/ANALYSIS

OF CHUA’SCIRCUIT

Junji KA WA TA1
lDept.

Yoshifumi

ABSTRACT
We have analyzed Chua’s circuit with a transmission
line,
where the parallel LC resonatorin original
Chua’s circuit is replaced
by a lossless or lossy transmission
line.
Some papers about Chua’s circuit with lossless transmission line (time-delayed
Chua’s circuit) have been reported,
but whose analysis is rather simple because the signal on
it is not affected an attenuation
due to the line loss. On
the other hand, the method of characteristics
is developed
for the simulationof high-speed
VLSI circuits containing
lossy transmission
lines. In this paper, we will apply thismethod to analyze Chua’s circuit with transmission
line.
The linear interpolationand the variable stepsize integration techniques are introduced
to get the accurate solution
around the breakpointof piecewise linear function.
Consequently we found from the numerical experiments
that
Chua’s circuit with lossy transmission
line has the complicate and interesting chaotic attractors compared to Chua’s
circuit with lossless line.

INTRODUCTION

Chua’s circuit [1] shown in Figure 1(a) is a simple oscillator
circuit which exhibits a variety of bifurcation
phenomena
and attractors.
The nonlinear resistor is a piecewise

(b)

(a)

Figure 1. (a) Chua’sof nonlinear
resistor.

circuit.

linear functionin Figure+~(ml‘m2)[lVR

+

Akio

LINE

USHIDA3and ElectronicEng., Tokushima
Bunri University
kawata@me.bunri-u.ac.jpof Electrical and ElectronicEng., Tokushima
University
nishioQee.tokushima-u.
ac.jp
of Electrical and ElectronicEng., Tokushima
University
ushida@ee. tokushima-u.ac.jp

3Dept.

as shown

NISH102

TRANSMISSION

of Mechanical

2Dept.

1.

WITH(b) v – i characteristic

%21

l(b)

–

and described

IVR

–

&21]

where m., m ~, mz are the slope in the each segments

by:

(1)
of thispiecewise linear function,
and Bpl, BPZ denote the breakpoints.
Many theoreticaland experimental
studies about
this circuit have been reported so far, some studies to utilize this circuit for novel applications
in engineering,e.g.
secure communicant ion, are also published recent] y.

Copyright (C) 1996 IEEE.

By replacing
parallel LC resonatorin original Chua’s
circuit by a short-circuitedtransmissionline, a circuit asshown in Figure 2 is obtained.
We will call the result-

Figure2. Chua’s

circuitwithtransmissionline.

ing circuit “Chua’s circuit with transmissionline”.
If thetransmission
line is Iossless, this circuit is also referred toas “time-delayed
Chua’s circuit”.
Recently,some papers
about the time-delayed
Chua’s circuit have been reported[5]- [10]. But the time-delayed
Chua’s circuit is analyzed
only for the special case of C = O in most of these literature[5]-[9]. Because the time-delayed
Chua’s circuit with C’ = O
can be governed by one-dimensional
difference equation, its
chaotic behavior can be analyzed by investigating the corresponding one-dimensional
map. On the other hand, Hosnyet aJ. [10] have presented an analytical method for nonlinear
lumped circuit with lossless transmission
line. As an example, the chaotic behavior of time-delayed
Chua’s circuit haveanalyzed for the case of c # O. The chaotic attractor andthe bifurcation
diagram have been also shown by computer
simulation.
Unfortunately,
this method can not be applied
to the analysis of lossy transmission
line. Our method canbe applied to any kind of lumped circuits with transmission lines. Note that for a lossy transmissionline, since thepropagation
signal is affected the attenuation,
time delay,
and reflect ions at both ends, the phenomena of it are much
more complicatecompared to lossless one.
It is the purpose of this paper to analyze Chua’s circuit
wit h lossless or lossy transmission
line. In the last decade,many approaches have been proposed for the simulation ofhigh-speed VLSI circuits, and we believe that the method ofcharacteristics[11], [13] are most suit able to analyze t ransmission lines. We will first apply the classical method ofcharacteristics[11] to Chua’s circuit with lossless transmission line, which can get the accurate solutions.
Next, thegeneralized
method of characteristics[12] is used to analyze Chua’s circuit with 10SSY transmission
line. To get theaccurate solution around the breakpointof piecewise linear
function, we introduce the linear interpolationand variable
stepsize integration techniques.
Numerical experiments
are given in section 3 for various
parameters.
It seems that the attractors become more complicate compared to the lossless transmissionline.

