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

Phase-Wave Propagation Phenomena in One-Dimensional Two-Layer CNNs
Kazuya TSURUTA, Zonghuang YANG, Masayuki YAMAUCHI,
Yoshifumi NISHIO and Akio USHIDA
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: tsuruta@ee.tokushima-u.ac.jp

1. Introduction
Dynamic property of networks of oscillatory andchaotic elements is one of the very lively studied topics. Many papers of international conferences and journals are devoted uniquely to studies of spatially extended
systems, active media, and coupled lattices showing important areas where studies of dynamic phenomena incoupled oscillators ﬁnd potential applications. Since Cellular Neural Networks (CNNs) were invented in 1988
[1], the CNN paradigm provides a ﬂexible framework(or universal model) to describe spatio-temporal dynamics in discrete space and - perhaps more importantly
from a practical point of view - allows for eﬃcient VLSI
implementation of analogue, array-computing structure.
Many nonlinear phenomena such as pattern formation,autowaves and etc, which come from many disciplines
ranging from biology to physics and chemistry, have been
reproduced in the CNNs, and the related literatures havebeen reported [8]-[11].
On the other hand, a lot of studies on phase relationship of mutually coupled oscillators or chaotic circuits
have been widely carried out. Endo and Mori [2]-[4] havestudied van der Pol oscillators as a ladder, ring or twodimensional array, and conﬁrmed that several modes ofsynchronization. We have also discovered very interesting wave propagation phenomena of phase-diﬀerence invan der Pol oscillators coupled by inductors as a ladder[5]. Unfortunately, the basic circuits of the above models contain inductive and high-order nonlinearly resistive
devices resulting in circuit sophistication, so that it is notsuitable to develop the implementation of the large-scale
arrays of these circuits. In this paper, we will investigate these dynamic behaviors encountered in a simpler
circuit structure. This structure is one-dimensional array of two-layer CNN cells, in which there exists resistive couple between cells and each cell has a well-known
piecewise-linear nonlinear output function at the output

stage. Depending on the application of couple coeﬃcients between cells, the initial conditions of cells andthe boundary conditions imposed on the array, we observed that many kinds of very interesting propagation
phenomena of phase diﬀerence, such as synchronization,phase-wave and phase-inversion-wave, can be regenerated in the one-dimensional two-layer CNN array.
2. One-Dimensional Two-Layer CNN Array
In this section, we describe the circuit array usedfor investigating the propagation phenomena of phase
diﬀerences between adjacent cells. This array is aone-dimensional two-layer CNN with constant template,with circuit topology simpler than those reported in literatures [2]-[5], which is developed from the original ChuaYang CNNs. In other words, the array consists of thecells of two-layer CNN with resistive couple. The state
equation of each cell is deﬁned as follows:ẋ1;i = −x1;i + A1 ∗ y1;i + B1 ∗ u1;i + C1 ∗ y2;i + I1ẋ2;i = −x2;i + A2 ∗ y2;i + B2 ∗ u2;i + C2 ∗ y1;i + I2
(1)
with output equationy1;i = 0.5(|x1;i + 1| − |x1;i − 1|)(2)y2;i = 0.5(|x2;i + 1| − |x2;i − 1|)
i = 1, 2, ..., N.
where A, B, C, and I are respectively the feedback template, control template, coupled template, and bias current, and the subscript 1 and 2 stand for ﬁrst layer andsecond layer of the two-layer CNN array. In this paper, we restrict our discussions to the following form oftemplate.

A1 = d1 a1 d1 , C1 = 0 c1 0 ,
A2 = d2 a2 d2 , C2 = 0 c2 0 ,-977-

B1 = B2 = 0, I1 = I2 = 0.

(3)

ykcelli

Oyk+1

celli+1O(a)ti (n)
n−1n−1nti+1 (n)tnt(b)
Figure 2: Deﬁnition of phase diﬀerence.

Figure 1: (a) Phase portrait in the x1;i –x2;i plane of thelimit cycle of the isolated two-layer CNN Cell. (b) Time and d2 = −0.1 is adopted. Analysis of this simulation
indicates that the anti-phase between adjacent cells isevolution of x1;i and x2;i .
transferred gradually from one end to the other end of
the CNNs array, and reﬂected at the end of array. This
It has been proved [6][7] that for a particular choice its transfer process of phase continuously exists in the CNN
parameters a1 , a2 , c1 , and c2 , this nonlinear second-order array, so called “phase-inversion-wave”. In the following
cell in absence of couple to its neighbor cells oscillates sections, we investigate the inﬂuence of the array boundwith a limit cycle centered to the origin and symmetric ary conditions, the cell initial states and the couplingwith respect to the phase plane axis, which is shown in parameters on the phenomena.
Fig. 1 by adopting a1 = 1.6, a2 = 0.7, c1 = −2, andc2 = 4.
x1;1x1;ix1;14
3. Phase-Wave Propagation Phenomena
3.1. Phase-Waves
Figure 3: A initial condition for one-dimensional array
The phase-waves were ﬁrstly introduced in literatureof 14 cells.
[5], which are observed from the changes of phase difference between adjacent cells in a ladder composed ofvan der Pol oscillators coupled by inductors. The phase
diﬀerence is deﬁned as follows:
φi,i+1 (n) = 2π ×ti (n) − ti+1 (n)ti (n) − ti (n − 1)(4)

