A MODIFIED CHUA’S CIRCUIT WITH AN ATTRACTION-REPULSION … · 2016-01-22 · f(y)=ay +byexp(cy2),...

Papers

International Journal of Bifurcation and Chaos, Vol. 18, No. 7 (2008) 1865–1888c© World Scientific Publishing Company

A MODIFIED CHUA’S CIRCUIT WITH ANATTRACTION-REPULSION FUNCTION*

RONG LI†, ZHISHENG DUAN and BO WANGState Key Laboratory for Turbulence and Complex Systems,

Department of Mechanics and Aerospace Engineering,College of Engineering, Peking University,

Beijing 100871, P. R. China†[email protected]

GUANRONG CHENDepartment of Electronic Engineering,

City University of Hong Kong,Hong Kong SAR, P. R. China

Received June 21, 2007; Revised August 6, 2007

In this paper, the original Chua’s circuit is modified by substituting its piecewise-linear functionwith an attraction-repulsion function. Some new complex dynamical behaviors such as chaosare observed through computer simulations. Basic properties of the new circuit are analyzedby means of bifurcation diagrams. Lagrange stability conditions of the circuit are derived. Acomparison between this modified Chua’s circuit with an attraction-repulsion function and themodified Chua’s circuit with a cubic nonlinear function is presented. Moreover, a generalizationof the new circuit that can generate multiple scrolls is designed and simulated. Finally, a physicalcircuit is built to visualize the new system, with some experimental observations reported.

Keywords : Attraction-repulsion function; chaos; bifurcation; Lagrange stability; Chua’s circuit.

1. Introduction

Chaotic dynamics in physical systems have beenobserved and analyzed for a long time in the non-linear science history, with the first physical chaoticsystem built in the laboratory by Chua and his col-leagues [Chua et al., 1986; Chua, 2002a].

Today, Chua’s circuit has been recognized as aclassical paradigm in the study of chaos and bifurca-tion theories, in which the piecewise-linear nonlin-earity plays an important role which was modifiedin several different ways particularly by a hys-teresis function [Kennedy & Chua, 1991], a cubic

function [Zhong, 1994], or a sine function [Tanget al., 2002], etc.

In this paper, the original Chua’s circuit is fur-ther modified with an attraction-repulsion function.The attraction-repulsion function is a nonlinearfunction, with long-range attraction and short-range repulsion, useful in the study of swarmingbehaviors in biological systems [Okubo, 1986; Gazi& Passino, 2002a, 2002b]. For further theoreticalstudy and practical applications of chaos theory tocircuits and swarms, it is worth investigating thepossible use of the attraction-repulsion function tochaos theory [Duan et al., 2005].

∗This work is supported by the City University of Hong Kong under the Research Enhancement Scheme and SRG grant7002134 and the National Science Foundation of China under grants 60674093, 60334030.†Author for correspondence

1865

1866 R. Li et al.

This paper reports the findings of the new mod-ified Chua’s circuit with the attraction-repulsionnonlinearity, including some new chaotic phenom-ena, period-doubling bifurcations, periodic orbits,etc., observed through computer simulations. More-over, the Lagrange stability (i.e. the boundedness ofall solutions) are analyzed since it is important inthe study of chaos [Duan et al., 2006].

The outline of this paper is as follows. In Sec. 2,the modified Chua’s circuit with an attraction-repulsion function is introduced, and its basic prop-erties including periodic orbits and period-doublingbifurcations are analyzed. In Sec. 3, the modifiedChua’s circuit in the dimensionless form is furtherstudied, with simulation results showing variouscomplex chaotic dynamics. In Sec. 4, the param-eter region of the Lagrange stability is given andanalyzed for the dimensionless modified Chua’s cir-cuit. In Sec. 5, some computer simulations on themodified Chua’s circuit with an attraction-repulsionfunction are presented, in comparison with thoseon the modified Chua’s circuit with a cubic func-tion. In Sec. 6, a generalization of the new circuit toone that can generate multiple scrolls is presented.In Sec. 7, a circuit realization of the new modifiedChua’s circuit is presented with some experimentalobservations. Finally, Sec. 8 concludes the paper.

2. The New Chaotic Circuit and ItsBasic Properties

The original canonical Chua’s circuit is shown inFig. 1, which is synthesized by using five linearelements (two capacitors, C1, C2, one inductor, L,and two resistors, R0, R1), along with one nonlin-ear resistor (NR) which can be built on off-the-shelfop-amps.

The circuit equations follow from Kirchoff’slaws:

C1dvC1

dt=

1R1

(vC2 − vC1) − g(vC1),

C2dvC2

dt=

1R1

(vC1 − vC2) + iL,

LdiLdt

= −vC2 − R0iL,

where vC1 and vC2 are the voltages across thecapacitors C1 and C2, respectively, iL denotes thecurrent through the inductance L, and the termg(vC1) represents the characteristic of the non-linear resistance. Here, g(vC1) can be expressed

Fig. 1. Chua’s circuit.

mathematically as

g(vC1) = m0vC1 + 0.5(m1 − m0)(|vC1 + b1|− |vC1 − b1|),

in which m0 and m1 are appropriate constants withm0 < 0 and m1 < 0 [Chua, 2002b], as shown inFig. 2.

