Clustering Phenomenon of Chaotic Circuits
Coupled Symmetrically by Mutual Inductors
Yuta KOMATSU, Yoko UWATE and Yoshifumi NISHIO
Department of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Phone:+81-88-656-7470, Fax:+81-88-656-7471
Email: {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
statistical information of the clustering for a six subcircuit case,such as sojourn time, cluster types, and so on.

I. I NTRODUCTION
Spatiotemporal phenomena observed from large-scale coupled chaotic networks have attracted many researchers’ attention and have been studied strenuously by many researchers.
The studies on coupled chaotic networks are classified into
two categories. One is discrete-time systems, the other is
continuous-time systems. A lot of studies have been carriedout for discrete-time mathematical models [1][2]. However,there have been a few studies on continuous-time real physicalsystems such as electrical circuits [3][4].
In this study, we particularly focus on the clustering phenomenon observed from continuous-time real physical systems
in spatiotemporal phenomena. We investigate the clusteringobserved from chaotic circuits coupled by mutual inductors indetail. By computer calculations, occurrences of the clusteringand chaotic changes of a synchronization state are confirmed.
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 statistical information of the clustering for a six
subcircuit case, such as sojourn time, cluster types, and so on.

Fig. 1.

Fig. 2.

II. C IRCUIT M ODEL
Figure 1 shows a circuit model. In the circuit, N identical chaotic circuits are coupled symmetrically by mutual
inductors. Each chaotic subcircuit is a symmetric version of
the circuit model proposed by Inaba et al. [5]. It consistsof three memory elements, one linear negative resistor and
one nonlinear resistor, and is one of the simplest autonomouschaotic circuits.
First, in order to implement circuit experiments, we consider
the simplified circuit model shown in Fig. 2. In the circuit, N
identical chaotic circuits are coupled by an inductor. Then,in order to implement computer calculations, we approximate
the i − v characteristics of the nonlinear resistor consisting ofdiodes by the following function.

1-4244-1342-7/07/$25.00 ©2007 IEEE

823

Circuit model.

Simplified circuit model.

√
vd (ik ) = 9 rd ik .

(1)

By using the following variables and parameters,t = (L1 − M ) C τ , a = 8 rdCd, “·”=,
L1 − MdτCC
xk , ik = a
yk , vk = azk ,Ik = aL1 − M
L1 − MC
M
L1 − M
, γ=
, β=r
,
α=
L2
L1 − M
L1(2)

the circuit equations are normalized and described asẋk = β(xk + yk ) − zk−Nγ{β(xj + yj ) − zj }1 + (N − 1)γ j=1(3)ẏk = α{β(xk + yk ) − zk − f (yk )}żk = xk + yk
where(k = 1, 2, · · · , N )√f (yk ) = 9 yk .

(4)

In the following computer calculations, (3) is calculated byusing the Runge-Kutta method with step size ∆t = 0.005.
III. C LUSTERING P HENOMENON(a) α = 20.0. β = 0.26. γ = 0.3.

We carried out computer calculations for the case of N =6. Figure 3 shows computer calculated results. From Fig.
3, we can confirm occurrences of a clustering phenomenon
and chaotic changes of a synchronization state. In Fig. 3(a),subcircuit 1 becomes in-phase synchronization state in almost
all of the time and becomes asynchronous state for subcircuits
3 and 6 at moments. While, for subcircuits 2, 4, and 5,
subcircuit 1 becomes anti-phase syncronization state in mostintervals and becomes asynchronous state occasionally. As
we can see from this figure, the cluster size of the clusterthat subcircuit 1 is a component is principally 3 under theseparameters.
By changing the parameter β, switching phenomena canbe confirmed at τ = 2500, 5000 as shown in Fig. 3(b).
In the interval [0, 2500], subcircuit 1 is almost synchronizedwith subcircuits 2, 4 and 5 at anti-phase, and almost becomes in-phase synchronization state for subcircuits 3 and6. Moreover, the cluster size of the cluster that subcircuit 1
belongs to is principally 3 in this interval. While, subcircuit
1 becomes in-phase synchronization state for subcircuit 2 in
the interval [2500, 5000] and for subcircuits 2 and 5 in theinterval [4000, 5000]. For subcircuits 3, 4 and 6, subcircuit
1 becomes anti-phase synchronization state in the interval[2500, 5000]. In this interval, the cluster size for subcircuit
1 is principally 3. Furthermore, subcircuit 1 is synchronizedwith subcircuits 3, 5 and 6 at anti-phase and becomes in-phasesynchronization state for subcircuits 2 and 4 in almost all of theinterval [5000, 10000]. As the other intervals, the cluster sizefor subcircuit 1 is also principally 3 in this interval. Namely,
we can consider that the cluster size for subcircuit 1 is easy
to become 3 for the case of N = 6.

(b) α = 20.0. β = 0.265. γ = 0.3.

