International Journal of Bifurcation and Chaos, Vol. 17, No. 5 (2007) 1785–1800c© World Scientific Publishing Company

A MODULE-BASED AND UNIFIED APPROACHTO CHAOTIC CIRCUIT DESIGN AND

ITS APPLICATIONS

SIMIN YUCollege of Automation,

Guangdong University of Technology,Guangzhou 510090, P. R. China

JINHU LÜ∗The Key Laboratory of Systems and Control, Institute of Systems Science,

Academy of Mathematics and Systems Science,Chinese Academy of Sciences, Beijing 100080, P. R. China

Department of Ecology and Evolutionary Biology,Princeton University, Princeton, NJ 08544, USA

jhlu@iss.ac.cn

GUANRONG CHENDepartment of Electronic Engineering,

City University of Hong Kong, Hong Kong SAR, P. R. Chinagchen@ee.cityu.edu.hk

Received December 15, 2005; Revised July 14, 2006

This paper proposes a module-based and unified approach to chaotic circuit design, where thedescription is based on the state equations without physical dimensions for simplicity of a generaldiscussion. The main design process consists of transformation of state variables, transformationfrom differential to integral operations, and transformation of the time-scale. The designedcircuit consists of anti-adder module integrator module, and inverter module. A novel 3-scrollChua’s circuit and a generalized Lorenz-like circuit are designed and implemented for verifyingthe effectiveness of this systematic circuit design methodology. Experimental observations areprovided for confirmation. Comparing with the traditional circuit design methods, this newdesign approach has the following typical characteristics: (i) module-based and unified design;(ii) independent adjustment of system parameters; (iii) adjustment of distribution regions forthe frequency spectra of chaotic signals; (iv) prominent observability.

Keywords : Chua’s circuit; multi-scroll; module-based and unified circuit design; time-scale.

1. Introduction

Over the last four decades, chaos has been inten-sively investigated within the nonlinear science,information science, and engineering communi-ties [Chen & Dong, 1998]. Aiming at real-world

applications, nonlinear circuit design has become akey issue in chaos-based technologies.

Remarkably, Chua’s circuit [Chua et al., 1986;Kennedy, 1993; Zhong et al., 2002] is a paradigmin the nonlinear circuit theory. Based on Chua’s

∗Author for correspondence

1785

1786 S. Yu et al.

circuit, many modified nonlinear circuits have beendesigned and implemented [Elwakil & Kennedy,2000], including the MCK circuit [Matsumoto et al.,1986], revised Chua’s circuit [Yin, 1997], and var-ious multi-scroll circuits [Han et al., 2005; Lü &Chen, 2006; Lü et al., 2004b; Lü et al., 2004c; Lüet al., 2006; Lü et al., 2003; Suykens & Vandewalle,1993; Yalcin et al., 2000; Yalcin et al., 2002]. Mostof these circuits are constructed by using capacitors,inductors and resistors along with Chua’s diode.In this paper, we propose a novel circuit designmethod in which the basic design idea is very dif-ferent from Chua’s circuit but can realize the samechaotic dynamics and even much more.

As we know now, there are various designapproaches of chaotic circuits by using various elec-tronic or logic devices reported in the literature. Forexample, Elwakil et al. [2003[ applied the so-calledcurrent feedback operational amplifiers (CFOAs)and digital logic operations to design the Lorenz-like circuit for implementing a four-wing butter-fly attractor; Zhong and Tang [2002] designed theChen circuit and Li et al. [2005] devised the hyper-chaotic Chen circuit both based on the dimension-less state equations of the circuits. Most of thecircuit design methods as above are not based ona common and unified framework and do not havethe universality and compatibility. In the follow-ing, based on the dimensionless state equationsof the circuit, a module-based and unified circuitdesign approach is then proposed. The designedcircuit consists of three different functional blocks:anti-adder module, integrator module and invertermodule. The main design process consists of trans-formation of state variables, transformation fromdifferential to integral operations, and transforma-tion of the time-scale. Comparing with the tradi-tional circuit design methods, such as those of theLorenz-like circuit [Elwakil et al., 2003], Chen cir-cuit [Zhong & Tang, 2002], and hyperchaotic Chencircuit [Li et al., 2005], this new method has the fol-lowing four typical characteristics: (i) module-basedand unified design; (ii) independent adjustment ofsystem parameters; (iii) adjustment of distributionregions for the frequency spectra of chaotic signals;(iv) prominent observability.

