2024 IEEE International Symposium on Circuits and Systems (ISCAS) | 979-8-3503-3099-1/24/$31.00 ©2024 IEEE | DOI: 10.1109/ISCAS58744.2024.10558633

Synchronizations in Oscillatory Networks with
Memristor Couplings as Ring Structure
Yukinojo Kotani, Yoko Uwate and Yoshifumi Nishio
Dept. Electrical and Electronic Eng.,
Tokushima University
Email: {kotani, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract—In this study, we propose the coupled oscillatory
networks with the memristor couplings as a ring structure.
We focus on the dynamics of the memristor couplings, andinvestigate synchronization phenomena. As a result, this study
observes three different synchronization state types: multi-phasesynchronization state, in-phase and anti-phase synchronization
state and amplitude death. In addition, we analyze the power
consumption of the memristor couplings to make clear the causeof phase sift.

II. O SCILLATORY N ETWORK M ODEL
Figure 1 (a) shows the schematic model of the memristor.
Resistance value of the memristor is memristance M (q).
Memristance is defined as the gradient of the charge q - flux
φ characteristics curve in Fig. 1 (b).

I. I NTRODUCTION
Synchronization phenomena observed from the coupledoscillatory networks have been studied. Many studies related tosynchronization phenomena in oscillatory networks have important roles: optimizing the power operation of microgrid [1],[2], realizing associative memory for patients with prosopagnosia [3], [4] and analyzing the characteristics of networks [5][9]. Besides, network structures affect data communication. A
ring structure has the property of preventing communication
disconnected compared to other structures. Therefore, synchronization phenomena in the oscillatory networks as a ring
structure need to be investigated.
A memristor is the fourth basic circuit elements; a resistor,
a capacitor and an inductor. It is introduced by L. O. Chua
in 1971 [10], and it was developed by Hewlett-Packard Lab
in 2008 [11]. Resistance value of a memristor is characterized
by a charge or a flux, whereas a resistor is characterized bya current or a voltage. The charge and the flux are defined as
the integral of the current and the voltage respectively. The
memristor has the dynamics compared to the resistor.
In previous studies, synchronization phenomena in the coupled oscillatory networks were investigated. The coupled oscillatory networks via resistors, time-varying resistors and fifthpower resistors were proposed [12]-[14]. Recently, various
coupled circuits with the memristor couplings were proposed[15]. Therefore, this study focuses on the dynamics of thememristor couplings in the oscillatory networks.
In this study, we focus on using the memristor couplings in
the oscillatory networks. We investigate synchronization statetypes analyzing the time-series of the voltages, the memristors
and the phase differences. In addition, we also investigate thecause and effect relationship between synchronization statetypes and the dynamics of the memristor by calculating thepower consumption of the memristor couplings.

(a)

(b)

Fig. 1: Memristor model. (a): Schematic model, (b): q − φcurve.
The q − φ is characterized by the piecewise-linear function
φ(q) as Eq. (1) [16].
dφ(q)M (q) =
=
dq

(
a
b

(|q| < 1)(|q| > 1)(1)

φ(q) = bq + 0.5(a − b)(|q + 1| − |q − 1|).
Figure 2 shows the N th-oscillatory network with the memristor couplings as a ring structure. r is the tiny resistors to avoid
L-loop.

Fig. 2: N th-oscillatory network with memristor couplings asa ring structure.

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Circuit equations are obtained from Kirchhoff’s laws as Eq.
(2). These equations need to be normalized to investigate thesynchronization states by using computer simulations.
dv C k = −iGk − iR,k − iL,k


dt



diR,k



2L
= vk − riR,k − M (qk )(iR,k + iL,k+1 )


dt



diL,k


 2L
= vk − riL,k − M (qk )(iR,k−1 + iL,k )
dt
(2)
dqk


=
i
+
i

R,k
L,k+1


dt

(



a
(|qk | < 1)
dφ(qk )=M (qk ) =


dqk
b
(|q

k | > 1)
φ(qk ) = bqk + 0.5(a − b)(|qk + 1| − |qk − 1|).
The variables and the parameters are defined as Eq. (3).
s
s
g1
g1 C
g1 C
vk =
xk , iR,k =yR,k , iL,k =yL,k , qk = zk ,
g3
g3 L
g3 L
r
r
r
r
√
L
C
g1
C
,γ =
,ζ = C.
t = LCτ, ε = g1
,η = r
C
L
g3
L
r

(3)

The normalized circuit equations are obtained by changing thevariables and the parameters.

