πσ, which is
in agreement with the homogeneous oscillating |W (x, t)| in Figs. 3(a2) and (b2) and
spatially-dependent and time-independent argument of W (x, t) in Figs. 3(a3) and (b3).
All of these demonstrate the presence of the twisted standing wave states at K = 3.
The winding numbers q of these two twisted states can be read from the variation of the
argument of Z(x, t) along space at an arbitrary time, for example q = 1 in Fig. 3(a3) and
q = 2 in Fig. 3(b3). The bottom two panels in Fig. 3 show the twist states with q = 1
and q = 2 at the strong coupling strength K = 10. Clearly, either the time sequences of
one of OA oscillators in Figs. 3(c1) and (d1) or the spatiotemporal evolutions of Z(x, t)
in Figs. 3(c2,c3) and (d2,d3) demonstrate that the twisted states are stationary.
Using the OA oscillators model Eq. (11), the twisted state at strong coupling strength
can be solved rigorously and their stabilities can be analyzed. Partially synchronized
twisted states with winding number q can be written as u1,2 (x, t) = b1,2 ei(qx+ξ1,2 t+ν1,2 )
where b1,2 > 0, ξ1,2 and ν1,2 are real numbers. Substituting the solution into Eq. (11),
we have
K
iξ1,2 b1,2 = (−γ ± iω0 )b1,2 + [Ze−i(qx+ξ1,2 t+ν1,2 )
2
2
i(qx+ξ1,2 t+ν1,2 )
−Zb1,2 e
],
(12)
where ξ1 = ξ2 is required. By letting ν1 − ν2 = φ, we have
= (−γ + iω0 )b1 +
b
K G(q)
[b1 + b2 e−iφ − (b1 + b2 eiφ )b21 ],
4
= (−γ − iω0 )b2 +
b
K G(q)
[b1 eiφ + b2 − (b1 e−iφ + b2 )b22 ]
4
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iξb1
(13)
R 2π
2πσq
b
with G(q)
= 0 G(x) cos(qx)dx = sin2πσq
. Equating real and imaginary parts of these
equations, we have two sets of solutions. One is a symmetrical solution with b1 = b2 and
φ < π/2, determined by
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iξb2
ξ
=
0,
0
=
b
ω0
γ
K G(q)
+
−
,
sin φ 1 + cos φ
2
0
=
(
b
ω0 2
γ
Kγ G(q)
) +(
)2 −
,
2
2
1+b
1−b
2(1 − b2 )
9
(14)
1
1
<|W|>
<|W|>
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q=1
q=3
q=1
0.6
0.8
10
15
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Figure 4: h|W |i of different stationary twisted states are plotted against the coupling strength at the
coupling radius σ = 0.035 in (a) and σ = 0.1 in (b). Numerical results are in black for N = 2000 and
blue for N = 10000, and theoretical results are in red. ω0 = 1.5, γ = 0.5.
and the other is an asymmetrical one with b1 6= b2 and π/2 < φ < 3π/2, determined by
0
=
0
= b1 b2 + cosφ,
b sin φ
K G(q)
(b22 − b21 ).
=
8 cos φ
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(15)
The stationary twisted states at strong coupling strength in Figs. 2 and 3 are symmetrical ones where φ is close to zero and their spatially-dependent order parameters
are frozen in space. For theseRstationary twisted states with different winding numbers
2π
q, h|W |i (defined as h|W |i = 0 |W (x, t)|dx) against the coupling strength K are presented in Figs. 4(a) and (b) for the coupling radius σ = 0.035 and σ = 0.1, respectively.
As a comparison, we also plot h|W |i acquired by simulating Eq. (1). We find that the
deviation of h|W |i between theoretical and numerical results is weak and increases with
K decrease. Moreover, we have carried out the numerical simulations for two different
N . As shown by Fig. 4, the results for N = 10000 are closer to the theoretical results
when compared with those for N = 2000. Thus, it can be inferred that the deviation
between theoretical and numerical results will go away as N goes to infinity.
