IEEE Workshop on Nonlinear Circuit Networks
December 12-13, 2014

Immobilization of Oscillation Groups in CNN
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.jpjAbstract—In our previous study, Cellular neural networks using three kinds of cloning templates have been researched. Somecomputer simulation results show that cells are immobilized. In
this study, immobilize of oscillation phenomena are investigated.

c(1,1)

I. I NTRODUCTIONc(1,2)

c(2,1)

There are many studies of coupled oscillatory systems. In
these systems, observed phenomena are influenced by the
network structure which is as ladder, ring, full-coupled andso on. Additionally, characteristics of the oscillator affect theobserved phenomena. Therefore, these two factors are very
important to study coupled chaotic systems.
On the other hand, Zou et al showed that CNN which
consists of three cells generates chaos [3]. By applying thisCNN to the element of coupled oscillatory systems, thecoupled system has a novel network structure and becomesa chaotic system.
In our previous study, we have proposed cellular neuralnetworks using three kinds of cloning templates [5]. Three,six, twelve, eighteen, twenty-four and thirty six cells areinvestigated. Some computer simulation results show thatcells are immobilized. In this study, immobilize of oscillation
phenomena are investigated.
By some computer simulations, relationships between thephenomena and the structure are investigated.

ic(1,3)c(2,2)

c(3,1)

c(2,3)

c(3,2)

c(i,1):αc(2,N)c(3,3)c(i,2)

c(M,1):βc(3,N)

c(i,j)

c(M,2)

c(i,N)

c(M,j)

c(M,N):γ

Fig. 1. Structure of cellular neural networks using three kind of cloning
templates.

Cell β:dxij= − xij + Iβ
dt∑+
Aβ (i, j; k, l)ykl

II. C ELLULAR NEURAL NETWORKS USING THREE KINDS

c(k,l)

OF CLONING TEMPLATESFigure 1 shows a structure of cellular neural networks using
three kinds of cloning templates. The system consists of threekinds of cells which name is Cell α, Cell β or Cell γ. Eachcells connects six neighbour cells. The difference of threekinds of cells is only values of cloning templates. Threekinds of cells are placed at uniformly. Cells are coupled as
triangle lattice. In this study, The boundary condition is set asa periodic condition. The state equations are shown as follows.
Cell α:dxij= − xij + Iα
dt∑+Aα (i, j; k, l)ykl(1)c(k,l)∑+Bα (i, j; k, l)ukl

c(1,N)+∑(2)Bβ (i, j; k, l)ukl

c(k,l)

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 ).

c(k,l)- 98 -(4)

wheref (x) = 0.5(|x + 1| − |x − 1|).

(5)

In order to keep a symmetric property, symmetric cloning
template is defined as follows.


k
l
1.24
k ,
Aα = l
k
l


m
k
1.1
m ,
Aβ = k
m
k
(6)


l
−m
1.0
l ,
Aγ = −ml−mFig. 4. Chaotic phenomena in the case of three cells.Vertical axis is Cell αc(1, 1) and horizontal axis is Cell β c(1, 2).
c(4,2)

c(1,3)

B α = B β = B γ = 0 andIα = Iβ = Iγ = 0.

c(1,1)

c(2,3)

Parameter k is coupling strength of Cell α and Cell β. Parameter l is coupling strength of Cell β and Cell γ. Parameter m
is coupling strength of Cell α and Cell γ.

Three cells are connected as shown in Figure 2. Figure 3 and4 shows a simulation result corresponding to Figure 2. Chaotic
phenomena are observed. Each cell oscillates in positive
area or negative area. Additionally, these two oscillations are
switching randomly. Switching phenomena occur almost sametime. Cell α c(1, 1) and Cell β c(1, 2) oscillate same area.

c(2,1)c(1,1)c(1,2)

c(2,1)c(1,1)

Fig. 2.

c(1,1)

c(2,1)c(1,2)c(1,1)c(1,2)

Cells are connected in the case of three cells.

c(1,3)c(2,2)

c(3,1)

c(4,3)

Fig. 5.

c(4,2)c(1,1)

c(2,3)

c(3,2)

c(4,1)

c(2,1)c(3,3)

c(4,2)

c(1,3)

c(3,1)

c(4,3)c(1,1)

c(4,1)c(1,2)

Cells are connected in the case of twelve cells ( M = 4, N = 3 ).