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2.

NUMERICALANALYSIS

The method of characteristics
has been proposed by Branin[11] for the transient analysis of a single lossless transmission line. The lossless transmission
line is modeled as disconnected two-port network (henceforthcalled the characteristic model), shown in Figure 3, consisted of the characteristic impedances
and the waveform generators.

two equations of (3) must be rewritten by the corresponding
state equations.
The transient responses can be calculatedby the numerical integration method such as Runge-Kutta
met hod. Observe that, since (3) cent ain the time-delayed
waveform generators el (t— T) and ez (t— T),their values in
the period -r need to be stored. To get the accurate solution,
we need to estimate the exact state-variables
and time passing through a breakpoint of piecewise linear resistor. Therefore, we have applied an interpolationand variable stepsize
integration techniques to get their values around the points.
In our examples,
we used 4th order Runge-Kutta
methodas the numerical integration method.
3.

(a)

(b)

Figure 3. (a) The transmission
acteristic
model.
Where el, ez are the waveformreflection and defined by:
cl(t)

=

ez(t)

=

line and (b) its char-

generators

to simulate

2vz(t) – ez(t – -r)
27J~(t) – el(t – ~).

Then Chang [12] has extended this method for a 10SSY
transmission
line. The 10SSY transmission
line is also modeled by characteristic
model shown in Figure 3.
Thecharacteristic
impedance
function 20 and the exponential
propagation
function e–o are synthesized as the equivalent
lumped circuits by Pad6 approximation(see [13] for detail).
The equivalent circuit for the exponential propagation
function consists of ideal delay line to simulate the time delay
and lumped-parameter
networks to simulate the attenuation loss.
For now on, we apply these methods to Chua’s circuitwith transmission
line.
The equivalent circuit of Chua’s
circuit with transmission
line is shown in Figure 4, where
the transmission
line is replaced by the characteristic
model.

I

I

Figure 4. The equivalent
with transmission
line.
Thus, for a lossless
circuit equations:
Vi(t)

1

I

circuittransmission=‘U2(t) =1

of Chua’soR ez(t–T) + Zo ‘vR(t)
R+ZO
~R(t)

V2 (t) –R— ~R(~R)cl(t)=2vz(t)– e2(t – 7)

e2(t)

=

–el(t

– 7).

20 = 424[Q], ~ = 0.05[msec],
C = 10[nF],= –0.75[mS],
ml = –0.41[mS],mz = 10[mS],
E&, = l[V], B,z = 8[V]where we have determined(3)

Note that for a 10SSYtransmissionline, since Zo and e–e aremodeled by the lumped circuit, the second equation and last

Copyright (C) 1996 IEEE.

ZO = m,

r = ~

1,1 =

I [m] by using the values L = 18[mH], Ch = 100[nF] used in[4]. With varying the value of linear resistor R, the computer simulation
is carried out. The period-doubling
phenomenon
and Double Scroll attractors
similar to original
Chua’s circuit, as shown in Figure 5, are observed by thetransient analysis.
Next, we continue the simulationby changing the parameters. The parameters as listed below are used because
the characteristic
impedance of a coaxial cable is normally

