Journal of Robotics, Networking and Artificial Life, Vol. 3, No. 2 (September 2016), 84-87

Analog Circuit Implementation and Full State Observation of Chua’s Circuit
Hong Niu, Dongcheng Tan, Yongjun Wu

College of Electronic Information and Automation, Tianjin University of Science & Technology,
80 Mailbox, Tianjin University of Science & Technology, No. 1038 Dagu Nanlu, Hexi District
Tianjin, 300222, China
E-mail: spots@163.com

Abstract
The inductance simulator and the nonlinear resistor, which are constructed by operational amplifiers and resistors,
are applied to complete the analog circuit implementation of Chua’s circuit as well as improve the accuracy ofcircuit parameters. The state variable z, which represents the product of the linear resistance and the current of theinductor, should be observed even if it is not an actual measurable physical quantity. It is found that z can beobtained via scaling of the voltage of the resistor in the inductance simulator. The real chaotic curves generated
from the analog Chua’s circuit are displayed on the oscilloscope clearly. It demonstrates that the realization ofChua’s circuit is correct.
Keywords: Chua’s Circuit; Inductance Simulator; Analog Circuit Implementation; Scaling; Full State Observation

1. Introduction

2. Chua’s Circuit

Chua's circuit was introduced in 1983 by Leon Ong
Chua.1 It is a three-dimensional autonomous electronic
circuit that exhibits classic chaos behavior. Because
Chua’s circuit has rich chaotic dynamics and a simple
circuit structure, it has been one of the best chaotic
circuits for experimental observation and study.2-6
In Refs. 7-8, hybrid realizations of Chua’s circuit, as
well as the circuit topologies proposed for the nonlinear
resistor and the inductor element, are presented.
However, there is no study on how to observe the statevariable z which represents the product of the linear
resistance and the current of the inductor. In this paper,
Chua’s circuit will be implemented by the inductancesimulator and the piecewise linear resistor consisting ofsix resistors and two operational amplifiers, as shown in
Ref. 8. Moreover, the state variable z will be obtained
via scaling of the voltage of the resistor in theinductance simulator to show the chaotic attractors ofChua’s circuit completely.

Fig. 1. Chua’s circuit

Chua's circuit consists of two capacitors, one inductor,
one linear resistor and one nonlinear resistor named
Chua’s diode, as shown in Fig. 1. The inductance,resistance and capacitances respectively are L = 21.83
mH, R = 1.875 kΩ, C1 = 10 nF and C2 = 100 nF. The
volt-ampere characteristics of Chua’s diode can beformulated asiN= G b v1 + 0.5 ( G a − G b )  v1 + E − v1 − E  ,where E = 1.1 V, Ga = -0.7576 mS and Gb = -0.4091
mS.

Published by Atlantis Press
Copyright: the authors
84

Hong Niu, Dongcheng Tan, Yongjun Wu

From Fig. 1, the state equation of Chua’s circuit canbe formulated as

dv1 1  v2 − v1=− iN  ,
dt C1  Rdv2
1  v1 − v2=+ iL  ,
dt C 2  R
diLv= − 2.
dt
L(1)

supply voltages are ±9V, so that the output saturation
voltage is 8.3V. The inductance simulator has noequivalent internal resistance, and its inductance is easy
to adjust.
Assume that the currents through the resistors R1,R2, R3, R4 and the capacitor C5 respectively are i1, i2, i3,
i4 and i5, whose reference directions are all from top tobottom. When the operational amplifiers U1A and U1B
are being operated in linear mode, it is obtained thati2 = i3 ,
i4 = i5 ,
i1R1 = −i2 R 2 ,Let x = v1, y = v2, z = RiL and t = (RC2)τ, andsubstitute them into Eq. (1). For convenience,
abbreviate dx/dτ to x . The dynamic equation of Chua’s
circuit is represented as=x α  y − x − h ( x )  ,y = x − y + z ,z = − β y,( 2)i3 R 3 = −i5Thus, the relationship between the currents i1 and i4 isexpressed as
RRC i4 = i1  jω 1 3 5  .
R2 

2

where α = C2/C1 = 10 and β = (R C2)/L = 16.1.
Compare Eq. (2) with Eq. (1), then,h ( x )= m1 x + 0.5 ( m0 − m1 )  x + E − x − E  ,where m0 = RGa = -1.4205 and m1 = RGb = -0.7671.
Let the initial values of the system (2) be (x0, y0, z0)= (0.1, 0.2, 0.5), then the Lyapunov exponents are
λ1=0.3193>0, λ2=-0.0001≈0 and λ3=-3.1683<0. It
indicates that the system (2) is chaotic. The simulation
phase portraits of Chua’s circuit are shown in Fig. 3(a1)-(a3).
3. Implementation of Chua’s Circuit
3.1. Inductance simulator
Because the internal resistance of an ordinary inductor
is too large and the inductance is difficult to alter, it is
not suitable to put an ordinary inductor in a chaotic
circuit which is very sensitive to the parameters. Hence,
the inductance simulator should be used to construct
Chua’s circuit instead of an ordinary inductor.6-8 The
equivalent circuit of the inductance simulator is shownin the left dashed box in Fig. 2, where R1 = R2 = R3 = 1kΩ, R4 = 2.183 kΩ and C5 = 10 nF. The operational
amplifiers used in this paper are TL082CP and the1.
jω C 5

According to the definition of two-terminal network
impedance, the equivalent impedance of the inductancesimulator isZ=RRR C
i4 R 4= jω 1 3 4 5 → L= 21.83mH.
R2
i1

As a result, the desired inductance is obtainable byadjusting the resistance of the resistor R4.
3.2. Chua’s diode
Chua’s diode consists of two operational amplifiers
U2A, U2B and six resistors, as shown in the right dottedbox in Fig. 2. The resistances are R6 = R7 = 22 kΩ, R8 =3.3 kΩ, R9 = R10 = 220 Ω and R11 = 2.2 kΩ. The design
procedure of this equivalent circuit can be found indetail in Ref. 6.
3.3. Realization of Chua’s circuit
Chua’s circuit is implemented by replacing the ordinary
inductor and Chua’s diode shown in Fig. 1 with theinductance simulator and the equivalent circuit ofChua’s diode. The analog Chua’s circuit is presented inFig. 2.

