2004 RISP International Workshop on Nonlinear
Circuit and Signal Processing (NCSP'04)
Hawaii, USA, Mar. 5-7, 2004.

Two Chaotic Self-Oscillatory Circuits Coupled by Diodes
Yasuteru Hosokawa†and Yoshifumi Nishio‡†Shikoku University
Furukawa, Ohjin, Tokushima, Japan
Phone:+08-88-665-1300
FAX:+08-88-665-9928
Email:hosokawa@keiei.shikoku-u.ac.jp

1. Introduction‡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

function. Each of diodes is approximated piecewise linear
function as shown in Fig. 3.
We define the parameters and variables:
pva1 = Eth xa , va2 = Eth ya , ia = Ethp Ca1 /La za ,vb1 = Eth xb , vb2 = Ethpyb , ib = Eth Ca1 /La Zb ,√Cτ,α=gt = Lpa a1a La /Ca1 , β = Ca1 /Ca2 ,γ = gb pLa /Cb1 , δ = Ca1 /Cb2 , ε = La /Lb ,ζ = gd La /Ca1

Many researchers have been investigating chaotic circuits.
For instance, proposing new chaotic circuit, analyzing generated phenomena by the chaotic circuit, development of its
applications and so on. In particular, many three-dimensional
chaotic self-oscillatory circuits are proposed[1][2]. Thesecircuits are investigated in detail and there are many its applications. Most of these circuits include a diode or piecewise linear function circuit as a nonlinear element. A threeNormalized equations are described as follows:
dimensional chaotic self-oscillatory circuit proposed by Shinriki et al[3] includes two diodes as nonlinear elements. Theẋa = αxa − ζf 0 (xa − ya ) + ζf 0 (yb − xa ),chaotic circuit can generate chaos in the case of disconnectẏa = β{−za + ζf 0 (xa − ya ) − ζf 0 (ya − xb )},ing one of diodes. In this study, we propose a system coupled ża = ya ,by disconnected diodes. This system consists of two threedimensional chaotic self-oscillatory circuit. By circuit experẋb = γxb − ζf 0 (xb − yb ) + ζf 0 (ya − xb ),iments and computer calculations, chaotic wandering and itsẏb = δ{−zb + ζf 0 (xb − yb ) − ζf 0 (yb − xa )},
control possibility are confirmed. Chaotic wandering control
can be expected by its application field.
żb = εyb ,where f 0 (X) = (X − 1 + |X − 1|)/2.

2. System Modeli

D4

Ca1Db

D2

D1-g ava1 Ca2 va2 L a(1)ia-g bD3
Cb1

Figure 1: System Model.

vb1 Cb1 vb2 L b-gC1

Dagdv
C2

Liib

Figure 2: chaotic selfoscillatory circuits proposed by Shinriki et al.

Vth1v

Figure 3: diode model.

Figure 1 shows a proposed system. The system consistsof two chaotic self-oscillatory circuits as shown in Fig. 2. In 3. Chaotic Wandering and its Control
this study, the diode Db of the circuit is utilized as a couplingelement. The diode is disconnected as shown in Fig. 2 and it
3.1. Experimental Results
is connected to the node which disconnected from the other
diodes. Therefore, the proposed system consists of two circuit
Some kind of chaotic attractors are observed in the cironly. Each of negative resistors is approximated as a linear cuit experiment. Figures 4, 5 and 6 show the circuit exper-- 503 -iment results. In Figs. 4,four attractors are observed. In thiscase, these results are observed on same parameters. The observed attractor depend on an initial state only. In Figs. 5,two attractors are observed. These results are also observedin same parameters. In another parameters, point symmetricattractors of Figs. 5 are observed. Figures 6 are observed attractors in another parameters. These results are observed indifferent parameters. In Figs. 6 (a), the orbit wanders chaotically among areas which are separated vertical and horizontal
axis. Additionally, we found the phenomena that changing
values of two negative resistors make it possible to control
the chaotic wandering. We investigate this phenomena in thecomputer calculations.
3.2. Computer Calculated Results
Equations (1) are used for the computer calculations. First,
we assume that parameters of two circuits is the same. In
this case, attractors similar into Fig. 4, 5 and 6 are observed.
However, controlled phenomena of chaotic wandering are notobserved. Next, we assume that parameters of two circuits
is not the same. Namely, β and δ are changed. β and δ
are corresponding to values of capacitors. In this case, same
phenomena as experimental results are observed. This resultshows that the asymmetry of two circuit parameters is important. Figures 7, 8 and 9 show the computer calculated results.
These results are corresponding to Fig. 4, 5 and 6, respectively. These results show that the system model is valid. Inorder to investigate the phenomena that changing values oftwo negative resistors make it possible to control the chaotic
wandering, this system model and its parameters are used.
Figures 10, 11 show the relationship among α, γ and theorbit existence rate per each quadrant. Parameters α and γ
are corresponding to values of two negative resistors in thesystem. Initial values are set first quadrant and third quadrant
as follows:
In Fig. 10 : xa1 = xa2 = xb1 = xb2 = 0.10.
In Fig. 11 : xa1 = xa2 = xb1 = xb2 = −0.10.
We also investigated cases that initial values are set secondand fourth quadrants. These results are similar to the case of
first and third quadrant, respectively. Therefor, we compare
Fig. 10 and Fig. 11. For α < 0.40 and γ < 0.40, the orbit
existence rate is 0. This means that the orbit staying in one
quadrant. Namely, the orbit existence rate is 0 or 100[%],accurately.
Now, we pay attention to each area in figures.

