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

Coexistence Phenomena of Chaotic and Non-Chaotic Oscillations
in a Three-Dimensional Chaotic Circuit

Masahiro WADA and Yoshifumi NISHIO
yzy Konan Universityz Tokushima University

8{9{1 Okamoto, Kobe, Japan

2{1 Minami-Josanjima, Tokushima, Japan

Phone/Fax: +81{78{435{2531

Email: nishio@ee.tokushima{u.ac.jp

Email: wada{m@konan{u.ac.jp

Abstract| In this study, chaotic behavior on a sim- change a characteristic of the negative resistor to a con-ple chaotic circuit with multi-driving points is investi-

tinuousgated. Modi ed chaotic circuit which have diodes with

posed system is symmetric with respect to the origin. Itve-segment piecewise-linear resistor. Our pro-both polarities and a piecewise-linear resistor with sev-

is similar to a three-dimensional chaotic system[5] at aeral segments are shown. A design scheme of a piecewise-

glance, however it is di erent from stability at the ori-linear element constructed by using electrical elements

gin, i.e., it is oscillatory at the origin because of negativeis proposed and shown.

resistance.

Both chaotic and non-chaotic

Generally speaking, it can be constructedoscillations in the same parameters can be con rmed si-a concept for complicated structure of chaotic attrac-

multaneously in the proposed chaotic circuit.

tors to make many equilibrium points.

Further

Piecewise seg-some experimental results and piecewise-linear charac-

ment design scheme constructed by electrical elements

teristic of the nonlinear element by PSPICE simulation

is proposed. Then, several computer simulation results

are also shown.

in the proposed circuit are shown. We can con rm si-1. Introduction

multaneously both chaotic and non-chaotic oscillationsin the same parameters. Further some experimental results and piecewise-linear characteristic of the nonlin-

Many types of chaotic systems and circuits have already been proposed and investigated.

They have in-teresting features, if the negative resistor as an active
element are assumed to be constructed in parallel orseries connection with nonlinear functions.

Complexand strange nonlinear structures yield a wide varietyof chaotic phenomena.

It is an important idea thathow does nonlinearity lead to various kinds of behaviors. There are famous chaotic attractors such as doublescroll[1],n{double scroll and scroll grid attractors[2].

They are also constructed by making several equilibrium
points.

A mechanism and search of chaotic regions inpiecewise-linear systems were investigated[3]. The purpose of our study is to clarify coexistent both phenomenaof chaotic and non-chaotic in chaotic systems with sev-ear element by PSPICE simulation are also shown. For
the future research, we make a proposal idea to realize
chaotic complex motions in coupled systems and to apply several chaotic applications.

2. Model Description
There are several chaotic circuits constructed by threedimensional autonomous systems.

One of the well-known chaotic circuit proposed by Inaba and Saito[4] is shown in Fig. 1. It is also well known
that it can be behave chaotic, and it shapes R
ossler typechaotic attractors. The circuit equations are describedby

eral equilibrium points.
In this study,

In this study weadopted a modi ed three-dimensional chaotic system.

we adopt a three-dimensional au-

tonomous chaotic circuit, and chaotic behavior on a simple chaotic circuit with multi-driving points is investigated. This circuit consists of three memory elements,

8 diL1>> L1>< didt = v
L2>> L2 dt = v 0 vd (iL2)>: C dv = 0(iL1 + iL2) 0 gv
dt

some diodes and designed negative resistors. It is wellknown that the chaotic circuit proposed by Inaba et al.[4]g is a linear negative conductance value of NR

is asymmetric with respect to the origin and it behaves

wherechaotic. In general, a linear region of the negative resis-

if we consider the negative resistor as an ideal.

tor realized by an OP amp on a circuit experiment is only

variableused. We modi ed this chaotic circuit to include diodes
with both polarities and with some minor changes. We(1)

Thevd (iL2) is a function depending on the current
through their diodes D in Fig. 1(a), which determines
their chaotic dynamics. The i 0 v characteristic of one-869-

diode is approximated by two-segment piecewise-linear
functions as shown in Fig. 2(a). The part of diodes withi L1

both polarities is constructed by some diodes, and their

Vd and 0Vd respec-threshold voltages can be set as +i L2L2+tively. Its characteristic is shown in Fig. 2(b) which isNRv

described as three-segment piecewise-linear functions byCL1-vd (iL2 ) =1fjrs iL2 + Vd j 0 jrs iL2 0 Vd jg

