MOVEMENT OF SMALL AMPLITUDE PARTS IN A COUPLED CHAOTIC SYSTEM.
Yasuteru Hosokawa

Yoshifumi Nishio

Department of Information Science,Shikoku University
123-1 Furukawa, Ohjin, Tokushima, Japan
Email: hosokawa@keiei.shikoku-u.ac.jp

Department of Electrical and Electronics
Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Email: nishio@ee.tokushima-u.ac.jp
V1ABSTRACT
We investigate the movement of small amplitude parts in a coupled chaotic system. The system consists of chaotic circuits using
CMOS ring oscillators and capacitors as coupling elements. In thecase of coupling 25 chaotic circuits, irregular movements of small
amplitude parts are observed for the same parameter and different
initial values. This is a kind of spatio-temporal phenomena. Wefocus attention on difference of the movement. Distributions of thesmall amplitude parts are investigated.

V2

V3

R1
C1

Vdd
M1

M2
VssV4

V5

V6
R2

1. INTRODUCTION
Figure 1: Chaotic circuit using ring oscillators.
In our past study, we have investigated a chaotic circuit using
CMOS ring oscillators[1]. The circuit consists of two CMOS ringoscillators and two diodes. A CMOS ring oscillator is used for theCMOS process performance test. This oscillator consists of a series of inverter circuits. It has a very simple structure. Therefore,
it is considered that the proposed circuit is realized very easily. In
the past study, we have shown the bifurcation scenario, have calculated the Lyapunov exponents, have explained physical mechanismof chaos generation and so on.
In this study, we investigate the movement of small amplitudeparts in a coupled chaotic system using the chaotic circuits. The
study of large coupled chaotic systems is very important not only
as models of nonlinear systems with many degrees of freedom but
also to understand biological information processing and for engineering applications. Especially, we consider that it is important touse continuous-time chaotic circuits, which is natural real physicalsystems, in these studies. On the other hand, the system consistsof many chaotic circuits using CMOS ring oscillators. Because
this system utilizes the circuits designed by the integrated circuit
technology, this study is very important for the analysis and applications of large coupled chaotic systems[2]–[5]. We consider thatlarge coupled systems should be realized as VLSI for future studies and future applications. The proposed system is very suitablefor integrated circuits. It means that VLSI realization of the system
is easily. In Sec. 2, the chaotic circuit and its simplified model isdescribed. In Sec. 3, the coupled system and the simulated results
are shown. Some concluding remarks are presented in Sec. 4.

structures. The basic problem is that the circuits are analog circuits
and include nonlinear elements. Therefore, integration techniques
are needed. One of the attraction of chaos is that a simple systemgenerates complex phenomena. We went back again to this attraction and a simple integrated circuit was proposed. We consider
utilizing a ring oscillator which has a simple structure on the chip.
A CMOS ring oscillator is one of the simplest circuit constructedon the chip. The oscillator consists of the odd number of CMOS
inverters. We designed a chaotic circuit using the ring oscillators.
Figure 1 shows the designed circuit. In order to control amplitudesof oscillators, R1 and R2 are connected. The frequency of theupper side oscillator are controlled by C1 . Two three-dimensional
autonomous oscillatory circuits are coupled by two diodes. We approximate M 1, M 2 and CMOS inverters in the circuit as follows.
all parasitic capacitor
associated with the input nodes

all parasitic capacitor
associated with the output nodes
gain of inverter
output resistance

input resistanceCo

previous stage

inverter

Ro

Ci

Ri

next stage

2. CHAOTIC CIRCUIT USING CMOS RINGOSCILLATORS

Figure 2: Model of a CMOS inverter in the ring oscillator.

Several chaotic circuits have been realized as integrated circuits.
These circuits are not complex so far as an electric circuit are concerned. However, the realized circuits on the IC chip have complex

We focus attention on one inverter as shown in Fig. 2. All parasitic
capacitors associated with the input node is connected parallel withall parasitic capacitors associated with the output node of the previ-0-7803-7762-1/03/$17.00 ©2003 IEEEIII-80

3. COUPLED SYSTEM OF THE CHAOTIC CIRCUITSGm VinVinRCVoutOsc. 1

Osc. 2

Osc. nFigure 3: Simplified model of a CMOS inverter in the ring oscillator.
ous stage inverter. The input resistor is connected parallel with theoutput resistor of the previous stage inverter. Correspondingly, All
parasitic capacitors associated with the output node is connected
parallel with all parasitic capacitors associated with the input nodeof the next stage inverter. The output resistor is connected parallelwith the input resistor of the next stage inverter. Therefore, we canobtain a simplified inverter model as shown in Fig. 3. M 1 and M 2Figure 5: Coupled system of the chaotic circuits.
The circuits are coupled by capacitors as shown in Fig. 5. Thecoupling points are V3 of Fig. 1. The value of all coupling capacitors is Cc . Coupled system equations are the same as Eq. (2)except the third equation. Figure 6 shows attractors of the singlev
D1i

i

D2



1-V thv

Vthi

v

1(a)

(b)

Figure 4: Diodes model.
have a characteristic of the same as the coupling diodes shown inFig. 4. Therefore, M 1 and M 2 are approximated as the followingthree-regions piecewise-linear function.

iD =

1


 rd (vD − Vth ) for vD > Vth ,0for −Vth ≤ vD ≤ Vth , 1 (vD + Vth ) for vD < −Vth .

