1ICSES 2006 I NTERNATIONAL C ONFERENCE ON S IGNALS AND E LECTRONIC S YSTEMSS EPTEMBER 17-20, 2006, Ł ÓD Ź , P OLAND

Comparison of Synchronization Phenomena on
Asymmetrical Global Coupled Chaotic Systems
Rysunek 1:

Yasuteru Hosokawa

Yoshifumi Nishio

Department of Management and Information Science,
Shikoku University
Furukawa, Ohjin-cho, Tokushima, 771-1192 Japan
Email: hosokawa@keiei.shikoku-u.ac.jp

Department of Electrical and Electronics Engineering,
Tokushima University
2-1 Minami-Johsanjima-cho, Tokushima, 770-8506 Japan
Email: nishio@ee.tokushima-u.ac.jp

Abstract— In this paper, three kinds of asymmetrical global
chaotic coupled systems are investigated. The asymmetries are realized by different coupling nodes or different parameters sets. In
these systems, an interesting phenomenon about synchronization
phenomena is observed. The phenomenon is that a ratio of thesynchronization time increases in spite of increasing parametermismatches in the system.

GG

subcircuit
AGG

G

GI. I NTRODUCTION

G

Many researchers have focused on engineering applications of chaos, for instance, chaotic communication systems,
chaotic control, chaotic synchronization and so on. Especially,
chaotic synchronization is very interesting phenomenon thatchaotic subsystems are synchronized in spite of different initial values[1]. Additionally, coupled chaotic systems generatevarious kinds of complex higher-dimensional phenomena suchas spatio-temporal chaotic phenomena, clustering phenomena
and so on. One of the most studied systems may be thecoupled map lattice proposed by Kaneko[2]. The advantageof the coupled map lattice is its simplicity. However, manyof nonlinear phenomena generated in nature would be not sosimple. Therefore, it is important to investigate the complex
phenomena observed in natural physical systems such as
electric circuits systems[3]-[4].
In this study, three kinds of asymmetrical global coupledchaotic systems are investigated. Especially, we paid attention to relationships between synchronization phenomena andsmall parameter mismatches. In all systems, an interesting
phenomenon is observed. The phenomenon is that a ratio of thesynchronization time increases in spite of increasing parametermismatches in the system.
In the Sect. 2, proposed systems and its circuit equations
are shown. In the Sect. 3, computer simulation results of eachsystems are shown. Relationships between synchronization
phenomena and small parameter mismatches are also shown.
Some concluding remarks is presented in the Sect. 5.

Gsubcircuit
AGG

subcircuit
BFig. 1.

subcircuit
Bsubcircuit
B

Proposed system.

iGdv

1i

VthFig. 2.

v

Diode model.

as m and n, respectively. The numbers of subcircuit A and B
are two and three. Actual systems are shown in following threesubsections. Additionally, in order to carry out the computer
simulations, the circuit equations are derived. In all systems,
the diode model is a piecewise linear function shown in Fig. 2.
Small parameter mismatches of Subcircuit A and B are pk andqk , respectively.
A. System 1

II. P ROPOSED S YSTEMS
A proposed system is shown in Fig. 1. This system consistsof two kinds of subcircuits and resistors as coupling elements.
Subcircuits are coupled globally. An asymmetry of the system
is realized by using two kinds of subcircuits ( subcircuit A andB ). In this study, the number of subcircuit A and B are shown87

The subcircuit of System 1 is shown in Fig. 3. This chaotic
circuit is a simple three-dimensional autonomous circuit proposed by Shinriki et al.[6]. By selecting one of two couplingnodes(A-node and B-node), an asymmetry of the system isrealized. Using following variables and parameters,

ICSES 2006 I NTERNATIONAL C ONFERENCE ON S IGNALS AND E LECTRONIC S YSTEMS
A-nodeB-nodei dkvk1

vk2

i k2

ik-g

C1

C2

Fig. 3.

Ł ÓD Ź , P OLAND

L-g

L2i k2

L1

Cvdvk

Subcircuit 1.
Fig. 4.