2D(2)Furthermore the circuit equation (1) can be rewritten
by changing the following variables and parameters asfollows:V x; i =V y;
L1 d p L2L1 dv = Vd z ; t = L1 C d ; \ 1 " = d ;rr CdLL1=; = g 1 ;  = rsL2CL1i L1 =gure of a chaotic circuit. A basicmodel was proposed by Inaba et al.

rCrCFigure 1: Schematicvv

VdVd(3)r d /2 = r srdi

The circuit equation can be normalized if we chosei

Vd for-Vda threshold voltage value of the diodes. Then the circuit
equations can be rewritten by(a)8>< x_ = z>: zy__ == 0((xz 0+ fy)(y0)) z(b)Figure 2: Examples of characteristic of a typical diodein (a) and a diode model with both polarities in (b).
(4)

whereNR2rNR

f (y) = 1 fjy + 1j 0 jy 0 1jg
2

In this study, we consider the negative resistor

D1

NR in-

cluded in the circuit as a piecewise-linear characteristic
shown in Fig.

3.

NR1

(5)

D2

Figure 3: Nonlinear resistor model

The piecewise-linear resistor can be

NR .

easily constructed by a connection of some components
in parallel as shown in Fig. 4(a){(c), and its compoundcharacteristic ofNR can be illustrated as shown in (d).

Let us consider that the part of negative resistance inEq.(4) replaces toh(z ), then it was rewritten by

8>< x_ = z>: yz__ == 0((zx0+fy()y0)) h(z)where(6)ear components, such as diodes with switching role andpiecewise-linear negative resistors with active elements

3. Simulation Results
Some simulation results are shown for the parameters:
 = 100, and construction of thepiecewise-linear characteristics form are realized by Bp1= 0.1, Bp2 = 0.3, m0 = 01:0, m1 = 1:0 and m2 = 01:0.
3 = 0.20 and

3z 2
f(m0 0 m1 )(jz 0 Bp2 j 0 jz + Bp2 j)+(m1 0 m2 )(jz 0 Bp1 j 0 jz + Bp1 j)g

ve equi-for their oscillations.

acteristic is described with a canonical form as follows:msymmetric with respect to the origin and it has

stable. A feature of this model is to possess two nonlin-h(z ) is a function of z, which is designed by ve-h(z )m k

3 is a basic parameter, hence valuesk( =3 . This circuit model is0,1,2) are the ratio to the value
librium points which three are unstable and others aresegment piecewise-linear as shown in Fig. 5. This char-3 +0

whereeters are(7)3 = 0 50 respectively. Other param-= 10.0,

It is noticed that threshold voltage of diodes with both

-870-

h(z)

ii

Bp3
Esat

Bp1v

-Bp1

-Esat

Bp2
-Bp3

m0

Esat

m1m2v

-Bp2

-Bp 2

-Esat

Bp1z

Bp2

-Bp 1

m0(a)

(b)
ii
Esat

Figure 5:
-Bp2

-Bp1

-Bp3

Bp1

Bp3

Bp1

-Bp1

Bp2v

Piecewise-linear characteristics of the non-linear resistor.

v
couple these systems as like as a network, a ring and so-Esaton. It is expected to be utilized for several applications
by these coupled systems.
(c)Figure 4:

The circuit experiment results by using PSPICE sim-(d)Design method forv {i characteristic of apiecewise-linear resistor in the chaotic circuit. (a) NR1 ,(b) NR2 and ideal diodes with both polarities, (c) ideal

ulator are also shown. In order to realize the non-linear
characteristic ofNR , we designed a piecewise-linear re-sistor constructed by using some OP amps(TL082CP)
and resistors. The details technique is explained in [6].

NR . The nonlinear

diodes and one resistor, (d) piecewise-linear character-Figure 7 shows a schematic design of

istic of NR: compound characteristic of (a) to (c) con-resistor is realized by two OP amps, some diodes andnected in parallel.

DC voltage which is used for setting threshold voltage
strictly, and resistors.

Figure 8 shows piecewise-linear

characteristic realized on the PSPICE simulation. Norpolarities including in the chaotic circuit is normalized

Vd = 1.0. Hence we chose Vd between breakpoints Bp2
Bp3 . Because the value of breakpoint Bp3 is larger
than enough Vd , it can be ignored in computer calculaas

andtion. Therefore

ve-segment region is only drawn in Fig.

