A MODIFIED CHUA S CIRCUIT WITH AN ATTRACTION-REPULSION FUNCTION *

Transcription

1 Papers International Journal of Bifurcation and Chaos, Vol. 8, No. 7 (8) c World Scientific Publishing Company A MODIFIED CHUA S CIRCUIT WITH AN ATTRACTION-REPULSION FUNCTION * RONG LI, ZHISHENG DUAN and BO WANG State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Aerospace Engineering, College of Engineering, Peking University, Beijing 87, P. R. China lirong@pku.edu.cn GUANRONG CHEN Department of Electronic Engineering, City University of Hong Kong, Hong Kong SAR, P. R. China Received June, 7; Revised August 6, 7 In this paper, the original Chua s circuit is modified by substituting its piecewise-linear function with an attraction-repulsion function. Some new complex dynamical behaviors such as chaos are observed through computer simulations. Basic properties of the new circuit are analyzed by means of bifurcation diagrams. Lagrange stability conditions of the circuit are derived. A comparison between this modified Chua s circuit with an attraction-repulsion function and the modified Chua s circuit with a cubic nonlinear function is presented. Moreover, a generalization of the new circuit that can generate multiple scrolls is designed and simulated. Finally, a physical circuit is built to visualize the new system, with some experimental observations reported. Keywords: Attraction-repulsion function; chaos; bifurcation; Lagrange stability; Chua s circuit.. Introduction Chaotic dynamics in physical systems have been observed and analyzed for a long time in the nonlinear science history, with the first physical chaotic system built in the laboratory by Chua and his colleagues [Chua et al., 986; Chua, a]. Today, Chua s circuit has been recognized as a classical paradigm in the study of chaos and bifurcation theories, in which the piecewise-linear nonlinearity plays an important role which was modified in several different ways particularly by a hysteresis function [Kennedy & Chua, 99], a cubic function [Zhong, 994], or a sine function [Tang et al., ], etc. In this paper, the original Chua s circuit is further modified with an attraction-repulsion function. The attraction-repulsion function is a nonlinear function, with long-range attraction and shortrange repulsion, useful in the study of swarming behaviors in biological systems [Okubo, 986; Gazi & Passino, a, b]. For further theoretical study and practical applications of chaos theory to circuits and swarms, it is worth investigating the possible use of the attraction-repulsion function to chaos theory [Duan et al., ]. This work is supported by the City University of Hong Kong under the Research Enhancement Scheme and SRG grant 74 and the National Science Foundation of China under grants 66749, 64. Author for correspondence 86

2 866 R. Li et al. This paper reports the findings of the new modified Chua s circuit with the attraction-repulsion nonlinearity, including some new chaotic phenomena, period-doubling bifurcations, periodic orbits, etc., observed through computer simulations. Moreover, the Lagrange stability (i.e. the boundedness of all solutions) are analyzed since it is important in the study of chaos [Duan et al., 6]. The outline of this paper is as follows. In Sec., the modified Chua s circuit with an attractionrepulsion function is introduced, and its basic properties including periodic orbits and period-doubling bifurcations are analyzed. In Sec., the modified Chua s circuit in the dimensionless form is further studied, with simulation results showing various complex chaotic dynamics. In Sec. 4, the parameter region of the Lagrange stability is given and analyzed for the dimensionless modified Chua s circuit. In Sec., some computer simulations on the modified Chua s circuit with an attraction-repulsion function are presented, in comparison with those on the modified Chua s circuit with a cubic function. In Sec. 6, a generalization of the new circuit to one that can generate multiple scrolls is presented. In Sec. 7, a circuit realization of the new modified Chua s circuit is presented with some experimental observations. Finally, Sec. 8 concludes the paper.. The New Chaotic Circuit and Its Basic Properties The original canonical Chua s circuit is shown in Fig., which is synthesized by using five linear elements (two capacitors, C, C, one inductor, L, and two resistors, R, R ), along with one nonlinear resistor (N R ) which can be built on off-the-shelf op-amps. The circuit equations follow from Kirchoff s laws: dv C C = (v C v C ) g(v C ), dt R dv C C = (v C v C )+i L, dt R Fig.. Chua s circuit. mathematically as g(v C )=m v C +.(m m )( v C + b v C b ), in which m and m are appropriate constants with m < andm < [Chua, b], as shown in Fig.. Now, substitute the piecewise-linear function g(v C ) with an attraction-repulsion function defined by f(y) =ay + by exp(cy ), () where a, b, andc are constants. For the y R case, two graphs of the function () are shown in Fig.. Obviously, one function has three real roots while the other has only one real root. With the attraction-repulsion function, the modified Chua s circuit is described by dv C = (v C v C ) f(v C ), dt C R C dv C dt = C R (v C v C )+ C i L, di L dt = L v C L R i L. () L di L dt = v C R i L, where v C and v C are the voltages across the capacitors C and C, respectively, i L denotes the current through the inductance L, and the term g(v C ) represents the characteristic of the nonlinear resistance. Here, g(v C ) can be expressed Fig.. Piecewise-linear v i function.