Subcircuit 2.

rvk1Lvk2
1,
yk =,
zk =ik ,VthVthVth C2√dC2,α=,t = LC2 τ, “ · ” =dτC1rrr
LL1Lβ=g, γ = Gd, δ=,C2C2R C2
xk =and variables,(1)

Normalized circuit equations are described as follows: Subcircuit A (1 ≤ k ≤ m):ẋk = αβxk − αγf (xk − yk )
m+nnXX+αδ{xi +yj − (m + n)xk },i=n+1j=1
ẏk = γf (xk − yk ) − zk , ż = (1 + p )y .
kk(2)

kvk1
xk =,
Vth√
t = La C2a τ,
r
La
β=g,
C2aC2aε=,C1bẏk =żk == αβxk − αγf (xk − yk ),= δ{m+nX

xi +i=n+1nXyj

j=1(3)−(m + n)yk } + γf (xk − yk ) − zk ,= (1 + qk )yk ,The nonlinear function f (x) corresponding to the characteristics of the diodes is described as follows:

r
1La
zk =,
Vth C2aC2aα=,
C1ar
La
δ=G,
C2aLa
η=.
Lb(4)

Normalized circuit equations are described as follows:
Subcircuit A (1 ≤ k ≤ m):ẋ = αβxk − αγf (xk − yk )
 k




(m+n)

X


+αδ
xi − (m + n)xk ,Subcircuit B (m + 1 ≤ k ≤ m + n):
ẋ

 k



 ẏk żkvk2yk =,Vthd“·”=,dτrLa
γ = Gd,C2aC2aζ=andC2bi=1(5)−zk + γf (xk − yk ),(1 + pk )yk ,Subcircuit B (m + 1 ≤ k ≤ m + n):ẋk = εβxk − εγf (xk − yk )




(m+n)

X



+εδ
xi − (m + n)xk , ẏk =





żk =(|x − 1| − |x + 1|).
f (x) = x +2

i=1(6)

ζ {−zk + γf (xk − yk )} ,
η(1 + qk )yk ,where,

B. System 2
The subcircuit of System 2 is same as the subcircuit ofSystem 1. However, only A-node is used in this system.
An asymmetry of the system is realized as a difference ofparameters. Namely, parameters of subcircuit A in Fig. 1 isdifferent from subcircuit B. Using the following parameters

88f (x) = x +(|x − 1| − |x + 1|).
2

C. System 3
The subcircuit of System 3 is shown in Fig. 4. This chaotic
circuit was proposed by Inaba et al.[7]. Using the following

ICSES 2006 I NTERNATIONAL C ONFERENCE ON S IGNALS AND E LECTRONIC S YSTEMS

parameters and variables,rr
L1a ik1
L1a ik2xk =, yk =,Ca VthCa Vth√dt = L1a Ca τ,,“·”=dτrr
L1aCa
β = ga,γ = rd,Ca
L1aL1aL1aε=,ζ=,
L1bL2bgb
and θ = .
gavk
zk =,
Vth
L1aα=,
L2ar
L1a
δ=G,
CaCa
η=,
Cb

6--x1x3(b) Subcircuit B (k = 3)

Fig. 6. Attractors of System 1. Horizontal axes are xk and vertical axes arezk .

(8)

Subcircuit B (m + 1 ≤ k ≤ m + n):ẋk = (1 + qk )zk
 ẏk = ζ {zk − f (yk )} ,żk = η[−xk − yk + βθzkm+nX+δ{zi − (m + n)zk , }]

(9)subcircuits in the case of System 2. Vertical axes show voltage
differences and horizontal axes show time. Namely, in thecase of synchronizing two subcircuits, the amplitude becomeszero. First graph shows the voltage difference between thetwo subcircuit A. Synchronizations and un-synchronized burst
appear alternately in a random way. The second graph shows
the voltage difference between subcircuit A and subcircuit B.
These are not synchronized at all. The third and fourth graphs
show the voltage differences between two subcircuit B. In theSystems 1 and 3, similar results are observed.
x1 − x2x2 − x3x3 − x4i=1

x4 − x5

where,
γyk + 1 − |γyk − 1|.
f (yk ) =2
III. C OMPUTER S IMULATION

Fig. 7. Voltage differences between two subcircuits in the case of System 2.

