2011 International Workshop on Nonlinear Circuits,Communication and Signal Processing
NCSP'11, Tianjin, China, March 1-3, 2011

Influences of a Kind of Coupled Elements on Synchronization Phenomena
in an Asymmetrical Coupled Chaotic Circuit
Yasuteru Hosokawa† and Yoshifumi Nishio‡†Shikoku University
Furukawa, Ohjin, Tokushima, Japan
Phone:+81-88-665-1300
FAX:+81-88-665-1318
Email: hosokawa@keiei.shikoku-u.ac.jp‡Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Phone:+81-88-656-7470
FAX:+81-88-656-7471
Email: nishio@ee.tokushima-u.ac.jp

Abstract

ChaosIt is very interesting to study about large-scale chaotic coupled systems using two kinds of elements. However, there
are not any study using two kinds of elements. In this study,
relationship between synchronization phenomena and parameter mismatches in coupling chaotic systems using two kindsof elements are investigated.

Chaos

Chaos

Chaos

1. Introduction

Chaos

Figure 1: System model.

There are studies about coupled chaotic circuit systems. In
these studies, chaos synchronization, spatio-temporal chaotic
phenomena, many kind of bifurcation phenomena are observed. These phenomena are investigated for engineer applications, understanding many kinds of natural phenomena,
relationship of signal processing of brain neurons. In recentyears, large-scale chaotic coupled systems have been investigated because a computer performance is high performance.
One of very famous study is Coupled Map Lattice (CML)
which is proposed by Kaneko. CML consists of simple difference equations. However, many interesting phenomena canbe observed and some of these phenomena are corresponding to natural phenomena. Therefore, the study of large-scalechaotic coupled systems becomes important.
On the other hands, there are many kinds of networks
which consist of two kinds of elements in the real world. For
instance, relationship between male and female in human society, information processing by servers and clients in computer networks, signal processing by pMOS and nMOS in
VLSI, relationship among male and female clusters of animals, meteorological phenomena caused by the relationshipbetween anticyclone and cyclone and so on. These phenomena can be caused by coupling two kinds of elements artificially or naturally. It cannot be observed in the case of one
kinds of elements. Although it is very interesting to study
about large-scale chaotic coupled systems using two kinds ofelements, there are not any study using two kinds of elements.

In this study, relationship between synchronization phenomena and parameter mismatches in coupling chaotic systems using two kinds of elements are investigated. Somemodels which include parameter mismatches are already proposed. The reason is that there are not identical things in
the real world, there are some differences between two things
which are considered as identical things. Positively, individual differences of elements are important factors for observed
phenomena in the case of coupling system which consists of
two kinds of elements.

2. System model
Figure 1 shows a system model in this study. The system
consists of two kinds of elements. Each element is coupledwith the others. Additionally, there are small parameter mismatches in one kind of elements. Normally, by increasing
parameter mismatches, synchronization rates of a system isdecrased. However, it is not always to observe the phenomenain this system model. In following section, computer simulation results are shown in the case that some kinds of chaotic
circuits are applied as the elements. In order to carry out
computer simulations, normalized system equations of eachsystem are derived as follows.

- 155 -Node Ag

Node B

vk1 L

C1

ik
C2

L2

vk2

g

ik1

ik2
L1

vdCvk

Figure 2: Chaotic circuit using System 1 and 2.
Figure 3: Chaotic circuit using System 3.
2.1. System 1

Subsystem A (1 ≤ k ≤ m):
Figure 1 shows a chaotic circuit as used System 1. By seẋ = αβxk − αγf (xk − yk ) klecting one of two coupling nodes (Node A and B), asymmetry of the system is realized. Normalized system equations
{m+n}
∑


are given as follows.
+αδxi − (m + n)xk ,Subsystem A (1 ≤ k ≤ m):i=1 ẏk = −zk + γf (xk − yk ),ẋ=αβx−αγf(x−y)
kkkk
m+nn∑∑+αδ{xi +yj − (m + n)xk },żk = (1 + pk )yk ,(1)i=n+1j=1
Subsystem B (m + 1 ≤ k ≤ m + n):ẏk = γf (xk − yk ) − zk , ż = (1 + p )y .
ẋk = εβxk − εγf (xk − yk )
kk k



{m+n}
∑



+εδ
xi − (m + n)xk ,Subsystem B (m + 1 ≤ k ≤ m + n):
ẋ

 k


 ẏk
 żk ẏk =
żk == αβxk − αγf (xk − yk ),= δ{m+n∑xi +i=n+1n∑

yj

j=1

−(m + n)yk } + γf (xk − yk ) − zk ,(2)f (x) = x +(|x − 1| − |x + 1|).
2(4)ζ {−zk + γf (xk − yk )} ,
η(1 + qk )yk ,where,= (1 + qk )yk ,where,

i=1(3)

f (x) = x +(|x − 1| − |x + 1|).
2

xk , yk and zk are corresponding to vk1 , vk2 and ik , respectively. Function f (x) is correspondinig to a characteristic ofdiodes.

xk , yk and zk are correspoinding to vk1 , vk2 and ik respec- 2.3. System 3
tively. Function f (x) is correspondinig to a characteristic of
Figure 3 shows a chaotic circuit which is as a element ofdiodes.
System 3. Normalized system equations are given as follows.
Subsystem A (1 ≤ k ≤ m): ẋk = (1 + pk )zk
2.2. System 2 ẏk = α {zk − f (yk )} ,żk = −xk − yk + βzk(5)
In this system, the circuit as shown in Fig. 2 is also applied.
m+n
∑

However, only Node A is used in this system. Normalized
+δ{zi − (m + n)zk , }system equations are given as follows.
i=1- 156 -x1 − x2

ik1
L1

L2
Cr

vkx2 − x3

ik2

x3 − x4x4 − x5vd

Figure 5: One of the simulation results of System 1.
3. Computer Simulations

Figure 4: Chaotic circuit using System 4.
Subsystem B (m + 1 ≤ k ≤ m + n):ẋk = (1 + qk )zk ẏk = ζ {zk − f (yk )} ,żk = η[−xk − yk + βθzkm+n∑+δ{zi − (m + n)zk , }]

3.1. Configuration of simulations

(6)

i=1

where,f (yk ) =γyk + 1 − |γyk − 1|.
2

xk , yk and zk are corresponding to ik1 , ik2 and vk . Function
f (x) is corresponding to a characteristic of diodes.