3 A Modified Chua s Circuit with an Attraction-Repulsion Function 867 f(y) a=.,b.,c=. a=.,b=.,c=..4 y Fig.. Two graphs of the attraction-repulsion function... Basic dynamical analysis: Equilibria The equilibria of the new circuit can be found by solving the following algebraic equations simultaneously: (v C v C ) f(v C )=, C R C (v C v C )+ i L =, () C R C L v C L R i L =. It follows from () that v C = (R + R )i L, v C = R i L, i L [R + ar R + ar + br (R + R )exp{c(r + R ) i L}] =. Obviously, S =[,, ] is one equilibrium. Furthermore, by setting ( q = c(r + R ) ln a ) b, b(r + R ) two other equilibria can be obtained, as S =[ (R + R ) q, R q, q] S =[(R + R ) q, R q, q]. and Remark. System () is symmetric with respect to the origin, which can be proved via the following transformation: (v C,v C,i L ) ( v C, v C, i L ). And obviously S and S are symmetric with respect to the origin. Remark. System () has three real equilibria when parameters a and b satisfy the following inequality: < a b <. (4) b(r + R ) Throughout this section, for numerical simulations, take the parameter values as C =., C =., R =., R =, L =.. Then, condition (4) becomes.497 < a < b.497 () Remark. Let matrix A be the linear part of system (). Then, system () can be rewritten as V = AV + Bf (v C ), (6) where V = v C v C i L a C R C C R, A = C R C R C, R L L b B = C, and f (v C )=v C exp(cv C ). It is easy to see that f is bounded when c<. Here, a is the only variable parameter of A and A is Hurwitz stable when a [.497, ). Under the conditions that f is bounded and A is Hurwitz stable, one can easily draw the conclusion that system () is Lagrange stable, i.e. all solutions of system () are bounded. When A is unstable, the Lagrange stability conditions are not satisfied. Computer simulation shows the existence of unbounded solutions of system (), as displayed in Fig. 4, with initial value x = (.,.9,.) T. The following investigation of dynamical behaviors of systems is under the Lagrange stability conditions.

