A DESIGN METHOD OF CHAOTIC CIRCUITSUSING AN OSCILLATOR AND A RESONATORyyz and Akio Ushiday .

Yasuteru Hosokawa , Yoshifumi Nishio

yDept. of E. E. Eng., Tokushima University, Japan.
zDept. of Communication Systems,Swiss Federal Institute of Technology Lausanne, Switzerland

hosokawa@ee.tokushima-u.ac.jp

ABSTRACTIn this study, a simple design method of chaotic circuits is
proposed, which utilizes an autonomous oscillator, a resonator and two diodes. By applying three kinds of oscillators and two types of resonators, six circuit models areobtained. It is con rmed that chaotic phenomena are observed in all of six models by both circuit experiments andcomputer calculations.
1.

For computer calculations, the coupling diodes are modeledwith piecewise linear function as shown in Fig. 4. When we
use the parallel type resonator, the voltage controlled type
diode model is used. While, the current controlled type
diode model is used for the series type resonator. Further,
we assume that the nonlinear resistors and input-output
characteristics of the operational ampli er in all oscillators
are described by linear functions, because we observed attractors only in their linear regions in circuit experiments.

INTRODUCTION

Recently, engineering applications using chaotic circuits attract many researchers' attentions such as chaos communication systems, chaos controls, chaos cryptosystems, chaosnoise generators, and so on. In those systems, individual
characteristics of chaotic circuits usually much in uence theperformance of the systems. Therefore, researchers have toselect a chaotic circuit appropriate to their systems. However, the number of known chaotic circuits is not so large
and it is not easy to design new chaotic circuits except special skillful people [1]-[5]. Hence, they have to abandon
to search the best circuit and compromise with relatively
better circuit. Therefore, simple techniques to design newchaotic circuits are necessary[8][10]. We have already proposed a simple design technique which utilizes two oscillators coupled by diodes and have com rmed we could produce several new chaotic circuits by circuit experiments andcomputrer calculations[6][7][9][11].
In this study, we propose a new design method to produce simpler chaotic circuits. The method utilizes an autonomous oscillator, a resonator and two diodes. By applying three kinds of oscillators and two types of resonators,
six circuit models are obtained. It is con rmed that chaotic
phenomena are observed in all of six models by both circuit
experiments and computer calculations.
2.

Oscillator

Resonator

Figure 1: System model.

Series Type van der Pol Oscillator+-Wien Bridge Oscillator
Parallel Type van der Pol Oscillator

Figure 2: Three kinds of oscillators.

SYSTEM MODELIn our design method, we use a system model shown inFig. 1, which consists of an oscillator, a resonator and twodiodes. In this study, three kinds of second order oscillators: two types of van der Pol oscillators ( parallel typeand series type ) and a Wien bridge oscillator as shown in
Fig. 2 and two types of resonators: parallel and series typeLC resonators as shown in Fig. 3 are applied to the system.

Parallel Type ResonatorIII-3730-7803-6685-9/01/$10.00©2001 IEEESeries Type Resonator

Figure 3: Two types of resonators.

By substituting the variables and parameters,idrrd-V thVthvd

1idvdrd

D1

Voltage Controlled TypeD2Vth(3)(1) is normalized asvd

= x1 0 x2 0 x4 ;= x1 ;= x3 ;= (x1 0 x2 0 xd );

8>
<x_ 1
x_ 2
x_ 3>
:x_ 4rdid-V thFigure 4: Two types of diode models.
Obviously, we can obtain six circuit models using theoscillators in Fig. 2 and the resonators in Fig. 3. Circuit
equations governing these six circuit models are describedby piecewise linear four-dimensional di erential equations.
For two of the six circuit models, we show the circuit equations in the followings.

xd = 1 fjx4 + 1j + j 0 x4 + 1jg:
2.2.

Model 2The Circuit Model 2, which is shown in Fig. 6, consists ofa series type van der Pol oscillator and a parallel type LCresonator.
D1

The Circuit Model 1, which is shown in Fig. 5, consists ofa parallel type van der Pol oscillator and a series type LCresonator.

i 2 L1

D1-R1

i4
i2

D2
C1

C1 dvldt
diL1 2dtdv>
>
C2 3>
>
dt>
>
>>
di4> L:2dt

whereD2C1v1i4
L2C2v3

L2

Figure 6: Circuit Model 2.

