Chua's circuit, which consists of two capacitors, one inductor, two resistors, one linear voltage-controlled voltage source (VCVS) and voltage-controlled current source (VCCS) type non-linear element has been realized by operational amplifiers. Various dynamical phenomena such as periodic orbit with periods T, 2T, and 4T, single scroll and double scroll similar to the Lorenz attractor have been observed experimentally by changing only the resistance value of the linear resistor R6. The experimental observations have been confirmed by computer simulations and also using harmonic balance analysis

Savacı, Ferit Acar

Yalçın, M. E.

English

DSpace@IZTECH Institutional Repository

ARI (1999) 51 : 169}174 ( Springer-Verlag 1999
ORIGINAL ARTICLE
M. E. Yaly1n ' F. A. Savac1
New realization of Chua’s circuit and verification of chaos
by harmonic balance analysis
Received: 30 September 1997/Accepted: 3 February 1998
Abstract Chua's circuit, which consists of two capacitors,
one inductor, two resistors, one linear voltage-controlled
voltage source (VCVS) and voltage-controlled current
source (VCCS) type non-linear element has been realized
by operational ampli"ers. Various dynamical phenomena
such as periodic orbit with periods „, 2„, and 4„, single
scroll and double scroll similar to the Lorenz attractor
have been observed experimentally by changing only the
resistance value of the linear resistor R
6
. The experimental
observations have been con"rmed by computer simula-
tions and also using harmonic balance analysis.
Key words Choas ) Chua's ) circuit )
Harmonic balance analysis
1 Introduction
With the new realization of the individual Chua circuit
presented in this paper, where the inductor in the Chua
circuit of Matsumoto (1985) is also implemented by op-
apms as in MorguK l (1995), the VLSI implementation of the
chaotic cellular neural networks of Chua's circuit intro-
duced in Kavaslar and GuK zelis, (1995) would be possible.
The proposed circuit model shown in Fig. 1, in which
M. E. Yaliin ( )
Electrical-Electronics Engineering Faculty,
I0 stanbul Technical University,
Maslak TR-80626, Turkey
Tel: #90 212 285 3618;
fax: #90 212 285 3679; e-mail: mey@ehb.itu.edu.tr
F. A. Savacm
I0 zmir Institute of Technology,
Electrical-Electronics Engineering Dept, hankaya,
I0 zmir, Turkey
Tel: #90 232 445 1724;
fax: #90 232 483 9598; e-mail: savaci@likya.iyte.edu.tr
Fig. 1 a VCVS-type realization of Chua's circuit. b The characteristic of
piecewise linear VCVS
parallel combination of the linear resistor R
6
and the
voltage-controlled current source (VCCS) is equivalent to
Chua's diode (Kennedy 1993), with its non-linear voltage-
controlled voltage source (VCVS) characteristics as in
Eq. 3 and with its CNN like topology also allows inhibit-
ory connections as well as excitatory connections. Since
the chaotic CNN o!ers rich dynamical behaviour, such as
single-scroll and double-scroll regimes, some pattern rec-
ognition applications would be possible, as in Kavaslar
and GuK zelis, (1995).
In the next sections we give a new realization of Chua's
circuit with our experimental results. These results and the
simulations have been con"rmed by the spectral tech-
niques given in Genesio et al. (1993). Our observation of
Fig. 2 Implemented circuit (C
1
"100 nF, C
2
"1 nF, ‚"68 mH, R"10 k), mN
0
"15
7
, mN
1
"12
7
, E
1
"1 <)
Fig. 3 a Periodic orbit with period „ observed with the circuit of Fig. 2 on the <
C1
!<
C2
plane. Horizontal scale 0.1 V/div, vertical scale 0.1 V/div.
b Power spectra of the capacitor <
C1
on P
VC1
( f )!f for period „
the double scroll also supports the conjecture made in
Genesio and Tesi (1992) and Genesio et al. (1993) to
predict chaos.
2 Realization and some experimental observations
The proposed circuit shown in Fig. 1 has the following
corresponding state equations
dX
dq
"A )X#b ) f (y(q)) (1)
y"C )X (2)
f (x)"m
1
x#1
2
) (m
0
!m
1
) ) ( Dx#E
1
D!Dx!E
1
D) (3)
where
A"
!a(1#d) a 0
1 !1 1
0 !b 0
,
b"
a
0
0
, C"[1 0 0], q" t
C
2
R
6
,
X"[<
C1
<
C2
i
L
]T, a"C2
C
1
, b"C2R26
C
1
, d"R
R
6
,
m
0
"mN 0
R
6
, m
1
"mN 1
R
6
.
170
Fig. 4a,b As for Fig. 3 but with period 2„
Fig. 5a,b As for Fig. 3 but with period 4„
The implemented circuit is shown in Fig. 2, where the
bifurcation parameter is chosen to be R
6
as given in
Yalimn et al. (1996). With this implementation the slopes
and break points of the characteristic can be independent-
ly adjusted, as in Kennedy (1993), the slopes and the
breakpoints are mN
0
"R21
R11
R4
R4`R3
#R22
R12
, mN
1
"R22
R12
and
E
1
"R11
R21
E
sat
. By adjusting the value of R
6
in the circuit of
Fig. 2, we have observed periodic orbit with period „ on
<
C1
!<
C2
plane for the resistor R
6
in the range of 7870)
to 8217), periodic orbit with period 2„ for the range of
