NOLTA 2006
11-14 september
Bologna - ITALY

Asymmetrical Chaotic Coupled System Using Two Different Parameter Sets.
Yasuteru Hosokawa† and Yoshifumi Nishio‡†Department of Information Science, Shikoku University
Furukawa, Ohjin-cho, Tokushima 771-1192 Japan‡Department of Electrical and Electronics Engineering, Tokushima University
2-1 Minami-Johsanjima-cho, Tokushima 770-8506 Japan
Email: hosokawa@keiei.shikoku-u.ac.jp† , nishio@ee.tokushima-u.ac.jp‡
Abstract—In our previous study, we observed the interesting phenomenon in an asymmetrical chaotic coupledsystem. It is that the synchronous rate of the one subcircuits group increases in spite of increasing parameter mismatches in the system.
In this study, in order to verify the this phenomenon using another system, an asymmetrical chaotic coupled system is proposed and investigated. Asymmetry of the system is realized by using two parameter sets. In the case of
five subcircuits, we confirmed synchronization phenomena
in computer calculations. Additionally, It was confirmedthat synchronous rates of subcircuits using one parameter
set are increased by increasing a parameter mismatch rateof the other subcircuits. We consider that this result is corresponding to results of previous study.

synchronization of one group decreases, the synchronization of the other group increases.
In this study, in order to verify the this phenomenon using another system, an asymmetrical chaotic coupled system is proposed and investigated. Asymmetry of the system is realized by using two parameter sets. In the case offive subcircuits, we confirm synchronization phenomena incomputer calculations. Additionally, It is confirm that synchronous rates of subcircuits using one parameter set areincreased by increasing a parameter mismatch rate of another subcircuits.
2. Asymmetrical Coupled Systems

G

1. Introduction
Coupled systems of chaotic subsystems generate various 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[1]. The advantage of the coupled map lattice is its simplicity. However,many of nonlinear phenomena generated in nature would
be not so simple. Therefore, it is important to investigate
the complex phenomena observed in natural physical systems such as electric circuits systems[2]-[6].
In our previous study[7], synchronization phenomena inan asymmetrical coupled system of chaotic circuits wereinvestigated. The system is coupled globally and the coupling elements are resistors. Each subcircuit has two coupling nodes and the asymmetrical coupled system is realized by selecting one of two coupling nodes. This system
was investigated as non-ideal system. The small parametermismatches were given to the subcircuits as an mismatchof the oscillation frequency. As a result of investigating
this system, interesting phenomenon was observed. Thephenomenon is that the synchronous rate of the one subcircuits group increases in spite of increasing parameter mismatches in the system. We suppose that the phenomenon
can be explained as follows. The synchronizations of theone subcircuits group and other subcircuits group are constricted each other. Therefore, in the case of decreasing the543

G

subcircuit

GG

G

subcircuit

subcircuit

GG
G

G

G
subcircuit

subcircuit

Figure 1: Asymmetrical coupled chaotic system.
A-nodeB-node

vk1

i dk

vk2
ik-g

C1

C2

L

Figure 2: Chaotic subcircuit.
The system is coupled globally and the coupling ele-ments are resistors as shown in Fig. 1. The coupled subcircuit is a chaotic circuit[8] as shown in Fig. 2. In ourprevious study, asymmetry is realized as selecting one of
two nodes in Fig. 2. In this study, only one node areused and asymmetry is realized by using two parameter
sets. The circuit using one parameter set is called as ”Asubcircuits”, and using the other parameter sets is called as”B-subcircuit”. Now, in order to carry out computer calculation, circuit equations are derived. Figure 3 shows a nonlinear resistor which consist of two diodes. Circuit equations are derived using Figure 3(b). We define m and n as
the number of circuits. The number of A-subcircuits is m,
and the number of B-subcircuits is n.

iv

dGdd-Vthi

1

Vthvdd(a)circuit model

variables,
vk1,Vth√t = La C2a τ,

xk =

β=g
ε=⎪⎪
⎪
⎪⎪
⎪ẏk =⎪⎪⎪⎪⎪⎪⎪⎪⎩żk =(b)characteristics

A-subcircuits(1 ≤ k ≤ m):dvk2⎪⎪⎪C2a⎪⎪dt⎪⎪⎪⎪di⎪⎩ L k3adt

i=1= −ik3 + Gd f (vk1 − vk2 ),⎪⎪⎪⎪⎪⎪ẏk =⎪⎪⎪⎪⎪⎪⎪⎪⎩żk =(1 + pk )vk2 ,(1)B-subcircuits(m ≤ k ≤ m + n):
⎧
dvk1
⎪
⎪
C1b⎪
⎪dt
⎪
⎪
⎪
⎪
⎪
⎪
⎨
dvk2
⎪
⎪
⎪
C
⎪
⎪ 2b dt⎪⎪⎪⎪⎪⎩ L dik3bdt