It should be especially pointed out that all statevariables in the forms of the original or inverse vari-ables input to the inverting terminals of the anti-adders and all noninverting terminals are connectedto the earth in this proposed approach. However,in most traditional circuit design methods based

on the dimensionless state equations of the circuits[Zhong et al., 2002; Zhong & Tang, 2002; Li et al.,2005], all state variables simultaneously input tothe inverting and noninverting terminals. Therefore,all parameters in our method are independentlyadjustable, however, all parameters in the tradi-tional approaches as above are coupled togetherand not independently adjustable. To verify theeffectiveness of this new approach, a novel 3-scrollChua’s circuit and a generalized Lorenz-like circuitare designed and implemented with experimentalobservations provided for confirmation. Moreover,the proposed circuit design method can be easilyand naturally generalized to the circuit designs ofother chaotic circuits.

The rest of this paper is organized as follows. InSec. 2, the new systematic circuit design approachis described and discussed. A novel 3-scroll Chua’scircuit and a generalized Lorenz-like circuit are thendesigned and implemented in Secs. 3 and 4, respec-tively, with experimental observations reported.Finally, some conclusions are drawn in Sec. 5.

2. A Module-Based and UnifiedCircuit Design Approach

This section proposes a module-based and unifiedcircuit design approach, in which the fundamentaldesign principle differs from those of the traditionalcircuit design methods.

This approach is based on the dimensionlessstate equations of the circuit. The main procedurecan be summarized into three key steps; that is,Step I: transformation of state variables; Step II:transformation from differential to integral opera-tions; Step III: transformation of the time-scale.

To start, consider a general system of n-dimensional state equations described by

dx1dτ

=n∑

i=1

a1ixi +n∑

j=1

n∑p=1

b1jpxjxp

+ · · · + f1(x1, x2, . . . , xn)dx2dτ

=n∑

i=1

a2ixi +n∑

j=1

n∑p=1

b2jpxjxp

+ · · · + f2(x1, x2, . . . , xn)· · · · · · · · ·

dxndτ

=n∑

i=1

anixi +n∑

j=1

n∑p=1

bnjpxjxp

+ · · · + fn(x1, x2, . . . , xn),

(1)

A Module-Based and Unified Approach to Chaotic Circuit Design and Its Applications 1787

where the first sum represents the linear termsand the other sums are all cross products, withfi(x1, x2, . . . , xn) (1 ≤ i ≤ n) being the differenttypes of nonlinear functions, which can be satu-rated function series, hysteresis function series, stepwave, sawtooth wave, triangular wave, transconduc-tor wave and exponent functions, and so on.

It is well known that the dynamic regions ofmost electronic devices are very limited. For exam-ple, the linear dynamic region of the operationalamplifier TL082 with electrical source of ±15 V isonly within ± 13.5 V. However, for many chaoticsystems, the dynamic regions of the state variablesin the dimensionless state equations are far exceed-ing the linear dynamic regions of the operationalamplifiers. To physically realize these chaotic sys-tems, one has to compress the dynamic regions of allthe state variables into the linear dynamic regionsof the operational amplifiers. To do so, one may letx′i = kxi(1 ≤ i ≤ N), where k ≤ 1 is the compressedratio. Then, according to Eq. (1), one has

dx′1dτ

=n∑

i=1

a1ix′i +

1k

n∑j=1

n∑p=1

b1jpx′jx

′p

+ · · · + kf1(

x′1k

,x′2k

, . . . ,x′nk

)

dx′2dτ

=n∑

i=1

a2ix′i +

1k

n∑j=1

n∑p=1

b2jpx′jx

′p

+ · · · + kf2(

x′1k

,x′2k

, . . . ,x′nk

)

· · · · · · · · ·dx′ndτ

=n∑

i=1

anix′i +

1k

n∑j=1

n∑p=1

bnjpx′jx

′p

+ · · · + kfn(

x′1k

,x′2k

, . . . ,x′nk

).

