2006 RISP International Workshop
on Nonlinear Circuits and Signal Processing (NCSP'06)
Waikiki Beach Marriott, Honolulu,Hawaii, USA, March 3-5, 2006

Analysis of an Asymmetrical Coupled System Using Chaotic Circuits and van der Pol Oscillators
Yasuteru Hosokawa† and Yoshifumi Nishio‡†Shikoku University
Furukawa, Ohjin, Tokushima, Japan
Phone:+81-88-665-1300 FAX:+81-88-637-1318
Email: hosokawa@keiei.shikoku-u.ac.jp

Abstract‡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(1) The synchronizations of the A-node circuits group andB-node circuits group are constricted each other.

In our previous study, we observed the interesting phenomenon. It is that the synchronous rate of the whole system (2) Decreasing the synchronization of one group decrease
an influence to the other group.
increases in spite of increasing parameter mismatches in thesystem.
(3) Therefore, the synchronization of the other group inIn this study, in order to verifier the this phenomenon,creases.
an asymmetrical coupled system is proposed and investigated. The system is realized by connecting chaotic subIn this study, the system which replaced the one side ofcircuits and van der Pol oscillators. In the case of five sub- chaotic circuits with van der Pol oscillators is investigated.
circuits, we confirmed synchronous phenomena in computer
calculations. Additionally, It was confirmed that synchronous
2. Asymmetrical Coupled Systems
rates of chaotic subcircuits are increased by increasing a parameter mismatch rate of van der Pol oscillators. We consider
that this result is corresponding to results of previous study.
GG

subcircuit

1. Introduction
Coupled systems of chaotic subsystems generate various
kinds of complex higher-dimensional phenomena such as
spatio-temporal chaotic phenomena, clustering phenomena,and so on. One of the most studied systems may be the coupled map lattice proposed by Kaneko[1]. The advantage ofthe coupled map lattice is its simplicity. However, many of
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[2]-[6].
In our previous study[?], synchronization phenomena in an
asymmetrical coupled system of chaotic circuits were investigated. 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 investigatedas non-ideal system. The small parameter mismatches weregiven to the subcircuits as an mismatch of the oscillation frequency. As a result of investigating this system, interesting
phenomenon was observed. The phenomenon is that the synchronous rate of the whole system increases in spite of increasing parameter mismatches in the system. We supposethat the phenomenon can be explained as follows:

GGG

subcircuit

subcircuit

GGG

GG

subcircuit

subcircuit

Figure 1: Asymmetrical coupled chaotic system.
The system is coupled globally and the coupling elements
are resistors as shown in Fig. 1. Coupled subcircuits arechaotic circuits[8] as shown in Fig. 2 and van der Pol circuits. In our previous study, only chaotic circuits are used andasymmetry is realized as selecting one of two nodes in Fig. 2.
In this study, circuits coupled with B-node are replaced vander Pol oscillators. Now, in order to carry out computer calculation, circuit equations are derived.

- 54 -A-node

B-nodei dkvk1

Function f1 (v) is characteristics of the nonlinear resistor
shown in Fig. 3.
van der Pol oscillator (m + 1 ≤ k ≤ m + n):
dvk1


= −ik3 − Gn f2 (vk1 )
C3

dt

( n)

X+Gvi1 − (m + n)vk1 ,(2)

k=1

 L2 dik3 = (1 + qk )vk1 ,dt

vk2ik-gC1C2L

Figure 2: Chaotic subcircuit.

iv

where,df2 (v) = v − |v + Vth | + |v − Vth |.
Gdd-Vthi1Vthv

Function f2 (v) is characteristics of the nonlinear negative resistor shown in Fig. 4.
Using the following parameters and variables,
rvk1
L1
vk2
1
xk1 =,
xk2 =,
xk3 =,VthVthVth C2√dC2,
α=t = L1 C2 τ, “ · ” =,dτCr
1rr(3)L1L1L, γ = Gd, δ=G,
β=gC2C2r C2C2L1L1ε=,ζ = Gn, η=,C3
C2L2dd(a)circuit model(b)characteristics

Figure 3: Nonlinear resistor which consist of two diodes.
i1

normalized equations are described as follows.
chaotic circuit(1 ≤ k ≤ m):0ẋk = αβxk − αγf1 (xk − yk )







(m+n)

X




+αδ
xi − (m + n)xk ,Gnv






ẏk =żk =Figure 4: Characteristics of the nonlinear negative resistor onvan der Pol oscillator.
chaotic circuit (1 ≤ k ≤ m):dvk1
 C1= gvk1 − Gd f1 (vk1 − vk2 )dt( n)

X

+Gvi1 − (m + n)vk1 ,

dvk2



C2


dt



 L dik3
1dt

k=1= −ik3 + Gd f1 (vk1 − vk2 ),=(1 + pk )vk2 ,where,
f1 (v) = v +|v − Vth | |v + Vth |−.
22

i=1(4)

0−zk + γf1 (xk − yk ),(1 + pk )yk ,where,
0f1 (x) = x +(|x − 1| − |x + 1|).
2(1) van der Pol oscillator(m + 1 ≤ k ≤ m + n):0ẋk = εζf2 (xk ) − εzk
(m+n)
X+δεxi − (m + n)xk ,



i=1


żk = (1 + qk )ηxk ,- 55 -(5)where,0f2 (x) = x − |x + 1| + |x − 1|.

