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

Spatio-Temporal Chaos in a Coupled System of Ring Oscillators
Yasuteru HOSOKAWA† and Yoshifumi NISHIO‡†

Department of Information Science
Shikoku University
123-1 Furukawa, Ohjin, Tokushima, Japan
Phone:+81-88-665-1300, Fax:+81-88-665-9928
Email: hosokawa@keiei.shikoku-u.ac.jp‡

Department of Electrical and Electronics Engineering
Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Phone:+81-88-656-7470, Fax:+81-88-656-7471
Email: nishio@ee.tokushima-u.ac.jp

1. Introduction

V1We have investigated a chaotic circuit utilizing CMOS
ring oscillators[1]. The circuit consists of two CMOS
ring oscillators and two diodes. A CMOS ring oscillator
is used for CMOS process performance test. This oscillator consists of a series of inverter circuits. It has a very
simple structure. Therefore, we consider that the proposed circuit is realized very easily. In the past studies,
we have shown the bifurcation scenario, have calculated
the Lyapunov exponents, have explained physical mechanism of chaos generation and so on.
In this study, we investigate spatio-temporal chaos in acoupled system of the ring oscillators. The system consists of many chaotic circuits utilizing CMOS ring oscillators. Because this system utilizes the circuits designed
by integrated circuit technology, this study is very important for analysis and applications of large coupledchaotic systems[2]–[5]. We consider that large coupledsystems should be realized as VLSI for future studies andfuture applications. The proposed system is very suitable for integrated circuits. It means that VLSI realization of the system is easily. In Sect. 2 and 3, the chaotic
circuit and the coupled system are described. In Sect. 4,
simulated results of the coupled system are shown. Someconcluding remarks are presented in the Sect. 5.

V2

V3

R1
C1

Vdd
M1

M2
Vss

V4

V5

V6
R2Figure 1: Chaotic circuit using ring oscillators.
oscillator is one of the simplest circuit constructed on thechip. The oscillator consists of the odd number of CMOS
inverters. We designed a chaotic circuit using ring oscillators. Figure 1 shows the designed circuit. In orderto control amplitudes of oscillators, R1 and R2 are connected. The frequency of the upper side oscillator arecontrolled 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. We focus attention on one inverter as shown
all parasitic capacitor
associated with the input nodes

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

input resistance

2. Chaotic Circuit Using CMOS Ring Oscillators
Chaotic circuits have been realized as integrated circuits. These circuits are not complexity so far as an electric circuit are concerned. However, realized circuits on
the IC chip are complexity structures. The basic problems are that the circuit is an analog circuit and included
nonlinear elements. Therefore, integration techniques
are needed. One of the attraction of chaos is that asimple system generate complex phenomena. We went
back again to this attraction, a simple integrated circuit
was proposed. We consider utilizing a ring oscillator
which is a simple structure on the chip. A CMOS ringCo

previous stageinverter

RoCi

Ri

next stage

Figure 2: Simplified model of the ring oscillator’s inverter.
in Fig. 2. All parasitic capacitors associated with theinput node is connected parallel with all parasitic capacitors associated with the output node of the previous-159-

whereVin

Gm VinRC1(v − v6 − Vth ) for v3 − v6 > Vth , rd 30 for −Vth ≤ v3 − v6 ≤ Vth ,
id =

1


(v3 − v6 + Vth ) for v3 − v6 < −Vth .
rdVoutFigure 3: Simplified model of the ring oscillator’s inverter.

By substituting the variables and the parameters,vn
Rd
1stage inverter. The input resistor is connected parallel
xn =t,,
yd =id ,τ=with the output resistor of the previous stage inverter.
VthVthRC
Correspondingly, All parasitic capacitors associated withCR
α = GmR,β=,γ=,
the output node is connected parallel with all parasitic
C + C1
R1R
R
capacitors associated with the input node of the nextδ=,ε=.
stage inverter. The output resistor is connected parallel
RdR2with the input resistor of the next stage inverter. Therefore, we can obtain a simplified inverter model as shown (2) and (3) are normalized asin Fig. 3. M 1 and M 2 have a characteristic of the sameẋ1 = −x1 − αx3 ,ẋ2 = −x2 − αx1 ,v ẋ3 = −β(γ + 1)x3 − αβx2 − βδyd ,D1i
ẋ4 = −x4 − αx6 ,
i D2
1


 ẋ5 = −x5 − αx4 ,-V thVth vv
 ẋ6 = −(ε + 1)x6 − αx5 + δyd ,
1i(3)

(4)

(5)

where x3 − x6 − 1 for x3 − x6 > 1,0for −1 ≤ x3 − x6 ≤ 1,yd =x3 − x6 + 1 for x3 − x6 < −1.

Figure 4: Diodes model.