Now, substitute the piecewise-linear func-tion g(vC1) with an attraction-repulsion functiondefined by

f(y) = ay + by exp(cy2), (1)

where a, b, and c are constants. For the y ∈ R1 case,two graphs of the function (1) are shown in Fig. 3.Obviously, one function has three real roots whilethe other has only one real root.

With the attraction-repulsion function, themodified Chua’s circuit is described by

dvC1

dt=

1C1R1

(vC2 − vC1) −1C1

f(vC1),

dvC2

dt=

1C2R1

(vC1 − vC2) +1C2

iL,

diLdt

= − 1L

vC2 −1L

R0iL.

(2)

Fig. 2. Piecewise-linear v–i function.

A Modified Chua’s Circuit with an Attraction-Repulsion Function 1867

−15 −10 −5 0 5 10 15−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

y

f(y)

a=−0.02,b−0.5,c=−0.5a=0.02,b=−0.5,c=−0.5

Fig. 3. Two graphs of the attraction-repulsion function.

2.1. Basic dynamical analysis:Equilibria

The equilibria of the new circuit can be foundby solving the following algebraic equationssimultaneously:

1C1R1

(vC2 − vC1) −1C1

f(vC1) = 0,

1C2R1

(vC1 − vC2) +1C2

iL = 0,

− 1L

vC2 −1L

R0iL = 0.

(3)

It follows from (3) that

vC1 = −(R0 + R1)iL, vC2 = −R0iL,

iL[R1 + aR1R0 + aR21

+ bR1(R1 + R0) exp{c(R1 + R0)2i2L}] = 0.

Obviously, S0 = [0, 0, 0] is one equilibrium. Further-more, by setting

q =1

c(R1 + R0)2ln

(−a

b− 1

b(R1 + R0)

),

two other equilibria can be obtained, as

S1 = [−(R1 + R0)√

q,−R0√

q,√

q] and

S2 = [(R1 + R0)√

q,R0√

q,−√q].

Remark 1. System (2) is symmetric with respect tothe origin, which can be proved via the following

transformation:

(vC1 , vC2 , iL) → (−vC1 ,−vC2 ,−iL).

And obviously S1 and S2 are symmetric withrespect to the origin.

Remark 2. System (2) has three real equilibria whenparameters a and b satisfy the following inequality:

0 < −a

b− 1

b(R1 + R0)< 1. (4)

Throughout this section, for numerical simula-tions, take the parameter values as

C1 = 0.1, C2 = 1.05, R0 = 0.01, R1 = 2,

L = 0.35.

Then, condition (4) becomes

−0.4975 < a < −b − 0.4975 (5)

Remark 3. Let matrix A be the linear part of system(2). Then, system (2) can be rewritten as

V = AV + Bf 1(vC1), (6)

where

V =

vC1

vC2

iL

, A =

−1C1R1

− a

C1

1C1R1

0

1C2R1

− 1C2R1

1C2

0 − 1L

−R0

L

,

B =

− b

C1

00

,

and f1(vC1) = vC1 exp(cv2C1

).It is easy to see that f1 is bounded when c < 0.

Here, a is the only variable parameter of A and A isHurwitz stable when a ∈ [−0.4975, ∞). Under theconditions that f1 is bounded and A is Hurwitz sta-ble, one can easily draw the conclusion that system(2) is Lagrange stable, i.e. all solutions of system(2) are bounded. When A is unstable, the Lagrangestability conditions are not satisfied. Computer sim-ulation shows the existence of unbounded solu-tions of system (2), as displayed in Fig. 4,with initial value x0 = (0.002, 0.009,−0.002)T .The following investigation of dynamical behav-iors of systems is under the Lagrange stabilityconditions.

1868 R. Li et al.

490 500 510 520 530 540 550−1

−0.5

0

0.5

1

1.5x 10

17

t

Vc1Vc2iL

Fig. 4. An unbounded solution of (2) with a = −0.4976, b =c = −0.5.

2.2. Stability of equilibria

By linearizing system (2) at S0, one obtains thecharacteristic equation

f(λ) = λ3 +(

1C1R1

+1C1

(a + b) +1

C2R1+

R0

L

)λ2

+[

a + b

C1C2R1+

R0

C1R1L+

(a + b)R0

C1L

Table 1.

a Eigenvalues of S0

0.002488 λ1 = 0, λ2,3 = −0.2648 ± 0.5442i0.011935 λ1 = −0.6241, λ2,3 = ±0.6434i0.40929 λ1 = −4.5977, λ2,3 = ±1.5555i

+1

C2L

(R0

R1+ 1

)]λ +

a + b

C1C2L

(R0

R1+ 1

)

+1

C1C2R1L. (7)

Let b = −0.5. Then, based on the Routh–Hurwitzcriterion, S0 is stable when 0.002488 < a <0.011935 or a > 0.40929. Table 1 shows the eigen-values of system (2) at S0 with respect to the vari-able parameter a. When 0.002488 < a < 0.011935or a > 0.40929, the system orbit is a fixed pointS0 (as shown in Figs. 5(a) and 5(d)). When a =0.40929, the system orbits constitute a cluster ofperiodic orbits surrounding S0 (Fig. 5(b)). When0.011935 < a < 0.40929, although S0 is an unstableequilibrium, system (2) is Lagrange stable and thesystem orbit is a stable periodic orbit [Fig. 5(c)].When a < 0.002488, based on condition (5), thesystem has a pitchfork bifurcation and produces twonew stable equilibria, S1 and S2, simultaneously.

