2015 International Symposium on Nonlinear Theory and its Applications
NOLTA2015, Kowloon, Hong Kong, China, December 1-4, 2015

Relationship between Synchronization Rates and Parameter Mismatches in
Coupled Chaotic Circuits Using Memristors
Yasuteru Hosokawa† and Yoshifumi Nishio‡†Department of Media and Information Systems, Shikoku University
Furukawa, Ohjin-cho, Tokushima 771-1192, Japan‡Department of Electrical and Electronic Engineering, Tokushima University
2–1 Minami-Josanjima, Tokushima 770-8506, Japan
Email: hosokawa@keiei.shikoku-u.ac.jp, nishio@ee.tokushima-u.ac.jp
Abstract—In our previous studies, the relationship between synchronization rates and small parameter mismatches on an asymmetrical coupled chaotic systems hasbeen investigated. As a result of investigating these systems, interesting phenomenon can be observed. We consider that the phenomenon can be observed in the case of
any kinds of chaotic oscillators.
In this study, a memristor based chaotic circuit are applied to our proposed system. By this investigation, ourhypothesis for the phenomenon observed in our system isreinforced.

searchers who are interested in a memristor developed theemulator circuits [5]-[8]. By using the emulator, somechaotic circuits are proposed and some applications areproposed.
In this study, a memristor based chaotic circuit [5] areapplied to our proposed system. By this investigation, ourhypothesis for the phenomenon observed in our system isreinforced.
2. System Model
2.1. Memristor Based Chaotic Circuit

1. Introduction
Many kinds of coupled chaotic systems have been proposed and investigated. In these system, most important
point is that various kinds of interesting phenomena suchas spatio-temporal chaotic phenomena, clustering phenomena, and so on can be observed.
In our previous studies [1]-[2], the relationship between
synchronization rates and small parameter mismatches onan asymmetrical coupled chaotic systems has been investigated. The system is coupled globally and coupling elements are resistors. Sub-circuits are divided into twogroups. The diﬀerence of two groups is circuit parameters, coupled nodes or circuits as coupled elements. The
small parameter mismatches were given to the sub-circuits
as an mismatch of the oscillation frequency. As a resultof investigating these systems, interesting phenomenon canbe observed. This phenomenon is that the synchronization
rates of the one sub-circuit increases in spite of increasing
parameter mismatches of the other group. We consider thatthe phenomenon can be observed in the case of any kindsof chaotic oscillators.
On the other hand, a memristor proposed by L. O. Chua[3] is called as the fourth fundamental circuit element. A
physical implementation was realized by a group of researchers from Hewlett-Packard laboratories in 2007 [4].
Many researchers consider that the memristor has a potential of application for computer science, neural networks
and so on. However, the physical implementation is nanoscale size and is not available on the market. Some re-Figure 1 shows a memristor based chaotic circuit(MBCC) proposed by B. Muthuswamy et. al. [5]. This circuit consists of two capacitors, one inductor and one memristor. Note that the memristor used in the circuit is an active element. By connecting a well-known passive memristor with a negative resistor in parallel, the active memristor
can be realized [9].
Figure 2 shows an implementation of the memristor.
This memristor is a flux-controlled memristor. Voltage vm
is corresponding to flux ϕ of the memristor. The characteristic of the memristor is defined as a cubic nonlinearity forthe q-ϕ function:q(ϕ) = m1 ϕ + m2 ϕ3 .

(1)

Therefore, the memductance function W(ϕ) is given by :W(ϕ) =dq= m1 + 3m2 ϕ2 .
dϕ

(2)

A circuit experimental result and a computer simulation
result are shown in Fig. 3. Chaotic attractors are observed.
2.2. System ModelFigure 4 is a proposed system. This system consists ofresistors R and MBCCs. Two kinds of MBCCs (which arecalled as Group A and Group B) are included in the system. A diﬀerence between Group A and Group B is avalue of circuit parameters. The number of circuit is defined as k = m + n. Namely, the numbers of Group A and- 471 -Figure 1: Memristor Based Chaotic Circuit [5].

