2023 RISP International Workshop on Nonlinear Circuits, Communications and Signal
Processing (RISP NCSP 2023), Feb. 28 - Mar. 3, 2023

Chimera States in a Two-Group Network of Kuramoto model
Toru Kumagawa, Yoko Uwate and Yoshifumi Nishio
Dept. Electrical and Electronic Engineering,
Tokushima University
2-1 Minami-Josanjima, Tokushima 7708506, Japan
Email:{kuma, uwate, nishio}@ee.tokushima-u.ac.jp

abstract

2. Kuramoto model

Synchronization phenomena can be seen in many aspectsof our daily life and has applications in a variety of fields,such as pacemakers and telecommunications. In recent years,
chimera states, where synchronous and asynchronous phenomena are mixed, have attracted much attention. The oscillators are phase-locked when synchronous, and incoherent
when asynchronous.
In this study, we proposed a network of oscillators divided
into two-group and coupled only in one part. We observed
the time variation of the synchronization rate in each groupand phase distribution of the oscillators. As a result, it wasobserved that the oscillators in two parts, a synchronous partand an asynchronous part.

The Kuramoto model is used in this study. We consider
a network of N coupled limit-cycle oscillators whose phaseare θi , i = 1, 2, ..., N . The equation of the Kuramoto model isshown in Eq. (1).
N∂θiK∑sin(θj − θi ).
= ωi +∂t
N j=1

(1)

At this time, the native frequencies ωi of the oscillators
are randomly distributed. Where K is the global couplingstrength.
These oscillators are located around a cycle and rotate attheir own frequencies to define a complex order parameter R.
N1. Introduction

Reiφ =

Networks of coupled system are very common in many
scientific and engineering fields. One very important phenomenon about this kind of network is the collective synchronization, where all the oscillators in a large-scale network
are locked to a common frequency or phase, although theirnative frequency are quite different and widely distributed.
Therefore, synchronizing a network of coupled oscillators isa very important issue for various mathematically describemodels, one of which is the famous Kuramoto model used
to study such collective synchronous phenomena in 1975. In
the Kuramoto model, oscillators are generally all-to-all coupled. Normally, if the oscillator is all-to-all coupled with acertain coupling strength, it is expected that it synchronise
completely with time. This study presents about a chimera
state in which an array of identical oscillators split into twodominants. Namely, one coherent and phase locked, the otherincoherent and asynchronized.
In this study, a system with two-group is proposed and synchronization state is investigated. By using computer simulations, we observe the phase state of the oscillators by changing coupling strength K.

1 ∑ iθje .
N j=1

(2)

Where φ indicates the average phase of the coupled oscillators, and the order parameter R (0 ≤ R ≤ 1) indicates ameasure of phase coherence. In the case of R = 1, it indicatesthat all oscillators come to a single tight clump. In the caseof R = 0, it indicates that oscillators are scattered uniformly
around the cycle. From Eq. (2), Eq. (3) is as follows.
∂θi= ωi − KR sin(θj − φ).
∂t(3)3. Proposed system
In conventional studies, all-to-all oscillators are coupled.
We proposed that oscillators are divided two parts. Oscillators are all-to-all coupled in the same group. Figure 1 showsa simplified version of coupling.
In this paper, ωi follows the Cauchy distribution. The number of oscillators N = 100 and the coupling strength K = 5.0
or 9.0. K is constant everywhere.

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2023 RISP International Workshop on Nonlinear Circuits, Communications and Signal
Processing (RISP NCSP 2023), Feb. 28 - Mar. 3, 2023

30'-'

t=0.0[s]25.0

percentage[%]2520.0
20

15.0
15

10.0
105.0
5:oscillaitor

0-3.0 -2.0 -1.0-3-2-10.0 1.001

2.0 3.023phase
30'-'

Figure 1: Simplified diagram of the couplingof the proposed system.

t=4.7[s]25.0

percentage[%]254. Simulation results

20.0
20

15.0

15

10.0

10

5.0

54.1 For Two-group case K = 5.0

0-3.0 -2.0 -1.0
-3

Figure 2 shows time evolution of R. R shows the overall
synchronization rate. R1 and R2 indicate the synchronization
rate of each group. R3 shows the synchronization rate of theoscillators connecting each group. R1 and R2 differ in theway they are synchronized. R2 is more slowly synchronizedthan R1.

-2-10.0 1.00

1

2.0 3.023

phase
Figure 3: Phase histogram in two-group.

