6th International Forum on Advanced Technologies
March 9th-10 th, 2020,Tokushima, Japan

Clustering Phenomena in Coupled Chaotic Circuits
Located in 3-Dimensional Space
Takumi Nara , Yoko Uwate and Yoshifumi Nishio
Tokushima University, Tokushima 770-8501, Japan
E-mail: (nara, uwate, nishio)@tokushima-u.ac.jp
1. Introduction
Synchronization phenomena are the most familiar phenomena that exist in nature and they have been studiedin various ﬁelds. Synchronization phenomena can be observed everywhere in our life. For example, we canconﬁrm ﬂashing ﬁreﬂy lights and so on. Especially, synchronization phenomena of oscillatory networks areinteresting. In addition, complex networks attract attention from various ﬁelds. The feature of networks is thedegree distribution, the path length and the clustering coefﬁcient. Therefore, we focus on the clustering
phenomena in this research.
In addition, clustering phenomena are one of interesting nonlinear phenomena observed from coupled chaoticcircuits. The clustering phenomena are to divide the set to be classiﬁed into subsets. Previously, many of thestudies for clustering have been carried out for discrete time model [1]-[2]. However, analysis of using acontinuous time model has not almost studied. Therefore, we focus on research on clustering phenomena using
electronic circuits in continuous time model.
On the other hand, coupled chaotic circuits that are electronic circuits can be observed various amusing
phenomena. In recent years, many methods are studied to apply to clustering and synchronization phenomena
observed in coupled chaotic circuits for natural sciences. At the same time, synchronization phenomena andclustering have been studied associated with the chaotic phenomena [3]-[4].
In this study, we focus on the clustering phenomena in the network of coupled chaotic circuits.
2. Circuit model
Figure 1 shows the circuit model which is called Shinriki-Mori circuit. This circuit consists of a negative
resistor, an inductor, two capacitors and dual-directional diodes.
The circuit equation of this circuit is given as follows:
𝑑𝑖𝐿 𝑑𝑡𝐿 = 𝑣2
𝐶1

𝑑𝑣1
𝑑𝑡
𝑑𝑣2

= 𝑔𝑣1 − 𝑖𝑑𝑛(1){ 𝐶2 𝑑𝑡 = 𝑖𝑑𝑛 − 𝑖𝐿 .
The nonlinear function 𝑖𝑑𝑛 corresponds to the i-v
characteristics of the nonlinear resistors consisting of the diodes
and are given as follows:
𝐺𝑑 (𝑣1 − 𝑣2 − 𝑉) (𝑣1 − 𝑣2 > 𝑉)0(|𝑣1 − 𝑣2 | ≤ 𝑉)
𝑖𝑑𝑛 = {(2)𝐺𝑑 (𝑣1 − 𝑣2 + 𝑉)(𝑣1 − 𝑣2 < 𝑉).

Fig. 1. Circuit model

By changing the variables and parameters such that
𝐶𝐶𝐿𝐿

12

2𝑖𝐿 = √ 𝐿2 𝑉𝑥, 𝑣1 = 𝑉𝑦, 𝑣2 = 𝑉𝑧, 𝛼 = 𝐶2 , 𝛽 = 𝐺𝑑 √𝐶 , 𝛾 = 𝑔√𝐶 , 𝑡 = √𝐿𝐶2 𝜏The normalized equation of chaotic circuit is given as follows:
𝑑𝑥=𝑧𝑑𝜏𝑑𝑦𝑑𝜏𝑑𝑧= 𝛼(𝛾𝑦 − 𝛽𝑓){ 𝑑𝜏 = 𝛽𝑓 − 𝑥.

(3)

6th International Forum on Advanced Technologies
March 9th-10 th, 2020,Tokushima, JapanThe nonlinear function f corresponds to the characteristics of the nonlinear resistor consisting of the diodes
and described as follows:𝑦 − 𝑧 − 1 (𝑦 − 𝑧 > 1)0(|𝑦 − 𝑧| ≤ 1) (4)𝑓={𝑦 − 𝑧 + 1 (𝑦 − 𝑧 < 1).
For the computer simulation, we set the parameters as α = 0.50, β = 20.00 and γ = 0.50.
3. Results and discussion
We investigate the case of 3-dimensional networks. Thirty chaotic circuits are located in 3-dimensional,
including the positional information. The location of 30 chaotic circuits is shown in Fig. 2 and Table I.