(2)

It is quite easy to obtain the integral form ofEq. (2), based on which one can further get the finalcircuit equation by using a transformation of thetime-scale. According to the above circuit equation,one can easily design a block circuit diagram, whichincludes three main modules: anti-adders moduleintegrators module and inverters module.

In the following, this circuit design approachis illustrated by working out two typical examples:a novel 3-scroll Chua’s circuit and a generalizedLorenz-like system.

3. A Novel 3-Scroll Chua’s Circuit

It is known that the piecewise linear function inthe original Chua’s circuit can be replaced by othernonlinear functions, such as the sine function [Tanget al., 2001], exponent function [Abdomerovic et al.,2000], and polynomial function [Tang & Man, 1998;Zhong, 1994], so as to generate various double-scrollor even multi-scroll chaotic attractors from the cir-cuit [Yu et al., 2004a; Yu et al., 2004b; Yu et al.,2005]. Here, we are especially interested in polyno-mial characteristic functions, such as ax+bx|x|, ax+bx3, a + bx + cx2 + dx3, etc. Notice that all thesepolynomial characteristic functions can only gener-ate double-scroll Chua’s attractors. To create morescrolls in Chua’s circuit, the polynomial character-istic function is first modified to be ax+bx|x|+cx3.Thus, Chua’s circuit equation becomes

dx

dτ= α(y − h(x))

dy

dτ= x − y + z

dz

dτ= −βy,

(3)

where α = 12.8, β = 19.1, h(x) = ax + bx|x| + cx3.When a = 0.472, b = −1, c = 0.47, the polyno-mial characteristic curve is shown in Fig. 1. Here,five equilibria, four turning points and five char-acteristic regions are denoted by xi(0 ≤ i ≤ 4),ei(1 ≤ i ≤ 4) and Di(0 ≤ i ≤ 4), respectively. Fig-ure 2 shows the numerical simulation results of the

−1.5 −1 −0.5 0 0.5 1 1.5−0.15

−0.1

−0.05

0

0.05

0.1

0.15

x

h(x)

D4

D2

D0

D1

D3

x4 x2 x0

x1

x3

e4 e2 e1 e3

Fig. 1. h(x) and its five characteristic regions.

1788 S. Yu et al.

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

x

y

(a) x − y plane

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x

z

(b) x − z planeFig. 2. Plane projection of the 3-scroll Chua’s attractor.

3-scroll Chua’s attractor with Lyapunov exponents:λ1 = 0.2, λ2 = 0, λ3 = −5.56.

Both theoretical analysis and numerical sim-ulations show that there are various bifurcationphenomena in system (3). Figures 3(a)–3(d) showthe bifurcation diagrams versus parameter a withα = 12.8, β = 19.1, b = −1, c = 0.47, parameterb with α = 12.8, β = 19.1, a = 0.472, c = 0.47,

parameter c with α = 12.8, β = 19.1, a = 0.472,b = −1, and parameter α with β = 19.1, a = 0.472,b = −1, c = 0.47, respectively. According to Fig. 3,there are some continuous chaotic regions for par-ameters α, a, b, c, which are very useful for hardwareimplementation.

When a = 0.472, b = −1, c = 0.47, system (3)has five equilibria: (xi, 0,−xi)(0 ≤ i ≤ 4), where

A Module-Based and Unified Approach to Chaotic Circuit Design and Its Applications 1789

(a)

(b)

Fig. 3. Bifurcation diagrams of parameters a, b, c, α.