We carry out computer calculations in the case of five subcircuits. The system consists of two chaotic circuits and threevan der Pol oscillators. Figure 5 shows computer calculatedresults using following initial values and parameters.
xk (0) = yk (0) = zk (0) = 0.100, (k = 1, 2, 3, 4, 5),
α = 0.400, β = 0.620, γ = 20.0, δ = 0.080,
 ε = 0.1, ζ = 0.3, η = 0.2
pk = 0.001(k − 1), qk = Q(k − 1).
(6)
The synchronization of van der Pol oscillators is very strong

computer calculated results. Left side figures 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. Right side figures show attractors of each subcircuits. The first graph shows the voltage difference between
the two chaotic circuits. We can see that synchronization andun-synchronized burst appear alternately in a random way.
The second graph shows the voltage difference between an
chaotic circuit and a van der Pol oscillator. This graph isdrawn on upper side and these two are not synchronized atall. The third and fourth graphs show the voltage differencesbetween the van der Pol oscillators. Periodic frequency differences are observed.
In the case of previous study, corresponding results areshown in Fig. 6. From comparing two results, we can see
as follows: Fig. 5(a) is similar to the case of higher negative
resistor value than Fig. 6. Figs. 5(b) are similar to Fig. 6.
In Figs. 5(c), synchronous rates are higher than Fig. 6. Additionally, voltage differences between van der Pol oscillators
are lower than Fig. 5(b). Now, in order to investigate relation-(a) Q = 0.10

Figure 6: One of the computer calculated result of previous
study.
(b) Q = 0.30

ship between synchronous rates and parameter mismatches,
we define the synchronous as shown in Fig. 7 and follows:synchronous(e) Q = 0.50

Figure 7: Definition of the synchronous.

Figure 5: Computer calculated results of proposed system.

|yk − yk+1 | < 0.01

(7)

against noise than synchronization of chaotic circuits. Ac- where k is number of subcircuits.
cordingly, by using larger difference of qk values than pre- Fig. 8 shows synchronous rate of chaotic circuits to Q which
vious study, an asynchronous state is realized. Figs. 5 show is corresponding to parameter mismatch rate. Fig. 9 shows- 56 -a result of previous study corresponding to Fig. 8. A-nodeof Fig. 8 is corresponding to chaotic circuits and B-node iscorresponding to van der Pol oscillators. In each of them,synchronous rate is increasing by increasing Q. We can consider to obtain the same phenomena. However, we can also
consider that this result shows influence decreasing small amplitudes of van der Pol oscillators.

increased the number of subcircuits. In the system, it is possible to investigate the relationship between synchronous rate
and the rate between numbers of chaotic circuits and van der
Pol oscillators.
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 inChaotic 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 8: Synchronous rate of chaotic circuits to the parame- [4] D. A. Likens, “Analytical solution of large numbersof mutually-coupled nearly-sinusoidal oscillators,” IEEEter mismatch of van der Pol oscillators.
Trans. Circuits and Systems, col. 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., 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, vol. 3, pp. 57–60, 2002.
[7] Y. Hosokawa, R. Tsujioka and Y. Nishio, “Relation between Synchronous Rate and Small Variations
on an Asymmetrical Coupled Chaotic System,” Proc.
NOLTA2005, pp. 166–169, 2005.

Figure 9: Synchronous rate of chaotic circuits to the parame[8] M. Shinriki, M. Yamamoto and S. Mori, “Multimode Oster mismatch in previous study.
cillations in a Modified van der Pol Oscillator Containing
a Positive Nonlinear Conductance,” Proc. IEEE, vol. 69,pp. 394–395, 1981.
3. Conclusions
In this study, in order to verifier the phenomenon of previous study, an asymmetrical coupled system is proposed andinvestigated. The system is realized by connecting chaotic
subcircuits and van der Pol oscillators. In the case of five subcircuits, we confirmed similar phenomena in computer calculations. Additionally, It was confirmed that synchronous ratesof chaotic subcircuits are increased by increasing a parameter mismatch rate of van der Pol oscillators. We consider that
this result is corresponding to results of previous study. However, we can also consider that this result shows influence ofsmall amplitudes of van der Pol oscillators. In order to clarify this phenomena, we will investigate on the system which- 57 -