• Area α = 0.40 and γ = 0.20-0.40.
The rate of first quadrant is less than third quadrant. Additionally, around these parameter area, the orbit can bealso moved to second and fourth quadrant.
• Area α = 0.10-0.20 and γ = 0.40-0.55.
in the case of Fig. 10, the orbit movement is biased tofirst and second quadrants. in the case of Fig. 11, theorbit movement is biased to third and fourth quadrants.
Additionally, both cases are biased to one of two quadrants by parameters.
• Area α = 0.40-0.55 and γ = 0.40-0.55.
The rates of second and fourth quadrants are 30[%] or
more, the rates of first and third quadrants are 20[%] orless.
• The others.
Some projecting points in first and third quadrants areobserved. As a result of investigating these points, thesepoints shows windows.
As a result, we can say that changing values of two negative
resistors make it possible to control the chaotic wandering insome measure.
4. Conclusions
In this study, we have proposed a system which is consistof two chaotic self-oscillatory circuits coupled by diodes. As
a result of circuit experiments and computer calculations, wecould confirmed that changing values of two negative resistors make it possible to control the chaotic wandering in somemeasure. In order to observe these phenomena, the asymmetry of two circuit parameters is important.
References[1] L. O. Chua, M. Komuro and T. Matsumoto, “The double
scroll family,” IEEE Trans. Circuits Syst., vol. 33, no. 11,
pp. 1073–1118, 1986.
[2] N. Inaba, S. Mori and T. Saito, “Chaotic Phenomena in
3rd Order Autonomous Circuits with a Linear Negative
Resistance and an Ideal Diode II,” Technical Report ofIEICE, Vol. 87, No. 271, pp.9-15, Nov. 1987.

[3] M. Shinriki M.Yamamoto and S. Mori, “Multimode oscillations in a modified van der Pol Oscillator Containing• Area α = 0.20-0.40 and γ = 0.40.
a Positive Nonlinear Conductance,” Proc. IEEE, vol. 69,
The rate of first quadrant is 40[%] or more, the rate ofpp. 394–395, 1981.
third quadrant is 40[%] or less. This result shows thatthe orbit move between first quadrant and third quadrant
and the rate of first quadrant is more than third quadrant.
On the other hands,- 504 -

(a)

(b)
(a)

(c)

(b)

(d)

(c)
(d)
Figure 4: Circuit experimental results 1. Horizontal axis:
va1 [0.5V/div]. Vertical axis: vb1 [0.5V/div]. Ca1 = Figure 7: Computer calculated results 1. Horizontal axis: x ,a98.90[nF], Cb1 = 103.02[nF], Ca2 = 69.77[nF], Cb2 = Vertical axis: x . α = 0.30, β = 1.47, γ = 0.20, δ = 1.30,
b69.68[nF], La = 51.32[mH], Lb = 52.27[mH], ga = ε = 1.00 and ζ = 10.0.
0.5489[mS] and gb = 0.6048[mS].