IV. S TATISTICAL A NALYSIS
A. Introduction of Phase Variables
In this section, we investigate the clustering phenomenon indetail. First, in order to discriminate whether each subcircuit
is synchronized or not, let us define the Poincaré section
as x1 < 0 and z1 = 0. Next, we introduce the following
independent variables from the discrete data of xk (n) andzk (n) on the Poincaré map. The values of ϕk (n) correspond

Fig. 3. Computer calculated results for N = 6. (k = 1, 2, 3, 4, 5, 6.)(x7 = x1 .)to the phase differences between subcircuit 1 and the others.
(Note that the argument of the point (x1 (n), z1 (n)) is always
π, because of the definition of the Poincaré map.) By using thevalues of ϕk (n), we make detailed investigation on statisticalinformation of the clustering for the case of N = 6. In thestatistical analysis, we carried out computer simulations during
one hundred thousand iterations of the Poincaré map under thefollowing parameters; α = 20.0, β = 0.265, and γ = 0.3.
zk (n)

π − tan−1xk (n)xk (n) ≥ 0







z (n)

 − tan−1 kxk (n)ϕk (n) =(5)xk (n) < 0 and zk (n) ≥ 0





zk (n)
2π − tan−1xk (n)xk (n) < 0 and zk (n) < 0





(k = 2, 3, 4, 5, 6.)824

corresponds to the in-phase synchronization state.

ϕ2(n)

−

ϕ3(n)

π
π
< ϕk (n) < .
6
6

(6)

Table I shows the probability distribution of the cluster sizeof the cluster that subcircuit 1 is a component. “Cluster size
= 1” means that subcircuit 1 is not synchronized with theothers. Moreover, “Cluster size = 2” means that subcircuit 1
is synchronized with one of five others and not synchronizedwith the others. As we can see from Tab. I, the probability of“Cluster size = 3” is the largest and the cluster size hardly
becomes more than 4.

ϕ4(n)
ϕ5(n)
2π

ϕ6(n) πTABLE I

0
5000

10000

→ n : iteration

Cluster size
1
2
3

Fig. 4. Time series ϕk (n) calculated from the data xk (n) and zk (n).

B. Probability distribution of ϕk (n)Figure 4 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 difference ofeach subcircuit for subcircuit 1. Switching phenomena occur atshort intervals and a synchronization state changes chaotically.
Moreover, we can see that subcircuit 1 becomes in-phaseor anti-phase synchronization state for five others with highfrequency.
0.2

Cluster size
4
5
6

0.1
0.05

0.7

0.7

0.6

0.6

0.5

0.5

0.4
0.3
0.2
0.1

0
3

4

5

6

7

8

0.3
0.2
0

1

2

0.4

0.1

0

1

Probability
0.000200.00000
0.00000

While, Fig. 6 shows the probability distributions of sojourntime. The slots in the horizontal axis of the figure denote theranges of the sojourn time and are summarized in Tab. II. For
example, compared Fig. 6(b) with Fig. 6(c), the probability
distributions of the slots 3, 4, and 5 of “Cluster size = 2” arelarger than those of “Cluster size = 3”. Furthermore, “Cluster
size = 3” disappears within 1 or 2 iterations of the Poincarémap. Clearly, it is confirmed that the larger cluster size is, thelarger sojourn time is as shown in Fig. 6, except the case of“Cluster size = 4”.

Probability

Probability

0.15

Probability
0.188690.300570.51054

Probability

0

P ROBABILITY DISTRIBUTION OF THE CLUSTER SIZE FORSUBCIRCUIT 1.

2

3

4

5

6

7

8

1

2

3

4

Slot

9 10 11 12(a) Cluster size = 1.

Slot

C. Probability distributions of sojourn time
We define that subcircuit 1 is synchronized with subcircuit
k if ϕk (n) satisfies (6). This definition of the synchronization

825

8

0.8

0.5

Probability

Figure 5 shows the probability distributions of ϕ2 (n). The
probabilities of slots 1, 6, 7, and 12 are larger than the others.
We can consider that the slots 1 and 12 correspond to inphase states and the slots 6 and 7 correspond to anti-phasestates. This means that each subcircuit becomes in-phase oranti-phase synchronization state with large probability. Furthermore, we have confirmed that the probability distributionsof ϕk (n) become about the same probability distributions. For
the reason to become in this way, our circuit model’s symmetry
can be considered.

7

1

0.6
Probability

interval [(m − 1)π/6, mπ/6].

6(b) Cluster size = 2.

0.7

Fig. 5. Probability distributions of ϕ2 (n). Horizontal slot m indicates

5
Slot

0.4
0.3
0.2

0.6
0.4
0.2

0.1
0

0
1

2

3

4

5

6

7

8

1

2

Slot

4

5

7

(d) Cluster size = 4.

Probability distributions of sojourn time.
TABLE IIR ANGES OF SLOTS IN F IG . 5.
Slot
1
2
3
4

6

Slot

(c) Cluster size = 3.
Fig. 6.

