IEEE Workshop on Nonlinear Circuit Networks
December 13-14, 2013

Chaotic Phenomena in Cellular Neural Networks
Using Three Kinds of Cloning Templates
Mana Tanaka

Yasuteru Hosokawa

Yoshifumi Nishio

Dept. Media and Information Syst.,
Shikoku University
Email: mana.8sk@gmail.com

Dept. Media and Information Syst.,
Shikoku University
Email: hosokawa@keiei.shikoku-u.ac.jp

Dept. Electrical and Electronic Eng.,
Tokushima University
Email: nishio@ee.tokushima-u.ac.jp

Abstract—In this study, chaotic phenomena in cellular neuralnetworks using three kinds of cloning templates are shown. Casesof number of cells is six, twelve and twenty-four are investigated.
Some interesting phenomena are observed.

I. I NTRODUCTION
There are many studies of coupled oscillatory systems. In
these systems, observed phenomena are influenced by thenetwork structures which are ladder, ring, full-coupled and soon. Additionally, characteristics of the oscillator affect the phenomena. Especially, chaotic systems can be generated manykinds of phenomena. These two factors are very important tostudy coupled oscillatory systems.
On the other hand, some interesting phenomena can beobserved in Cellular Neural Networks (CNN) [1][2] using two
kinds of cloning templates in [4]. The study showed that thesystem is one of novel coupled oscillatory system.
In this study, cellular neural networks using three kinds ofcloning templates are proposed. This system consists of threekinds of cells which have different values of cloning templates.
In [3], CNN using three cells which have different cloning
templates generates chaos is reported. Therefore, the proposedsystem is one of chaotic coupled oscillatory system.
The element of the system is coupled as triangle lattice.
By applying symmetric cloning template, the system becomeshomogeneous coupled system. By some computer simulations,relationships between phenomena and the number of cells orparameters are investigated.

Fig. 1. Structure of cellular neural networks using three kind of cloningtemplates.

Cell α:dxij= − xij + Iα
dt∑+Aα (i, j; k, l)ykl
c(k,l)+∑(1)Bα (i, j; k, l)ukl

c(k,l)

II. C ELLULAR NEURAL NETWORKS USING THREE KINDS
Cell β:OF CLONING TEMPLATESFigure 1 shows a structure of cellular neural networks using
three kinds of cloning templates. Cells are coupled as triangle
lattice. The system consists of three kinds of cells which nameis Cell α, Cell β or Cell γ. The difference of three kindsof cells is only values of cloning templates. The boundary
condition is set as a periodic condition. Namely, this system
has a torus structure. The state equations are shown as follows.

- 30 -dxij= − xij + Iβ
dt∑+
Aβ (i, j; k, l)ykl
c(k,l)+∑c(k,l)Bβ (i, j; k, l)ukl(2)

Fig. 2.

Chaotic phenomena in the case of three cells.
Fig. 3. Waveforms in the case of six cells. k = −1.06, l = −1.07 andm = −1.47.

Cell γ:dxij= − xij + Iγ
dt∑+Aγ (i, j; k, l)ykl
c(k,l)+∑(3)

Bγ (i, j; k, l)ukl

c(k,l)

where, A{αβγ} (i, j; k, l)ykl , B{αβγ} (i, j; k, l)ukl , I{αβγ}show feedback value, input value, bias value, respectively. The
output function is shown as follows.
yij = f (xij ).

(4)f (x) = 0.5(|x + 1| − |x − 1|).

(5)whereIn order to keep a symmetric property, symmetric cloning
template is defined as follows.
kl
1.24k ,Aα = lkl−mk
1.1−m ,A β = k−mk(6)lm
1.0l ,A γ = mlm
B α = B β = B γ = 0 andIα = Iβ = Iγ = 0.

Fig. 4. Waveforms in the case of six cells. k = −1.05, l = −1.07 andm = −1.47.