(1)rdNormalized circuit equations are described as follows:

ẋ1


 ẋ

2




ẋ3ẋ4




ẋ5



 ẋ6where

(===
=
=
=−x1 − αx3 ,−x2 − αx1 ,−β(γ + 1)x3 − αβx2 − βδyd ,−x4 − αx6 ,−x5 − αx4 ,−(ε + 1)x6 − αx5 + δyd ,for x3 − x6 > 1,
for −1 ≤ x3 − x6 ≤ 1,
for x3 − x6 < −1.

(3)

Rd1
id ,
τ =t,VthRCCR
α = GmR,β=,γ=,
C + C1
R1RR
δ=,
ε=.
Rd
R2(4)

yd =xn =x3 − x6 − 1
0x3 − x6 + 1(2)

vn,
Vth

yd =

Figure 6: Attractor of the chaotic circuit using the ring oscillators.
α = 4, β = 0.1, γ = 4.5, δ = 70 and ε = 2.4
chaotic circuit using the ring oscillators. Chaotic attractor is observed in Fig. 6 (a). Periodic orbit is observed in Fig. 6 (b). The
same parameters are used, but different initial values are used. The
coupled system is simulated using the same parameters as Fig. 6.
The number of the coupled chaotic circuits is 25. At first, we investigate time series of x3 for various sets of initial values. Figures 7–
11 show simulated results of the coupled system. The horizontal
axis shows time. The vertical axis shows xk3 (k = 1, 2, · · · 25) ofeach chaotic circuit.
We found that the system produce one or more small amplitude parts consisting of the attractor in Fig. 6 (b). One small amplitude part is observed in Fig. 7. The position of this part is changing
irregularly. In Fig. 8, all chaotic circuits generate the small amplitudes. Three small amplitude parts are observed in Fig. 9. These
parts do not move. In the case that the number of coupled chaotic
circuits is 25, four or more small amplitude parts are not observed.
This result means that about seven chaotic circuits are needed inorder to produce one small amplitude part. Two small amplitudeparts are observed in Fig. 10. Basically, this state is almost stable.
However, the position of these parts rarely moves irregularly as
shown in Fig. 11. In the case that initial values of all chaotic circuits are the same, one small amplitude part is observed as Fig. 7
via no small amplitude part state. Therefore, it is considered thatthe system wishes to produce the small amplitude parts. Fromthese results, we can consider that each small amplitude part hates
getting close.
Next, the number of the local maximum of the small amplitude
are counted. This counting means calculating the distributions ofIII-81

the small amplitude parts of each oscillator. Figures 12–15 show
the numbers of the local maximum of the small amplitude parts
corresponding to Figs. 7–11. The horizontal axis shows the number of the chaotic circuits. The vertical axis shows the numbers of
the local maximum of the small amplitude parts. Figure 12 showsthat the small amplitude part is almost between no. 11 and no. 15.
Figure 13 shows that the small amplitude part is pervasive. Figure 14 shows that three small amplitude parts are observed. Thecenter part is symmetric. However, both side parts are asymmetric.
Figure 15 shows that two small amplitude parts are observed. Themain parts of the small amplitude are no. 6 and no. 20. Between
no. 6 and no. 20, only few counts are observed. We think that thiscounts corresponds to the irregular movement as shown in Fig. 11.
This result shows that the irregular movement as is very rare.

  

4. CONCLUSIONSIn this study, we have investigated a coupled chaotic system utilizing CMOS ring oscillators. The system can be constructed bysimple CMOS elements. We carried out computer simulation formany sets of initial values. As a result, the characteristics of thissystem are described as follows. The system wishes to produce
one or more small amplitude parts. The existence of one small amplitude part needs about seven oscillators. Each small part hates
getting close. We observed movements of the small amplitudepart. We consider that the chaotic behavior of each circuit makeschanging the position of the small amplitude parts. However, bythe reason that one small amplitude part needs about seven chaotic
circuits and each part hates getting close, in the case of many small
amplitude parts we could not observed changing the position of theparts. These results contribute to the studies on VLSI realization
and applications of large coupled chaotic systems.

Figure 7: Simulated result in the case that one small amplitude part
is observed.

5. REFERENCES[1] Y. Hosokawa, Y. Nishio and A. Ushida, “Analysis of ChaoticCircuit Using Two Simple Ring Oscillators Coupled byDiodes,” Proc. of NOLTA2001, vol. 1, pp. 203–206, Oct.
2001.

  Figure 8: Simulated result in the case that all oscillators generatesmall amplitudes as Fig. 6 (b).

[2] K. Kaneko, “ Theory and Applications of Coupled Map Lattices,” John Wiley & Sons, 1993.
[3] A. L. Zheleznyak and L. O. Chua, “Coexistence of Low- andHigh-Dimensional Spatiotemporal Chaos in a Chain of Dissipatively Coupled Chua’s Circuits,” Int. J. Bifurcation andChaos, vol. 4, no. 3, pp. 639–674, 1994.
[4] Y. Nishio and A. Ushida, “Spatio-Temporal Chaos in Simple
Coupled Chaotic Circuits,” IEEE Trans. on Circuits and Systems I, vol. 42, no. 10, pp. 678–689, Oct. 1995.
[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.

  

Figure 9: Simulated result in the case that three small amplitudeparts are observed.

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Figure 10: Simulated result in the case that two small amplitudeparts moves irregularly.

Figure 13: Number of local maximum of small amplitude in thecase that all oscillators generate small amplitudes as Fig. 6 (b).

  

Figure 11: Simulated result in the case of two small amplitudeparts.(continued from Fig. 10)

Figure 12: Number of local maximum of small amplitude in thecase that one small amplitude part is observed.

Figure 14: Number of local maximum of small amplitude in thecase that three small amplitude parts are observed.

Figure 15: Number of local maximum of small amplitude in thecase that two small amplitude parts moves irregularly.

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