Relationship between a Synchronization Rate and a Dopant Mobility of a Memristor
on Chaotic Circuits Coupled by Two Memristors
Kana Kobayashi1 , Yasuteru Hosokawa2 , Yoko Uwate1 , and Yoshifumi Nishio1
1Tokushima University
2-1 Minami-Josanjima-Cho, Tokushima 770-8506 Japan
Phone:+81-88-656-7470
E-mail: {kobayashi,uwate,nishio}@ee.tokushima-u.ac.jp

Abstract
In our previous study, a relationship between observed phenomena and the coupling strength in coupled chaotic circuits
including memristors was investigated. However, there was
no influence from the memristor. Basically, a memristor developed by HP reacts from a few Helz to a few hundred Helz.
A main frequency band of chaotic circuits in our previous
study is kilo Helz. Therefore, a behavior of the memristor
become a resistor. In this study, the relationship between a
synchronization rate and a dopant mobility of a memristor on
chaotic circuits coupled by two memristors are investigated.
1. Introduction

2

Shikoku University
Furukawa, Ohjin-Cho, Tokushima 771-1192 Japan
Phone:+81-88-665-1300
E-mail: hosokawa@keiei.shikoku-u.ac.jp

phenomena and the coupling strength in coup lied chaotic circuits including memristors was investigated. However, there
was no influence from the memristor. Basically, a memristor
developed by HP reacts from a few Helz to a few hundred
Helz. A main frequency band of chaotic circuits in our previous study is kilo Helz. Therefore, a behavior of the memristor
become a resistor.
In this study, the relationship between a synchronization
rate and a dopant mobility of a memristor on chaotic circuits coupled by two memristors are investigated. In Sect. 2,
the system model is shown. Computer simulation results areshown in Sect. 3. Finally, concluding remarks are mentionedin Sect. 4.
2. System Model

Recently, a memristor, which is called as the fourth fundamental passive element of the electric circuit, has been remarked by many researchers. The memristor was proposedby L. O. Chua in 1971 [1]. In 2008, D. B. Strukov et al.
implemented the memristor as a semiconductor element [2].
The memristor has a simple structure which sandwiches theTiO2 between two electrodes of Pt. Additionally, the memristor can keep a value as a resistance. These features havebeen attracted many researchers.
On the other hand, chaos is studied in various fields suchas mathematics, physics, engineering, economics, chemistry
and so on. In the field of electrical engineering, chaotic
phenomena observed in electric circuits are investigated.
Shinriki-Mori circuit [3], which is one of the chaos circuits,
has been studied by many researchers. Some of them reported
about coupled these circuits. In these coupled systems, various kinds of interesting phenomena are observed [4]-[6].
Gambuzza et al. applied memristors to coupled chaotic
circuits [7]. Two memristors are coupled in antiparallel as
the coupling element. The system coupling multiple chaotic
circuits have been investigated. However, a relationship between coupling strength and observed phenomena was not investigated.
In our previous study, the relationship between observed

2.1 Memristor Model
Figure 1 shows the memristor model proposed by Strukovof Hewlett-Packard et al. [2]. The voltage applied to the memristor is v, and the current flowing is i. The equation of relationship between the voltage v and the current i is describedby the following equation [7].
v(t) = (RON ω(t) + ROF F (1 − ω(t)))i(t),(1)

where RON has a low resistance value and ROF F has a highresistance value. The variable ω is described by the followingequation.
ω=w.
D(2)

The w is the width of the doped region in the memristor. Thevariable ω is normalized by the maximum width D of thememristor. The variable ω means the proportion occupied byRON . Therefore, ω is limited to a value between 0 and 1.
The memristor model is described by the following equation.

- 475 -µv RONdω(t)=ηF(ω(t),i(t)) i(t),dt
D2(3)

(a)

(b)

Figure 3: Bidirectionally coupled diode model. (a)Circuit diagram. (b)The voltage-current characteristic.