-R1C2v3

Figure 5: Circuit Model 1.
Circuit equations of the Circuit Model 1 are described asfollows:
8>
>
>
>>
>>>
>
<(5)

2Model 1v 1 L1(4)whereCurrent Controlled Type

2.1.

r

L1 i2v
L1 i4v; x3 = 3 ; x4 =;x1 = 1 ; x2 =Vth
C1 VthVth
C1 Vthxd = vd ; \ 1 " = d ;  = p 1 t;
Vthdt
L1 C1rr1LCL111= R C ; = C ; = L ;  = rd CL1 ;1
122
1

1= R1 v1 0 i2 0 i4 ;1= v1 ;= i4 ;

C1 dvldt
diL1 2dt
dv>
>
C2 3>
>
dt>
>
>
>
di4>
: L2dt
8>
>
>
>
>
>
>
>
>
<(1)

= 0i2 0 id ;= v1 + R1 i2 ;= 0i4 + id ;(6)

= v3 ;

where= v1 0 v3 0 vd ;novd = 1 rd i4 + Vth + 0i4 + Vth2rdrd

Circuit equations of the Circuit Model 2 are describe as follows:

id =(2)

III-3741 nv 0 v 0 1 (jv 0 v 0 V jrd 1 3 2 1 3 tho0 jv1 0 v3 + Vth j) :(7)

By substituting the variables and parameters,r

3.2.

rv1
L1 i2v
L1 i4; x2 =; x3 = 3 ; x4 =;Vth
C1 VthVth
C1 Vthxd = vd ; \ 1 " = d ;  = p 1 t;
Vthdt
L1 C1rr= R1 CL1 ; = CC1 ; = LL1 ;  = r1 CL1 ;
1221dx1 =(8)(6) is normalized as8>
<x_ 1x_ 2x_ 3>:
x_ 4= x1 0 x2 0 "yd ;= x1 ;= ( x3 0 x4 + "yd );= x3 ;
2

3.
3.1.

Figure 8 shows circuit experimental results (a), computer
calculated results (b) and it's Poincare maps (c) of the Circuit Model 2. We can also observe the Hopf bifurcationfrom periodic orbit (1) to quasi-periodic orbit (2). Further,chaotic attractors are observed after torus-breakdown (3).
(1)

(9)
(2)

where

yd = x1 0 x3 0 1 (jx1 0 x3 0 1j 0 jx1 0 x3 + 1j):

Circuit Model 2(10)ANALYSIS(3)

Circuit Model 1

Figure 7 shows circuit experimental results (a), computer
calculated results (b) and it's Poincare maps (c) of the Circuit Model 1. We can observe the Hopf bifurcation from periodic orbit (1) to quasi-periodic orbit (2). Further, chaotic
attractors are observed after torus-breakdown (3).

(1)

(a)
(b)
(c)
Figure 8: Attractors of Circuit Model 2. (a) circuit experimental results, (b) computer calculated results and (c) it'sPoincare maps.
Circuit experiment: Horizontal axis is v1 [0.5v/div]. Vertical axis is v3 [0.5v/div]. C1 = 33.0 [nF], C2 = 33.0 [nF], L1= 50.0 [mH], L2 = 20.0 [mH], (1) R1 = 0.472 [k ], (2) R1= 0.464 [k ] and (3) R1 = 0.757 [k ].
Computer calculation: = 0.33, = 2.50,  = 30.0, (1)= 0.20, (2) = 0.26 and (3) = 0.50.
3.3.

(2)

Other circuit models

We carried out circuit experiments and computer calculations for the other four circuit models. Figures 9, 10, 11 and12 show the four circuit models (a) and examples of chaotic
attractors obtained from the corresponding circuit models.
We con rmed that all of the circuit generated chaotic attractors for relatively wide range of circuit parameters.

(3)

D1

D2

(a)
(b)
(c)
Figure 7: Attractors of Circuit Model 1. (a) circuit experimental results, (b) computer calculated results and (c) it'sPoincare maps.
Circuit experiment: Horizontal axis is v1 [0.5v/div]. Vertical axis is v3 [0.5v/div]. C1 = 33.0 [nF], C2 = 100.0 [nF],
L1 = 50.0 [mH] and L2 = 20.0 [mH], (1) R1 = 2.43 [k ],(2) R1 = 2.23 [k ] and (3) R1 = 1.80 [k ].
Computer calculation: = 0.33, = 2.50,  = 30.0, (1)= 0.20, (2) = 0.25 and (3) = 0.50.