Furthermore, we study the linear stability of stationary twisted states. We introduce
perturbation v(x, t) to a stationary twisted state with the winding number q such that
u1,2 = [b + v1,2 (x, t)]ei(qx+ν1,2 ) . Inserting u1,2 into Eq. (11), we have the evolution of v1,2
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2ω0 +
M
=
ξ
155
b sin φ
4γ
K G(q)
[1 + 3 cos2 φ −
],
b
4 cos φ
K G(q)
4γ
1 + cos2 φ −
− b21 − b22 ,
b
K G(q)
0
10
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q=1
q=2
q=3
0.5
9
6
3
12
15
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K
Figure 5: The amplitude |b| as a function of the coupling strength K for stationary twisted state
solutions of Eq. (14) with q = 0, 1, 2 and 3. Their stabilities are determined by Eq. (18). Solid: stable;
dashed: unstable. Pink dashed arrows show the critical coupling strength. Here, K = 4, 4.6, 8.9 for
q = 0, 1, 2, respectively, which agree well with the results shown in Fig. 1(b2). Other parameters:
ω0 = 1.5, γ = 0.5, σ = 0.1.
v̇1 (x, t)
K
−η1 v1 (x, t) +
4
Z 2π
+ e−iφ )
G(x − y)v1 (y, t)ei(y−x)q dy −
=
ED
[(1
M
as
PT
b2 (1
v̇2 (x, t)
CE
[(1
+
eiφ )
0
2π
Z
0
G(x − y)v 1 (y, t)e−i(y−x)q dy],
K
−η2 v2 (x, t) +
4
Z 2π
+ eiφ )
G(x − y)v2 (y, t)ei(y−x)q dy −
=
0
b2 (1
+
e−iφ )
Z
0
2π
G(x − y)v 2 (y, t)e−i(y−x)q dy]
∓iφ
b
+ b2 + b2 e±iφ ).
with η1,2 = K
4 G(q)(1 + e
Then, we look for solutions of Eq. (16) with the form v1,2 (x, t) = v1,2 eiqx eλt . Substituting this into Eq. (16) and after some algebras, we have the spectrum following
q
1
λ± (q) = (trB1,2 ± (trB1,2 )2 − 4detB1,2 ),
(17)
2
AC
170
(16)
11
ACCEPTED MANUSCRIPT
(1 − b2 )M1
Imη1 + K
(1 − b2 )H1
−Reη1 + K
8
8
B1 =
,
2
2
−Imη1 − K
−Reη1 + K
8 (1 + b )H1
8 (1 + b )M1
2
2
−Reη2 + K
Imη2 + K
8 (1 − b )M2
8 (1 − b )H2
B2 =
K
K
−Imη2 − 8 (1 + b2 )H2 −Reη2 + 8 (1 + b2 )M2
and
M1 = (1 + cos φ)h+ − ih− sin φ,
H1 = h+ sin φ + ih− (1 + cos φ),
M2 = (1 + cos φ)h+ + ih− sin φ,
H2 = −h+ sin φ + ih− (1 + cos φ),
b + q0 ) + G(q
b − q0 ),
h+ (q, q0 ) = G(q
AN
US
b + q0 ) − G(q
b − q0 ).
h− (q, q0 ) = G(q
CR
IP
T
where
The boundary of the stable stationary twisted states can be determined by
00
Re[λ+ (0)] = 0
00
M
180
with the second derivative of λ+ (q) with respect to its argument at q = 0.
As shown in Fig. 5, the critical coupling strength for the onset of stable stationary
twisted states increases with q and no stable twisted state with q > 2 exists in the
parameter regimes investigated here. Comparing with Fig. 1 where the critical values
of K for different stationary twisted states are marked by blue dashed line, theoretical
results agree well with the simulations. On the other hand, similar stability analysis can
be made on the asymmetrical twisted states Eq. (15) and we find that these states are
always unstable.
4. Conclusion
PT
AC
190
In summary, we studied a ring of nonlocally coupled phase oscillators with bimodal
natural frequency distribution and zero mean natural frequency. We found the existence of partially coherent twisted states. Depending on the coupling strength, there
are two types of twisted states, the twisted standing wave states for intermediate coupling strength which consist of two synchronous clusters oscillating in opposite directions
and the twisted states for strong coupling strength which consist of only one synchronous
clusters oscillating at the mean natural frequency. It is worthy to mention that, when the
frequency distribution of the oscillators is unimodal [16], partially synchronized twisted
states can be found whereas the twisted standing wave states do not exist. Moreover,
these twisted states are investigated with the assistance of OA ansatz and the stability
of the stationary twisted states are theoretically analyzed.