4 868 R. Li et al.. x 7 Vc Vc Table. a Eigenvalues of S.488 λ =, λ, =.648 ±.44i.9 λ =.64, λ, = ±.644i.499 λ = 4.977, λ, = ±.i t Fig. 4. An unbounded solution of () with a =.4976, b= c =.... Stability of equilibria By linearizing system () at S, one obtains the characteristic equation ( f(λ) =λ + + (a + b)+ + R ) λ C R C C R L [ a + b + + R C C R C R L + (a + b)r C L + ( C R C J = R + R + ( a + R + R + ( )] R + λ + a + b ( ) R + C L R C C L R + C C R L. (7) Let b =.. Then, based on the Routh Hurwitz criterion, S is stable when.488 < a <.9 or a>.499. Table shows the eigenvalues of system () at S with respect to the variable parameter a. When.488 <a<.9 or a>.499, the system orbit is a fixed point S (as shown in Figs. (a) and (d)). When a =.499, the system orbits constitute a cluster of periodic orbits surrounding S (Fig. (b)). When.9 <a<.499, although S is an unstable equilibrium, system () is Lagrange stable and the system orbit is a stable periodic orbit [Fig. (c)]. When a <.488, based on condition (), the system has a pitchfork bifurcation and produces two new stable equilibria, S and S, simultaneously. By linearizing system () at S (or S ), one obtains the Jaccobian ) ln ( a )) b b(r + R ) C R C R C R C L R L, by setting p = /(R + R ) + (a + (/ R + R )) ln( (a/b) (/b(r + R ))), one obtains the following characteristic equation: f(λ) =λ + [ + ( p + + R C R C C R L p C C R + R + C L + ( R R + C R L pr C L )] p λ C C L ) λ ( ) R + R C C R L. (8) By some tedious numerical calculations, one can verify that S and S are stable equilibria for a (.6,.) or a (.497,.49679) when b =.. Let a =.6, a = Then, by calculation again, one can see that the conjugate roots of Eq. (8) satisfy the following conditions of Hopf bifurcation: Re λ(a) a=a,a =, Im λ(a) a=a,a, dre λ(a) da, a=a,a

5 A Modified Chua s Circuit with an Attraction-Repulsion Function 869 x x 4 Vc... Vc x 4 Vc x x Vc.... (a) a =.6 (b) a =.499 Vc... Vc... x.. Vc Vc... (c) a =. (d) a =. Fig.. Solutions of () with b = c =.. where a,a are critical values, implying that system () has a Hopf bifurcation. With respect to the variable parameter a, the bifurcation diagram and the Lyapunov-exponent spectrum (see Figs. 6(a) and 6(b)) show the transition from periodic orbits to chaos, when a [.,.6]. When system () is chaotic, it has a positive Lyapunov exponent. Some periodic orbits appear when the maximum Lyapunov exponent equals zero. The corresponding periodic orbits and Lyapunov-exponent spectrum versus time t are shown in Figs. 6(c) 6(k). Here, the initial conditions are chosen near the origin. Remark 4. Under the conditions of Lagrange stability, the range of the parameter c (c <) is comparatively large for system () to produce bifurcations and chaos. Throughout, set c =.. Remark. When parameters a, b, c are all negative, it is clear from condition () that one only needs to analyze the constant item of Eq. (7) in the investigation of the stability of S.Whenb<.497, system () has three equilibria and S is unstable according to the Routh Hurwize criterion. When b>.497, S can change from stable to unstable with respect to the variable parameters a, b, and system () can produce a pitchfork bifurcation simultaneously. For example, when b =.4 and.97 <a<.88, system () has only one real and stable equilibrium S [see Fig. 7(a)]. As the parameter a decreases, two new stable equilibria S and S appear from pitchfork bifurcation. Although system () can also produce chaos with respect to the parameters a and b [see Fig. 7(b)], the region of chaos is very small.

6 87 R. Li et al... Lyapunov exponents of Dynamics... Vc... Lyapunov exponents a x a x (a) Bifurcation diagram of v C : a (.,.6) (b) Lyaponov-exponent spectrum: a (.,.6) Vc..... Vc Vc Vc.... (c) a =.6 (d) a = Vc Vc.... (e) a =.77 Fig. 6. Evolution process as a is changing with b = c =..