Figure 1: Memristor model.

follows:

Figure 2: System model.
where η (η = 1 or η = −1) is the memristor polarity, µv isthe dopant mobility and D is thickness of the memristor madeby TiO2 . The current i(t) is given by Eq. (1) as follows:i(t) =v(t),RON M(4)

where M = ω(t) + β(1 − ω(t)) with β = ROF F /RON .
F(ω(t),i(t)) is the Biolek window function [8]. The Biolek
window function is given by:2F(ω(t),i(t)) = 1 − (ω(t) − stp(−i(t))) ,(5)

dv1,1


C1,1

dt







dv

 C2,1 2,1

dt



diL,1


L1


dt
dv

1,2

 C1,2dt

dv2,2


C2,2

dt


diL,2
L2
dtdω1dtdω2
dt=−id1 + g1 v1,1+(M1 + M2 )(v1,2 − v1,1 ),R(M1 + M2 ) + RON M1 M2=

id1 + iL,1 ,=−v2,1 ,=−id2 + g2 v1,2+(M1 + M2 )(v1,1 − v1,2 ),R(M1 + M2 ) + RON M1 M2=

id2 + iL,2 ,=−v2,2 ,=η1 F(ω1 )=(6)µv RON v1,2 − v1,1 + vR,D2
RON M1µv RON v1,2 − v1,1 + vR
η2 F(ω2 ),D2
RON M2where stp(i) = 1 when i ≥ 0, and stp(i) = 0 when i < 0.
In this study, two memristors are coupled in antiparallel as where M1 , M2 are described as follows:a coupling element.
{M1 = ω1 + β(1 − ω1 ),(7)M2 = ω2 + β(1 − ω2 ).

2.2 System Model
Figure 2 shows a system model in this study. It is amodel combining the nodes of the negative resistance of two
Shinriki-Mori circuits with a resistor and two memristors
coupled in antiparallel. g1 and g2 are negative resistors.
Figure 3 shows a diode model in the system model. Figure 3 (a) shows the circuit diagram, and Fig. 3 (b) shows thevoltage-current characteristic. This model is a piecewise linear model.
The circuit equation derived from these model is shown as

The voltage of the resistor vR is described as follows:
RvR = RonR

(M1 + M2 )(v1,1 − v1,2 )Ron (M1 + M2 ) + M1 M2.

(8)

By using the following variables and parameters, the nor-- 476 -malized circuit equation is derived.
x1 =v1,1v2,1, x2 =, x3 =v0v0

where m1 , m2 are described as follows:{m1 = x7 + β(1 − x7 ),√√L1 iL,1,C2,1 v0

m2 = x8 + β(1 − x8 ),v1,2L1 iL,2v2,2, x5 =, x6 =,v0v0
C2,1 v0
dx7 = ω1 , x8 = ω2 ,= ” · ”,dt√
1L1t, α = Gτ=√,C2,1C2,1 L1x4 =ROF F
L1,, γ=RON
L2C2,1C2,1C2,1δ=, ε=, ζ=,C1,1C1,2C2,2√L1
η1 = 1, η2 = −1, θ1 = g1,C2,1√√µv
L1θ2 = g2, ι = C2,1 L1 2 v0 ,C2,1D√
1L1RVthk=, λ=and ξ =.
RON C2,1RONv0β=

and iM1 , iM2 are described as follows:(x4 − x1 )m1 m2 iM1 = m {λ(m + m ) + m m } ,
1
121 2

(x−x)mm411 2

iM2 =.
m2 {λ(m1 + m2 ) + m1 m2 }
(9)

(11)

(12)

In this study, simulations are carried out using these normalized circuit equations.
3. Simulation Results
Figures 4 and 5 show simulation results in this study. Figures 4 and 5 are time series waveforms. From the top, verticalaxes show voltages of two circuits, the voltage difference between two circuits, variables w1 and w2 of memristors, andthe difference between variables w1 and w2 . Horizontal axes
show time.
Initial values and parameters are set as follows: x1 =0.001, x2 = 0.001, x3 = 0.001, x4 = 0.001, x5 = 0.001,
x6 = 0.0011, x7 = 0.0, x8 = 0.0, α = 20.0, β = 100.0,
γ = 1.0001, δ = 3.3, ε = 3.3, ζ = 1.0, η1 = 1.0, η2 = −1.0,θ1 = 0.19, θ2 = 0.19, k = 5.423, λ = 310.0, ξ = 0.6,
where γ is added as the parameter mismatch 0.01%. Parameter λ and ι show the resistor R as a coupling strength and the
dopant mobility µv of the memristor, respectively.
In Figure 4, parameter ι is set as ι = 1.0. Movements ofvariables x7 and x8 are very slow comparing with oscillationsof two circuits. In this case, the system behaves two chaotic
circuit coupled by a resistor.