By linearizing system (2) at S1 (or S2), oneobtains the Jaccobian

J =

− 1C1R1

+1C1

(1

R0 + R1+ 2

(a +

1R0 + R1

)ln

(−a

b− 1

b(R0 + R1)

))1

C1R10

1C2R1

− 1C2R1

1C2

0 − 1L

−R0

L

,

by setting p = 1/(R0 + R1) + 2(a + (1/R0 + R1)) ln(−(a/b)−(1/b(R0 + R1))), one obtainsthe following characteristic equation:

f(λ) = λ3 +(

1C1R1

− p

C1+

1C2R1

+R0

L

)λ2

+[ −p

C1C2R1+

R0

C1R1L− pR0

C1L

+1

C2L

(R0

R1+ 1

)]λ − p

C1C2L

(R0

R1+ 1

)

+1

C1C2R1L. (8)

By some tedious numerical calculations, onecan verify that S1 and S2 are stable equilibria fora ∈ (−0.00226, 0.0025) or a ∈ (−0.4975,−0.49679)when b = −0.5. Let a0 = −0.00226, a1 = −0.49679.Then, by calculation again, one can see that theconjugate roots of Eq. (8) satisfy the following con-ditions of Hopf bifurcation:

Reλ(a)|a=a0,a1 = 0,Im λ(a)|a=a0,a1 �= 0,

dRe λ(a)da

∣∣∣∣a=a0,a1

�= 0,


−2

−1

0

1

2

x 10−3

−4

−2

0

2

4

x 10−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

x 10−3

Vc1iL

Vc2

−0.01

−0.005

0

0.005

0.01

−5

0

5

x 10−3

−4

−3

−2

−1

0

1

2

3

4

x 10−3

iLVc1

Vc2

(a) a = 0.6 (b) a = 0.40929

−4

−2

0

2

4

−3

−2

−1

0

1

2

3−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

iLVc1

Vc2

0.02−0.015

−0.01−0.005

00.005

0.010.015

−0.01−

−0.005

0

0.005

0.01−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

x 10−3

Vc1iL

Vc2

(c) a = 0.015 (d) a = 0.01

Fig. 5. Solutions of (2) with b = c = −0.5.

where a0, a1 are critical values, implying that sys-tem (2) has a Hopf bifurcation.

With respect to the variable parameter a, thebifurcation diagram and the Lyapunov-exponentspectrum (see Figs. 6(a) and 6(b)) show the tran-sition from periodic orbits to chaos, when a ∈[−0.011,−0.006]. When system (2) is chaotic, ithas a positive Lyapunov exponent. Some periodicorbits appear when the maximum Lyapunov expo-nent equals zero. The corresponding periodic orbitsand Lyapunov-exponent spectrum versus time t areshown in Figs. 6(c)–6(k). Here, the initial conditionsare chosen near the origin.

Remark 4. Under the conditions of Lagrange stabil-ity, the range of the parameter c (c < 0) is compar-atively large for system (2) to produce bifurcationsand chaos. Throughout, set c = −0.5.

Remark 5. When parameters a, b, c are all negative,it is clear from condition (5) that one only needs toanalyze the constant item of Eq. (7) in the inves-tigation of the stability of S0. When b < −0.4975,system (2) has three equilibria and S0 is unstableaccording to the Routh–Hurwize criterion. Whenb > −0.4975, S0 can change from stable to unsta-ble with respect to the variable parameters a, b,and system (2) can produce a pitchfork bifurcationsimultaneously. For example, when b = −0.4 and−0.0975 < a < −0.0881, system (2) has only onereal and stable equilibrium S0 [see Fig. 7(a)]. As theparameter a decreases, two new stable equilibria S1

and S2 appear from pitchfork bifurcation. Althoughsystem (2) can also produce chaos with respect tothe parameters a and b [see Fig. 7(b)], the region ofchaos is very small.

1870 R. Li et al.

−11 −10.5 −10 −9.5 −9 −8.5 8 −7.5 −7 −6.5 −6

x 10−3

−0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

a

Vc2

−11 −10 −9 −8 −7 −6 −5

x 10−3

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1 Lyapunov exponents of Dynamics

a

Lya

pu

no

v ex

po

nen

ts(a) Bifurcation diagram of vC2 : a ∈ (−0.011, −0.006) (b) Lyaponov-exponent spectrum: a ∈ (−0.011, −0.006)

0.05

0.1

0.15

0.2

0.25

−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02−0.015

−0.01

−0.005

0

0.005

0.01

0.015

Vc1iL

Vc2

00.05

0.10.15

0.20.25

0.30.35

−0.15

−0.1

−0.05

0

0.05−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

Vc1iL

Vc2

(c) a = −0.006 (d) a = −0.0074

00.05

0.10.15

0.20.25

0.30.35

−0.15

−0.1

−0.05

0

0.05−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

Vc1iL

Vc2

(e) a = −0.0077

Fig. 6. Evolution process as a is changing with b = c = −0.5.


