2023 International Symposium on Nonlinear Theory and Its Applications
NOLTA2023, September 26-29, 2023, Catania and Online

Various Synchronizations on Coupled van der Pol Oscillatorswith Memristor Synapse
Yukinojo Kotani† , Yoko Uwate† and Yoshifumi Nishio††Dept. of Electrical and Electronic Engineering,
Tokushima University, 2-1 Minami-Josanjima, Tokushima, Japan
Email: {kotani, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract— In this study, various synchronization phenomena are observed on the coupled circuit model of two vander Pol oscillators with the memristor synapse. In-phasesingle mode, double-mode oscillation and anti-phase single
mode are confirmed by changing the memristor characteristics. In addition, this study investigates synchronizationin dependence on the characteristics of the memristor andthe coupling factor.
1. Introduction
The human brain has high information processing ability. All the details of the human brain is not clear, so many
researchers have been studied using several methods. Inelectrical engineering, the memristor has been focused on
because of dynamics. There is a possibility to realize thesynape of the human brain.
The memristor is the forth basic circuit element: the resistor, the capacitor and the inductor. Basic physical variables are the voltage, the current, the charge and the flux.
Ohm’s law correlates the voltage and the current. Faraday’s law of electromagnetic induction also correlates thevoltage and the flux. The charge conservation is based on
the current and the charge. The relationship between thevoltage and the charge is described on Gauus’s law of thecharge. Ampére’s circuital law associates the current withthe flux. There are the basic circuit elements correspond tothese laws. However, the law of the relationship between
the charge and the flux had not been observed. L. O. Chua
had expected that there are the law and the basic circuit
element related the charge and the flux. So, the memristor was theoretically introduced from symmetry arguments
by Chua in 1971 [1]. The most interesting properties ofthe memristor is resistance value depends on the chargeor the flux. In 2008 [2], Hewlett-Packard fabricated thememristor, which is a nano-size solid-state device based on
TiO2 and has the hysteretic current-voltage characteristics.
The memristor characteristics are modeled using the piecewise linear function of the charge and the flux [3]. Hence,
many studies investigated synchronization phenomena on
the several models with the memristor synapse [4-6].

Synchronization is a typical nonlinear phenomenon.
It is one of the most interesting phenomena in nature,
which is that many objects behave same motion such as themovement of the pendulums, fireflies luminescence, living
things cry and so on. Many researchers have been studying
synchronization phenomena in several fields. Heartbeat isthat myocardial cells are synchronized with each other inthe field of medical science. Belousov-Zhabotinsky reaction is popular as chemical oscillation In the human brain,each neuron produces a small current using the electric potential diﬀerence between the inside and outside of the cell
membrane. A large number of neurons synchronize and release electric current at the same time to create a large electrical signal. It is one of the synchronization phenomena.
Therefore, synchronization phenomena have been studied
actively to analyze and realize the human brain behavior.
In previous study [7], synchronization phenomena werealso observed on the coupled circuit model of two vander Pol oscillators via the nonlinear resistor with fifthpower. In-phase and anti-phase synchronizations were observed on the previous model and investigated time wavesof phase diﬀerence, synchronization in dependence on thenonlinearity, synchronization in dependence on the coupling strength and amplitude of van der Pol oscillator. Also,
various synchronization phenomena were investigated on
the coupled circuit model of two van der Pol oscillators
via the time-varying resistor [8]. As a result, various synchronization phenomena were confirmed by changing thecharacteristics of the time-varying resistor. However, othersynchronization phenomena such as double-mode oscillation was not obtained on the coupled van der Pol oscillators
via the resistor.
In this study, we proposed the coupled circuit modelof two van der Pol oscillators with the memristor synapse.
The most interesting point is that the memristor has dynamics for the resistor because the charge is defined as the current. Also, various synchronization phenomena such as inphase single mode, double-mode oscillation and anti-phasesingle mode were researched on the proposed model. Inaddition, this study investigated synchronization in dependence on the characteristics of the memristor and the coupling factor.

ORCID iDs Yukinojo Kotani:0009-0003-2837-600X, Yoko Uwate:0000-0002-2992-8852, Yoshifumi Nishio:
0000-0002-0247-0001

This work is licensed under a Creative Commons
Attribution-NonCommercial-NoDerivatives 4.0 International.

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Figure 4 shows the circuit model. In this model, two vander Pol oscillators are coupled with the memristor.