The number of elements is 5. The number of Subsystem A
and B is 2 and 3, respectively. Figure 5 shows one of simulation results of system 1. This figure shows a difference between two elements like x1 −x2 . The amplitude is zero meansa synchronization state. Phenomena changing between synchronization state and asynchronization state are observed.
Additionally, rates of synchronization state and asynchronization state is changed by changing the parameter mismatches.
In order to investigate the rates, we define synchronization
state as follows:|xk − xk−1 | < 0.01
Furthermore, parameter mismatches qk is set as follows:

2.4. System 4

qk = Q(k − 1).

Figure 4 shows a chaotic circuit which is as a element of
System 4. Normalized system equations are given as follows.
Subsystem A (1 ≤ k ≤ m):ẋk = αa xk + zk ẏk = (1 + pk )β (zk − f (yk )) ,{m+n }
(7)
∑
z,ż
=−x−y−
δ(m+n)+δikk
 ki=1Subsystem B (m + 1 ≤ k ≤ m + n):ẋk = αb xk + zk ẏk = (1 + qk )β (zk − f (yk )) ,[{m+n }]∑zi , żk = ε −xk − yk − δ(m + n) + δ(9)(10)

where k is the element number and Q is corresponding to parameter mismatches of Subsystem B. Rates of synchronization states are calculated by computer simulations. Each simulation result is shown as following subsections.
3.2. System 1
Figure 6 shows a relationship of a rate of the synchronization time and small parameter mismatches in the case of System 1. By increasing Q, a rate of the synchronization time ofSubsystem A is increased. The circuit as shown in Fig. ?? cangenerate double-scroll-type attractors. However, we could
not observed the phenomena in the parameter range which
double-scroll-type attractors are observed in.

i=1(8)
where,f (yk ) =|γyk + 1| − |γyk − 1|.
2

xk , yk and zk are corresponding to ik1 , ik2 and vk . Function
f (x) is corresponding to a characteristic of diodes.

3.3. System 2
Figure 7 shows a relationship of a rate of the synchronization time and small parameter mismatches in the case of
System 2. This result is similar to Fig. 6. In this case, thephenomena is observed in the parameter range which doublescroll-type attractors are observed in.

- 157 -Figure 6: Relationship between synchronization rates andsmall parameter mismatches in the case of System 1. m = 2,
n = 3, pk = 0.001(k − 1), α = 0.400, β = 0.500,
γ = 20.0 and δ = 0.070

Figure 8: Relationship between synchronization rates andsmall parameter mismatches in the case of System 3. m = 2,
n = 3, pk = 0.001(k − 1), α = 6.00, β = 0.400, γ =100.0, δ = 0.060, ε = 1.4, ζ = 7, η = 0.5, and θ = 1.2.

Figure 7: Relationship between synchronization rates andsmall parameter mismatches in the case of System 2. m = 2,
n = 3, pk = 0.001(k − 1), α = 0.600, β = 0.500,
γ = 20.0, δ = 0.070, ε = 0.6, ζ = 1.5 and η = 0.5.

Figure 9: Relationship between synchronization rates andsmall parameter mismatches in the case of System 4. m = 2,
n = 3, pk = 0.001(k − 1), αa = 0.41, αb = 0.43,
β = 3.50, γ = 470, δ = 0.130 and ε = 0.7.
on an Asymmetrical Coupled Chaotic System,” Proc.
NOLTA’05, pp. 166–169, 2005.

3.4. System 3 and 4

Figure 8 and 9 shows a relationship of a rate of the synchronization time and small parameter mismatches in the case of [2] Y. Hosokawa and Y. Nishio, “Synchronization PhenomSystem 3 and 4, respectively. These results are also similar toena on Asymmetrical Globally Coupled Chaotic Systems
Fig. 6.
Using Circuits or Maps,” Proc. of NDES’07, pp. 30–33,2007.
4. Conclusions[3] M. Shinriki, M. Yamamoto and S. Mori, “Multimode Oscillations in a Modified van der Pol Oscillator Containing
In this study, a relationship between synchronization phea Positive Nonlinear Conductance,” Proc. IEEE, vol. 69,nomena and parameter mismatches in coupling chaotic syspp. 394–395, 1981.
tems using two kinds of elements have been investigated.
As a result, following common points are mentioned. Two [4] N. Inaba, T. Saito and S. Mori, “Chaotic Phenomena insubsystems are not synchronized each other. By increasing
a Circuit with a Negative Resistance and an Ideal Switch
synchronization rate of Subsystem B, synchronization rate ofof Diodes,” IEICE Trans. fundamentals, vol. E-70, no. 8,Subsystem A is incrased. Additionally, the phenomena canpp. 744–754, 1987.
be observed in the case of chaotic states only.
[5] Y. Nishio, N. Inaba, S. Mori and T. Saito, “RigorousAnalyses of Windows in a Symmetric Circuit,” IEEE
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