−0.4−0.3

−0.2−0.1

00.1

0.20.3

−0.2

−0.1

0

0.1

0.2−0.03

−0.02

−0.01

0

0.01

0.02

0.03

Vc1iL

Vc2

−0.3−0.2

−0.10

0.10.2

0.30.4

−0.2

−0.1

0

0.1

0.2−0.03

−0.02

−0.01

0

0.01

0.02

0.03

Vc1iL

Vc2

(f) a = −0.00931 (g) a = −0.00945

−0.5

0

0.5

−0.4

−0.2

0

0.2

0.4−0.06

−0.04

−0.02

0

0.02

0.04

0.06

Vc1iL

Vc2

(h) Chaotic solution of system (2): a = −0.02 (i) Lyapunov-exponents spectrum when a = −0.02

−30−20

−100

1020

30

−15

−10

−5

0

5

10

15−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Vc1iL

Vc2

−0.495 −0.4948 −0.4946 −0.4944 −0.4942 −0.494 −0.49380

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

a

Vc2

(j) Chaotic solution of system (2): a = −0.4942 (k) Bifurcation diagram of vC2 : a ∈ (−0.495,−0.494)

Fig. 6. (Continued )

1872 R. Li et al.

0 10 20 30 40 50 60−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

t

Vc1 Vc2iL

−15−10

−50

510

15

−10

−5

0

5

10−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Vc1iL

Vc2

(a) a = −0.095, c = −0.5 (b) a = −0.488, c = −0.5

Fig. 7. Solution of system (2) with initial value x(0) = (0.016, 0.018,−0.015).

3. Numerical Analysis of theNormalized Circuit

For convenient numerical analysis of system (2),rescale the parameters with x1 = vC1 , x2 =vC2 , x3 = R1iL, t = (1/C2R1)τ, α = C2/C1, β =C2R

21/L, γ = R0/R1, k = R1 and redefine τ with

t. Then, the normalized equations of the modifiedChua’s circuit are given by

x1 = α(x2 − x1 − kf(x1)),x2 = x1 − x2 + x3,

x3 = β(−x2 − γx3),(9)

where f(x) = ax + bx exp(cx2). Fix a = −0.02, b =c = −0.5. Then, the dynamics of system (9) dependonly on the parameters α, β, γ and k. Let p =1/c ln(−(1 + kα(1 + γ))/(kb(1 + γ))). One obtainsthree equilibria of system (9):

S0 = [0, 0, 0], S1 =[√

p,γ√

p

(1 + γ),

−√p

(1 + γ)

],

S2 =[−√

p,−γ

√p

(1 + γ),

√p

(1 + γ)

].

Obviously, these equilibria are symmetric withrespect to the origin. Similarly to system (2), system(9) is Lagrange stable if its linear part is Hurwitzstable. Several simulations have been carried out,with some new findings summarized as follows:

3.1. Fixed β = 12, γ = 0.005, k = 2

When 0 < α < 8.1, the orbit of system (9) is a fixednonzero point; when 8.1 < α < 9.2 or α > 10.5,

the system has a periodic orbit. From the anal-ysis of the parameter region of Lagrange stabil-ity in Sec. 4 (see Fig. 13), it can be seen that, inthe present case, the parameters are located in theLagrange stable region. As shown in Fig. 8(a), sys-tem (9) evolves to chaos through a series of period-doubling bifurcations when α ∈ [9.2, 10.5]. One partof Fig. 8(a) is magnified and displayed in Fig. 8(b),which shows the transitions from chaos to non-chaos and from nonchaos to chaos of system (9)with a periodic segment splitting a region of chaosinto several parts. Interestingly, system (9) displaysa one-scroll chaotic orbit when α ∈ [9.52, 9.69].Moreover, the system phase portrait evolves toanother cyclic orbit when α ∈ [9.731, 9.844],with the corresponding portraits are shown inFigs. 8(c)–8(j).

3.2. Fixed α = 10.5, β = 12, k = 2

In this subsection, the dynamics of system (9) inthe region of 0 < γ < 24 are studied.

When γ > 24, system (9) does not satisfy theLagrange stability conditions (see Fig. 12) and itescapes to infinity (see Fig. 9(b)). When 0.051 <γ < 24, the system orbit converges to a fixedpoint; when 0 < γ < 0.0034, the system orbitis a periodic orbit. Figure 9(a) shows the inversebifurcation diagram of system (9) in the regionof 0.0034 < γ < 0.06, where the system hasvery rich dynamical behaviors. As in Sec. 3.1, theinverse bifurcation diagram also shows the transi-tions from chaos to nonchaos and from nonchaos to


(a) Bifurcation diagram of x2: α ∈ (9.2, 10.5) (b) Bifurcation diagram of x2: α ∈ (9.7, 9.9)

0.050.1

0.150.2

0.250.3

0.350.4

−0.5

−0.4

−0.3

−0.2

−0.1

0−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

X(1)X(3)

X(2

)

00.1

0.20.3

0.40.5

−0.6

−0.4

−0.2

0

0.2−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

X(1)X(3)

X(2

)

(c) α = 9.2 (d) α = 9.4

00.1

0.20.3

0.40.5

−0.6

−0.4

−0.2

0

0.2−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

X(1)X(3)

X(2

)

00.1

0.20.3

0.40.5

−0.6

−0.4

−0.2

0

0.2−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

X(1)X(3)

X(2

)

(e) α = 9.48 (f) α = 9.508

Fig. 8. Solutions and bifurcation diagrams of system (9) versus increasing α.