2. Proposed Models
Figure 1 shows the memristor model. From Ohm’s law,
the equation of between the current i and the voltage v isdescribed as Eq. (1)Figure 1: Memristor Model.
v = M(q)i(1)where memristance M(q) is depend on the charge q. It
is defined as Eq. (2) and characterized by piecewise linear
function φ(q) in Fig. 2. Horizontal axis is the charge q andvertical axis is the flux φ. The φ−q curve is given as Eq. (3).

Figure 4: Circuit Model.
The circuit equations are given as Eq. (4).
dv1
= −iG1 − i1Cdt
di1


L
= v1 − M(q)(i1 + i2 )

dt

 dv2= −iG2 − i2C

dt

di2


L
= v1 − M(q)(i1 + i2 )


dt



dq



 dt = i1 + i2(4)The iGn and vn characteristics of the nonlinear resistor ofvan der Pol oscillator is defined as Eq. (5).
iGn = −g1 vn + g3 v3nFigure 2: q-φ curve.
dφ(q)dqφ(q) = bq + 0.5(a − b)(|q + 1| − |q − 1|)
M(q) =(2)(3)

where a is the slope of φ(q) when the absolute value of thecharge q is less than one. b is also the slope of φ(q) when
the absolute value of the charge q is greater than one.
Figure 3 shows the van der Pol oscillator. It is consist on acapacitor, an inductor and the nonlinear resistor with thirdpower.

Figure 3: van der Pol Oscillator.

(g1 , g3 > 0)(5)

By changing the variables and parameters,√√√g1
g1 Cvk =xk , ik =yk , q = z, t = LCτ
gg L√ 3√3√LCg1
ε = g1, γ=, ζ =C(k = 1, 2).
CL
g3
where τ is the scaling time, ε is the strength nonlinearity,
γ is the coupling strength and ζ is the coupling factor. The
normalized circuit equations are obtained as Eq. (6).
dx1



= ε(1 − x12 )x1 − y1


dτ




dy1



= x1 − γM(z)(y1 + y2 )

dτ


 dx2= ε(1 − x22 )x2 − y2(6)


dτ



dy2

= x2 − γM(z)(y1 + y2 )dτdz dτ = ζ(y1 + y2 )where memristance M(z) represents the slope of the function φ(z), so M(z) is redefined as Eq. (7){dφ(z)
a (|z| < 1)
=M(z) =(7)b (|z| > 1)
dz

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3.1. Various Synchronizations
In this section, this study investigated various synchronizations on the coupled circuit model with the memristor synapse. In this simulation, the parameters of the normalized circuit equations Eq. (6) were set up to ε = 0.1,
γ = 1.0, ζ = 1.0, τ = 20000 and step size of Runge-Kuuta
method h = 0.01.
This study confirmed that in-phase single-mode and antiphase single-mode coexist at the same parameter values
and were determined by the diﬀerent initial values. The
parameters of the characteristics of the memristor were set
up to a = 0.01 and b = −0.1. In-phase single-mode synchronization state is shown in Fig. 5. Besides, Fig. 6 shows
anti-single synchronization state.

Also, this study confirmed that double-mode oscillation
was obtained by changing the parameters of the memristor. The parameters of the characteristics of the memristor
were set up to a = −0.1 and b = 10. Figure 7 shows thesimulation results of double-mode oscillation.

(d)

(a)

(e)

Figure 7: Double-mode oscillation. (a) First circuit attractor (x1 - y1 ). (b) Second circuit attractor (x2 - y2 ). (c)Phase diﬀerence (x1 - x2 ) (d) Time wave forms of x. (e)Time wave form of M. Initial values x1 = 1.0, y1 = 2.0,x2 = −1.1, y2 = −2.1 and z = 1.0.

(c)

(b)

(c)

(b)

(a)

3.2. Synchronization in Dependence on a and b(e)(d)

Figure 5: In-phase single-mode synchronization. (a) Firstcircuit attractor (x1 - y1 ). (b) Second circuit attractor (x2 y2 ). (c) Phase diﬀerence (x1 - x2 ). (d) Time wave formsof x. (e) Time wave form of M. Initial values x1 = 1.0,
y1 = 2.0, x2 = 1.1, y2 = 2.1 and z = 1.0.

(b)

(a)

(d)

In this section, this study researched synchornization in
dependence on the parameters of the memristor under thecondition of double-mode oscillation in Fig. 7. In this simulation, the parameters of the normalized circuit equations
Eq. (6) were set up to ε = 0.1, γ = 1.0, ζ = 1.0, τ = 20000
and step size of Runge-Kuuta method h = 0.01.
Figure 8 shows synchronization in dependence on the parameters of the memristor.