Cell γ c(2, 1) oscillates different area. A double-scroll typeattractor is observed in Figure 4. By changing parameter k,some periodic orbits, bifurcation phenomena and chaos areobserved.
B. Twelve cells
In this case, two pattern are considerated as shown in Figure 5 and 6. Figure 7 shows a simulation result corresponding
to Figure 5. Cell α c(1, 1) and Cell β c(1, 2) oscillate same
area. Cell γ c(1, 3) oscillates different area. This phenomena
is same as case of three cells. Cell α c(1, 1), Cell β (1, 2)
and Cell γ c(1, 3) is immobilized as one set. In this case, one
set consists of one Cell α, one Cell β and one Cell γ. These
sets consist of same row cells. Namely, this case has four sets.
Each set does not concern each other.
Figure 8 shows a simulation result corresponding to Figure 6. This case has no set. Additionally, switching phenomena
disappeared.
C. Eighteen cells

c(1,1) 0c(1,2)

c(2,1)c(1,2)

A. Three cells

c(2,1)c(1,2)c(3,3)

III. C OMPUTER SIMULATIONSIn this section, some computer simulations are shown.
Initial values are set randomly. Parameter k, Parameter l andParameter m are refered to [3].

c(1,2)

c(4,1)

c(4,3)

0c(2,1) 0

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

In this case two pattern considerated as shown in Figure 9
and 10. Figure 11 shows a simulation result corresponding to
Figure 9. In this case, cells are immobilized as six sets. Sets
which consists of same row cells are observed. Each set does
not concern each other. This phenomena is same as case of
twelve cells (M = 4, N = 3) .
Figure 12 shows a simulation result corresponding to Figure 10. In this case, cells are immobilized as six sets. Sets

- 99 -

c(2,1)c(2,6)c(2,2)

c(2,3)

c(2,4)

c(2,5)

c(2,6)c(1,1)c(1,2)

c(1,6)c(1,1)

c(2,6)c(1,2)

c(2,1)c(1,1)

c(1,3)c(2,2)c(1,2)

c(1,4)

c(2,3)

c(1,3)

c(1,5)

c(2,4)

c(1,4)

c(1,6)

c(2,5)

c(1,5)c(1,1)

c(2,6)

c(1,6)

Cells are connected in the case of twelve cells ( M = 2, N = 6 ).

Fig. 6.
c(1,1)0c(1,2)0c(1,3)

0c(2,1)0c(2,2)0c(2,3)

0c(3,1)0c(3,2)0c(3,3)

0c(4,1)

0c(4,2)

0c(4,3)00c(1,3)0c(1,4)0c(1,5)

0c(2,1)c(1,1)0c(1,6)

0c(2,1)0c(2,2)0c(2,3)0c(2,4)

0c(2,5)

0c(2,6)

0Fig. 8. Waveforms in the case of twelve cells( M = 2, N = 6 ). k = −1.20,
l = −1.30 and m = −2.50.
c(6,1)

c(1,3)

c(6,2)c(1,1)

c(2,3)

c(6,3)c(1,2)

c(2,1)c(3,3)

Fig. 7. Waveforms in the case of twelve cells( M = 4, N = 3 ). k = −1.06,
l = −1.07 and m = −1.47.

c(6,1)

c(1,3)c(2,2)

c(3,1)

c(4,3)

c(2,3)

c(3,2)

c(4,1)

c(5,3)

c(2,1)c(3,3)

c(4,2)

c(5,1)

c(6,3)

which consists of same column cells are observed. Each set
does not concern each other.

c(1,1)

c(3,1)

c(4,3)

c(4,1)

c(5,2)

c(6,1)

c(1,3)

c(5,3)

c(6,2)c(1,1)

c(5,1)

c(6,3)c(1,2)

c(6,1)

c(1,3)