(a)

(b)

(a)
(b)
Figure 5: Circuit experimental results 2. Horizontal axis:va1 [0.5V/div]. Vertical axis: vb1 [0.5V/div]. Ca1 = Figure 8: Computer calculated results 2. Horizontal axis: xa ,98.90[nF], Cb1 = 103.02[nF], Ca2 = 69.77[nF], Cb2 = Vertical axis: xb . α = 0.20, β = 1.47,γ = 0.40, δ = 1.30,
69.68[nF], La = 51.32[mH], Lb = 52.27[mH], ga = ε = 1.00 and ζ = 10.0.
0.3486[mS] and gb = 0.8070[mS].

(a)(b)(a) α = 0.40, γ = 0.40 (b) α = 0.50, γ = 0.45
Figure 6: Circuit experimental results 3. Horizontal axis:
va1 [0.5V/div]. Vertical axis: vb1 [0.5V/div]. Ca1 =98.90[nF], Cb1 = 103.02[nF], Ca2 = 69.77[nF], Cb2 = Figure 9: Computer calculated results 3. Horizontal axis: xa ,69.68[nF], La = 51.32[mH] and Lb = 52.27[mH]. (a): ga = Vertical axis: xb . β = 1.47, δ = 1.30, ε = 1.00 and ζ =0.7407[mS] and gb = 0.7407[mS]. (b): ga = 0.8953[mS] 10.0.
and gb = 0.8540[mS].

- 505 -(a) fourth quadrant

0.5

50
4030
rate
20
10
040-50
30-40
20-30
10-20
0-10

0.55

0.4

0.3

0.35

0.2

0.54

(c) third quadrant

0.5

0.55

0.4

0.3

0.35

0.2

0.3

alpha

0.45

0.48
gamma 0.42
0.36
0.1

0.5

40-50
30-40
20-30
10-20
0-10

0.55

0.4

0.45

0.3

0.35

0.2

0.25

0.1

0.15

40-50
30-40
20-30
10-20
0-10

alpha

0.25

50
40
30
rate
20
10
0

0.48

0.3

50
40
30
rate
20
100(b) first quadrant

0.54
gamma 0.42
0.36

0.25

0.1

0.3

alpha

0.45

0.48
gamma 0.42
0.36

0.15

0.5

0.54

0.55

0.4

0.3

0.35

0.2

0.25

0.1

0.15

0.3

0.45

0.48
gamma 0.42
0.36

40-50
30-40
20-30
10-20
0-10

0.15

50
40
30
rate
20
1000.54alpha(d) second quadrant

Figure 10: Relationship among two negative resistance values and the orbit existence rate per each quadrant. (initial values:first quadrant). horizontal axis: xa . Vertical axis:xb . β = 1.47, δ = 1.30, ε = 1.00 and ζ = 10.0.

(a) fourth quadrant

0.550
4030
rate
2010040-50
30-40
20-30
10-20
0-10

0.55

0.4

0.3

0.35

0.2

40-50
30-40
20-30
10-20
0-10

alpha

40-50
30-40
20-30
10-20
0-10

0.54

(c) third quadrant

0.5

0.55

0.4

0.3

0.35

0.2

0.3

alpha

0.45

0.48
gamma 0.42
0.36
0.25

50
40
30
rate
20
10
0

0.1

0.5

0.4

0.3

0.35

0.2

0.25

0.1

0.15

0.3

0.45

0.48
gamma 0.42
0.36

50
40
30
rate
20
100(b) first quadrant

0.55

0.54

0.25

0.3

alpha

0.45

0.48
gamma 0.42
0.36
0.1

0.5

0.54

0.55

0.4

0.30.35

0.2

0.25

0.1

0.3

0.15

gamma 0.42
0.36

0.45

0.48

0.15

0.54

40-50
30-40
20-30
10-20
0-10

0.15

50
4030
rate
20100alpha(d) second quadrant

Figure 11: Relationship among two negative resistance values and the orbit existence rate per each quadrant. (initial values:
third quadrant). Horizontal axis: xa . Vertical axis: xb . β = 1.47, δ = 1.30, ε = 1.00 and ζ = 10.0.

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