1790 S. Yu et al.

(c)

(d)

Fig. 3. (Continued)

A Module-Based and Unified Approach to Chaotic Circuit Design and Its Applications 1791

x0 = 0, x1,2 = ±0.7068, x3,4 = ±1.4209. Moreover,the Jacobians of system (3) evaluated at its equilib-ria (xi, 0,−xi)(0 ≤ i ≤ 4) are given by

J(xi) =

−αdh(x)

dx

∣∣∣∣x=xi

α 0

1 −1 10 −β 0

for 0 ≤ i ≤ 4. Their corresponding eigenvaluesare: γ0 = −7.4606, σ0 ± jω0 = 0.2095 ± j3.9273,γ1,2 = 4.3504, σ1,2 ± jω1,2 = −1.1570 ± j3.4629,and γ3,4 = −7.5171, σ3,4±jω3,4 = 0.2066±j3.9328,respectively. Obviously, x0, x3, x4 are the equilib-ria with index 2, which can generate scrolls in theattractor; on the other hand, x1, x2 are the equi-libria with index 1, which can connect neighboringscrolls in the attractor.

In the following, a circuit diagram is designed tophysically realize the above 3-scroll Chua’s attrac-tor. According to the circuit design method intro-duced in Sec. 2, one firstly gets the correspondingEq. (3) after transformation of state variables, asfollows:

du

dτ= α

[v −

(au +

b

ku|u| + c

k2u3

)]

dv

dτ= u − v + w

dw

dτ= −βv,

(4)

where k is the compressed ratio. From Fig. 2, onecan see that the dynamic regions of all the statevariables of system (3) belong to the linear dynamicregions of the operational amplifiers. In this case,one may simply let k = 1.

Thus, the integral form of system (4) isgiven by

u =∫

[−a11(−v) − a12h(u)]dτ

v =∫

[−a21(−u) − a22v − a23(−w)]dτ

w =∫

−a31vdτ,

(5)

where aij(1 ≤ i, j ≤ 3) are system parametersand a11 = a12 = α = 12.8, a13 = a32 = a33 =0, a21 = a22 = a23 = 1, a31 = β = 19.1.h(u) = au+(b/k)u|u|+(c/k2)u3, a = 0.472, b = −1,c = 0.47, k = 1. Based on system (5), one

can design a block circuit diagram to physicallyrealize the 3-scroll Chua’s attractor, as shownin Fig. 4.

All operational amplifiers shown in Fig. 4 areTL082, the supply voltages of the positive andnegative electrical sources are ±15 V. Moreover,for convenient adjustment and higher precision, allresistors are precisely adjustable resistors or poten-tiometers. In addition, in Fig. 4, the operationalamplifiers OP1, OP4, OP7 are anti-adder mod-ules; the operational amplifiers OP2, OP5, OP8are anti-integrator modules; the operationalamplifiers OP3, OP6, OP9 are inverter modules.Figures 4(a) and 4(b) are the basic Chua’s cir-cuit and polynomial signal generator for h(u) =au + (b/k)u|u| + (c/k2)u3, respectively. Here, theabsolute-value circuit is shown in the dashed-lineblock of Fig. 4(b). From the generalized super-position principle, the relationship between inputand output of the absolute-value circuit satisfiesu0 = −u − 2u1. For u < 0, the diodes D1 and D2are connected and u1 = 0. Thus, u0 = −u > 0. Foru > 0, the diodes D1 and D2 are disconnected andu1 = −u. Thus, u0 = −u + 2u = u > 0. Therefore,u0 = |u|.

According to Fig. 4(b), the relationshipbetween the input and the output of the polyno-mial signal generator is described by

h(u) =RnRa

u − Rn10Rb

u|u| + Rn100Rc

u3 (6)

where a = Rn/Ra, b/k = −(Rn/10Rb), c/k2 =Rn/100Rc, k = 1. When Rn is fixed, one can adjustthe system parameters a, b/k, c/k2 of polynomialh(u) by adjusting the resistors Ra, Rb, Rc, respec-tively.

It follows from Fig. 4(a) that the state equationof the nonlinear circuit is given by

u =1

R0C0

∫ [− Rf

R11(−v) − Rf

R12h(u)

]dt

v =1

R0C0

∫ [− Rf

R21(−u) − Rf

R22v − Rf

R23(−w)

]dt

w =1

R0C0

∫ [− Rf

R31v

]dt.

(7)

Let τ = t/R0C0. Here, 1/R0C0 is the integralconstant of the integrators shown in Fig. 4(a), whichis also the transformation factor of the time-scale.