D. twenty four cells
Twenty four cells are connected as shown in Figure 13. Figure 14 shows a simulation result corresponding to Figure 13.
In this case, there are no immobilized sets which are observedin cases of twelve ( M = 4, N = 3 ), eighteen ( M = 3, N = 6) and eighteen ( M = 6, N = 3 ) cells. This case is observed
clustering phenomena. We consider that Cell α c(1, 1), Cell β(1, 3) and Cell γ c(4, 5) have similar waveforms in rigth sideof Figure 14. And Figure 14 shows that all cells switchingoccur almost same time.
E. Thirty six cells
Thirty six cells are connected as shown in Figure 15. Figure 16 shows a simulation result corresponding to Figure 15.
Each cells move separately. However, clustering phenomena
are observed. Cell γ c(3, 5) and Cell α c(3, 6) have similar
waveforms in centre area of Figure 16.
F. Consideration
Computer simulation results show that observed phenomena
can be divided into two phenomena by the structure as follows.
In cases of twelve ( M = 4, N = 3 ), eighteen ( M = 6,
N = 3 ) and eighteen ( M = 3, N = 6 ), one set consists of
one Cell α, one Cell β and one Cell γ. Same row or column
cells are immobilized. Namely, three cells of a set are coupled
as a loop.
In cases of twelve ( M = 2, N = 6 ), twenty four (M = 4, N = 6 ) and thirty-six ( M = 6, N = 6 ), theloop does not exist. Therefore, the set could not be observed
and immobilized.

Fig. 9. Cells are connected in the case of eighteen cells ( M = 6, N = 3 ).

We consider that existence of a loop which consists ofCell α, Cell β and Cell γ makes oscillation sets.
IV. C ONCLUSIONSIn this study, cellular neural networks using three kinds ofcloning templates are investigated. Some computer simulations
were carried out. In cases of twelve ( M = 4, N = 3 ), eighteen( M = 6, N = 3 ) and eighteen ( M = 3, N = 6 ) cases can besome sets which have a Cell α, Cell β and Cell γ. However,these sets are disappeared in case of twelve ( M = 2, N = 6), twenty four and thirty six cells.
In the future work, twelve ( M = 2, N = 6 ), twenty fourand thirty six cells are investigated in detail.
R EFERENCES[1] L. O. Chua and L. Yang, “Cellular Neural Networks: Theory,” IEEE
Trans. Circuits Syst., vol. 35, no. 10, pp. 1257–1272, 1988.

c(3,1)

c(1,6)

c(3,2)c(1,1)

c(2,6)c(1,2)

c(2,1)

c(3,6)- 100 -

c(3,4)

c(1,3)c(2,2)

c(3,1)

c(1,6)

Fig. 10.
).

c(3,3)

c(1,4)

c(2,3)

c(3,2)c(1,1)

c(3,5)

c(1,5)

c(2,4)c(3,3)c(1,2)

c(3,6)

c(1,6)

c(2,5)

c(3,4)

c(1,3)

c(3,1)c(1,1)

c(2,6)

c(3,5)

c(1,4)

c(2,1)

c(3,6)

c(1,5)

c(3,1)

c(1,6)

Cells are connected in the case of eighteen cells ( M = 3, N = 6

c(6,1)c(1,1)c(1,2)
c(1,3)
c(2,1)c(2,2)
c(2,3)
c(3,1)
c(3,2)c(3,3)
c(4,1)
c(4,2)
c(4,3)
c(5,1)
c(5,2)
c(5,3)
c(6,1)
c(6,2)
c(6,3)0c(1,6)

c(3,1)
c(3,2)c(3,3)
c(3,4)
c(3,5)
c(3,6)

c(6,3)c(1,2)

c(6,4)

c(1,3)

c(6,5)

c(1,4)

c(6,6)

c(1,5)

c(6,1)c(1,1)

c(1,6)00c(2,6)

c(2,1)c(2,2)

c(2,3)

c(2,4)

c(2,5)

c(2,1)

c(2,6)00c(3,6)

c(3,1)

c(3,2)c(3,3)

c(3,4)

c(3,5)

c(3,1)

c(3,6)00c(4,6)