The normalized circuit equation is described as follows:[{1 ẋ1 = δ −α (x1 − x2 ) + (|x1 − x2 − ξ|

2


−|x−x+ξ|)}+θ
121 x1](m1 + m2 )+k(x−x),41λ(m1 + m2 ) + m1 m2


{
1
ẋ
=α(x1 − x2 ) + (|x1 − x2 − ξ|22
−|x−x+ξ|)}+ x3 ,
12ẋ3 = −x2 ,

[{
1

ẋ
=ε−α(x4 − x5 ) + (|x4 − x5 − ξ|
 42(10)−|x4 − x5 + ξ|)} + θ2 x4
](m1 + m2 )+k(x−x),14
λ(m1 + m2 ) + m1 m2



[ {

1


ẋ
=
ζα (x4 − x5 ) + (|x4 − x5 − ξ|
5
2




−|x−x+ξ|)}+x6 ] ,
45




ẋ6 = −γx5 ,Figure 4: Simulation result. The parameter mismatch is[]
0.01%,
ι = 1.0, λ = 310.0. Time series waveform.
2ẋ7 = η1 ι 1 − {x7 − stp (−iM1 )} iM1 ,()
By increasing parameter ι, the memristor behaves as a
 ẋ = η ι 1 − [x − stp {− (−i )}]2 i ,828M2M2memristor. In Figure 5, parameter ι is set as ι = 70.0. In- 477 -605040Rate [%]this case, the resistance of the memristor is changed by current flow of the memristor. Increasing parameter ι makes increasing the current of the memristor. In case of ι > 70.0, thememristor current value does not have a reality.
Interesting relationship between oscillations of two circuitand memristor variables is observed in Fig. 5. When two
memristor variables become saturated states, oscillation biases of two circuits become the opposite side. When two
memristor variables are stables, two circuits are synchronizing. When two memristor variables move intensely, oscillation biases of two circuits become the same side and twocircuits are not synchronizing. These three states can be observed randomly. By some computer simulations, it was considered that there are some relationship between these threestates and parameter ι.

3020Synchronization rate
Asynchronization rate
Synchronization of switching phenomena rate

10012

4

8

1632

64

128

256

512

Iota
Figure 6: Relationship between three state and parameter ι.
The parameter mismatch is 0.01%, λ = 310.0.
coupled by two memristors has been investigated. As a result,the most effective value of a dopant mobility of a memristor
for synchronization phenomena of the coupled system. As
a future work, investigating the influence of memristors using the most effective value to the coupled chaotic circuits isconsidered.
Acknowledgment

Figure 5: Simulation result. The parameter mismatch is This work was partly supported by JSPS Grant-in-Aid for
Scientific Research 16K06357.
0.01%, ι = 70.0, λ = 310.0. Time series waveform.
In order to investigate the relationship between these statesand ι in detail, three states are defined as follows. Synchronization: the extreme value of the difference of x1 − x4 andthe previous extreme value are in the range of ±0.05. Asynchronization: x1 · x4 is minus. Synchronization of switching
phenomena: without synchronization and asynchronization.
Using this definition, the relationship between three states andι are investigated.
Figure 6 shows relationship between three state and parameter ι. The horizontal axis shows ι and the vertical axis shows
the rate of each state.
When ι = 16, synchronization state becomes the lowest
value and synchronization of switching state and asynchronization state become the highest value. Therefore, ι = 16
is the most effective value for synchronization phenomena ofthe coupled system.

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4. Conclusions
In this study, the relationship between a synchronization
rate and a dopant mobility of a memristor on chaotic circuits- 478 -