Similar to Fig. 3 and 4 shows the distribution of phase. Onegroup is made up of 50 oscillators, each of which is found tobe locked phase. On the other hand, oscillators locked another group of phases are also seen.
R

phase

t=4.7

t[s]

Figure 2: Time evolution of R in two-group.

Number of Oscillators

Figure 4: Phase diagram in two-group.
Figure 3 shows phase histogram. When t=0.0, the oscillators are falling apart. It can be seen that the phase is trying
to remain constant over time. However, two peaks are seen.
This could be due to the division of the oscillators into twogroup.
1592023 RISP International Workshop on Nonlinear Circuits, Communications and Signal
Processing (RISP NCSP 2023), Feb. 28 - Mar. 3, 2023

4.2 For original Kuramoto model K = 5.0
The fact that the network is one-group is the same as in
the original Kuramoto model. Figure 5 shows time evolutionof R. The synchronization rate has increased compared toFig. 2.

t=5.0[s]RFigure 7: Phase diagram in one-group.

t[s]Figure 5: Time evolution of R in one-group.
Figure 6 shows the frequency distribution of phases as inFig. 3. When t = 0.0, the phases of all oscillators are scattered.
However, unlike the two-group case, only one peak are seen.

4.3 For Two-group case K = 9.0
The case of increased bond strength is shown below Here,coupling strength K = 9.0. Figure 8 shows that both groups
are synchronised and that the groups are synchronised witheach other.

Figure 8: Time evolution of R in two-group K = 9.0.

Figure 6: Phase histogram in one-group
Figure 7 shows the phase of each oscillator as in Fig. 4. It
shows that the phases are trying to become constant. Phase islocked some extent, but asynchronous phase changing can beseen.
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2023 RISP International Workshop on Nonlinear Circuits, Communications and Signal
Processing (RISP NCSP 2023), Feb. 28 - Mar. 3, 2023

Phase

Figure 9 shows the temporal variation of the frequency disFigure 10 shows phase diagram as Fig. 4 and Fig. 7. The
tribution. Unlike Fig. 3 and Fig. 6, Fig. 9 shows the time vari- phase is reached a more constant value than in Fig. 4. As ination of the frequency distribution in 0.1 second increments. Fig. 4, many phases are found to be locked two values.
Several oscillators are found to be freely phase-shifting. Most
t=5.0[s]phases are clustered in one place, but some oscillators are
asynchronous.

30

percentage[%]'-'

t=4.7[s]

25.0
25

20.0
20

Number of Oscillator

15.0
15Figure 10: Phase diagram in two-group K = 9.0.

10.0
10

5.0
50-3-2-1-3.0 -2.0 -1.0

0.0 1.0
phase 012.0

3.0

2330percentage[%]'-'

t=4.8[s]25.0

5. Conclusions

2520.020

15.0
1510.0105.050-3.0 -2.0 -1.0-3-2-1

0.0 1.0
phase 01

2.0

3.0

2330

In this study, we proposed two-group Kuramoto model.
The phase was seen to be fixed and freely changing. In otherwords, a mixture of synchronised and asynchronous oscillators states was observed. However, it was difficult to observe
even the specific behaviour of each oscillator. In the future,
We would like to investigate the time variation of the synchronised percentage and the phase of each oscillators. These
make the observation of the chimera states even clearer.

percentage[%]'-'t=4.9[s]25.0References

2520.020[1] Daniel M Abrams, Steven H Strogatz, ”Chimera
States for Coupled Oscillators” Physical Review letters,
vol.93,num.17, 174102, 2004.

15.0
1510.0105.0
50-3.0 -2.0 -1.0-3-2-1

0.0 1.0
phase 012.0

3.0

2330

percentage[%]'-'t=5.0[s]25.0[2] Hanquing Wang, Xiang Li, ”Synchronization andchimera states of frequency-weighted KuramotoOscillator network” Physical Review, E 83, 066214,
2011.

25

[3] Murray Shanaham, ”Metastable Chimera States in
Community-Structured Oscillator Networks” Chaos, 20,013108, 2010,(https://doi.org/10.1063/1.330545).

20.0
20

15.0
15

10.0
10

5.0
50-3-2-1-3.0 -2.0 -1.0

0

10.0 1.0
phase 22.03

3.0[4] Naziru M Awai,Domenico Bullara,and Irving R Epstein,”The smallest chimera:Periodicity and chaos in a pair ofcoupled chemical oscillators” Chaos, 29, 013131, 2019.

Figure 9: Phase histogram in two-group K = 9.0.

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