1874 R. Li et al.

−0.10

0.10.2

0.30.4

0.50.6

−0.6

−0.4

−0.2

0

0.2−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

X(1)X(3)

X(2

)

−0.5

0

0.5

−0.5

0

0.5−0.05

0

0.05

X(1)X(3)

X(2

)

(g) α = 9.55 (h) α = 9.7

−0.5

0

0.5

−0.5

0

0.5−0.05

0

0.05

X(1)X(3)

X(2

)

−0.5

0

0.5

−0.5

0

0.5−0.05

0

0.05

X(1)X(3)

X(2

)

(i) α = 9.8432 (j) α = 9.8432


chaos of the system with periodic segments split-ting the region of chaos into several parts. And,with decreasing γ, some periodic orbits and somechaotic orbits of system (9) are shown in Figs. 9(c)–9(j). It can be easily seen that although these orbitslook different, they share many similarities in termsof shape and location. Figures 9(k) and 9(l) showtwo symmetric periodic orbits generated by a tinychange of γ.

3.3. Fixed α = 10.5, γ = 0.005, k = 2

Now, fix α = 10.5, γ = 0.005, k = 2 and let param-eter β vary. When 0 < β < 2.87, the orbit of system(9) first wanders around and then escapes to infin-ity. As shown in Fig. 13, in this case, the param-eters are out of the Lagrange stable region. Whenβ > 17.1, the system orbit converges to a fixed pointprovided that the initial value is small (as shownin Fig. 10(b)). Inverse period-doubling bifurcation

of system (9) in the region of 11.95 < β < 14 isshown in Fig. 10(a). One can see that the bifur-cation diagram is similar to Fig. 8(a). A chaoticorbit of system (9) is shown in Fig. 10(c). As βincreases, the orbit evolves to a three-loop-like peri-odic one through inverse bifurcation, as shown inFig. 10(d). It is clear from Figs. 9(c)–9(l) that,although the two orbits look different, they sharesome similarities in terms of shape and location.Furthermore, two quite different cyclic orbits areshown in Figs. 10(e) and 10(f).

Furthermore, fix γ = 0.005 and k = 2 andlet parameters α and β vary. Figure 11 shows alocal parameters division (9.1 ≤ α ≤ 10.5, 10.5 ≤β ≤ 18). In Fig. 11, there are mainly four regions:blue, green, yellow, and red divisions, represent-ing convergent areas (F), periodic strip (P), one-scroll chaotic areas (S) and double-scrolls chaoticareas (C). Pale yellow represents the transitionareas (T). It can be easily seen that the dynamics


0 500 1000 1500 2000 2500 3000−0.5

0

0.5

1

1.5

2

2.5x 10

4

t

x

x1x2x3

(a) Inverse bifurcation from chaos to periodic orbit (b) γ = 26

−0.5

0

0.5

−1

−0.5

0

0.5

1−0.05

0

0.05

X(1)X(3)

X(2

)

−0.5

0

0.5

−1

−0.5

0

0.5

1−0.05

0

0.05

X(1)X(3)

X(2

)

(c) γ = 0.0244 (d) γ = 0.024

−0.5

0

0.5

−1

−0.5

0

0.5

1−0.05

0

0.05

X(1)X(3)

X(2

)

−0.5

0

0.5

−1

−0.5

0

0.5

1−0.05

0

0.05

X(1)X(3)

X(2

)

(e) γ = 0.0234 (f) γ = 0.02331

Fig. 9. Solutions and bifurcation diagrams of system (9) versus varying γ.

1876 R. Li et al.

−0.5

0

0.5

−0.5

0

0.5−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

X(1)X(3)

X(2

)

−0.5

0

0.5

−0.5

0

0.5−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

X(1)X(3)

X(2

)

(g) γ = 0.0144 (h) γ = 0.01417

−0.5

0

0.5

−1

−0.5

0

0.5

1−0.06

−0.04

−0.02

0

0.02

0.04

0.06

X(1)X(3)

X(2

)

−0.5

0

0.5

−1

−0.5

0

0.5

1−0.06

−0.04

−0.02

0

0.02

0.04

0.06

X(1)X(3)

X(2

)

(i) γ = 0.0111 (j) γ = 0.0106

−0.5

0

0.5

−1

−0.5

0

0.5

1−0.05

0

0.05

X(1)X(3)

X(2

)

−0.5

0

0.5

−1

−0.5

0

0.5

1−0.05

0

0.05

X(1)X(3)

X(2

)

(k) γ = 0.01485 (l) γ = 0.01486



00.05

0.10.15

0.20.25

0.30.35

−0.6

−0.4

−0.2

0

0.2−0.04

−0.02

0

0.02

0.04

0.06

0.08

X(1)X(3)

X(2

)(a) Inverse bifurcation from chaos to periodic orbit (b) β = 19.3

−0.5

0

0.5

−0.5

0

0.5−0.05

0

0.05

X(1)X(3)

X(2

)

−0.5

0

0.5

−0.5

0

0.5−0.05

0

0.05

X(1)X(3)

X(2

)

(c) β = 12.897 (d) β = 12.9

−0.5

0

0.5

−0.5

0

0.5−0.05

0

0.05

X(1)X(3)

X(2

)

−0.5

0

0.5

−0.5

0

0.5−0.05

0

0.05

X(1)X(3)

X(2

)

(e) β = 12.479 (f) β = 12.6

Fig. 10. Solutions and bifurcation diagrams of system (9) versus varying β.