50[Q].
ZO = 50[f)], T = 14.25 [nsec], C=

m. = –0.8[mS],
ml = –0.4[mS],
BP, = l[V], BP2 = 8[V]l[PF],
mz = 10[mS],

The trajectories
in phase plane are shown in Figure 6. We
have observed more complex bifurcation
phenomenon
andDouble Scroll attractors
compared to the above examples.
Finally, we carry out the simulation
for Chua’s circuit
wit h lossy transmission
line. We used the following parameters:
Rd = 0.55[Q/m],
Gd = O.l[mS/m],& = 0.2375 [pH/m],
1 = 3[m], C = l[PF],

mo

ml

=

–0.9[ms],

circuitline, we have the following

RESULTS

mO

the

(2)

SIMULATION

We first simulate Chua’s circuit with lossless transmission
line by using the classical method of characteristics[11] anduse the following parameters:=–0.5[4,cd=

95[pF/m],~pl = l[VIwhere Rd, Ld, cd, Gd denote the per-unit-lengthline parameters and the outer segments with negative slope m ~ of non= *CO).
linear resistor are extended to infinity (i.e., +BP2
Note that this transmission
line is relatively close to the distortiordessline satisfying the conditionof (R&’d/LdGd=
22
In this case, 4th order Pad6 approximationof Z. ande:)
has good agreement[14].
The trajectories
in phaseplane are shown in Figure 7.
Chua’s circuit with 10SSY
transmission
line has more complex
Double Scroll attractors compared to lossless one.
4.

CONCLUSION

we have presented
an efficient
analysis
In this paper,
method for a chaotic circuit with both transmission
line andpiecewise linear element.
Chua’s circuit with transmission
line was efficiently analyzed by the proposed
method.
We

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I
0.6
1.0,

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y

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continued.

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(d)

-0.8
-1.0 1Figure 5. Phaseless transmission
R=

1760[Q],planes
line

(c)

R =

of Chua’s circuit withfor (a) R =1800[0],1745[0],(d)

R =loss(b)1720[Q],(e) R = 1690[fl].

-4.0

-2.0

0.0

2.0
VR[v]

4.0

..

b.u

(c)

Figure 6. Phase planes of Chua’s circuit withless transmission
line for (a) R =2000[Q],R = 1800[0],(c) R = 1300[fl].

Copyright (C) 1996 IEEE.

..

8.U

All Rights Reserved.

loss(b)have confirmed the generation of various chaotic phenomena
including Double Scroll attractor by computer simulation.
In future, we intend to confirm these results experimentally. Further, we are going to simulate this circuit in detailby varying the circuit parameters
and try the qualitative
analysis of the bifurcationroute to chaos and stability of
attractors.
Chua’s circuit with transmission
line may also
has the different types of chaotic phenomena depending on
the line length. Also, we intend to confirm the generation of
period-adding
phenomenon
in time-delayed
Chua’s circuit
without C.

2.0

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1.0
0.5
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g

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.05

REFERENCES

-1,0
-1.5~
.20
-10.0

[1]L. O. Chua,

o

I
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0 (1

5.0

1(1.0

15.0

VR[V]

1.5

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from Chua’s cirVO1.31, pp.1055-[3] T. Matsumoto,

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L. O. Chua, and M. Komuro, “The double scroll,” IEEE Trans. Circuits Syst., VO1.32, pp.797818, Aug. 1985.

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of Chua’s[5] A. N. Sharkovsky

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[2] T. Matsumoto,

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g

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0.0

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I
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100

1S.o

[10] E. A. Hosny and M. I. Sobhy, “Analysis

VR[V]

(c)
2.0

I

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mission
1967.

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-05
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:~
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[11] F. H. Branin, Jr., “Transient

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g

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-5.0

0.0

S.o

10.0

150

VR[VJ

(d)
Phase
planesof Chua’s
circuitwith
Figure7.
lossy transmission
line for (a) R = 1830[0],(b)R = 1820[Q],(c) R = 1810[0],(d) R = 1700[fl],(e) R = 1600[Q].

Copyright (C) 1996 IEEE.

line,”

Proc.

IEEE,

of chaotic beapplied to theTrans. Circuits

analysis of lossless transVO1.55, pp. 2012-2013, Nov.

[12] F. Y. Chang, “Waveform
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