La,
C2a

γ = Gdζ=C2aC2b

δ=Gand η =

La,
C2aLa.
Lb

(3)

(4)

i=1


−zk + γf (xk − yk ),(1 + pk )yk ,B-subcircuits(m + 1 ≤ k ≤ m + n):⎧ẋk = εβxk − εγf (xk − yk )⎪⎪⎪⎪⎪⎪⎪⎪m+n⎪⎪⎪⎪⎪⎪+εδxi − (m + n)xk ,⎨= gvk1 − Gd f (vk1 − vk2 )
m+n


+G
vi1 − (m + n)vk1 ,=C2a,C1bzk =Normalized equations are described as follows.
A-subcircuits(1 ≤ k ≤ m):⎧⎪ẋ = αβxk − αγf (xk − yk )⎪⎪ k⎪⎪⎪⎪⎪⎪m+n⎪⎪⎪⎪⎪⎨+αδxi − (m + n)xk ,Figure 3: Nonlinear resistor which consist of two diodes.

⎧dvk1
⎪⎪
C1a⎪⎪dt⎪⎪⎪⎪⎪⎪⎨

La,
C2aLa
1,
Vth C2aC2a,
α=C1avk2,
Vthd“·”=,dτyk =(5)i=1ζ −zk + γf (xk − yk ) ,η(1 + qk )yk ,where,= gvk1 − Gd f (vk1 − vk2 )m+n+Gvi1 − (m + n)vk1 ,f (x) = x +(|x − 1| − |x + 1|).
2i=1= −ik3 + Gd f (vk1 − vk2 ),= (1 + qk )vk2 ,(2)where,
f (v) = v +|v − Vth | |v + Vth |−.
22

Function f (v) is characteristics of the nonlinear resistor
shown in Fig. 3. Parameters pk and qk shows small parameter mismatches. Using the following parameters andWe carry out computer calculations in the case of five subcircuits. The system consists of two A-subcircuits and three
B-subcircuits. Figures 4 and Figure 5 show computer calculated results using following initial values and parameters. Figure 4(a) show a attractor of A-subcircuit (k = 1).
Figure 4(b) show a attractor of B-subcircuit (k = 3). Horizontal axes are xk and vertical axes are zk . Double scroll
type attractors are observed. Figures 5 show voltage differences between each subcircuits. Vertical axes show voltage differences and horizontal axes show time. Namely,in the case of synchronizing two subcircuits, the amplitude
becomes zero. Value Q is corresponding to parameter mismatch rate of B-subcircuits.

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lows: Fig. 5(a) is similar to the case of lower Q value thanFig. 6. Fig. 5(b) is similar to Fig. 6. In Fig. 5(c), synchronous rate is higher than Fig. 6. Now, in order to iny1 − y 2y2 − y3(a)A-subcircuity3 − y4(b)B-subcircuity4 − y5

Figure 4: Attractors of A-subcircuit (k = 1) and Bsubcircuit (k = 3). Horizontal axes are xk and verticalaxes are zk . In the case of Eq. (6) and Q = 0.05.
⎧xk (0) = yk (0) = zk (0) = 0.100, (k = 1, 2, 3, 4, 5),⎪⎪⎨m = 2, n = 3, α = 0.600, β = 0.500, γ = 20.0,
δ = 0.070, ε = 0.6, ζ = 1.5, η = 0.5⎪⎪⎩
pk = 0.001(k − 1) and qk = Q(k − 1).
(6)
The first graph shows the voltage difference between two

Figure 6: One of the computer calculated result of previousstudy.
vestigate the relationship between synchronous rates andparameter mismatches, we define the synchronization asshown in Fig. 7 and follows:synchronizationx1 − x2
x2 − x3
x3 − x4
Figure 7: Definition of the synchronization.

x4 − x5
x1 − x2(a) Q = 0.001|xk − xk+1 | < 0.01

where k is number of subcircuits.
Fig. 8 shows synchronous rate averages between two subcircuits to Q which is corresponding to parameter mismatch rate. The synchronous rate is defined as a ratio ofsynchronous time and total time. Fig. 9 shows a result ofprevious study corresponding to Fig. 8. A-node of Fig. 9 iscorresponding to A-subcircuit, and B-node is corresponding to B-subcircuit. In Fig. 8 and Fig. 9, the synchronous
rates of one group are increasing by increasing Q. We
can consider to obtain the same phenomena. Additionally,thirty subcircuits case is investigated. Figures 10 and 11
shows the result. We can also observed the same result.
Namely, in spite of increasing parameter mismatch rate,synchronous rate of A-subcircuit increasing.

x2 − x3
x3 − x4x4 − x5
x1 − x2(7)(b) Q = 0.005

x2 − x3
x3 − x4
x4 − x5(c) Q = 0.010
Figure 5: Voltage difference between two subcircuits. In
the case of Eq. (6).

