2005 International Symposium on Nonlinear
Theory and its Applications (NOLTA2005)
Bruges, Belgium, October 18-21, 2005

Relation between Synchronous Rate and Small Variations
on an Asymmetrical Coupled Chaotic System
Yasuteru Hosokawa† , Ryosuke Tsujioka‡ 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 this study, the relation between synchronous rate and small variations on an asymmetrical coupled chaotic system is investigated. The system is realized
by connecting one of two nodes in each chaotic subcircuit.
The small variations are given to the subcircuit as an errorof the oscillation frequency. In a computer calculation, aninteresting phenomenon is observed. The phenomenon isthat the synchronous rate of the whole system increases inspite of increasing parameter variations in the system.

2. System Model
The proposed system is shown in Fig. 1. This system
consists of chaotic circuits as shown in Fig. 2 and resistors as coupling elements. The chaotic circuit is a simple
three-dimensional autonomous circuit proposed by Morigroup [7][8]. In order to realize the asymmetrical couplingsystem, we use one of the two coupling nodes. The circuit
connected by the node vk1 is called as ”A-node circuit”, andthe node vk2 is called as ”B-node circuit”. The numbers ofA-node circuits and B-node circuits are denoted by m andn, respectively. Now, in order to carry out computer calcu-1. Introduction

In this study, synchronization phenomena in an asymmetrical coupled system of chaotic circuits are investigated.
The system is coupled globally and the coupling elements
are resistors. This system is the modified version of thesymmetrical coupled chaotic system proposed by Miyamura et al.[6]. Each subcircuit has two coupling nodes andselecting one of two coupling nodes realizes the asymmetrical coupled system. The small variation is given to thesubcircuit as an error of the oscillation frequency. Namely,
the small variation is small parameter mismatches. In computer calculation, the influences of small variation to thesystem are investigated.

CCR

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 that chaotic subsystems synchronized in spite ofdiﬀerent initial values [1]. Additionally, coupled systemsof chaotic subsystems generate various kinds of complex
higher-dimensional phenomena such as spatio-temporalchaotic phenomena, clustering phenomena, and so on. Oneof the most studied systems may be the coupled map latticeproposed by Kaneko[2]. 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 [3]-[5].

R

C
C

R

R

R

C
C

R

R
R

R

R

C
C

C
C

Figure 1: Asymmetrical coupled chaotic system.

A-nodeAvk1

BCC

B-nodei dkvk2
ik-r

C1

C2

L

Figure 2: Chaotic subcircuit.
lation, normalized circuit equations are derived. Using the166

following variables and parameters,
vk1
xk =,
a
√
t = LC2 τ,
L
1β=,
r C2

L
vk2
1ik ,
yk =,
zk =aa C2
dC2“·”=,,
α=dτ
C1

LL
1
γ = Gd, δ=,C2
R C2(1)the circuit equations of A-node and B-node circuits groupare described as follows:
A-node circuit equations:⎧⎪ẋk = αβxk − α f (xk − yk )⎪⎪⎪⎪⎪⎪m+nn⎪⎪⎪⎪⎪x+y j − (m + n)xk },+αδ{⎪i⎨(2)⎪
i=n+1j=1⎪⎪⎪⎪⎪⎪⎪ẏk = f (xk − yk ) − zk ,⎪⎪⎪⎪⎪⎩ ż = (1 + p )y .
kk k
B-node circuit equations:⎧⎪ẋk = αβxk − α f (xk − yk ),
⎪⎪
⎪⎪
⎪
⎪
m+nn⎪


⎪⎪⎪
⎪
=
δ{
x+yjẏ
⎪
i⎨ k⎪
i=n+1
j=1⎪
⎪
⎪⎪
⎪
⎪
⎪
−(m + n)yk } + f (xk − yk ) − zk ,⎪⎪⎪⎪⎪⎩ ż = (1 + q )y ,kk k(3)Figure 3: Computer calculated result of the asymmetricalcoupled system in the case of m = 3 and n = 2. α = 0.4,
β = 0.5, γ = 20, δ = 0.0702, and pk = qk = 0.001(k − 1).
becomes zero. The first graph (y1 − y2 ) shows the voltage diﬀerence between the two B-node circuits. We cansee that synchronization and un-synchronized burst appear
alternately in a random way. The second graph (y2 − y3 )shows the voltage diﬀerence between an A-node circuit anda B-node circuit. We can see that these two are not synchronized at all. The third and fourth graphs (y3 −y4 and y4 −y5 )show the voltage diﬀerences between the A-node circuits.
The synchronous rate is larger than the first graph.
Figure 6 shows the synchronous rate of the system tosmall variations qk of the B-node circuits group. We define
the synchronous as shown in Fig. 4 and follows:

where k = 1, 2, 3, · · · , m + n i.e. , the number of A-node circuits is m, the number of B-node circuits is n, and pk and qk
are introduced to give small variations of the oscillation frequencies. The nonlinear function f (xk − yk ) corresponding
to the characteristics of the diodes is described as follows:synchronous

Figure 4: Definition of the synchronous.

f (xk − yk ) =⎧⎪γ(xk − yk − 1) (xk − yk > 1),⎪⎪⎪⎨0(|xk − yk | ≤ 1),⎪⎪⎪⎪⎩ γ(x − y + 1) (x − y < −1).
kkkk|yk − yk+1 | < 0.01
(4)