Figure 4: System Model.

Group B circuits are m and n, respectively. Additionally,small parameter mismatches of Group A and Group B areincluded in the system. Circuits are coupled globally byresistors. This system is asymmetrically coupled chaoticsystem. Now, in order to carry out computer simulations,Circuit equations are derived.
Group A (1 ≤ k ≤ m):

Figure 2: Implementation of a Memristor.

dϕk



dt






dvk1



C1a



dt






dvk2

C2a


dt

















di

 La kdt(a)===vk1
Rm C m1(vk2 − vk1 ) − (m1 + 3m2 ϕ2k )vk1
Ra1
(vk1 − vk2 ) − ikRam+n1∑+ v−(m+n)v
i2k2
R−(3)i=1=(1 + pk )vk2(b).

Figure 3: Circuit experimental result and computer simulation result of a MBCC. (a) Horizontal axis is ik1 . Verticalaxis is vk2 . Rk1 = 1800 [Ω], Ck1 = 6.8 [nF], Ck2 =68 [µF],Lk1 = 20 [mF], Rm = 10 [kΩ], Cm = 10 [nF], Rr = 1000[kΩ], R1 = 2 [kΩ], R2 = 2 [kΩ], R3 = 1.48 [Ω], R4 = 30[kΩ] and R5 = 30 [kΩ]. (b) Horizontal axis is xk4 . Vertical axis is xk3 . α = 0.037, βa = 11, βb = 10, γa = 0.28,
γb = 0.29, M1 = −0.379, M2 = 0.065, δ = 0.09, ε = 1.1
and ζ = 1.0.

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Group B (m + 1 ≤ k ≤ m + n):
dϕk


dt



dvk1


C1b


dt



 C dvk22bdt
di
 Lb kdt===vk1
Rm C m1(vk2 − vk1 ) − (m1 + 3m2 ϕ2k )vk1
Rb1
(vk1 − vk2 ) − ikRbm+n1∑vi2 − (m + n)vk2 + 

R

−i=1

=

(1 + qk )vk2(4)By using the following variables and parameters,√La
xk1 = ϕk , xk2 = vk1 , xk3 = vk2 , xk4 =ik ,C2a√
1dC2a La
C2at, “ · ” =, α=, βa =,τ= √dτ
Rm C m
C1aC2a La√√C2a1La1Laβb =, γa =, γb =,C1bRa C2aRb C2a√C2aLa
1La, ε=, ζ= ,
δ=R C2aC2bLb√√
La
La
M1 = m1and M2 = 3m2.
C2aC2a(5)the normalized circuit equations of Group A and Group B
circuits are described as follows:
Group A (1 ≤ k ≤ m):
ẋk1 = −αxk2 ,



{}

2
ẋk2 = βa γa (xk3 − xk2 ) − (M1 + M2 xk1)xk2 ,m+n∑ẋ=γ(x−x)−x+δx−(m+n)x,

k3ak2
k3
k4
i3k3

i=1 ẋ = (1 + p )x .
k4
k k3(6)
Group B (m + 1 ≤ k ≤ m + n):ẋk1 = −αxk2 ,




{}
2

ẋk2 = βb γb (xk3 − xk2 ) − (M1 + M2 xk1)xk2 ,
m+n∑
x−(m+n)xγ(x−x)−x+δẋ
=ε,
i3k3bk2
k3
k4
k3





i=1


 ẋ = ζ(1 + q )x .
k4k k3(7)
where k = 1, 2, 3, · · · , m+n, i.e. the number of Group A circuits is m, the number of Group B circuits is n, and pk andqk are introduced to give parameter mismatches of the oscillation frequencies. By using these equations, computer
simulations are carried out.
3. Computer Simulations
Figure 5 shows the voltage diﬀerences between two
MBCCs in the case of using following initial values andparameters.
xk1 (0) = 0.0, xk2 (0) = 0.1 + 0.1 · k, xk3 (0) = 0.0,