8218) to 8277 ), periodic orbit with period 4„ for the
range of 8278 ) to 8291), double-scroll orbit for the
range of 8400) to 10400). In Figs 3}6, periodic orbit
with periods „, 2„ and 4„, double-scroll orbit and their
spectra have been given. The observed power spectra
con"rm a period-doubling route to chaos "rst discovered
by Feigenbaum in the parameter range 0.806(d(0.9,
and the "rst period doubling is a leading indicator for the
existence and the nearness of chaotic regions in the para-
meter space.
3. Frequency domain analysis
From the state equations in given by Eqs. 1 and 2 and
assuming the system output of the form
y(t)"A#B sin(wt) (4)
and the corresponding non-linearity output
f (y(t))"N
0
(A, B) )A#N
1
(A, B) )B ) sin(wt) (5)
and then using the well-known harmonic balance method,
the frequency domain equations have been obtained as
171
Fig. 6 a Double scroll observed with the circuit of Fig. 2 on the <
C1
!<
C2
plane. Horizontal scale 2 </div, vertical scale 5 V/div. b Power spectra of
the capacitor <
C1
) on P
VC1
( f )!f for double scroll
Fig. 7a+c For period „: a the phase portrait (<
C1
!<
C2
!i
L
plane); b <
C1
(t) versus t; c Nyquist plots of ‚( jw) and 1
N (A (B),B
follows
A(1!N
0
(A, B)‚ (0))"0 (6)
1!N
1
(A, B)‚ ( jw)"0 (7)
where the frequency response of the linear part of the
circuit
‚( jw)"C( jwI!A)~1B
" a ((b!w2)#jw)
[ab(1#d)!(1!a(1#d))w2]#[(b#ad)w!w3] j
(8)
and the dual-input describing functions
N
0
(A, B)"(m0!m1) (1!A)sin~1(
1~A
B
)
nA
#(m0!m1)BJ1!(1~AB )2#n2(m0!m1#A(m0#m1))
nA
(9)
and
N
1
(A, B)"(m0!m1)(1!A)J1!(1~AB )2
nB
#0.5Bn(m0#m1)#B (m1!m0)sin~1(1~AB )
nB
(10)
With the foregoing equations and using plots of ‚ ( jw) and
1
N1(A(B),B)
, experimental observations and simulation re-
sults have been con"rmed for the following cases in which
the stability of the limit cycles has been decided according
to the Loeb criterion in Atherton (1975). This is a well-
known method which considers Eq. 7 where N
1
(A, B) is
expressed as function of B from Eq. 6. Then the stability of
the limit cycle depends on the features of the point where
the loci of ‚ ( jw) and of 1/N (A(B), B) intersect in the
Nyquist plane and on the way in which this occurs.
Figures 7}9 have been obtained by simulations and
172
Fig. 8a+c As Fig. 7 for period 2„
Fig. 9a+c As Fig. 7 for double scroll
indicate stable limit cycles. (The motion of the loci of
‚( jw) with increasing frequency and the motion of the
1/N(A (B), B) with increasing the amplitude B is shown by
arrows).
Periodic orbit with period „ for a"9, b"14.28,
d"0.801. From Fig. 7c, the intersection point corres-
ponds to A"4.8, B"!4.1 and w"2.45rad/sn. Since for
these values the stable limit cycle y(t) is not interacting
with the unstable equilibrium point &the origin', chaotic
behaviour is not expected, as the conjecture of Genesio
and Tesi (1992) and Genesio et al. (1993) suggests.
Periodic orbit with period 2„ for a"9, b"14.28,
d"0.806. In Fig. 8c the intersection point corresponds to
A"4.8, B"!4.0 and w"2.45rad/sn. Once again, for
these values the stable limit cycle y(t) is not interacting
with the unstable equilibrium point, &the origin', and so
chaotic behaviour is not expected.
Double-scroll behaviour for a"9, b"14.28, d"0.900:
In Fig. 9c the intersection point corresponds to A"2.1,
B"!2.7 and w"2.5rad/sn. For these values, the stable
limit cycle y(t) and the other symmetrically located stable
limit cycle are interacting with the unstable equilibrium
point, &the origin', and since ‚ ( jw) satis"es the medium
"ltering e!ect, chaotic behaviour has been observed and
hence the conjecture given in Genesio et al. (1993) is
supported. For this case we have also computed the lar-
gest Liapunov exponent as 0.2428 using the standard
QR-algorithm (Eckmann et al. 1986) which con"rms the
chaotic trajectories in Figs 6a and 9a.
4 Conclusion
In a similar way to the realizations in Kennedy (1993)
using synthesis techniques, we have given op-amp imple-
mentations for the piecewise-linear voltage-controlled
voltage source in the chaotic CNN. Various dynamical
phenomena as shown in Figs 3a}6a and a chaotic spec-
trum which has a continuous, broad-band nature (Fig. 6b)
have been observed. The experimental observations and
173
theoretical results based on the harmonic balance analysis
have justi"ed the simple model for predicting the existence
of chaotic motion given in Genesio and Tesi (1992) and
Genesio et al. (1993).
References
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174


New realization of Chua's circuit and verification of chaos by harmonic balance analysis

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New realization of Chua's circuit and verification of chaos by harmonic balance analysis

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