1792 S. Yu et al.

Then, one obtains

u =∫ [

− RfR11

(−v) − RfR12

h(u)]

dτ

v =∫ [

− RfR21

(−u) − RfR22

v − RfR23

(−w)]

dτ

w =∫ [

− RfR31

v

]dτ.

(8)

According to (5) and (8), all system parame-ters are given by a11 = Rf/R11, a12 = Rf/R12,a21 = Rf/R21, a22 = Rf/R22, a23 = Rf/R23,a31 = Rf/R31. Let Rf = 100k, R11 = 7.8k, R12 =7.8k, R21 = 100k, R22 = 100k, R23 = 100k, R31 =5.3k. Then, one has a11 = a12 = 12.8, a21 =a22 = a23 = 1, a31 = 19.1. Since there arethree anti-adder modules in Fig. 4, for a fixed

Rf , one can independently adjust various sys-tem parameters, aij(1 ≤ i, j ≤ 3), by tuningthe corresponding resistors Rij(1 ≤ i, j ≤ 3) inFig. 4, respectively. Therefore, this independentlyadjustable characteristic is one of the useful fea-tures of the modular circuit design. Figure 5 showsthe experimental observations of the 3-scroll Chua’sattractor.

From (7), the transformation factor of thetime-scale is completely determined by the integralresistor R0 and integral capacitance C0. Compar-ing with the traditional design methods, such asthat of Chua’s circuit, this module-based and uni-fied design approach can change the distributionregion of the frequency spectrum of a chaotic sig-nal as required by tuning integral resistor R0 orintegral capacitance C0 for real-world applications.That is, when R0 (or C0) is decreasing, one can

(a)

(b)

Fig. 4. Circuit diagram for realizing the 3-scroll Chua’s attractor.

A Module-Based and Unified Approach to Chaotic Circuit Design and Its Applications 1793

(a) x − y plane, where x = 0.5 V/div, y = 1.0 V/div

(b) x − z plane, where x = 1V/div, z = 0.6 V/div

(c) y − z plane, where y = 1V/div, z = 0.6V/divFig. 5. Experimental observations of the 3-scroll Chua’s attractor.

1794 S. Yu et al.

extend the distribution region of the frequency spec-trum of the high-frequency end of a chaotic signal.However, when R0 (or C0) is increasing, one canreduce the distribution region of the frequency spec-trum of the high-frequency end of a chaotic signal.Therefore, comparing with the fixed capacitanceand inductance in Chua’s circuit, this adjustingcharacteristic is very useful for real circuit designand practical engineering applications.

Here, two typical examples are used to show theeffectiveness of this proposed design method. Fig-ure 6 shows the waveforms and power spectrumsof the time domain of the variable x for two dif-ferent cases: (I) C0 = 33nF, R0 = 50 kΩ and(II) C0 = 33nF, R0 = 10 kΩ, respectively, wherethe other parameters are given in Fig. 4. Our exper-imental observations are consistent with the numer-ical observations in Fig. 6.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

t/s

x

(a) Waveform of x, C0 = 33 nF, R0 = 50 kΩ

200 400 600 800 1000 1200 1400 1600 1800 20000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency/Hz

Uni

tary

Spe

ctru

m

(b) Power spectrum of x, C0 = 33 nF, R0 = 50 kΩ

Fig. 6. Numerical simulations of the waveforms and power spectrums of variable x.

A Module-Based and Unified Approach to Chaotic Circuit Design and Its Applications 1795

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

t/s

x

(c) Waveform of x, C0 = 33 nF, R0 = 10 kΩ

500 1000 1500 2000 2500 3000 3500 4000 4500 50000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency/Hz

Uni

tary

Spe

ctru

m

(d) Power spectrum of x, C0 = 33 nF, R0 = 10 kΩ

Fig. 6. (Continued)

It is well known that almost all design meth-ods of chaotic circuit are not beauideal. Comparingwith the traditional design approaches, such asthat of Chua’s circuit, the main disadvantage ofthe proposed design method is that it is likelyto need more electronic devices. More electronicdevices may increase the total hardware errors.However, one can minimize the total hardware

errors by using the electronic devices with highprecision.

