IEEE Workshop on Nonlinear Circuit Networks
December 8-9, 2023

Analysis for Synchronization in Dependence onMemristor Behavior in Coupled Oscillators
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 investigate synchronization phenomena in coupled oscillatory networks with the memristor
couplings as a ring structure. We focus on the dynamics of amemristor, and our proposed method provides the dynamicsfor the couplings. This study confirmed that three different
synchronization state types: multi-phase synchronization state,in-phase and anti-phase synchronization state and amplitude
death were obtained by depending on the power consumptionof the memristor couplings.

II. O SCILLATORY N ETWORK M ODEL
Figure 1 shows the N th oscillatory network with the memristor couplings as a ring structure. r is the tiny resistors toavoid L-loop for calculating equations.

I. I NTRODUCTION
Synchronization phenomena observed from the coupledoscillatory networks have been studied. Many studies related
to synchronization phenomena in oscillatory networks haveimportant roles: analyzing the characteristics of networks [1].
Therefore, various synchronization phenomena in oscillatory
networks 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 [2], and it was developed by Hewlett-Packard Lab in2008 [3]. Resistance value of a memristor is characterized bya charge or a flux, whereas a resistor is characterized by acurrent or a voltage. The charge and the flux are defined as
the integral of the current and the voltage respectively. Thememristor has the dynamics compared to the resistor.
In previous studies, synchronization phenomena in thecoupled oscillatory networks were investigated. The coupledoscillatory networks via resistors were proposed, and various
synchronization phenomena were investigated [4]. In addition, the memristor coupling were proposed in several circuit
models, and synchronization phenomena were investigated.
Therefore, this study focuses on the memristor couplings in
the oscillatory network as ring structures.
In this study, we focus on the memristor coupling comparedto the conventional coupling methods of the other circuit
elements depended on the current and the voltage. We investigate synchronization state types analyzing the time-seriesof the voltages, the memristors and the phase differences.
In addition, we investigate the cause and effect relationshipbetween synchronization state types and the dynamics ofthe memristor by calculating the power consumption of thememristor couplings. 1
1 The extended version of the study is being reviewed for ISCAS’24 [5].

-8-Fig. 1: Oscillatory network with memristor couplings as a ringstructure.
Circuit equations are obtained from Kirchhoff’s laws. Theseequations need to be normalized to investigate the synchronization states by using computer simulations.
The normalized circuit equations are obtained by changing
the variables and the parameters as Eq. (1).
dxk
= ε(1 − x2k )xk − yR,k − yL,kdτdyR,k1= (xk − ηyR,k − M (z)(yR,k + yL,k+1 ))dτ
2dyL,k1= (xk − ηyL,k − M (z)(yR,k−1 + yL,k ))dτ2dzk= ζ(yR,k + yL,k+1 )dt(a(|zk | < 1)dφ(zk )M (zk ) ==dqb(|zk | > 1)

(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.

III. R ESULTS
The normalized circuit equations were calculated by theRunge-Kutta method with step size h = 0.01. The parameters
were set to τ = 20, 000, ε = 0.1, γ = 1, ζ = 0.1 andη = 0.001. The parameters of the memristors a = 0.1 andb = 10.
In this section, we analyzed the time-series of x, the timesereis of M and the phase differences. The phase differences
are calculated by using Poincaré maps. The methods ofcalculating the phase differences is described as follows.
1) A van der Pol oscillator with third-power has a stable
limit cycle, so Poincaré section is defined as the planethat x(τ ) > 0 and y(τ ) = 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),each of the relative phase difference for first oscillator
is calculated. Count is the number of the movement of
the solutions x1 and y1 from fourth quadrant to firstquadrant.
Figure 2 shows the simulation results. We confirmed threedifferent synchronization state types: three-phase synchronization state, in-phase and anti-phase synchronization state andamplitude death.

(1)

(2)

Next, we investigated the average power consumption. The
total of the average power consumption Pall in the oscillatory
network with the memristor couplings can be calculated by Eq.
(2). T is a long period. Therefore, Table I summarize the average power consumption compared N th-phase synchronization,
in-phase and anti-phase synchronizations and amplitude death.
T

Pall =1 X
γM (zk )(yR,k + yL,k+1 )2 .
T

(2)

k=1

TABLE I: Average Power Consumption (N = 3).
Power
P1,2
P2,3
P3,1PallThree-phase
0.01280.01250.01280.0382Synchronization State Types
In-phase and Anti-phase Amplitude Death
0.04520.00000485
0.0000585
0.01270.0000574
0.01280.04530.0255In these simulations, for the case of N th-phase synchronization, each of the average power consumption was the samevalues because all memristances equaled to b. For the case ofin-phase and anti-phase synchronizations state, even number
pairs of anti-phase components were canceled each other andthe other component were not canceled, so the total average
power consumption was much larger than the case of N thphase synchronization. However, for the case of amplitude
death, even number pairs of anti-phase components were also
canceled each other and the amplitude of the other componentwas dead, so the total average power consumption of thecase of amplitude death was less than the case of N th-phasesynchronization. Therefore, our simulation results show thatin-phase and anti-phase synchronizations state and amplitude
death is caused by the dynamics of the memristor couplings.
IV. C ONCLUSIONSThis study proposed the memristor couplings in coupledoscillatory networks as a ring structure. This study confirmedthat three different synchronization state types: multi-phasesynchronization state, in-phase and anti-phase synchronization
state and amplitude death were obtained by depending on thepower consumption of the memristor couplings.

(3)

(a)

(b)

R EFERENCES(c)

Fig. 2: Simulation results. (1): Three-phase, (2): In-phase andanti-phase, (3): Amplitude death, (a): Time-series of x, (b):Time-series of M , (c): Phase differences.
In these simulation results, for the case of (1), N th-phasesynchronization state were observed because all memristances
assumed b constantly. For the cases of (2) and (3), in-phaseand anti-phase synchronization state and amplitude death wereconfirmed because M1 assumed a and the other memristances
assumed b. Therefore, in-phase and anti-phase synchronization
states and amplitude death were arisen from the dynamics ofthe memristor couplings.

-9-[1] G. Yang, D. Tong, Q. Chen and W. Zhou,“Fixed-Time Synchronizationand Energy Consumption for Kuramoto-Oscillator Networks With Multilayer Distributed Control”, IEEE Trans. Circuits Syst. II Regul. Pap,
vol. 70, no. 4, April 2023.
[2] L. O Chua, “Memristor-The Missing Circuit Element”, IEEE Trans. onCircuit Theory, vol. CT-18, no. 5, pp. 507-519, Sep. 1971.
[3] D. B. Strukov, G. S. Snider, D. R. Stewart and R. S. Williams, “Themissing memristor found”, Nature Lett., vol. 453, pp. 80-83, May 2008.
[4] Y. Setou, Y. Nishio and A. Ushida, “Synchronization Phenomena in
Resistively Coupled Oscillators with Different Frequencies”, IEICE
Trans. Fundam., vol. E79-A, no. 10, pp. 1575–1580, Oct. 1996.
[5] Y. Kotani, Y. Uwate and Y. Nishio, “Synchronizations in Oscillatory
Networks with Memristor Couplings as Ring Structure”, Proc. of Int.
Conf. ISCAS’24, 2024. (being submitted)