社団法人 電子情報通信学会
THE INSTITUTE OF ELECTRONICS,
INFORMATION AND COMMUNICATION ENGINEERS

信学技報
TECHNICAL REPORT OF IEICE.

Research on Clustering of Coupled Chaotic Circuitswith Mutual Inductors
Yuta KOMATSU† , Yoko UWATE† , and Yoshifumi NISHIO†† Department of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
E-mail: †{komatsu, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract In this study, the clustering phenomenon observed from chaotic circuits coupled by mutual inductors
is investigated. In order to analyze the phenomena, dependent variables corresponding to phases of solutions for
subcircuits are introduced. By using this dependent variables, we make detailed investigation on the statisticalinformation of the clustering for a six subcircuit case, such as sojourn time of a certain size of clusters, the numberof clusters, and so on.
Key words Chaotic Circuit, Clustering, Mutual Inductor

1. Introduction
Clustering is one of the most interesting nonlinear phenomena observed from a large number of coupled chaotic
systems. A lot of studies on the clustering have been carriedout for discrete-time mathematical models [1][2]. However,there have been a few studies on clustering of continuoustime real physical systems such as electrical circuits [3][4].
In our previous study, quasi-synchronization phenomena
observed from chaotic circuits coupled by mutual inductors
were investigated [5]. We could observe various kinds of synchronization phenomena of chaos by carrying out computer
calculations and circuit experiments for two or three subcir-

Fig. 1 Circuit model.

cuit cases. However, we could not investigate the complex
phenomena observed from higher dimensional circuits such

inductors. Each chaotic subcircuit is a symmetric version ofas clustering, chaotic wandering, pattern switching, and so

the circuit model proposed by Inaba et al. [6]. It consistson.

of three memory elements, one linear negative resistor andIn this study, we focus on the clustering phenomenon. We

one nonlinear resistor, and is one of the simplest autonomous

investigate the clustering observed from chaotic circuits cou-chaotic circuits. First, we approximate the i − v character-pled by mutual inductors in detail. By computer calcula-

istics of the nonlinear resistor consisting of diodes by thetions, the generation of clustering and chaotic changings offollowing function.

the synchronization states are confirmed. In order to ana-√
vd (ik ) = 9 rd ik .

lyze the phenomena, dependent variables corresponding tophases of solutions for subcircuits are introduced. By using(1)

By changing the variables and parameters,

this dependent variables, we make detailed investigation on
the statistical information of the clustering for a six subcir-t=(L1 − M ) C τ , a = 8 rd

cuit case, such as sojourn time of a certain size of clusters,
the number of clusters, and so on.

r
Ik = a2. Circuit Model
Figure 1 shows the circuit model. In the circuit, N identical chaotic circuits are coupled symmetrically by mutual

r

p

α=

C
xk , ik = a
L1 − M

L1 − M
, β=r
L2

r

r

d
C
, “·”=,
L1 − MdτCyk , vk = azk ,L1 − MCM, γ=.
L1 − M
L1(2)—1—

the circuit equations are normalized and described asẋk = β(xk + yk ) − zkX
γ{β(xj + yj ) − zj }1 + (N − 1)γ
N

−

j=1

(3)

ẏk = α{β(xk + yk ) − zk − f (yk )}żk = xk + yk

(k = 1, 2, · · · , N )where√f (yk ) = 9 yk .

(4)

In the following computer calculations, (3) is calculated byusing the Runge-Kutta method with step size ∆t = 0.005.
0

3. Clustering Phenomenon

5000

10000

(a) α = 20.0. β = 0.26. γ = 0.3.

We carried out computer calculations for the case of N = 6.
Figure 2 shows computer calculated results. Upper and lower
figures in Fig. 2 show phase differences. While, middle figures show attractors. From Fig. 2, we can confirm the generation of the clustering phenomenon and chaotic changingsof the synchronization states. In Fig. 2(a), subcircuit 1 issynchronized at the in-phase in almost all of the times andsometimes becomes the asynchronous state for subcircuits 3
and 6. While, for subcircuits 2, 4, and 5, subcircuit 1 becomes the anti-phase syncronization state in almost all of
the times and the asynchronous state occasionally. As we
can see from this figure, a size of the cluster for subcircuit 1
is principally 3 under these parameters.
By changing the parameter β, the switching phenomenon
can be confirmed at τ = 2500, 5000 as shown in Fig. 2(b).

