International Symposium on Nonlinear Theory and its Applications
Xi’an, PRC, October 7–11, 2002

Statistical Analysis of Clustering in Globally Coupled Chaotic Circuits
Manabu MIYAMURA† , Yoshifumi NISHIO† and Akio USHIDA†† Dept. of Electrical and Electronic Eng.,
Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Phone:+81-88-656-7470, Fax:+81-88-656-7471
Email: miyam@ee.tokushima-u.ac.jp

Abstract:
In this study, a globally coupled systemof chaotic circuits with small variations in their oscillation frequencies is investigated. In our previous study,
by computer simulations, the generation of clusteringand chaotic change of the cluster size are conﬁrmed for
ﬁve and thirty circuits case. Further, we conﬁrmed thatphenomenon from experimental results for ﬁve circuits
case. In order to understand the phenomena correctly,
we make detailed investigation on the statistical information of the clustering for thirty circuits case, such as
survival time of a certain size of clusters.

and one nonlinear resistor of two diodes. The chaotic
circuit is three-dimensional autonomous and generates
chaotic attractor shown in Fig. 2. In our coupled system, chaotic circuits with the same circuit parameters
except small variations in their oscillation frequencies
are globally (namely all-to-all) coupled by resistors R.

1. Introduction
Clustering is one of the most interesting nonlinear phenomena observed from a large number of coupled chaoticsystems. A lot of studies on the clustering have been
carried out for discrete-time mathematical models [1][2].
However, there have been a few studies on clustering ofcontinuous-time real physical systems such as electricalcircuits.
In this study, we analyze clustering phenomenon observed from globally coupled chaotic circuits. The
chaotic circuit used in this study is a simple threedimensional autonomous circuit proposed by Mori et al.
[3][4]. We consider the case that the chaotic circuits havesmall variations in their oscillation frequencies. In ourprevious study [5], by computer simulations and circuit
experiments, the generation of clustering and chaotic
change of the cluster size were conﬁrmed. In order toFigure 1: Circuit model.
understand the phenomena correctly, we make detailedinvestigation on the statistical information of the clusAt ﬁrst, we approximate the v −i characteristics of thetering, such as survival time of a certain size of clusters. nonlinear resistor of the diodes by the following piecewise
linear function as shown in Fig. 3.

2. Circuit Model Gd (vk1 − vk2 − a) (vk1 − vk2 > a)0(|vk1 − vk2 | ≤ a)(1)
idk =
Figure 1 shows the circuit model for the case that 5
Gd (vk1 − vk2 + a) (vk1 − vk2 < −a).
chaotic circuits are coupled. Each chaotic circuit consistsof three memory elements, one linear negative resistor-155-

By using the following variables and parameters,f (xk − yk ) = γ(xk − yk − 1) (xk − yk > 1)0(|xk − yk | ≤ 1)γ(xk − yk + 1) (xk − yk < −1).

(4)

3. Clustering

Figure 2: Example of chaotic attractors. α = 0.4, β =0.5 and γ = 20.

Figure 4: Computer simulated results for N = 5. α =0.4, β = 0.5, γ = 20, δ = 0.067 and qk = 0.001(k − 1).

Figure 3: Approximation of the nonlinear resistor.

vk1vk2,yk =,
xk =aa√d,t = LC2 τ, “· =dτ1 L
Lβ=, γ = Gd,
r C2C21zk =a

Lik ,
C2C2,
C1

1L
δ=,R C2α=(2)the circuit equations for the case of N chaotic circuits
are given as
N



ẋ
=αδ
(xj − xk ) + αβxkk

j=1
(3)
−αf (xk − yk )ẏk = f (xk − yk ) − zkżk = (1 + qk )yk
where k = 1, 2, 3, · · · N and qk is the coeﬃcients to givesmall variations of the oscillation frequencies. The nonlinear function f (xk − yk ) corresponds to the characteristics of the nonlinear resistor of the diodes and is Figure 5: Computer simulated results for N = 30. α =described as0.5, β = 0.5, γ = 20, δ = 0.0105 and qk = 0.0003(k − 1).