Using these equations, computer calculations are carriedout in the next section.
3. Computer calculation
3.1. The case of using five chaotic circuits
Figure 3 shows the computer calculated result of theasymmetrical coupled system in the case of m = 3 andn = 2. The voltages of A-node circuits are y3 , y4 and, y5 .
The voltages of B-node circuits are y1 and y2 . The horizontal axis is time, and the vertical axes show the voltage
diﬀerences between two chaotic circuits. Namely, in thecase of synchronizing two chaotic circuits, the amplitude(5)

where k is number of subsystems.
Figure 5 shows the influence of the number of B-nodecircuits to the synchronous rates of A-node circuits groupand B-node circuits group. The horizontal axis is the number of B-node circuits, and the vertical axis is the synchronous rate of the groups.
As the number of the B-node circuits increases, the synchronous rate of the whole system decreases. Therefore,
we can say that the B-node circuits group is more diﬃcult
to be synchronized than the A-node circuits group.
Next, the influence of the synchronous rate between Anode circuits group and B-node circuits group is investigated. Note that only qk is varied and pk of the A-nodecircuit group is constant. The horizontal axis is Q corresponding to the variation of the B-node circuits group,and the vertical axis is the synchronous rate of the system.
When the variations Q increases, the synchronous rate of167

Figure 5: Synchronous rate of the groups to the number ofthe B-node circuits. α = 0.4, β = 0.5, γ = 20, δ = 0.07,
and pk = qk = 0.001(k − 1).
the B-node circuits group decreases. However, in spite ofincreasing the variations, the synchronous rate of the Anode circuits group increases.

Figure 6: Synchronous rate of the groups to the variationof the B-node circuits. α = 0.4, β = 0.5, γ = 20, δ = 0.07,
pk = 0.001(k − 1), and qk = Q(k − 1).
between two chaotic circuits. From the first to the fourthgraphs show voltage diﬀerences between two B-node circuits. The fifth graph shows the voltage diﬀerence between
an A-node circuit and a B-node circuit. The other graphs
show the voltage diﬀerences between two A-node circuits.

3.2. Consideration
In the case of all B-node circuit, low synchronization rate
is observed as line first of Fig. 3 and, in the case of all Anode circuit, very high synchronization rate is observed asline third of Fig. 3. Additionally, these two type circuits do
not synchronous each other as Fig. 3. Therefore, interesting phenomenon that in spite of increasing variations whichconstrict the synchronization of the whole system, the synchronous rate of the subgroup increased can be explained
as follows:(1) The synchronizations of the A-node circuits group andB-node circuits group are constricted each other.
(2) Decreasing the synchronization of one group decrease
an influence to the other group.
(3) Therefore, the synchronization of the other group increases.
3.3. The case of using thirty chaotic circuits
In order to reveal the relation among the small variations,the ratio of the number of A-node and B-node circuits, andthe synchronous rate, the coupled system of thirty chaotic
circuits is investigated. Figure 7 shows one example of thesimulated result in the case of m = 25 and n = 5. The
viewpoint of the figure is the same as Fig. 3, the horizontal
axis is time, and the vertical axes are the voltage diﬀerence

Figure 7: Simulated result of the asymmetrical coupledsystem in the case of m = 25 and n = 5. α = 0.5, β = 0.5,
γ = 20, δ = 0.01, and pk = qk = 0.0001(k − 1).
Figure 8 shows the relation among the synchronous rateof the A-node circuits group, the small variations Q of
the B-node circuits, and the number of the B-node circuitsn. The synchronous rate of the A-node group approaches
to the complete synchronization as the variations and thenumber of the B-node circuits increase. Drastic increase of
the synchronous rate is observed around Q = 0.01. Further,
for the cases that the number of the B-node circuits is less168

than five, there are few influences on the synchronizationof the A-node circuits group.

[3] T. Endo, and S. Mori, “Mode Analysis of a Ring ofa Large Number of Mutually Coupled van der Pol
Oscillators,” IEEE Trans. Circuit and Systems, vol.
CAS-25, no. 5, pp. 7–18, 1978.
[4] D. A. Likens, “Analytical solution of large numbers of mutually-coupled nearly-sinusoidal oscillators,” IEEE 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.

Figure 8: Synchronous rate of the A-node group to the variations Q and the number n of the B-node circuits.

4. CONCLUSION
In this study, the relation between synchronous rate andsmall variations on an asymmetrical coupled chaotic system is investigated. In spite of increasing variations whichconstrict the synchronization of the whole system, the synchronous rate of the subgroup increased. This interesting
phenomenon can be explained as follows:(1) The synchronizations of the A-node circuits group andB-node circuits group are constricted each other.
(2) Decreasing the synchronization of one group decrease
an influence to the other group.
(3) Therefore, the synchronization of the other group increases.
Furthermore, the synchronous rate of the A-node circuits
group was investigated as changing the small variations andthe number of the B-node circuits for the case of thirty
chaotic circuits. We found that there are some limits forthe groups to be synchronized. The detailed analysis of theinfluence of the synchronous state of each node group is a
future research subject.
References[1] L. M. Pecora and T. L. Carrol, “Synchronization inChaotic Systems,” Physical Review Letters, vol. 64,pp. 821–824, 1990.
[2] K. Kaneko, “Clustering, Coding, Switching, Hierarchical Ordering, and Control in a Network of ChaoticElements,” Physica D, vol. 41, pp. 137–172, 1990.

169[7] 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.
[8] N. Inaba, T. Saito and S. Mori, “Chaotic Phenomenain a Circuit with a Negative Resistance and an Ideal
Switch of Diodes,” IEICE Trans., vol. E-70, pp. 744–754, 1987.