where ti (n) is time when the state x1;i crosses 0[V]at n-th period. In all of the following Figs.4, 5 and6, the vertical axes are the sum of the ﬁrst layer outputs of adjacent cell, and the horizontal axes are time.
Hence, the diagrams show qualitatively how phase diﬀerences between adjacent cells change as time goes, wherewhite regions correspond to the state that two adjacent cells/oscillators are anti-phase synchronization, andblack regions to the in-phase synchronization. Figure4(a) shows wave propagation phenomenon of phase differences, which is observed from one-dimensional array,composed of 14 two-layer CNN cells with zero-ﬂux condition. The initial condition of the ﬁrst layer of this arrayis set as Fig.3, where the black squares denote cells having value 1 and the white squares value −1. The initialcondition of the second layer is zero. The parameter
set: a1 = 1.2, a2 = 1.1, c1 = −2, c2 = 4, d1 = 0.1,

Figure 4: Phase-inversion-wave in a one-dimensional array of 14 cells.

3.2. Influence of Coupling Parameters
In this section, the inﬂuence of coupling parameters
between adjacent cells on the propagation of phase-waves
is investigated in a one-dimensional array composed nine
two-layer CNN cells with zero-ﬂux boundary condition.
The initial state of the ﬁrst layer of array is shown in-978-Fig.5, and the second layer initial state is zero. The
parameter set is the same with the last simulation except parameters d1 and d2 . We respectively carry out
three simulations by adopting parameters d1 = 0.1 andd2 = −0.1, d1 = 0.15 and d2 = −0.15, and d1 = 0.2
and d2 = −0.2. The simulation results are respectively
shown in Figs.6 (a), (b) and (c). As we have seen, thepropagation speed of phase-wave is directly proportional
to the coupling parameters (coupling conductance) d1
and d2 between adjacent cells.
x1;1x1;ix1;1x1;ix1;1x1;ix1;1x1;ix1;18(a)x1;18(b)x1;18x1;9(c)Figure 7: Three initial states for one-dimensional array
Figure 5: A initial condition for one-dimensional array of 18 cells.
of 9 cells.
a2 = 1.1, c1 = −3, c2 = 5, d1 = −0.03, and d2 = 0.03
is adopted. Figure 7 shows three diﬀerent initial conditions. We obtain three diﬀerent patterns of wave propagation phenomena shown in Fig.8. The analysis of thesimulation indicates that the wave reﬂections take placeat both ends of array or the positions that two phasewaves collide with each other.
(a)3.4. Influence of Boundary Conditions(b)

In this section, two simulations are carried our to observe what phenomena arise with adopting two diﬀerent
boundary conditions – zero-ﬂux and ring in the same
circuit array composed of 14 cells. The parameter set isthe same with the last simulation. Two diﬀerent phaseinversion-waves are shown in Fig.9, where (a) is for zeroﬂux boundary and (b) is for ring boundary. We foundthat the phase-wave don’t be reﬂected at the both endsof array, but pass across the boundary and appear fromthe opposite side again in the case of ring boundary condition.

(c)

4. Conclusions

In this study, we have proposed a simpler circuit strucFigure 6: Inﬂuence of couple parameters on phase- ture – one-dimensional arrays of two-layer CNN cells
for investigating the phase-wave propagation phenominversion-wave
ena. We found that the propagation phenomena ofphase-wave, especially phase-inversion-wave, can be reproduced in the CNN array composed of almost numbers
3.3. Influence of Initial Statesof cells. Moreover, we have investigated qualitatively theWe investigate the inﬂuence of diﬀerent initial states inﬂuences of the coupling parameters, the initial states,on the phase-wave propagation in a one-dimensional ar- and the boundary conditions on these propagation pheray composed 18 two-layer CNN cells with zero-ﬂux nomena. In future works, we will try to ﬁnd out theboundary condition. The parameter set: a1 = 1.1, patterns of these dynamics and clarify their mechanism.

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(a)

(a)

(b)

(b)
Figure 9: Inﬂuence of boundary conditions on phaseinversion-wave(c)[6] F. Zou and J. A. Nossek, “A chaotic attractor withcellular neural networks,” IEEE Trans. Circuits &Syst., vol.38, no.7, pp.811-812, 1991.

Figure 8: Inﬂuence of three kinds of initial conditions onphase-inversion-wave[7] F. Zou and J. A. Nossek, “Bifurcation and chaos incellular neural networks,” IEEE Trans. Circuits &Syst., vol.40, no.3, pp.166-173, 1991.

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