5 and this canonical form ofve-segment piecewise-linear

function was also only described in (7). We can change
values3 and

mk as control parameters. In order to beaccuracy calculation, the boundary of switching of thediodes is calculated by using bisection method. Figure
6 shows some attractors when the initial conditions arechanged. Bothgures (a-1) and (a-2) does not behave:mally, the voltage of capacitor

C oscillates in the area ofVd (Vd = 5[V]).

the threshold voltage between around 6

A schematic design of a whole circuit on the SPICE simulation is shown in Fig. 9. Circuit parameters are all illustrated on a schematicgure. Chaotic and non-chaotic

attractors corresponding to theFig.

10.

gure 6 are shown inWe can also con rm that both chaotic andnon-chaotic attractors are simultaneously observed in
the same circuit parameters.

4. Conclusions

3 = 0 20. We can only ob-

In this study, we have investigated an chaotic behav-

serve one periodic oscillations in this parameter. On theior and coexisting phenomena of both chaotic and non-other hand, we can con rm that chaotic and non-chaotic

chaotic oscillations in the same parameters on a chaotic

oscillations are coexisting for the parametercircuit whose resistor has a piecewise linear characteris-chaotic for the parameter:

3 = 0 50.

Chaotic attractors were con rmed on such parameters.

tic. Namely, we could observe multi-driving of chaotic

Simultaneously we can also observe the non-chaotic os-

and non-chaotic oscillations.

cillation in the same parameters.

system has been proposed and some simulation results

This reason is that

Design of a new chaotic

solution of oscillation behaves just like a van der Pol os-

have been shown.

cillator because oscillation acts on a region around thethis study are expected to yield new chaotic phenomena

area jin several coupled chaotic systems. We have proposed

proposed circuit, i.e., a chaotic oscillation and a non-a chaotic system with nonlinear resistors connected inz j  0:2 Two oscillatory phenomena coexist in thisThese chaotic behavior observed inchaotic oscillation. This feature is very interesting be-parallel, however we expect to con rm very strange andcause variety of chaotic phenomena was emerged if wecomplex behavior on coupled such nonlinear elements in-871-

200

33k

4.7k

TL082CP

TL082CP

12k

330

1.23k

1.47k

3V

(a{1)

3V

(a{2)
Figure 7: Schematic design for non-linear resistor
the proposed chaotic circuit.

(b{1)
Figure 6: TheNR in(b{2)

z 0 x plane projection of attractors on thechaotic circuit for the parameter3 is 0.20 (a{1, a{2)and 0.50 (b{1, b{2) respectively. The other parameters

(a)

(b)

:  = 100, Bp1 = 0.1, Bp2 = 0.3, m0 = 01:0,m1 = 1:0 and m2 = 01:0. Both chaotic and non-chaotic

Figure 8:

initial conditions are just only di erent.

used in simulations is TL082CP. (a) the whole charac-are= 10 0,attractors have all the same parameters, however thePiecewise-linear characteristic of the non-linear resistor

NR realized by several circuit elements onthe PSPICE simulator. The model type of an OP amp
teristic, (b) enlargement area between 05 and +5[V].

series because it has been considered that coupled con200

33k

tinuous nonlinear elements yield a variety of complex
characteristics.
VReferences

C68.3nF[1] L.O.Chua, et al., \The Double Scroll Family," IEEEL210mH

I
L1100mH

5V

TL082CP

4.7k

TL082CP

12k

330

1.23k

1.47k

5V

Trans. Circuits Syst. I, vol. 33, no. 11, pp. 1072{

3V1118, 1986.

3V[2] M.E.Yalcin, J.A.Suykens, et al., \Families of Scroll
Grid Attractors," Int. J. of Bif. and Chaos, vol. 12,no. 1, pp. 23{41, 2002.
[3] T.Yang, L.O.Chua, \Piecewise{Linear Chaotic Systems with a Single Equilibrium Point," Int. J. ofFigure 9: Schematic design of a whole chaotic circuit on
the PSPICE simulation.

Bif. and Chaos, vol. 10, no. 9, pp. 2015{2060, 2000.
[4] N.Inaba, T.Saito,et al., \Chaotic Phenomena ina Circuit with a Linear Negative Resistance andan Ideal Switch of Diode," IEICE Trans. Circ. andSyst., vol. E70, no. 8, pp. 744{754, 1987.
[5] M.Kojima, Y.Nishio,et al., \Simultaneous Asyn-

chronous Oscillation of Chaos and Torus in Coupled
Chaotic Circuits," Proc. of NOLTA2000, vol. 2, pp.
567{570, 2000.
[6] M.P.Kennedy, \Robust OP Amp Realization ofChua's Circuit," Frequenz, vol. 46, pp. 66{80, 1992.

-872-(a)(b)

Figure 10: PSPICE simulation results.