4. Circuit Implementation of aGeneralized Lorenz-like System

A generalized Lorenz-like system was proposed in[Lü & Chen, 2002; Lü et al., 2002], which is

1796 S. Yu et al.

(a) Original attractor

(b) Compressed attractor

Fig. 7. Numerical simulations of the generalized Lorenz-like system.

described by

dx

dτ=

ab

a + bx − yz

dy

dτ= −ay + xz + d

dz

dτ= −bz + xy.

(9)

When a = 10, b = 5, d = 5, system (9) has a Lorenz-like chaotic attractor [Lü et al., 2004a], as shown inFig. 7(a).

Obviously, the dynamic regions of the statevariables x, y, z of system (8) are far exceeding thelinearly dynamic regions of the operational ampli-fiers. To physically realize system (9), one has tocompress the dynamic regions of the state variablesx, y, z. To do so, let u = kx, v = ky,w = kz, where

A Module-Based and Unified Approach to Chaotic Circuit Design and Its Applications 1797

k = 0.1. Then, one gets

du

dτ=

ab

a + bu − vw

dv

dτ= −av + 10uw + 10d

dw

dτ= −bw + 10uv.

(10)

When a = 10, b = 5, d = 5, system (10) has achaotic attractor as shown in Fig. 7(b). Obviously,

the dynamic regions of the state variables x, y, zof system (10) are compressed to be withinthe linearly dynamic regions of the operationalamplifiers.

Similarly, based on the circuit design princi-ple proposed in Sec. 2, one can construct a circuitdiagram, as shown in Fig. 8, to physically realizesystem (10). Figure 9 shows the experimentalobservations of the 4-scroll generalized Lorenz-likechaotic attractor.

Fig. 8. Circuit diagram for realizing the generalized Lorenz-like system.

(a) x − y plane, x = 1.0 V/div, y = 0.6 V/divFig. 9. Experimental observations of the generalized Lorenz-like system.

1798 S. Yu et al.

(b) x − z plane, x = 1.0 V/div, z = 1.0V/div

(c) y − z plane, where y = 1.0V/div, z = 1.2 V/divFig. 9. (Continued)

5. Conclusions

We have introduced a module-based and unifiedapproach to chaotic circuit design. This methodis based on the dimensionless state equations,and the main design process consists of trans-formation of state variables (which extends theparameter ranges in hardware implementation ofnonlinear circuits), transformation from differen-tial to integral operations, and transformationof the time-scale. The designed circuit includes

three different function blocks: anti-adder moduleintegrator module and inverter module. Compar-ing with the traditional circuit design methods,this systematic approach has the following fourtypical characteristics: (i) module-based and uni-fied design; (ii) independent adjustment of sys-tem parameters; (iii) adjustment of distributionregions for the frequency spectra of chaotic sig-nals; (iv) prominent observability. To measurethe current of the inductance of Chua’s circuit,an additional circuit is needed to perform the

A Module-Based and Unified Approach to Chaotic Circuit Design and Its Applications 1799

current-voltage transformation. Moreover, a novel3-scroll Chua’s circuit and a generalized Lorenz-likecircuit have been designed and implemented to ver-ify the effectiveness of the new design methodology,furthermore confirmed by experimental observa-tions. It is believed that this module-based andunified circuit design approach will have good appli-cations in practice.

Acknowledgments

This work was supported by the National Nat-ural Science Foundation of China under GrantsNo. 60304017, No. 6022130, No. 20336040 andNo. 60572073, the Scientific Research Startup Spe-cial Foundation on Excellent PhD thesis and thePresidential Award of Chinese Academy of Sciences,Natural Science Foundation of Guangdong Provinceunder Grants No. 32469 and No. 5001818, Sci-ence and Technology Program of Guangzhou Cityunder Grant No. 2004J1-C0291, Important Direc-tion Project of Knowledge Innovation Program ofChinese Academy of Sciences under Grant KJCX3-SYW-S01, and the City University of Hong Kongunder the SRG grant 7002134.

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