1878 R. Li et al.

Fig. 11. Local parameter regions of system (9).

of system (9) are varied following the trace. Theline “a → h” begins with the periodic strip (P),crosses the transition areas (T) and then reachesthe one-scroll chaotic areas (S). Subsequently, thesystem begins with the periodic strip (P) again, butat this time the shapes of circuits being differentfrom previous ones, crosses transition areas (T) andreaches double-scrolls chaotic areas. Finally, Fig. 11shows the processes of the system dynamics fromchaos to nonchaos and then from nonchaos to chaos,with periodic segments splitting the region of chaosinto several parts.

4. Parameter Regions of LagrangeStability

Lagrange stability is very important in chaosstudy since chaotic orbits are all bounded. Whenthe Lagrange stable conditions are not satisfied,unbounded solutions of a chaotic system may exist[Duan et al., 2006]. In what follows, correspond-ing to the analysis of the dynamics of system(9) in Sec. 3, an analysis of parameter regionsof Lagrange stability of the system is carried outin detail.

Take parameter c < 0 in system (9). The linearpart of system (9) can be written as

A =

−α − akα α 0

1 −1 10 −β −βγ

. (10)

As pointed out in Remark 3, system (9) is Lagrangestable if A is Hurwitz. The characteristic equation

of (10) is

f(λ) = λ3 + (1 + βγ + (1 + ka)α)λ2

+ (β + βγ + (1 + ka)αβγ + kaα)λ+ αβ(1 + ka(1 + γ)) (11)

Letd1 = 1 + βγ + (1 + ka)α,

d2 = β + βγ + (1 + ka)αβγ + kaα,

d3 = αβ(1 + ka(1 + γ)),d4 = d1d2 − d3.

(12)

Then, system (9) is Lagrange stable when d1, d2, d3

and d4 are all positive. Obviously, condition (12) isindependent of the parameter b. In the following,parameter regions of the Lagrange stability are dis-cussed for two cases.

Case I. Fixed k = 2, a = −0.02, β = 12

From condition (12), d1 is positive when parametersα and γ satisfy α > −1 − 12γ/0.96; while d3 is pos-itive when α > 0 and γ < 24, or α < 0 and γ > 24.Take the following coordinates transformation:

α = cos θξ − sin θη,

γ = sin θξ + cos θη.

It is easy to see that d2 = 0 is a hyperbola on theα–γ plane with asymptotes γ = 0.0035 and α =−1.0417. For the variable γ, d4 = 0 can be seen asa cluster of parabolas with respect to parameter α.In Fig. 12, the Lagrange stable region of system (9)is shown in the range surrounded by the blue colorcurve.

Case II. Fixed k = 2, a = −0.02, γ = 0.005

Similarly to Case I, from condition (12), d1 ispositive when parameters α and β satisfy α >−1 − 0.005β/0.96, and d3 is positive when α and

24

Lagrange stableregion

γ

α0.0035

−0.0833 17.1645 152.72950.0044

Fig. 12. α–γ plane.


Lagrange stable region

β

α

8.3333

2.87

10.53982.7147E005

−1.0417

−0.01

Fig. 13. α–β plane.

β have the same sign. Through the same coordi-nates transformation as in Case I, d2 = 0 repre-sents a hyperbola on the α–β plane with asymptotesβ = 8.3333 and α = −209.375. And, for the variableβ, d4 = 0 can be seen as a cluster of parabolas withrespect to α. In Fig. 13, the Lagrange stable regionof system (9) is shown in the range surrounded bythe blue color curve.

Compared with the Lagrange stability analysison the smooth Chua’s circuit [Duan et al., 2006],the analysis of the Lagrange stable region for (9) isharder because the parameters are coupled togetherin (12).

5. Attraction-Repulsion FunctionVersus Cubic Function

It is well known that Chua’s circuit with a cubicfunction can generate a large variety of chaos andbifurcation phenomena, and is structurally the sim-plest but dynamically the most complex, hencemore frequently being used in analysis and designof chaotic circuits [Altman, 1993a, 1993b; Huanget al., 1996; Zhong, 1994].

In this section, some computer simulations onthe modified Chua’s circuit with the attraction-repulsion function are presented, in comparison

with those on the modified Chua’s circuit with thecubic function.

Consider the dimensionless Chua’s circuit usedin [Tsuneda, 2005]:

x = kα(y − x − f(x)),

y = k(x − y + z),

z = k(−βy − γz),

(13)

where f(x) = ax3 + bx and k = ±1.Let a = ka, b = kb, γ = βγ. Then, system (9)

can be rewritten as

x1 = kα(x2 − x1 − f(x1)),

x2 = k(x1 − x2 + x3),

x3 = k(−βx2 − γx3),

(14)

where f(x) = ax + bx exp(cx2) and k = ±1.Although systems (13) and (14) are similar

in form, they have different nonlinear parts. Sincec is a fixed negative constant, f(x) and f(x)have two variable parameters, respectively. Becausebx exp(cx2) is bounded, the linear part ax is the pri-mary term of f(x) as x tends to infinity. However,for f(x), the nonlinear part ax3 is the primary termas x tends to infinity. So, the speed of f(x) is fasterthan f(x) as they tend to infinity.