7 A Modified Chua s Circuit with an Attraction-Repulsion Function..... Vc. Vc Vc (f) a =.9... Vc (g) a = Vc Vc (h) Chaotic solution of system (): a =. (i) Lyapunov-exponents spectrum when a = Vc Vc a Vc (j) Chaotic solution of system (): a =.494 Fig. 6. (k) Bifurcation diagram of vc : a (.49,.494) (Continued )

8 87 R. Li et al..8.6 Vc Vc Vc t Vc (a) a =.9,c=. (b) a =.488,c=. Fig. 7. Solution of system () with initial value x() = (.6,.8,.).. Numerical Analysis of the Normalized Circuit For convenient numerical analysis of system (), rescale the parameters with x = v C,x = v C,x = R i L,t=(/C R )τ, α = C /C,β = C R /L, γ = R /R,k= R and redefine τ with t. Then, the normalized equations of the modified Chua s circuit are given by ẋ = α(x x kf(x )), ẋ = x x + x, ẋ = β( x γx ), (9) where f(x) =ax + bx exp(cx ). Fix a =., b= c =.. Then, the dynamics of system (9) depend only on the parameters α, β, γ and k. Let p = /c ln( ( + kα( + γ))/(kb( + γ))). One obtains three equilibria of system (9): [ γ p ] S =[,, ], S = p, ( + γ), p, ( + γ) S = [ γ p p, ( + γ), p ( + γ) ]. Obviously, these equilibria are symmetric with respect to the origin. Similarly to system (), system (9) is Lagrange stable if its linear part is Hurwitz stable. Several simulations have been carried out, with some new findings summarized as follows:.. Fixed β =,γ =., k = When <α<8., the orbit of system (9) is a fixed nonzero point; when 8. <α<9.orα>., the system has a periodic orbit. From the analysis of the parameter region of Lagrange stability in Sec. 4 (see Fig. ), it can be seen that, in the present case, the parameters are located in the Lagrange stable region. As shown in Fig. 8(a), system (9) evolves to chaos through a series of perioddoubling bifurcations when α [9.,.]. One part of Fig. 8(a) is magnified and displayed in Fig. 8(b), which shows the transitions from chaos to nonchaos and from nonchaos to chaos of system (9) with a periodic segment splitting a region of chaos into several parts. Interestingly, system (9) displays a one-scroll chaotic orbit when α [9., 9.69]. Moreover, the system phase portrait evolves to another cyclic orbit when α [9.7, 9.844], with the corresponding portraits are shown in Figs. 8(c) 8(j)... Fixed α =., β =,k = In this subsection, the dynamics of system (9) in the region of <γ<4 are studied. When γ>4, system (9) does not satisfy the Lagrange stability conditions (see Fig. ) and it escapes to infinity (see Fig. 9(b)). When. < γ < 4, the system orbit converges to a fixed point; when < γ <.4, the system orbit is a periodic orbit. Figure 9(a) shows the inverse bifurcation diagram of system (9) in the region of.4 < γ <.6, where the system has very rich dynamical behaviors. As in Sec.., the inverse bifurcation diagram also shows the transitions from chaos to nonchaos and from nonchaos to

9 A Modified Chua s Circuit with an Attraction-Repulsion Function 87 (a) Bifurcation diagram of x : α (9.,.) (b) Bifurcation diagram of x : α (9.7, 9.9) X() X()..4. (c) α =9. (d) α = X() X()..4. (e) α =9.48 (f) α =9.8 Fig. 8. Solutions and bifurcation diagrams of system (9) versus increasing α.

10 874 R. Li et al X() X(). (g) α =9. (h) α = X().. X() (i) α =9.84 (j) α =9.84 Fig. 8. (Continued ) chaos of the system with periodic segments splitting the region of chaos into several parts. And, with decreasing γ, some periodic orbits and some chaotic orbits of system (9) are shown in Figs. 9(c) 9(j). It can be easily seen that although these orbits look different, they share many similarities in terms of shape and location. Figures 9(k) and 9(l) show two symmetric periodic orbits generated by a tiny change of γ... Fixed α =., γ =., k = Now, fix α =., γ=., k= and let parameter β vary. When <β<.87, the orbit of system (9) first wanders around and then escapes to infinity. As shown in Fig., in this case, the parameters are out of the Lagrange stable region. When β>7., the system orbit converges to a fixed point provided that the initial value is small (as shown in Fig. (b)). Inverse period-doubling bifurcation of system (9) in the region of.9 <β<4 is shown in Fig. (a). One can see that the bifurcation diagram is similar to Fig. 8(a). A chaotic orbit of system (9) is shown in Fig. (c). As β increases, the orbit evolves to a three-loop-like periodic one through inverse bifurcation, as shown in Fig. (d). It is clear from Figs. 9(c) 9(l) that, although the two orbits look different, they share some similarities in terms of shape and location. Furthermore, two quite different cyclic orbits are shown in Figs. (e) and (f). Furthermore, fix γ =. and k = and let parameters α and β vary. Figure shows a local parameters division (9. α.,. β 8). In Fig., there are mainly four regions: blue, green, yellow, and red divisions, representing convergent areas (F), periodic strip (P), onescroll chaotic areas (S) and double-scrolls chaotic areas (C). Pale yellow represents the transition areas (T). It can be easily seen that the dynamics