CE
185
ED
175
(18)
Acknowledge
195
This work was supported by the National Natural Science Foundation of China under
Grants No. 11575036 and No. 11505016.
12
ACCEPTED MANUSCRIPT
References
References
AC
CE
PT
ED
M
AN
US
CR
IP
T
[1] Barreto E, Hunt B, Ott E, So P. Synchronization in networks of networks: The onset of coherent
collective behavior in systems of interacting populations of heterogeneous oscillators. Phys Rev E
2008;77:036107.
[2] Ren L, Ermentrout B. Phase locking in chains of multiple-coupled oscillators. Physica D 2000;143:5673.
[3] Pikovsky A, Rosenblum M, Kurths J. Synchronization: a universal concept in nonlinear sciences.
Cambridge University Press; 2003.
[4] Abrams DM, Strogatz SH. Chimera states in a ring of nonlocally coupled oscillators. Int J Bifurcation Chaos 2006;16:21-37.
[5] Baesens C, Guckenheimer J, Kim S, MacKay RS. Three coupled oscillators: mode-locking, global
bifurcations and toroidal chaos. Physica D 1991;49:387-475.
[6] Strogatz S. From Kuramoto to Crawford: exploring the onset of synchronization in populations of
coupled oscillators. Physica D 2000;143:1-20.
[7] Kuramoto Y. International symposium on mathematical problems in theoretical physics. Lect Notes
Phys 1975;39:420.
[8] Kuramoto Y, Battogtokh D. Coexistence of coherence and incoherence in nonlocally coupled phase
oscillators. Nonlinear Phenom Complex Sys 2002;5:380-5.
[9] Laing CR. Fronts and bumps in spatially extended Kuramoto networks. Physica D 2011;240:196071.
[10] Laing CR. The dynamics of chimera states in heterogeneous Kuramoto networks. Physica D
2009;238:1569-88.
[11] Panaggio MJ, Abrams DM. Chimera states: Coexistence of coherence and incoherence in networks
of coupled oscillators. Nonlinearity 2015;28:R67.
[12] Omel’chenko OE. Coherence-incoherence patterns in a ring of non-locally coupled phase oscillators.
Nonlinearity 2013;26:2469-98.
[13] Wiley DA, Strogatz SH, Girvan M. The size of the sync basin. Chaos 2006;16:015103.
[14] Girnyk T, Hasler M, Maistrenko Y. Multistability of twisted states in non-locally coupled Kuramototype models. Chaos 2012;22:013114.
[15] Laing CR. Travelling waves in arrays of delay-coupled phase oscillators. Chaos 2016;26:094802.
[16] Omel’chenko OE, Wolfrum M, Laing CR. partially synchronized twisted states in arrays of coupled
phase oscillators. Chaos 2014;24:023102.
[17] Ott E, Antonsen TM. Low dimensional dehavior of large systems of globally coupled oscillators.
Chaos 2008;18:037113.
[18] Ott E, Antonsen TM. Long time evolution of phase oscillator systems. Chaos 2009;19:023117.
[19] Kuramoto Y. Chemical oscillations, waves and turbulence. Dover, editor. Berlin: Springer Berlin
Heidelberg; 1984.
[20] Pazó D. Thermodynamic limit of the first-order phase transition in the Kuramoto model. Phys Rev
E 2005;72:046211.
[21] Basnarkov L, Urumov V. Phase transitions in the Kuramoto model. Phys Rev E 2007;76:057201.
[22] Kloumann IM, Lizarraga IM, Strogatz SH. Phase diagram for the Kuramoto model with van Hemmen interactions. Phys Rev E 2014;89:012904.
[23] Martens EA, Barreto E, Strogatz SH, Ott E, So P, Antonsen TM. Exact results for the Kuramoto
model with a bimodal frequency distribution. Phys Rev E 2009;79:026204.
[24] Montbrio E, Kurths J, Blasius B. Synchronization of two interacting populations of oscillators. Phys
Rev E 2004;70:056125.
[25] Wang HB, Han WC, Yang JZ. Synchronous dynamics in the Kuramoto model with biharmonic
interaction and bimodal frequency distribution. Phys Rev E 2017;96:022202.
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