-156-In our previous study, we conﬁrmed the generation of
the clustering phenomenon and chaotic change of thecluster size such as Figs. 4 and 5. Figure 4 shows
computer simulated results for the case of N = 5. In
the ﬁgures, the vertical axes are the diﬀerences between
the voltages of the two chaotic circuits. Namely, if thetwo chaotic circuits synchronize, the value of the graphshould be almost zero. During the time interval ← c →in the ﬁgure, all of the values are almost zero. Thismeans that all of the 5 chaotic circuits almost synchronize, namely the number of the clusters is 1. While,during ← b →, only y3 − y4 oscillates with certain amplitude. This means that y1 , y2 and y3 almost synchronize
and that y4 and y5 almost synchronize. In this case, thenumber of the clusters is 2. As time goes, the numberof the clusters changes chaotically as 1 → 2 → 3 → 1
during ← a → in the ﬁgure.
As well as Fig. 4, we conﬁrmed the clustering phenomenon from circuit experiments for the case of N = 5
and from computer simulations for the case of N = 30
which is shown in Fig. 5.
4. Statistical Analysis

Figure 6: Probability of out-of-phase state between twocircuits during 0 ≤ τ ≤ 50,000. N = 30, Dth = 0.06, α =0.5, β = 0.5, γ = 20, δ = 0.0105 and qk = 0.0003(k − 1).
We conﬁrmed from Fig. 6, in our previous study, thatduring any two chaotic circuits, the rate of out-of-phaseis almost the same. Figure 6 shows how many times twochaotic circuits are out-of-phase during a certain timeinterval. We checked whether if the values of |yk − yk+1 |are larger than Dth or not for 0 ≤ τ ≤ 50,000. The

horizontal axis is the percentage that the condition |yk −yk+1 | > Dth is satisﬁed.
Now, in this study, in order to understand the phenomena correctly, we make detailed investigation on thestatistical information of the clustering for N =30. In
statistical analysis, we regard out-of-phase as the case|yk − yk+1 | > 0.06 as shown in Fig. 7. Furthermore, wechoose parameters as follows: α = 0.5, β = 0.5, γ = 20,
δ = 0.0105.

Figure 7: Decision of synchronization.
Figure 8 shows the distribution of the number of theclusters during a certain time interval. In this case, wechecked in-phase or out-of-phase every τ =1 and total
1,000,000 times. Then, we counted the number of thecluster. The holizontal axis is the number of the cluster
and vertical axis is the probability of the generation. In
this parameter, we could not observe the clusters withthe size of 27 or more.

Figure 8: Distribution of the number of the clusters during a certain time interval.
Figure 9 shows the distribution of the survival timeof the state such that all of the chaotic circuits are inphase. We checked 10,000 all-in-phase synchronization
states. In this ﬁgure, the numbers 1, 2, · · · indicate thesurvival time. Namely, 65% of all-in-phase synchronization disappear immediately. A part of “more than 5” is
5%, and the maximum survival time is τ = 4518.

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(a) Cluster size = 5.

Figure 9: Distribution of the survival time of the statesuch that all of the chaotic circuits are in-phase.
Finally, Fig. 10 shows the survival time of various
sizes of the clusters. We conﬁrmed that the larger thecluster size is, the shoter survival time is.

(b) Cluster size = 10.

5. Conclusions
In this study, we analyzed clustering phenomenon observed from a large number of coupled chaotic systems.
We made detailed investigation on the statistical information of the clustering.
References(c) Cluster size = 15.

[1] K. Kaneko, “Clustering, Coding, Switching, Hierarchical Ordering, and Control in a Network of ChaoticElements,” Physica D, vol. 41, pp. 137-172, 1990.
[2] Y. Maistrenko, O. Popovych and M. Hasler, “OnStrong and Weak Chaotic Partial Synchronization,”
Int. J. Bifurcation Chaos, vol. 10, no. 1, pp. 179-203,2000.
[3] M. Shinriki, M. Yamamoto and S. Mori, “MultimodeOscillations in a Modiﬁed van der Pol Oscillator Containing a Positive Nonlinear Conductance,” Proc. ofIEEE, vol. 69, pp. 394-395, Mar. 1981.

(d) Cluster size = 20.

[4] N. Inaba, T. Saito and S. Mori, “Chaotic Phenomena
in a Circuit with a Negative Resistance and an Ideal
Switch of Diodes,” Trans. IEICE, vol. E-70, pp. 744754, Aug. 1987.
[5] 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.

(e) Cluster size = 25.
Figure 10: Distribution of the survival time of various
sizes of the clusters.

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