In the following, some comparisons are pre-sented on the solution orbits of the two Chua’scircuits. In Table 1 of [Tsuneda, 2005], 20 sets ofparameter values for system (13) were given. Here,select several sets of the given parameter values (asshown in Table 2) in simulations (see Figs. 14–17).

For Chua’s circuit with the attraction-repulsionfunction, some periodic and chaotic orbits areobtained by using the same parameters α, β andγ as used in the circuit with the cubic function,by adjusting the parameters a and b. Here, bychoosing k = 1 (see Table 3), one can observesome similarities and some differences betweenthe dynamics of the two Chua’s circuits. Somecorresponding computer simulations are shownin Fig. 18.

Table 2.

No. α = α β = β γ = γ a = a b = b k = k Figure

C-2 3.7091002664 24.0799705758 −0.8592556780 0.4530092443 −2.9315446532 1.0 Fig. 14C-8 4.006 54.459671 −0.93435708 −0.0375582129 −0.8415410391 −1.0 Fig. 15C-11 15.6 28.58 0 0.0659179490 −1.1671315463 1.0 Fig. 16C-16 12.141414 95.721132 −0.8982235 −0.0375582129 −0.8415410391 −1.0 Fig. 17

1880 R. Li et al.

−3 −2 −1 0 1 2 3−4

−3

−2

−1

0

1

2

3

4

y

f(y)

cubicattraction–repulsion

(a) Two functions

−4−2

02

4

−20

−10

0

10

20−3

−2

−1

0

1

2

3

XZ

Y

(b) Cubic function

−1

0

1

2

−10

−5

0

5

10−1.5

−1

−0.5

0

0.5

1

1.5

X(1)X(3)

X(2

)

(c) Attraction-repulsion

Fig. 14. Nonlinear functions and solutions of system C-2.

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

y

f(y)


(a) Two functions

−1

0

1

2

3

−30

−20

−10

0

10

20−3

−2

−1

0

1

2

3

XZ

Y

(b) Cubic function

00.2

0.40.6

0.81

−1

−0.5

0

0.5−0.04

−0.03

−0.02

−0.01

0

0.01

X(1)X(3)

X(2

)




−3 −2 −1 0 1 2 3−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

y

f(y)


(a) Two functions

−3−2

−10

12

3

−4

−2

0

2

4−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

XZ

Y

(b) Cubic function

−0.10

0.10.2

0.30.4

0.50.6

−1

−0.5

0

0.5−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

X(1)X(3)

X(2

)



−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

y

f(y)


(a) Two functions

−1

0

1

2

3

−15

−10

−5

0

5

10

15−1.5

−1

−0.5

0

0.5

1

1.5

XZ

Y

(b) Cubic function

0

0.5

1

1.5

−1

−0.5

0

0.5−0.06

−0.04

−0.02

0

0.02

X(1)X(3)

X(2

)



1882 R. Li et al.

Table 3.

No. α β γ a b c Figure

C-2′ 3.7091002664 24.0799705758 −0.8592556780 0.9060184886 −5.8630893064 −0.5 Fig. 18(A-1)C-2′ 3.7091002664 24.0799705758 −0.8592556780 0.4060184886 −2.9315446532 −0.5 Fig. 18(A-2)C-2′ 3.7091002664 24.0799705758 −0.8592556780 0.4260184886 −2.9315446532 −0.5 Fig. 18(A-3)C-2′ 3.7091002664 24.0799705758 −0.8592556780 0.4960184886 −2.9315446532 −0.5 Fig. 18(A-4)C-8′ 4.006 54.459671 −0.93435708 −0.59 −3.883082 −0.5 Fig. 18(B-1)C-8′ 4.006 54.459671 −0.93435708 −0.685 −3.883082 −0.5 Fig. 18(B-2)C-11′ 15.6 28.58 0 −0.12 −1.0 −0.5 Fig. 18(C-1)C-11′ 15.6 28.58 0 −0.14 −1.0 −0.5 Fig. 18(C-2)C-16′ 12.141414 95.721132 −0.8982235 −0.852 −1.6830820782 −0.5 Fig. 18(D-1)C-16′ 12.141414 95.721132 −0.8982235 −0.904 −1.6830820782 −0.5 Fig. 18(D-2)

−3−2

−10

12

3

−20

−10

0

10

20−3

−2

−1

0

1

2

3

X(1)X(3)

X(2

)

−1−0.5

00.5

11.5

2

−10

−5

0

5

10−1.5

−1

−0.5

0

0.5

1

1.5

X(1)X(3)

X(2

)

(A-1) (A-2)

−1−0.5

00.5

11.5

2

−10

−5

0

5

10−1.5

−1

−0.5

0

0.5

1

1.5

X(1)X(3)

X(2

)

−1−0.5

00.5

11.5

2

−10

−5

0

5

10−1.5

−1

−0.5

0

0.5

1

1.5

X(1)X(3)

X(2

)

(A-3) (A-4)

Fig. 18. Chua’s circuit with an attraction-repulsion function.