dxk


= ε(1 − x2k )xk − yR,k − yL,k


dτ



1
 dyR,k= (xk − ηyR,k − γM (zk )(yR,k + yL,k+1 ))

 dτ
2



dyL,k
1



= (xk − ηyL,k − γM (zk )(yR,k−1 + yL,k ))
dτ
2
(4)
dzk


=
ζ(y
+
y
)

R,k
L,k+1


dτ

(



a
(|zk | < 1)dφ(zk )
M (zk ) =
=


dzk
b
(|zk | > 1)φ(zk ) = bzk + 0.5(a − b)(|zk + 1| − |zk − 1|).
Where τ is the scaling time, ε is the nonlinearity, γ is thecoupling strength, ζ is the coupling factor and η is the factorof the tiny resistor.
The instantaneous power p(t) of the memristor is defined as
Eq. (5). i(t) is the current through the memristor.
p(t) = M (q(t))i(t)2 .

k=1

(1)

(2)

(3)

(a)

Pall = P1,2 + P2,3 + ... + PN −1,N + PN,1N

A. Synchronization Phenomena
In this section, we analyzed the time-series of x andthe phase differences to research synchronization state types
changing the initial values of x, y and z. The phase differences
are calculated by using Poincaré maps. The methods ofcalculating the phase differences are described as follows.
1) The van der Pol oscillator with third-power has a stable
limit cycle, so Poincaré section is defined as the planethat x > 0 and y = 0 in phase plane.
2) If x1 and y1 move from fourth quadrant to first quadrant(x1 (τ − 1) > 0, y1 (τ − 1) < 0 and y1 (τ ) > 0), eachof the relative phase difference for first oscillator is
calculated. Count is the number of the times that thesolutions x1 and y1 move from fourth quadrant to first
quadrant.
Also, we analyzed the time-series of M to investigate
the cause and effect relationship between the dynamics ofmemristors and synchronization state types. Figure 3, 4, 5 and6 show the simulation results. We confirmed three different
synchronization state types: N th-phase synchronization state,in-phase and anti-phase synchronization state and amplitude
death.

(5)

The average power consumption is described as follows. T isa long period of the oscillator. Pi,j means the average power
consumption of the memristor between the ith-oscillator andjth-oscillator Therefore, the total of the average power consumption Pall in the oscillatory network with the memristor
couplings can be calculated by Eq. (6).

1 X=
γM (zk )(yR,k + yL,k+1 )2 .
T

III. R ESULTS
In this study, the normalized circuit equations were calculated by the Runge-Kutta method with step size h = 0.01. The
scaling time τ = 20, 000. The parameters were set to ε = 0.1,
γ = 1, ζ = 0.1 and η = 0.001. We carried out the simulations
for the cases of N = 3, 5, 7 and 9. For the cases of N = 3, 5
and 7, the parameters of the memristors a = 0.1 and b = 10.
For the case of N = 9, a = 0.05 and b = 10.

(6)

(b)

(c)

Fig. 3: Simulation results (N = 3). (1): Three-phase, (2): Inphase and anti-phase, (3): Amplitude death, (a): Time-seriesof x, (b): Time-series of M , (c): Phase differences.

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In these simulation results, for the case (1), N th-phasesynchronization state were observed because all memristances
assumed b constant. This tendency had already been confirmedin the N th-oscillatory network with the resistors as a ring
structure. In the cases (2) and (3), in-phase and anti-phasesynchronization state and amplitude death were confirmed
because M1 assumed a and the other memristances assumedb. Therefore, in-phase and anti-phase synchronization states
and amplitude death were arisen from the dynamics of thememristor couplings.

(1)

(2)

(3)

(a)

(b)

(c)

Fig. 4: Simulation results (N = 5). (1): Five-phase, (2): Inphase and anti-phase, (3): Amplitude death, (a): Time-seriesof x, (b): Time-series of M , (c): Phase differences.

B. Power Consumption
In Sec. III A, we reported that three synchronization statetypes; N th-phase synchronization, in-phase and anti-phasesynchronizations and amplitude death were caused by the
dynamics of the memristors. In this section, we analyzed
the power consumption of the memristors to make clear thecause of that in-phase and anti-phase synchronization state andamplitude death. Figure 7, 8, 9 and 10 show the time-seriesof the instantaneous power of the memristors and the total
instantaneous power for each synchronization type in the N th
oscillatory networks (N = 3, 5, 7, 9).

(1)

(c)

(b)

(a)

Fig. 7: Time-series of instantaneous power of memristors(N = 3). (a): Three-phase, (b): In-phase and anti-phase, (c):Amplitude death.

(2)

(a)

(b)

(c)

Fig. 5: Synchronization results (N = 7). (1): Seven-phase, (2):In-phase and anti-phase, (a): Time-series of x, (b): Time-seriesof M , (c): Phase differences.