L1

C1-R1

L2

C2(a)
(b)
Figure 9: (a) Circuit Model 3 and (b) chaotic attractor (circuit experimental result). Horizontal axis is v1 [0.5v/div].
Vertical axis is v3 [0.5v/div]. C1 = 68.0 [nF], C2 = 33.0[nF], L1 = 100.0 [mH], L2 = 20.0 [mH] and R1 = 1.34 [k ].

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5.

D1

L1-R1D2L2C1C2(a)(b)

Figure 10: (a) Circuit Model 4 and (b) chaotic attractor(circuit experimental result). Horizontal axis is v1 [0.5v/div].
Vertical axis is v3 [0.5v/div]. C1 = 33.0 [nF], C2 = 100.0[nF], L1 = 50.0 [mH], L2 = 20.0 [mH] and R1 = 0.68 [k ].

D1

D2CaCaRa2Ra1+-L2Ra3C2Ra4(a)(b)

Figure 11: (a) Circuit Model 5 and (b) chaotic attractor(circuit experimental result). Horizontal axis is v1 [0.5v/div].
Vertical axis is v3 [0.5v/div]. Ra1 = 2.10 [k ], Ra2 = 1.00[k ], Ra3 = 1.00 [k ], Ra4 = 2.00 [k ], Ca = 33.0 [nF], C2= 33.0 [nF] and L2 = 10.0 [mH].

D1D2Ca
Ca

Ra2

Ra1

L2+-C2Ra3

Ra4(a)(b)

Figure 12: (a) Circuit Model 6 and (b) chaotic attractor(circuit experimental result). Horizontal axis is v1 [0.5v/div].
Vertical axis is v3 [0.5v/div]. Ra1 = 0.680 [k ], Ra2 = 1.00[k ], Ra3 = 1.00 [k ], Ra4 = 2.00 [k ], Ca = 33.0 [nF], C2= 68.0 [nF] and L2 = 20.0 [mH].
4.

REFERENCES[1] Makoto Itoh, \Chaotic Attractors in Nonlinear Circuits," Technical Report of IECE, Vol. 85, No. 261,
pp.19-24, Jan. 1986.
[2] Naohiko Inaba, Shinsaku Mori and Toshimichi Saito,\Chaotic Phenomena in 3rd Order Autonomous Circuits with a Linear Negative Resistance and an Ideal
Diode II," Technical Report of IEICE, Vol. 87, No. 271,
pp.9-15, Nov. 1987.
[3] Toshimichi Saito, \An Approach Toward Higher Dimensional Hysteresis Chaos Generators," IEEE Trans.
Circuits Syst., Vol. 37, No. 3, pp.399-409, Mar. 1990.
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[5] Takashi Suzuki and Toshimichi Saito, \On Fundamental Bifurcations from a Hysteresis Hyperchaos Generator," IEEE Trans. Circuits Syst., Vol. 41, No. 12,
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[6] Yasuteru Hosokawa, Yoshifumi Nishio and Akio
Ushida, \RC-Transistor Chaotic Circuit using Phase
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[7] Yasuteru Hosokawa, Yoshifumi Nishio and Akio
Ushida, \Chaotic Circuit Using Two Simple Ring Oscillators Coupled by Diodes," Proc. NOLTA'99, Vol. 1,pp.107-110, Nov. 1999.
[8] A. S. Elwakil and M. P. Kennedy, \Systematic Realization of a Class of Hysteresis Chaotic Oscillators,"Proc. NOLTA'99, Vol. 1, pp.111-114, Nov. 1999.
[9] Yasuteru Hosokawa, Naoto Mori, Yoshifumi Nishio andAkio Ushida, \Analysis of Chaotic Phenomena in Two
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NDES2000, pp. 263-267, May 2000.
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CONCLUSIONSIn this study, we have proposed a new design method ofchaotic circuits utilizing an autonomous oscillator, a resonator and two diodes. We investigated six kinds of circuit
models and con rmed that chaotic phenomena were generated in all of the six circuit models by both circuit experiments and computer calculations. We consider that furtherinvestigation on designing methods of chaotic circuits including this study could contributes for increase of possible
engineering applications of chaotic systems.

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