Table. I.
The location of chaotic circuits in the 3-dimensional space.

Fig. 2. Location of 30 chaotic circuits in 3-dimensional space.
All circuits connected each other by resistors.
We consider the coupled chaotic circuits:𝑑𝑥𝑖𝑑𝜏
𝑑𝑦𝑖𝑑𝜏𝑑𝑧𝑖{𝑑𝜏= 𝑧𝑖= 𝛼(𝛾𝑦𝑖 − 𝛽𝑓 − ∑𝑁𝑖,𝑗=1 𝑟𝑖,𝑗 (𝑦𝑖 − 𝑦𝑗 ))(5)= 𝛽𝑓 − 𝑥𝑖 .

The nonlinear function f corresponds to the i-v characteristics of the nonlinear resistors consisting of the diodes
and are given as follows
𝑦𝑖 − 𝑧𝑖 − 1 (𝑦𝑖 − 𝑧𝑖 > 1)0(|𝑦𝑖 − 𝑧𝑖 | ≤ 1)𝑓={𝑦𝑖 − 𝑧𝑖 + 1 (𝑦𝑖 − 𝑧𝑖 < 1).

(6)where, i in the equation represents the circuit itself, and j is the coupling with other circuits. The parameter r
represents the coupling strength between the circuits. In this simulation, we set the coupling parameter value ri,jto correspond the distance between the circuits by the following equation:
𝑞𝑟𝑖,𝑗 =(7)(𝑑𝑖,𝑗 )2
di,j represents the Euclidean distance between the i−th and the j−th circuits. Further, the parameter q is the weightparameter that determines the coupling strengths. In this case, we set parameter q = 0.004724.

6th International Forum on Advanced Technologies
March 9th-10 th, 2020,Tokushima, Japan
Figure 3 shows the computer simulation results obtained from the 30 chaotic circuits located as shown in Fig.
2. From these results, we conﬁrm that the 1st and the 2nd circuits are synchronized at in-phase state. However,the 1st chaotic circuit and the 13th chaotic circuit are not synchronized. Also the 1st chaotic circuit and the 18thchaotic circuit are not synchronized. Similarly, between the 13th and the 14th chaotic circuit are synchronized,between the 18th and the 19th chaotic circuit are synchronized. However, between the 13th and the 18th chaotic
circuit are not synchronized. From these results, the circuits can form 3 clusters deﬁned by chaotic
synchronization as shown in Fig. 4.

Fig. 3. Location of 30 chaotic circuits in 3-dimensional space.

Fig. 4. The clustering result of 30 chaotic circuits.

4. Conclusion
In this study, we investigated synchronization phenomena when the chaotic circuits are located in 3dimensional space. Synchronization phenomena were seen between circuits at near distance, andsynchronization phenomena could not be seen between circuits at far distance. With these results, it was
conﬁrmed that the chaotic circuits were different from synchronization phenomena by distance information andthe clustering phenomena were observed.
In the future works, we would like to increase the number of chaotic circuits and the cluster. Moreover, weconsider that we would like to increase the dimensional space. We also would like to investigate changes in theclusters by changing the value of q.
References[1] K. Kaneko, “Clustering, Coding, Switching, Hierarchical Ordering, and Control in a Network of ChaoticElements”, Physical D, vol. 41, pp. 137-172, 1990.
[2] L. Angelini, F. D. Carlo, C. Marangi, M. Pellicoro and S. Stramaglia, “Clustering Data by Inhomogeneous
Chaotic Map Lattice”, Phys. Rev. Lett., 85, pp. 554-557, 2000.
[3] Y. Takamaru, H. Kataoka, Y. Uwate and Y. Nishio, “Clustering Phenomena in Complex Networks of ChaoticCircuits”, Proceedings of IEEE International Symposium on Circuits and Systems (ISCAS’12), pp. 914917,
May 2012.
[4] T. Chikazawa, Y. Uwate, Y. Nishio. “Investigation of Spreading Chaotic Behavior in Coupled Chaotic Circuit
Networks with Various Features”, Proceedings of RISP International Workshop on Nonlinear Circuits,Communications and Signal Processing (NCSP’17), pp. 337-340, Feb. 2017.