(b)

(a)

(c)

Fig. 8: Time-series of instantaneous power of memristors(N = 5). (a): Five-phase, (b): In-phase and anti-phase, (c):Amplitude death.

(1)

(2)

(a)
(a)

(b)

(c)

Fig. 6: Synchronization results (N = 9). (1): Nine-phase, (2):In-phase and anti-phase, (a): Time-series of x, (b): Time-seriesof M , (c): Phase differences.

(b)

Fig. 9: Time-series of instantaneous power of memristors(N = 7). (a): Seven-phase, (b): In-phase and anti-phase.

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TABLE IV: Average Power Consumption (N = 9).
Power

(a)

(b)

Fig. 10: Time-series of instantaneous power of memristors(N = 9). (a): Nine-phase, (b): In-phase and anti-phase.
For the case of in-phase and anti-phase synchronizations
state, even number pairs of anti-phase components werecanceled each other and the other component are notcanceled. The amplitude of the instantaneous power wasmuch larger than the case of N phase synchronization. For
the case of amplitude death, even number pairs of anti-phasecomponents were canceled each other and the amplitude of
the other component was dead. Therefore, the amplitude of
the instantaneous power was much smaller than the case of
N th-phase synchronization.
Next, Table I, II, III and IV summarize the average power
consumption compared N th-phase synchronization, in-phaseand anti-phase synchronizations and amplitude death.
TABLE I: Average Power Consumption (N = 3).
Power
P1,2
P2,3
P3,1
Pall

Three-phase
0.01280.01250.01280.0382Synchronization State types
In-phase and Anti-phase Amplitude death
0.04520.00000485
0.0000585
0.01270.0000574
0.01280.04530.0255TABLE II: Average Power Consumption (N = 5).
Power
P1,2
P2,3
P3,4
P4,5
P5,1
Pall

Five-phase
0.01240.01270.01290.01310.01330.0643

Synchronization State Types
In-phase and Anti-phase Amplitude death
0.103
0.000117
0.00155
0.006590.01480.01550.00173
0.01570.002180.005400.110
0.0434

TABLE III: Average Power Consumption (N = 7).
Power
P1,2
P2,3
P3,4
P4,5
P5,6
P6,7
P7,1
Pall

Synchronization State Types
Seven-phase In-phase and Anti-phase
0.007270.106
0.007540.00170
0.007330.00167
0.007880.001890.008310.002070.00857
0.000236
0.008730.000447
0.0556
0.116

P1,2
P2,3
P3,4
P4,5
P5,6
P6,7
P7,8
P8,9
P9,1
Pall

Synchronization State Types
Nine-phase In-phase and Anti-phase
0.00492
0.06340.005120.001470.005330.00192
0.00557
0.002530.005830.003280.006090.00401
0.006390.00476
0.00669
0.005210.006970.00186
0.05290.0887In these simulations, for the case of N th-phase synchronization, each of the average power consumption was thealmost same values because all memristances equaled to b.
For the case of in-phase and anti-phase synchronizations state,even number pairs of anti-phase components were canceledeach other and the other component were not canceled, so
the total average power consumption was much larger thanthe case of N th-phase synchronization. However, for thecase of amplitude death, even number pairs of anti-phasecomponents were also canceled each other and the amplitudeof the other component was dead, so the total average power
consumption of the case of amplitude death was less than thecase of N th-phase synchronization. Therefore, our simulation
results show that in-phase and anti-phase synchronizations
state and amplitude death in the N th-oscillatory networks asa ring structure is caused by the dynamics of the memristor
couplings.
IV. C ONCLUSIONSIn this study, we have proposed the coupled oscillatory
networks with the memristor couplings as a ring structure.
We investigated synchronization state types by analyzing thetime-series of the voltage, the memristors and the phase differences. As a result, in-phase and anti-phase synchronization
and amplitude death were obtained by the dynamics of thememristors. In addition, we investigated the cause and effectrelationship between the synchronization state types and the
dynamics by calculating the power consumption of the memristor. For the case of in-phase and anti-phase synchronizations
state, the power consumption of the memristors was larger
than the case of multi-phase synchronizations. On the other
hand, for the case of amplitude death, the power consumptionof the memristors was smaller than the case of multi-phasesynchronizations. Therefore, we confirmed that the power
consumption of the memristor couplings affects the synchronization states. Also, we confirmed that synchronization states
depend on the dynamics of the memristor couplings.
For the future works, we would like to use theoreticalanalysis and circuit simulations. These methods provides thefundamental evidences for the cause and effect relationshipbetween the dynamics of the memristor couplings and thesynchronization state types.

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