 xk4 (0) = 0.0, m = 2, n = 3, α = 0.037, βa = 11,βb = 10, γa = 0.28, γb = 0.29, δ = 0.09, ε = 1.1,
 ζ = 1.0, pk = 0.01(k − 1) and qk = Qb (k − 1).
(8)
Vertical axes show voltage diﬀerences and horizontal axes
show time. Namely, in the case of synchronizing twoMBCCs, the amplitude becomes zero. First graph x1 − x2
shows a voltage diﬀerence between the two MBCCs ofx1 − x2x2 − x3x3 − x4x4 − x5

Figure 5: Voltage diﬀerences between two circuits. α =0.037, βa = 11, βb = 10, γa = 0.28, γb = 0.29, δ = 0.09,
ε = 1.1, ζ = 1.0 and Qb = 0.005.
x1 − x2
x2 − x3
x3 − x4x4 − x5(b) Qb = 0.020
x1 − x2x2 − x3x3 − x4x4 − x5(c) Qb = 0.060
Figure 6: Voltage diﬀerences between two circuits. α =0.037, βa = 11, βb = 10, γa = 0.28, γb = 0.29, δ = 0.09,
ε = 1.1 and ζ = 1.0.
Group A. Synchronizations and un-synchronized burst appear alternately in a random way. Note that this synchronization is called as a quasi-synchronization. The secondgraph x2 − x3 shows a voltage diﬀerence between a MBCCof Group A and a MBCC of Group B. These are not synchronized at all. The third and fourth graphs x3 − x4 andx4 − x5 show voltage diﬀerences between two MBCCs ofGroup B.
By increasing Qb which is corresponding to small parameter mismatches of Group B as shown in Fig. 6, unsynchronized bursts of Group A and B are decreased andincreased, respectively.
In order to investigate this relationship, the synchronization is defined as following inequality.
|xk2 − x(k+1)2 | < 0.3(9)Figure 7 shows a relationship between synchronization
rates and Qb . The synchronization rate is defined as a ratioof synchronization time and total time of the calculation.
In Qb = 0, it means parameter mismatches free, a synchro-- 473 -Figure 7: Relationship between synchronization rates and small parameter mismatches Qb . Horizontal axis is Qb , Verticalaxis is a synchronization rate. α = 0.037, βa = 11, βb = 10, γa = 0.28, γb = 0.29, δ = 0.09, ε = 1.1 and ζ = 1.0. Initialvalues are set as xk1 = 0.0, xk2 = 0.1 + 0.1 ∗ k xk1 = 0.0 and xk1 = 0.0. Simulation time of each parameter is 100000.

nization rate of Group A is about 0.4, a synchronization rate
between Group A and B is about 0.1 and a synchronization
rate of Group B is about 0.8. In Qb = 0.001, MBCCs become periodic oscillation states. Therefore, very high or
low rates are observed. Namely, the case of synchronizing
two MBCCs perfectly becomes very high rate, and the caseof phase shifted synchronization like an anti-phase synchronization becomes very low rate. By increasing Qb fromQb = 0.001 to Qb = 0.083, a synchronization rate of GroupA is increased and synchronization rates of Group B aredecreased. In a area between Qb = 0.084 and Qb = 0.090,
MBCCs become periodic oscillation states. In the case of
Qb = 0.090 and more, MBCCs do not oscillate.
In previous studies, similar results are obtained in othersystems. We have hypothesized that this phenomenon canbe observed all kinds of chaotic coupled element of thesame couple system. This result reinforces this hypothesis.

4. Conclusions

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In this study, in order to verifier the phenomenon of previous studies, relationship between synchronization rates
and parameter mismatches in coupled chaotic circuits using
memristors. In the case of five circuits, we confirmed a phenomenon similar to our previous studies in computer simulations. By this result, our hypothesis for the phenomenon
observed in our system is reinforced.

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