−2−1

01

23

45

−40

−20

0

20

40−5

0

5

X(1)X(3)

X(2

)

−5

0

5

−40

−20

0

20

40−5

0

5

X(1)X(3)

X(2

)

(B-1) (B-2)

−0.20

0.20.4

0.60.8

−1.5

−1

−0.5

0

0.5−0.1

−0.05

0

0.05

0.1

X(1)X(3)

X(2

)

−1

−0.5

0

0.5

1

−1.5

−1

−0.5

0

0.5

1

1.5−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

X(1)X(3)

X(2

)

(C-1) (C-2)

−5

0

5

−40

−20

0

20

40−3

−2

−1

0

1

2

3

X(1)X(3)

X(2

)

−10

−5

0

5

10

−3

−2

−1

0

1

2

3−30

−20

−10

0

10

20

30

X(1)X(2)

X(3

)

(D-1) (D-2)


6. Generalization to a Multi-ScrollCircuit

It is well known that Chua’s circuit with differ-ent nonlinear functions, such as multi-breakpointpiecewise-linear function, sine function and hys-teresis function, can generate complex multi-scroll

chaotic attractors [Lu & Chen, 2006] and referencestherein. In order to study the complex dynamicsof the modified Chua’s circuit (9), in this sectionthe attention is focused on the generation of multi-scroll attractors [Wang, 2007]. For this purpose,the attraction-repulsion function in system (9) is

1884 R. Li et al.

−5 −4 −3 −2 −1 0 1 2 3 4 5−1.5

−1

−0.5

0

0.5

1

1.5

(a) Function f(x)

−4 −3 −2 −1 0 1 2 3 4−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

x

y

(b) Projection on (x, y)-plane

(c) Attractor in (y, x, z)-space

Fig. 19. The 3-scroll attractor generated by system (9) withfunction (16) and initial value x(0) = (0.1, 0.1, 0.1).

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

x

f(x)

(a) Function f(x)

−1.5 −1 −0.5 0 0.5 1 1.5−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

x

y





−1.5 −1 −0.5 0 0.5 1 1.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

x

f(x)

(a) Function f(x)

−4 −3 −2 −1 0 1 2 3 4−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

x

y




further modified, as

f(x) = f1(x) − f2(x) · · · + fn−1(x) − fn(x) + · · · ,(15)

where fi(x) = aix + bixecix2, i = 1, 2, . . . , n. Obvi-

ously, function (15) is obtained by overlapping somedifferent attraction-repulsion functions. And it maybe simplified as follows:

f(x) = ax + b1xec1x2 − b2xec2x2 · · · + bn−1xecn−1x2

− bnxecnx2 · · · . (16)

By choosing appropriate parameters, the num-ber of equilibria of function (16) can be increased.Substituting function (16) into system (9), corre-sponding multi-scroll attractors may be obtained.For example, by letting parameters n = 2, a =−0.35, b1 = 0.8, b2 = 1.3, c1 = −0.1, c2 = −1, α =15, β = 20, γ = 0 and k = 1, a 3-scroll attractoris generated in system (9) with function (16), asshown in Fig. 19.

Moreover, by choosing parameters n = 3, a =−0.4, b1 = 1, b2 = 2, b3 = 2, c1 = −1, c2 =−5, c3 = −12, α = 15, β = 20, γ = 0 andk = 1, a 4-scroll attractor is obtained, as shownin Fig. 20.

As one more example, when choosing param-eters n = 4, a = −0.72, b1 = 1.6, b2 = 2.8, b3 =2, b4 = 3, c1 = −0.1, c2 = −0.55, c3 = −1.4, c4 =−3.8, α = 17, β = 25, γ = 0 and k = 1,a 5-scroll attractor is generated, as shown inFig. 21.

7. Circuit Design and ExperimentalObservations

In this section, a simplified physical circuit is builtto visualize the realization of the new system.

Since the attraction-repulsion function is anexponential function, as is well known, a truncationof the exponential function is necessary in build-ing a real circuit for its realization. So, an approxi-mate circuitry was built, as shown in Fig. 22, for thepurpose of visualization but not accurate circuitrydesign.

The attraction-repulsion function of system (2)is thus replaced by its approximation f(x) =−0.52x + 0.25x3, and the generated one-scrollchaotic orbits and double-scroll chaotic orbits areshown in Fig. 23, obtained by tuning the adjustableresistor.

1886 R. Li et al.

Fig. 22. Implementation of the chaotic circuit (2), where all the active components are supplied by ±12 volts.

Projection on the x1–x2 plane Projection on the x2–x3 plane

Fig. 23. Experimental observations of system (2).


Projection on the x1–x2 plane Projection on the x1–x3 plane


8. Conclusions

In this paper, a new modified Chua’s circuit with anattraction-repulsion function has been presented.Basic dynamical properties have been analyzed,including pitchfork bifurcation, Hopf bifurcationand various chaotic behaviors of the circuit, underLagrange stability conditions. Some chaotic behav-iors have also been verified by a simplified electroniccircuit. This paper confirms that the attraction-repulsion function can play a similar role as thepiecewise-linear function and the cubic function inrealizing Chua’s circuits.

Acknowledgments

The authors wish to thank Prof. Wenbo Liu forher assistance in circuit implementation of the newsystem.

References

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A MODIFIED CHUA’S CIRCUIT WITH AN ATTRACTION-REPULSION … · 2016-01-22 · f(y)=ay +byexp(cy2),...

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