3. Conclusions

A-subcircuits. We can see that synchronization and unsynchronized burst appear alternately in a random way.
The second graph shows the voltage difference between anA-subcircuit and a B-subcircuit. These are not synchronized at all. The third and fourth graphs show the voltage
differences between two B-subcircuits. These results are
similar than first graph. However these are not the same. In
the case of previous study, corresponding results are shownin Fig. 6. From comparing two results, we can see as fol-In this study, in order to verifier the phenomenon of previous study, an asymmetrical coupled system is proposed
and investigated. Asymmetry of the system is realized byusing two parameter sets.
In the case of five subcircuits, we confirmed similar phenomena in computer calculations. Additionally, It was confirmed that synchronous rates between circuits using one
parameter set are increased by increasing a parameter mismatch rate of the other parameter set. We consider that this545

Figure 8: Synchronous rate of chaotic circuits to the parameter mismatch of B-subcircuit in the case of five subcircuits. m = 2, n = 3, α = 0.600, β = 0.500, γ = 20.0,
δ = 0.070, ε = 0.6, ζ = 1.5 and η = 0.5.

Figure 9: Synchronous rate of previous study corresponding to the parameter mismatch of B-subcircuit in the caseof five subcircuits. m = 2, n = 3, α = 0.400, β = 0.500,
γ = 20.0, and δ = 0.07.
result is corresponding to results of previous study. Furthermore, we also confirmed the same phenomena in thecase of thirty subcircuits. Therefore, we consider that thesephenomena are not the phenomena generated on a specialsystem.
References[1] K. Kaneko, “Clustering, Coding, Switching, Hierarchical Ordering, and Control in a Network of Chaotic Elements,” Physica D, vol. 41, pp. 137–172, 1990.
[2] L. M. Pecora and T. L. Carrol, “Synchronization in
Chaotic Systems,” Physical Review Letters, vol. 64, pp.
821–824, 1990.
[3] T. Endo, and S. Mori, “Mode Analysis of a Ring of aLarge Number of Mutually Coupled van der Pol Oscillators,” IEEE Trans. Circuit and Systems, vol. CAS-25,
no. 5, pp. 7–18, 1978.

Figure 10: Voltage difference between two subcircuits inthe case of thirty subcircuits. m = 12, n = 18, α = 0.600,
β = 0.500, γ = 20.0, δ = 0.013, ε = 0.6, ζ = 1.5 andη = 0.5.

Figure 11: Synchronous rate of chaotic circuits to the parameter mismatch of B-subcircuit in the case of thirty subcircuits. m = 12, n = 18, α = 0.600, β = 0.500,
γ = 20.0, δ = 0.013, ε = 0.6, ζ = 1.5 and η = 0.5.
[4] D. A. Linkens, “Analytical solution of large numbers ofmutually-coupled nearly-sinusoidal oscillators,” IEEE
Trans. Circuits and Systems, vol. CAS-21, pp.294–300, Mar, 1974.
[5] H. Sekiya, S. Moro, S. Mori, and I. Sasase, “Synchronization Phenomena on Four Simple Chaotic Oscillators Full-Coupled by Capacitors,” IEICE Trans. Fundamentals, vol. J82-A, No. 3, pp.375–385, 1999.
[6] M. Miyamura, Y. Nishio and A. Ushida, “Clustering in
Globally Coupled System of Chaotic Circuits,” IEEE
Proc. ISCAS’02, vol. 3, pp. 57–60, 2002.
[7] Y. Hosokawa, R. Tsujioka and Y. Nishio, “Relation
between Synchronous Rate and Small Variations onan Asymmetrical Coupled Chaotic System,” Proc.
NOLTA’05, pp. 166–169, 2005.
[8] M. Shinriki, M. Yamamoto and S. Mori, “Multimode Oscillations in a Modified van der Pol Oscillator
Containing a Positive Nonlinear Conductance,” Proc.
IEEE, vol. 69, pp. 394–395, 1981.

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