0c(4,1)

c(4,2)

c(4,3)

c(4,4)

c(4,5)

c(4,1)

c(4,6)

0c(5,6)

0c(5,1)

c(5,2)

c(5,3)

c(5,4)c(5,5)

c(5,1)

c(5,6)

0c(6,6)

0c(6,1)

c(6,2)

c(6,3)

c(6,4)

c(6,5)

c(6,1)

c(6,6)0c(1,6)0c(1,1)c(1,2)

c(1,3)

c(1,4)

c(1,5)

c(1,6)

00
0

Fig. 15.
).

0Fig. 11. Waveforms in the case of eighteen cells( M = 6, N = 3 ).
k = −1.06, l = −1.07 and m = −1.47.

c(1,1)c(1,2)
c(1,3)
c(1,4)
c(1,5)
c(1,6)
c(2,1)c(2,2)
c(2,3)
c(2,4)
c(2,5)
c(2,6)

c(6,2)c(1,1)

0
0
0
0
0
0
0
0
0
0
0
0
0

Cells are connected in the case of thirty six cells ( M = 6, N = 6

c(1,1)c(1,2)
c(1,3)
c(1,4)
c(1,5)
c(1,6)
c(2,1)c(2,2)
c(2,3)
c(2,4)
c(2,5)
c(2,6)
c(3,1)
c(3,2)c(3,3)
c(3,4)
c(3,5)
c(3,6)
c(4,1)
c(4,2)
c(4,3)
c(4,4)
c(4,5)
c(4,6)
c(5,1)
c(5,2)
c(5,3)
c(5,4)c(5,5)
c(5,6)
c(6,1)
c(6,2)
c(6,3)
c(6,4)
c(6,5)
c(6,6)

00
0

Fig. 16. Waveforms in the case of thirty six cells. k = −1.26, l = −1.28
and m = −2.50.

00Fig. 12. Waveforms in the case of eighteen cells( M = 3, N = 6 ).
k = −1.06, l = −1.07 and m = −1.47.

c(4,5)

c(4,6)c(1,1)

c(1,6)

c(4,1)c(1,2)

c(2,1)

c(2,6)c(2,2)

c(3,1)

c(3,6)

c(4,2)

c(1,3)

c(2,3)

c(3,2)

c(4,1)

c(4,6)c(1,2)

c(4,3)

c(1,4)

c(2,4)c(3,3)

c(4,2)

c(1,3)

c(4,4)

c(1,5)

c(2,5)

c(3,4)

c(4,3)

c(1,4)

c(4,5)

c(1,6)

c(2,6)

c(3,5)

c(4,4)

c(1,5)c(1,1)

c(2,1)

c(3,6)

c(4,5)

c(1,6)

c(3,1)

c(4,6)c(1,1)

c(4,1)[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 Neural
Networks,” 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 Neural
Networks Using Two Kinds of Template Sets,” Proc. of NOLTA’07,
pp. 23–26, 2007.
[5] M. Tanaka, Y. Hosokawa and Y. Nishio, “Relationship between a Numberof Cells and Phenomena in Cellular Neural Networks Using Three Kindsof Cloning Templates,” RISP Proc. NCSP’14, pp533–536, 2014.

c(1,2)

Fig. 13. Cells are connected in the case of twenty four cells ( M = 4,
N = 6 ).

c(1,1)c(1,2)
c(1,3)
c(1,4)
c(1,5)
c(1,6)
c(2,1)c(2,2)
c(2,3)
c(2,4)
c(2,5)
c(2,6)
c(3,1)
c(3,2)c(3,3)
c(3,4)
c(3,5)
c(3,6)
c(4,1)
c(4,2)
c(4,3)
c(4,4)
c(4,5)
c(4,6)

Fig. 14. Waveforms in the case of twenty four cells. k = −1.20, l = −1.25
and m = −2.50.

- 101 -