(6)

as the coupling diodes shown in Fig. 4. Therefore, M 1 3. Coupled system of the chaotic circuits
and M 2 are approximated as the following three-regions
piecewise-linear function.
1(v − Vth ) for vD > Vth , rd D
Osc. 1
Osc. 2
Osc. n0for −Vth ≤ vD ≤ Vth ,iD =(1)1(vD + Vth ) for vD < −Vth .
rdCircuit equations are described as follows:1Gmdv1
= −v1 −v3 ,dtRCC

dv21Gm= −v2 −v1 ,



dtRCC



dv3
R + R1


= −v3


RR1 (C + C1 )

 dt
Gm
id−v3 −,C + C1
C + C1 dv41Gm
 dt = − RC v4 − C v6 ,
dv51Gm
= −v5 −v4 ,dtRCC dv6 = − R + R2 v6 − Gm v6 + id ,dt
RR2 CCCFigure 5: Coupled system of the chaotic circuits
The circuits are coupled by capacitors as shown inFig. 5. The coupling points are V3 of Fig. 1. The valueof all coupling capacitors is as Cc . Coupled system equations are same as Eq. (2) except the third equation.
ẋk1 = −xk1 − αxk3 ,

(2)

 ẋk2 = −xk2 − αxk1 , ẋk4 = −xk4 − αxk6 ,(7)ẋk5 = −xk5 − αxk4 ,ẋk6 = −(ε + 1)xk6 − αxk5 + δyk ,f or1 ≤ k ≤ n,-160- ẋ13= ẋk3

βF(n) G(n)ζ½=F(n−k+1)¾(8) x3 − x6 − 1 for x3 − x6 > 1,0
for −1 ≤ x3 − x6 ≤ 1,yd =x3 − x6 + 1 for x3 − x6 > −1.

(9)βH(n−k+1) + ẋ(k−1)3ζ

whereζ=
F(1) =

F(k) =
 F(n) =Cc.
C + C1(10)1,ζ +11, f or 2 ≤ k ≤ n − 1,
ζ + 2 − F(k−1)
1.
ζ + 1 − F(n−1)G(k) =−αxk2 − (γ + 1)xk3 − δyk ,f or 1 ≤ k ≤ n.

(11)

(12)

kinds of initial values. Figures 7–10 show simulated results of the coupled system. Horizontal axis shows time.
Vertical axes shows amplitudes xk3 (k = 1, 2, · · · 25) ofeach oscillators. Attractors of small amplitude parts are
similar of Fig. 6 (b). One small amplitude part is observed in Fig. 7. The position of this part is changing
irregularly. In Fig. 8, all oscillators generate small amplitudes like Fig. 6 (b). Three small amplitude parts areobserved in Fig. 9. These parts do not move. In thecase that the number of coupled oscillators is 25, fouror more small amplitude parts are not observed. This
result means that about seven oscillators are needed inorder to make existing one small amplitude part. Twosmall amplitude parts are observed in Fig 10. Basically,
this state is quasi-stable. However, the position of theseparts sometimes move irregularly as shown in Fig. 11.
In the case that initial values of all oscillators are thesame, one small amplitude parts is observed as a Fig. 7
via no small amplitude part state. Therefore, it is considered that the system wishes to make existing smallamplitude parts. From these results, we can consider
that each small amplitude part hates closing.
5. Conclusions H(1) =
H =
 (k)

G(n)
G(n−k+1) + F(n−k+1) H(k−1)
f or 2 ≤ k ≤ n.

4. Spatio-Temporal Chaos(a)(b)

In this study, we have proposed coupled chaotic sys(13) tems utiling CMOS ring oscillators. It is the physicalfeature that the system is constructed by simple CMOS
elements. This system was simulated using many kindsof initial values. As a result, the characteristics of thissystem are shown as follows: The system wishes to makeexisting small amplitude parts. The existence of one
small amplitude part needs about seven oscillators. Eachsmall part hate closing. We consider that the chaotic
characteristic makes changing the position of a smallamplitude part. However, by the reason that one small
amplitude part needs about seven oscillators and eachpart hates closing, in the case of many small amplitudeparts we could not observed changing the position of apart. These results make studies of large coupled chaoticsystems wide. Namely, these results contribute studies
on VLSI realization and applications of large coupledchaotic systems.
References

Figure 6: Attractor of the chaotic circuit using ring os- [1] Y. Hosokawa, Y. Nishio and A. Ushida, “Analysis ofcillators. α = 4, β = 0.1, γ = 4.5, δ = 70 and ε = 2.4
Chaotic Circuit Using Two Simple Ring Oscillators Coupled by Diodes,” Proc. of NOLTA2001, vol. 1, pp. 203–Figures 6 shows attractors of the single chaotic circuit
206, Oct. 2001.
using ring oscillators. Chaotic attractor is observed in [2] K. Kaneko, “ Theory and Applications of Coupled MapFig. 6 (a). Periodic orbit is observed in Fig. 6 (b). Same
Lattices,” John Wiley & Sons, 1993.
parameters are used, different initial values are used. [3] A. L. Zheleznyak and L. O. Chua, “Coexistence of LowThe coupled system is simulated using the same parameand High-Dimensional Spatiotemporal Chaos in a Chainters of Fig 6. The number of coupled oscillators is 25. Weof Dissipatively Coupled Chua’s Circuits,” Int. J. of Bifurcation and Chaos, vol. 4, no. 3, pp. 639–674, 1994.
investigate time series of the amplitude x3 , about some-161-

[4] Y. Nishio and A. Ushida, “Spatio-Temporal Chaos inSimple 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. ofISCAS’02, vol. 3, pp. 57–60, May 2002.

  

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

  

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

  

Figure 10: Simulated result in the case that two small
amplitude parts moves irregulary.

  

Figure 8: Simulated result in the case that all oscillators
generate small amplitudes like Fig. 6 (b).

  

Figure 11: Simulated result in the case of observed two
small amplitude parts.(continued of Fig. 10)-162-