At first, each results of the computer simulations are shown.
Figures 5 are examples of the computer simulation results on
System 2. Double scroll type attractors are observed on theeach subcircuits. In the case of System 1, similar attractors
can be observed. Figures 6 are examples of the computer
simulation results on System 3. Rossler type attractors are observed. Figure 7 shows the voltage differences between eachz36(a) Subcircuit A (k = 1)

i=1



z1(7)

Normalized circuit equations are described as follows:
Subcircuit A (1 ≤ k ≤ m):ẋk = (1 + pk )zkẏ k = α {zk − f (yk )} ,żk = −xk − yk + βzkm+nX+δ{zi − (m + n)zk , }Ł ÓD Ź , P OLAND

Next, the relationship between the synchronization andsmall parameter mismatches are investigated. In order to investigate it, the synchronization is defined as following equationand figure.

synchronization

Fig. 8.

Definition of the synchronization.

|xk − xk+1 | < 0.01

  (a) Subcircuit A (k = 1)(b) Subcircuit B (k = 3)

Fig. 5. Attractors of System 1. Horizontal axes are xk and vertical axes arezk .

89

(10)

Figure 9 shows ratios of the synchronization time and total
time in the case of System 1. Q is shown as following
equation.
qk = Q(k − 1)
(11)
Q is corresponding to small parameter mismatches qk ofsubcircuit B group. By increasing small parameter mismatchICSES 2006 I NTERNATIONAL C ONFERENCE ON S IGNALS AND E LECTRONIC S YSTEMS

Fig. 9. Relationship of the ratio of the synchronization time and small
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

Ł ÓD Ź , P OLAND

Fig. 11. Relationship of the ratio of the synchronization time and small
parameter mismatches in the case of System 3. m = 2, n = 3, pk =0.001(k − 1), α = 0.600, β = 0.400, γ = 100.0, δ = 0.060, ε = 1.4,
ζ = 7, η = 0.5, and θ = 1.2.

IV. C ONCLUSION
In this study, three kinds of asymmetrical global chaotic
coupled systems are proposed and investigated. In the caseof five subcircuits, we confirmed synchronization phenomena.
Additionally, It was confirmed that the synchronization time
ratio of one subcircuit group are increased by decreasing thesynchronization time ratio of the other subcircuit group. We
suppose that the phenomenon can be explained as follows.
The synchronizations of the one subcircuit group and theother subcircuit group are constricted each other. Therefore,
in the case of decreasing the synchronization of one group,
the synchronization of the other group increases.
Fig. 10. Relationship of the ratio of the synchronization time and small
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.

of subcircuit B group, the synchronization time of subcircuit
A group is increased. Namely, in spite of increasing small
parameter mismatches of the system, the synchronization timeof subcircuit A group is increased. Figure 10 and Figure 11
show the case of System 2 and 3, respectively. In these case,periodic orbits are observed on some value Q. In particular,
Q = 0.004 and Q = 0.005 in the case of System 2 and Q =0.08 in the case of System 3. Excepting periodic orbits, thesynchronization time of subcircuit A group is also increased
by increasing small parameter mismatch of subcircuit B group.
On all systems, we can also observe the case of decreasing
the synchronization time when increasing small parametermismatches of subcircuit B group in other parameters ofsubcircuit A group and B group. We suppose that the phenomenon can be explained as follows. The synchronizationsof the one subcircuit group and the other subcircuit group areconstricted each other. Therefore, in the case of decreasing thesynchronization of one group, the synchronization of the othergroup increases.

90

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[6] M. Shinriki, M. Yamamoto and S. Mori, “Multimode Oscillations ina Modified van der Pol Oscillator Containing a Positive NonlinearConductance,” Proc. IEEE, vol. 69, pp. 394–395, 1981.
[7] N. Inaba, T. Saito and S. Mori, “Chaotic Phenomena in a Circuit witha Negative Resistance and an Ideal Switch of Diodes,” IEICE Trans.
fundamentals, vol. E-70, no. 8, pp. 744–754, 1987.