3

Sojourn time (n)
1−2
3−5
6−10
11−20

Slot
5
6
7
8

Sojourn time (n)21−5051−100101−200
201−500

8

D. Probability distribution of various cluster types.
The probability distribution of specific cluster types for
subcircuit 1 is shown in Tab. III. For example, the probabilityof the cluster type “1-2-3-4-5-6” is 2.04 × 10−3 . This cluster
type means that all subcircuits become asynchronous states
and the number of clusters is 6. Then, we can confirm that theprobability of the cluster type “12-3-4-5-6” is 1.89 × 10−3 .
This cluster type means that subcircuit 1 is syncronized with
subcircuit 2 and the others are not synchronized and thenumber of clusters is 5. Moreover, the cluster type “12-34-5-6”
means that subcircuit 1 is synchronized with subcircuits 2 andsubcircuit 3 is synchronized with subcircuits 4 and subcircuits
5 and 6 independently oscillate and the number of clusters is
4.
While, the probability distribution of cluster types is shownin Tab. IV. For example, the probability of the cluster type “21-1-1-1” is 3.06 × 10−2 . This cluster type means that any two
subcircuits are synchronized and the others are asynchronous
states and the number of clusters is 5. Furthermore, thecluster type “2-2-1-1” means that any two subcircuits aresynchronized and one cluster is formed and any two others offour subcircuits are synchronized and another cluster is also
formed and the others are independently oscillate.

investigation on statistical information of the clustering, suchas sojourn time, cluster types, and so on.
ACKNOWLEDGEMENTS
This work was partly supported by Yazaki Memorial Foundation for Science and Technology.
R EFERENCES[1] K. Kaneko, “Clustering, Coding, Switching, Hierarchical Ordering, andControl in a Network of Chaotic Elements,” Physica D, vol.41, pp.137172, 1990.
[2] Y. Maistrenko, O. Popovych and M. Hasler, “On Strong and Weak Chaotic
Partial Synchronization,” Int. J. Bifurcation Chaos, vol.10, no.1, pp.179203, 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] N. Inaba and S. Mori, “Chaotic Phenomena in Circuits with a Linear
Negative Resistance and an Ideal Diode,” Proc. of MWSCAS’88, pp.211214, Aug. 1988.

V. D ISCUSSIONFirst, we considered the probability distribution of specificcluster types shown in Tab. III as experimental values. Then,
we considered that the statistical data shown in Tab. IV
could be calculated as theoretical figures only by using theinformation shown in Tab. III. For example, the probability
P1 of the cluster type “2-1-1-1-1” can be calculated by using
the probability of the cluster type “12-3-4-5-6” as follows.
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.
(8)
P2 = 5 C1 × 4 C2 × 0.00359 = 0.1077.
Moreover, the other various probabilites can be calculated byusing the probability information shown in Tab. III. These
results are summarized in Tab. V.
The relative error between experimental values and theoretical figures for cluster types is shown in Tab. VI. We canconfirm that the relative errors of the cluster types “2-1-1-1-1”,“2-2-2”, “3-1-1-1”, “4-1-1”, and “4-2” are relatively large. We
have to investigate a reason to become in this way in detail.
However, if we could calculate various probabilites shown in
Tab. IV by using a little information shown in Tab. III, it would
be quite effective for the analysis when a circuit scale is large.
VI. C ONCLUSIONSIn this study, we investigated the clustering phenomenon observed from chaotic circuits coupled by mutual inductors. By
carrying out computer calculations, we confirmed occurrencesof the clustering and chaotic changes of a synchronization
state for a six subcircuit case. Moreover, we made detailed

826

TABLE IIIP ROBABILITY DISTRIBUTION OF SPECIFIC CLUSTER TYPES FORSUBCIRCUIT 1.
Cluster type1-2-3-4-5-6
12-3-4-5-6
12-34-5-6
12-34-56
123-4-5-6
123-45-6

Probability
0.00204
0.00189
0.003590.00055
0.00467
0.01491

Cluster type123-456
1234-5-6
1234-56
12345-6
123456

Probability
0.028450.00001
0.000020.00000
0.00000

TABLE IVP ROBABILITY DISTRIBUTION OF CLUSTER TYPES .
Cluster type1-1-1-1-1-1
2-1-1-1-1
2-2-1-1
2-2-2
3-1-1-1
3-2-1

Probability
0.00204
0.03006
0.10004
0.00597
0.085810.51865

Cluster type
3-3
4-1-1
4-2
5-1
6

Probability
0.256620.00033
0.000480.00000
0.00000

TABLE VT HEORETICAL FIGURE OF THE PROBABILITY DISTRIBUTION OF
CLUSTER TYPES .
Cluster type1-1-1-1-1-1
2-1-1-1-1
2-2-1-1
2-2-2
3-1-1-1
3-2-1

Probability
0.00204
0.00945
0.107700.01650
0.046700.44730

Cluster type
3-3
4-1-1
4-2
5-1
6

Probability
0.284500.000100.000200.00000
0.00000

TABLE VIR ELATIVE ERROR BETWEEN EXPERIMENTAL VALUES AND
THEORETICAL FIGURES FOR CLUSTER TYPES .
Cluster type2-1-1-1-1
2-2-1-1
2-2-2
3-1-1-1

Relative error
2.18095
0.071120.638180.83747

Cluster type
3-2-1
3-3
4-1-1
4-2

Relative error
0.15951
0.09800
2.30000
1.40000