0

5000

10000

In the interval [0, 2500], subcircuit 1 is almost synchronized(b) α = 20.0. β = 0.265. γ = 0.3.

with subcircuits 2, 4 and 5 at the anti-phase, and almost

Fig. 2 Computer calculated results for N = 6. Upper figures: x1

becomes the in-phase synchronization state for subcircuits 3
and 6. Moreover, the size of the cluster that subcircuit 1
belongs to is principally 3 in this interval. While, subcircuit
1 becomes the in-phase synchronization state for subcircuit
2 in the interval [2500, 5000] and for subcircuits 2 and 5 invs. xk+1 . (x7 = x1 .) Middle figures: xk vs. zk . Lower
figures: τ vs. x1 − xk+1 . (k = 1, 2, 3, 4, 5, 6.)

4. Statistical Analysis
4. 1 Introduction of Phase Variables

the interval [4000, 5000]. For subcircuits 3, 4 and 6, subcir-

In this section, we investigate the clustering phenomenon

cuit 1 becomes the anti-phase synchronization state in thein detail. First, in order to discriminate whether each subcir-

interval [2500, 5000]. In this interval, the size of the cluster

cuit is synchronized or not, let us define the Poincaré section

for subcircuit 1 is principally 3. Furthermore, subcircuit 1

as x1 < 0 and z1 = 0 Next, we introduce the following inde-

is synchronized with subcircuits 3, 5 and 6 at the anti-phasependent variables from the discrete data of xk (n) and zk (n)and becomes the in-phase synchronization state for subcir-

on the Poincaré map. The values of ϕk (n) correspond tocuits 2 and 4 in almost all of the interval [5000, 100000]. As

the phase differences between subcircuit 1 and the others.

the other intervals, the size of the cluster for subcircuit 1 is(Note that the argument of the point (x1 (n),z1 (n)) is always

also principally 3 in this interval. Namely, we can consider

π, because of the definition of the Poincaré map.) By us-that the size of the cluster for subcircuit 1 is easy to become

ing the values of ϕk (n), we make detailed investigation on

3 for N = 6.

the statistical information of the clustering for N = 6. In—2—the statistical analysis, we carried out computer simulations(n)2during one hundred thousand iterations of the Poincaré map
under the following parameters: α = 20.0, β = 0.265, and
γ = 0.3.

(n)3

−1 zk (n) π − tan xk (n)
xk (n) >

=0






zk (n)
− tan−1


x
k (n)

ϕk (n) =xk (n) < 0 and zk (n) >
=0






z (n)

 2π − tan−1 k


x
k (n)




x
(n)
<
0
and
zk (n) < 0
k







(n)

4

(n)

5

(5)

2


(n)

6

0
0

5000

10000


n : iteration

Fig. 3 Time series ϕk (n) calculated from the data xk (n) andzk (n).

(k = 2, 3, 4, 5, 6.)

4. 2 Probability distribution of ’k (n)Figure 3 shows the time evolution of ϕk (n) calculated fromthe data xk (n) and zk (n). Horizontal axis is the iterationof the Poincaré map. Vertical axis is the phase differenceof each subcircuit for subcircuit 1. The switching phenomena occurs at short intervals and the synchronization states(a) Subcircuit 2change chaotically. Moreover, we can see that subcircuit 1
becomes the in-phase or anti-phase synchronization state forthe others with high frequency.
Figure 4 shows the probability distributions of ϕk (n). The
slots in the horizontal axis of the figure denote the rangesof ϕk (n) and are summarized in Tab. 1. We can confirm
that all subcircuits become about the same probability dis-(b) Subcircuit 3(c) Subcircuit 4(d) Subcircuit 5(e) Subcircuit 6

tributions. For the reason to become in this way, our circuit model’s symmetry can be considered. Furthermore, theprobabilities of slots 1, 6, 7, and 12 are larger than the others. We can consider that the slots 1 and 12 correspond to
the in-phase states and the slots 6 and 7 correspond to theanti-phase states. This means that each subcircuit becomesthe in-phase or anti-phase synchronization state with large

Fig. 4 Probability distributions of ϕk (n).

probability.

Table 1 Ranges of slots in Fig. 4.