B. Six cells
In the case of six cells ( i = 2, j = 3 ), clustering phenomena depend on initial values are observed. Additionally,cells divided into two groups as shown in Fig. 3 as areobserved. In this case, c(1, 1) and c(1, 2) have very similar
waveforms and inverse waveform of c(1, 3) is also similar toc(1, 1). Similarly, c(2, 1), c(2, 2) and c(2, 3) have also similar
waveforms. Namely, each group consists of same row cells inFig. 1. Note that these are not synchronized. By changing aparameter k or l, sojourn time of high or low waveforms ischanged. Figure 4 shows a case of increasing a parameter k.
Sojourn time of high or low waveforms is increased. On theother hand, decreasing a parameter k decreases sojurn time asshown in Fig. 5.
C. Twelve cells

III. C OMPUTER SIMULATIONS
A. Three cells
In this section, some computer simulations are shown. In
the case of three cells, chaotic phenomena were confirmed by[3]. Figure 2 shows a simulation result in the case of threecells. A double-scroll type attractor is observed. By changing
parameter k, some periodic orbits, bifurcation phenomena andchaos are observed.

Figure 6 shows a simulation result in the case of twelve
cells ( i = 4, j = 3 ). In this case, four groups which consistsof same row cells are observed. Each group does not concern
each other. By changing a parameter k or l, sojourn time ofhigh or low waveforms is changed as shown in Fig. 7.
D. twenty four cells
Figure 8 shows a simulation result in the case of twenty fourcells ( i = 4, j = 6 ). In this case, there are no groups which
are observed in cases of six and twelve cells. Two kinds of- 31 -Fig. 5. Waveforms in the case of six cells. k = −1.095, l = −1.07 andm = −1.47.

Fig. 7. Waveforms in the case of twelve cells. k = −0.95, l = −1.07 andm = −1.47.

In the future work, a relationship between phenomena andthe number of cell, the reason of group exist and so on will
be investigated.
R EFERENCES

Fig. 6. Waveforms in the case of twelve cells. k = −1.06, l = −1.07 andm = −1.47.

[1] L. O. Chua and L. Yang, “Cellular Neural Networks: Theory,” IEEE
Trans. Circuits Syst., vol. 35, no. 10, pp. 1257–1272, 1988.
[2] L. O. Chua and L. Yang, “Cellular Neural Networks: Applications,” IEEE
Trans. Circuits Syst., vol. 35, no. 10, pp. 1273–1290, 1988.
[3] F. Zou, and J. A. Nossek, “Bifurcation and chaos in cellular neuralnetworks,” IEEE Trans. Circuits Syst. I, vol. 40, no. 3, pp. 166–173,
1993.
[4] J. Fujii, Y. Hosokawa and Y. Nishio, “Wave Phenomena in Cellular NeuralNetworks Using Two Kinds of Template Sets,” Proc. of NOLTA’07,
pp. 23–26, 2007.

oscillatory phenomena ( left side and right side of Fig. 8 ) areobserved alternately. These switching is almost same timing.
However, these waveforms are not synchronized. By changing
a parameter k or l, sojourn time of high or low waveforms ischanged as shown in Fig. 9 and 10.
E. Consideration
Each group which observed in six and twelve cases consistsof a Cell α, Cell β and Cell γ. These two cases show that eachgroup consists of cells which is same number of i. Namely,
three cells of a group are coupled as a loop. Coupling strength
between cells of one group becomes higher than other cells.
We consider that loops is the reason groups exist.
In the case of twenty four cells, a loop which consist of
three cell can not exist. Therefore, groups which observed in
six and twelve cases are disappeared.
IV. C ONCLUSIONSIn this study, cellular neural networks using three kindsof cloning templates have been proposed. Some computer
simulations were carried out. In cases of six cells and twelvecells, cells can be some groups which have a Cell α, Cell β
and Cell γ. However, these groups are disappeared in case oftwenty-four cells.

- 32 -Fig. 8.

Waveforms in the case of twenty four cells. k = −1.26, l = −1.20 and m = −2.50.

Fig. 9.

Waveforms in the case of twenty four cells. k = −1.10, l = −1.20 and m = −2.50.

Fig. 10.

Waveforms in the case of twenty four cells. k = −1.25, l = −1.20 and m = −2.50.

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