Published by Atlantis Press
Copyright: the authors85

Analog Circuit Implementation andFig. 2. Analog Chua’s circuit

4. Phase Portraits of Chua’s Circuit
4.1. Observation of state variable z
For the observation of all variables of the system(2), the state variable z = RiL should be observed
even if it is not an actual measurable physicalquantity in Chua’s circuit. The observation
circuit of z consists of four operational
amplifiers U3A, U3B, U4A, U4B and somerelated resistors, as shown in Fig. 2. The
resistances are R12 = R13 = R14 = R15 = R16 = R18= 10 kΩ, and the resistance of the resistor R17 isvariable.
The relationship between the voltage uR1 andthe state variable z can be described asiL =−i1 → z =RiL =−RR
uR1 =− 17 uR1.
R1R16Thus, the design procedure of the observation
circuit can be divided into three parts. Firstly,
U3A and U3B are two input voltage followers
used to get the potentials of both ends of R1.
Secondly, U4A and the resistors R12-R15
constitute the differential input proportion

operational circuit used to work out the voltage
uR1. Finally, the state variable z can be obtained
from the output of the inverse proportion
operational circuit consisting of U4B and theresistors R16 and R17. Furthermore, theoperational amplifiers U3A and U3B can also
insulate the main Chua’s circuit from the impactof the observation circuit.
It is found that some peak (valley) values ofz are larger than the output saturation voltage ofthe operational amplifier by solving the Eq. (2),such that the cut-off (saturation) distortion will
occur when observing the experiment phase
portraits of x-z and y-z on the oscilloscope.
Hence, scaling of z should be performed. Adjust
the resistance of the resistor R17 to R17 = 14.77
kΩ, then the distortions will no longer occur.
Consequently, the experiment value of z is about
0.8 times its simulation value. For a convenientcomparison, the simulation value of z is adjusted
to 0.8 times its original value in Fig. 3 (a2)-(a3).
4.2. Comparison of simulationexperiment phase portraits

andThe experiment phase portraits of Chua’s circuit
are shown in Fig. 3 (b1)-(b3). The scales of x, yPublished by Atlantis Press
Copyright: the authors
86

Hong Niu, Dongcheng Tan, Yongjun Wu

and z are 2 V/Div, 0.5 V/Div and 2 V/Div,respectively. By comparing (b1)-(b3) with (a1)(a3) in Fig. 3, it can be confirmed that here is agood qualitative agreement between thenumerical simulation and the experimentalrealization. It demonstrates that the realization ofChua’s circuit is correct.
5. Conclusions
The hybrid analog circuit implementation ofChua’s circuit, which exploits the circuit
topologies for the inductor element and thenonlinear resistor, is proposed in this paper. The
observation circuit for the directly non-measurable variable z is also designed forcomprehensive investigation of the chaotic
attractors of Chua’s circuit. It is necessary toobserve all chaotic state variables, especially for
full state chaos synchronization. For example, insynchronization physical test of two identicalthree-dimensional
chaoticcircuits,
theassumption that all three states wouldsynchronize as long as two of themsynchronized may be incorrect, so that all of thestates generated from the circuits should beinvestigated to verify the synchronization
phenomena. Therefore, the study in this paper
has some engineering significance.

Fig. 3. Comparison of phase portraits: Simulation-Experiment: (a1)-(b1) x-y; (a2)-(b2) x-z; (a3)-(b3) y-z

References
1. T. Matsumoto, A chaotic attractor from Chua's circuit,
IEEE Trans. on Circuits and Systems 31(12) (1984)
1055–1058.
2. Z. Yang, J. Zhang, Y. Ma, Y. Bai and S. Ma, Design andrealization of Chua's circuit based on current conveyers,
Acta Physica Sinica 59(5) (2010) 3007–3016.
3. B. Bao, W. Hu, J. Xu, Z. Liu and L. Zou, Analysis andimplementation of memristor chaotic circuit, ActaPhysica Sinica 60(12) (2011) 63–70.
4. E. Lv and S. Huang, The equivalent circuit design ofChua's circuit and application thereof, Chinese Journal of
Electron Devices 37(5) (2014) 891–895.
5. X. Zhang, H. Sun, J. Zhao, J. Liu, Y. Ma and T. Han,
Equivalent circuit in function and topology to Chua’scircuit and the design methods of these circuits, ActaPhysica Sinica 63(20) (2014) 95–102.
6. S. Yu, Chaotic Systems and Chaotic Circuits: Principle,Design and Its Application in Communications (XiDian
University Publishing House, Xi’an, 2011).
7. R. Kiliç, A comparative study on realization of Chua’scircuit: Hybrid realizations of Chua’s circuit combining
the circuit topologies proposed for Chua’s diode andinductor elements, International Journal of Bifurcation
and Chaos 13(6) (2003) 1475–1493.
8. H. Liu, F. Jiang, B. Xia and L. Yue, The experimentstudy of multiscroll chaotic circuit based on a stimulated
inductor circuit, Journal of Northeast Normal University(Natural Science Edition) 39(2) (2007) 55–59.

Published by Atlantis Press
Copyright: the authors
87