4. 3 Probability distributions of sojourn time
We define that subcircuit 1 is synchronized with subcircuit
k if the value of ϕk (n) satisfies (6). This definition of thesynchronization corresponds to the in-phase synchronization
state.
ππ− < ϕk (n) <
6
6

Slot

ϕk (n)Slot

ϕk (n)

1π
0<= ϕk (n) < 6π <π= ϕk (n) <

7

7π
π<
= ϕk (n) < 67π <4π= ϕk (n) <

2
3
4

(6)

5
6

6
3
π <
ϕ (n) < π2
3 = k
π <
ϕ (n) < 2π
2 = k
3
5π
2π <
ϕ
(n)
<
k
=
3
6
5π <
ϕ (n) < π
6 = k

8
9
10
11
12

6
3
4π <
ϕ (n) < 3π
3 = k
2
3π <
ϕ (n) < 5π
2 = k
3
11π
5π <
ϕ
(n)
<
k
=
3
6
11π <
ϕ (n) < 2π
6 = k

Table 2 shows the probability distribution of cluster sizes
for subcircuit 1 are from 1 to 6. “Cluster size = 1” means

with one of the other five subcircuits and not synchronizedthat subcircuit 1 is not synchronized with the others. Then,with the others. Moreover, “Cluster size = 4” means that

“Cluster size = 2” means that subcircuit 1 is synchronized

subcircuit 1 is synchronized with other three subcircuits. As—3—

Table 2 Probability distribution of cluster sizes for subcircuit 1.
Cluster size Probability Cluster size Probability

Table 4 Probability distribution of specific cluster types for subcircuit 1.

1

0.1886940.00020

Cluster type Probability Cluster type Probability

20.3005750.00000

1-2-3-4-5-6

0.00204

123-456

0.0284530.5105460.00000

12-3-4-5-6

0.001891234-5-6

0.00001

12-34-5-6

0.003591234-560.0000212-34-56

0.00055

12345-60.00000

123-4-5-6

0.004671234560.00000

123-45-6

0.01491

Table 5 Probability distribution of cluster types.
Cluster type Probability Cluster type Probability(a) Cluster size = 1.

(b) Cluster size = 2.

(c) Cluster size = 3.

(d) Cluster size = 4.

Fig. 5 Probability distributions of sojourn time.

Table 3 Ranges of slots in Fig. 5.

1-1-1-1-1-1

0.002043-3

0.256622-1-1-1-1

0.03006

4-1-1

0.00033

2-2-1-1

0.100044-2

0.000482-2-2

0.005975-1

0.00000

3-1-1-1

0.0858160.00000

3-2-1

0.51865

Table 6 Probability distribution of the number of clusters.
] of clusters Probability ] of clusters Probability
10.00000

40.1858520.2571050.03006

3

0.52495

60.00204

Slot Sojourn time (n) Slot Sojourn time (n)
1

1−2

5

21−50

2

3−5

6

51−1003

6−10

7101−200

4

11−20

8

201−500

cluster type means that all subcircuits become asynchronous
states and the number of the clusters is 6. This value is relatively large, which is quite interesting in the sense that all
subcircuits are connected to the same core. Then, we canconfirm that the probability of the cluster type “12-3-4-5-6”
is 1.89 × 10−3 . This cluster type means that subcircuit 1

we can see from Tab. 2, the probability of “Cluster size = 3”
is the largest and the cluster size hardly becomes more than4.

becomes synchronization state for subcircuit 2 and the othersubcircuits are not synchronized and the number of the clusters is 5. Moreover, the cluster type “12-34-5-6” means that

While, Fig. 5 shows the probability distributions of sojourn time. The slots in the horizontal axis of the figure
denote the ranges of the sojourn time and are summarizedin Tab. 3. For example, comparing Fig. 5(c) with Fig.
5(b), the probability distributions of the slots 3, 4, and 5of “Cluster size = 3” are larger than those of “Cluster size
= 2”. Furthermore, “Cluster size = 3” disappears within 1
or 2 iterations of the Poincaré map. Clearly, it is confirmedthat the larger cluster size is, except the case of “Cluster size
= 4”, the larger sojourn time is as shown in Fig. 5.

subcircuit 1 is synchronized with subcircuits 2 and subcircuit
3 is synchronized with subcircuits 4 and subcircuits 5 and 6
independently oscillate and the number of the clusters is 4.
While, the probability distribution of cluster types isshown in Tab. 5. For example, the probability of the cluster
type “2-1-1-1-1” is 3.06 × 10−2 . This cluster type means that
subcircuit 1 becomes synchronization state for subcircuit 2
and the others are not synchronized each other and the number of the clusters is 5. Furthermore, the cluster type “2-21-1” means that subcircuit 1 becomes synchronization state
for subcircuit 2 and any two of subcircuits except subcircuit

4. 4 Probability distribution of various cluster
types.
The probability distribution of specific cluster types for
subcircuit 1 is shown in Tab. 4. For example, the probability of the cluster type “1-2-3-4-5-6” is 2.04 × 10−3 . This

1 and 2 are synchronized and the others are independently
oscillate and the number of the clusters is 4. The probability
distribution of the number of clusters is summarized in Tab.
6. This table shows that the number of clusters is easy to
become 3 for N = 6.