(c)

(a)

(e)

Figure 6: Anti-phase synchronization. (a) First circuit attractor (x1 - y1 ). (b) Second circuit attractor (x2 - y2 ). (c)Phase diﬀerence (x1 - x2 ). (d) Time wave forms of x. (e)Time wave form of M. Initial values x1 = 1.0, y1 = −2.0,
x2 = −1.1, y2 = 2.1 and z = 1.0.

(b)

Figure 8: Synchronization in dependence on the parametersof the memristor. (a) Dependence on a (b = 10). (b) Dependence on b (a = −0.1). Initial values x1 = 1.0, y1 = 1.1,
x2 = −2.0, y2 = −2.1 and z = 1.0.
In Fig. 8 (a), anti-phase single mode was obtained in arange from −0.3 to −0.2 and from −0.1 to 0.3. Doble-modeoscillation was also confimed in the regions of −0.2 ≦ a ≦−0.1. In Fig. 8 (b), double-mode oscillation was constantlyobtained in the regions of 1.0 ≦ b ≦ 10. Besides, in-phasesingle mode was confirmed in the regions of b ≦ 0.0 because both of a and b were zero or negative.

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3.3. Synchronization in Dependence on ζ4. Conclusions

In this section, this study investigated synchronization independence on the coupling factor under the condition ofdouble-mode oscilation. In this simulation, the parametersof the normalized circuit equations Eq. (6) were set up toε = 0.1, γ = 1.0, τ = 20000 and step size of Runge-Kuuta
method h = 0.01. The parameters of the characteristics ofthe memristor were set up to a = −0.1 and b = 10.
Figure 9 shows synchronization in dependence on ζ.

This study proposed the coupled circuit model of two
van der Pol oscillators with the memristor synapse. First,
in-phase and anti-phase synchronization states coexist at
the same parameter values in the diﬀerent initial values.
Second, double-mode oscillation was confirmed by changing the characteristics of the memristor. Third, this study
confirmed that synchronization depends on the characteristics of the memristor by analyzing the relationship between phase diﬀerence and the parameters of the memristor. Finally, we investigated synchronization in depencenceon the coupling factor. As a result, in-phase single mode,
double-mode oscilattion and anti-phase single mode wereobtained by changing the coupling factor.
In the future, we would like to research the causes ofdouble-mode oscillation on proposed model using theoretical analysis, and compare to the simulation results.

(a)

(b)

(c)

References
(c)

(e)

(f)

Figure 9: Phase diﬀerence (x1 - x2 ). (a) ζ = 0.1. (b) ζ =0.7. (c) ζ = 0.8. (d) ζ = 1.0 (e) ζ = 1.2. (f) ζ = 1.5.
Initial values x1 = 1.0, y1 = 2.0, x2 = −1.1, y2 = −2.1 andz = 1.0.
In-phase single mode was confirmed in Fig. 9 (a) when wechose ζ = 0.1. Double-mode oscillation was obtained inwhen we chose ζ = 0.7, 0.8, 1.0 and 1.2. Also, anti-phasesingle mode was obtained in whne we chose ζ = 1.2. Next,
we researched synchronization in dependence on the coupling factor ζ. Figure 10 shows synchronizations in dependence on ζ in the regions of 0.0 ≦ ζ ≦ 2.0.

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[5] K. Kobayashi, Y. Hosokawa, Y. Uwate and Y. Nishio,“Relationship between a Synchronization Rate anda Dopant Mobility of a Memristor on Chaotic Circuits Coupled by Two Memristors,” Proc. of Int. Conf.
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Figure 10: Synchronization in dependence on ζ. Initial values x1 = 1.0, y1 = 1.1, x2 = −2.0, y2 = −2.1 and z = 1.0.
In Fig. 10, In-phase single mode was confirmed in theregions of 0.0 ≦ ζ ≦ 0.6. Also, double-mode oscillation
was obtained in the regions of 0.7 ≦ ζ ≦ 1.2. In the regionsof 1.3 ≦ ζ ≦ 2.0, anti-phase synchronization was observed.

[7] Y. Uwate and Y. Nishio, “Synchronization Phenomena in van der Pol Oscillators Coupled by Fifth-Power
Nonlinear Resistor,” Proc. of Int. Conf. NOLTA’07,
pp. 232–235, Sep. 2007.
[8] Y. Uwate and Y. Nishio, “Synchronization Phenomena in van der Pol Oscillators Coupled by A TimeVarying Resistor,” Int. J. Bifurcation and Chaos,vol. 17, no. 10, pp. 3565–3569, Mar. 2006.

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