—4—

Table 7 Theoretical figure of the probability distribution of clus-

5. Discussion

ter types.

We considered that the probability distribution of cluster
types as shown in Tab. 5 could be calculated only by using
the information shown in Tab. 4 as theoretical figures. For
example, the probability P1 of the cluster type “2-1-1-1-1”

Cluster type Probability Cluster type Probability
1-1-1-1-1-1

0.002043-3

0.284502-1-1-1-1

0.009454-1-10.000102-2-1-1

0.107704-20.000202-2-20.016505-10.00000

can be calculated by using the probability of the cluster type3-1-1-1

0.0467060.00000“12-3-4-5-6” as follows.

3-2-1

0.44730

P1 = 5 C1 × 0.00189 = 9.45 × 10−3 .

(7)As the same way, the probability P2 of the cluster type “2-21-1” can be calculated by using the probability of the cluster
type “12-34-5-6” as follows.
P2 = 5 C1 × 4 C2 × 0.00359 = 0.1077.

(8)

Table 8 Theoretical figure of the probability distribution of thenumber of clusters.
] of clusters Probability ] of clusters Probability
10.00000

40.1544020.2847050.0094530.4639060.00204

Moreover, the other various probabilites can be calculated byusing the probability information shown in Tab. 4. Those
results are summarized in Tab. 7. In addition, the theoretical figure of the probability distribution of the number ofclusters is summarized in Tab. 8.

Table 9 Relative error between statistical data and theoreticalfigures for cluster types.
Cluster type Relative error Cluster type Relative error
2-1-1-1-1

2.18095

3-2-1

0.15951

The relative error between statistical data and theoretical

2-2-1-1

0.071123-3

0.09800

figures for cluster types is shown in Tab. 9. We can confirm

2-2-2

0.63818

4-1-1

2.30000

that the relative errors of the cluster types “2-1-1-1-1”, “2-

3-1-1-1

0.837474-21.400002-2”, “3-1-1-1”, “4-1-1”, “4-2” are quite large. We have toinvestigate a reason to become in this way in detail. However, if we could calculate various probabilites as shown inReferences

Tab. 5 by using a little information as shown in Tab. 4, it[1] K. Kaneko, “Clustering, Coding, Switching, HierarchicalOrdering, and Control in a Network of Chaotic Elements,”
Physica D, vol.41, pp.137-172, 1990.
[2] Y. Maistrenko, O. Popovych and M. Hasler, “On Strong andWeak Chaotic Partial Synchronization,” Int. J. Bifurcation
Chaos, vol.10, no.1, pp.179-203, 2000.
[3] M. Miyamura, Y. Nishio and A. Ushida, “Clustering in
Globally Coupled System of Chaotic Circuits,” Proc. of ISCAS’02, vol. 3, pp.57-60, May 2002.
[4] Y. Nishio and A. Ushida, “Chaotic Wandering and its Analysis in Simple Coupled Chaotic Circuits” IEICE Transactions on Fundamentals, vol. E85-A, no. 1, pp. 248-255, Jan.
2002.
[5] Y. Komatsu, Y. Uwate and Y. Nishio, “Synchronization in
Chaotic Circuits Coupled by Mutual Inductors,” Proc. of
NCSP’06, pp.293-296, Mar. 2006.
[6] N. Inaba and S. Mori, “Chaotic Phenomena in Circuits witha Linear Negative Resistance and an Ideal Diode,” Proc. of
MWSCAS’88, pp.211-214, Aug. 1988.

would be excellent.

6. Conclusions
In this study, we investigated the clustering phenomenon
observed from chaotic circuits coupled by mutual inductors.
By carrying out computer calculations, we confirmed thegeneration of the clustering phenomenon and chaotic changings of the synchronization states for a six subcircuit case.
Moreover, we make detailed investigation on the statisticalinformation of the clustering, such as sojourn time of a certain size of clusters, the number of clusters, and so on.

Acknowledgements
This work was partly supported by Yazaki Memorial Foundation for Science and Technology.

—5—