11 A Modified Chua s Circuit with an Attraction-Repulsion Function 87. x 4 x x x. x.. t (a) Inverse bifurcation from chaos to periodic orbit (b) γ = X(). X() (c) γ =.44 (d) γ = X(). X() (e) γ =.4 (f) γ =. Fig. 9. Solutions and bifurcation diagrams of system (9) versus varying γ.

12 876 R. Li et al X().. X() (g) γ =.44 (h) γ = X(). X() (i) γ =. (j) γ = X(). X() (k) γ =.48 (l) γ =.486 Fig. 9. (Continued )

13 A Modified Chua s Circuit with an Attraction-Repulsion Function X().... (a) Inverse bifurcation from chaos to periodic orbit (b) β = X().. X() (c) β =.897 (d) β = X().. X() (e) β =.479 (f) β =.6 Fig.. Solutions and bifurcation diagrams of system (9) versus varying β.

14 878 R. Li et al. of () is f(λ) =λ +(+βγ +(+ka)α)λ +(β + βγ +(+ka)αβγ + kaα)λ + αβ( + ka( + γ)) () Let d =+βγ +(+ka)α, d = β + βγ +(+ka)αβγ + kaα, () d = αβ( + ka( + γ)), d 4 = d d d. Fig.. Local parameter regions of system (9). of system (9) are varied following the trace. The line a h begins with the periodic strip (P), crosses the transition areas (T) and then reaches the one-scroll chaotic areas (S). Subsequently, the system begins with the periodic strip (P) again, but at this time the shapes of circuits being different from previous ones, crosses transition areas (T) and reaches double-scrolls chaotic areas. Finally, Fig. shows the processes of the system dynamics from chaos to nonchaos and then from nonchaos to chaos, with periodic segments splitting the region of chaos into several parts. 4. Parameter Regions of Lagrange Stability Lagrange stability is very important in chaos study since chaotic orbits are all bounded. When the Lagrange stable conditions are not satisfied, unbounded solutions of a chaotic system may exist [Duan et al., 6]. In what follows, corresponding to the analysis of the dynamics of system (9) in Sec., an analysis of parameter regions of Lagrange stability of the system is carried out in detail. Take parameter c< in system (9). The linear part of system (9) can be written as α akα α A =. () β βγ As pointed out in Remark, system (9) is Lagrange stable if A is Hurwitz. The characteristic equation Then, system (9) is Lagrange stable when d,d,d and d 4 are all positive. Obviously, condition () is independent of the parameter b. In the following, parameter regions of the Lagrange stability are discussed for two cases. Case I. Fixed k =,a=., β= From condition (), d is positive when parameters α and γ satisfy α> γ/.96; while d is positive when α>andγ<4, or α<andγ>4. Take the following coordinates transformation: α = cos θξ sin θη, γ =sinθξ +cosθη. It is easy to see that d = is a hyperbola on the α γ plane with asymptotes γ =. and α =.47. For the variable γ, d 4 =canbeseenas a cluster of parabolas with respect to parameter α. In Fig., the Lagrange stable region of system (9) is shown in the range surrounded by the blue color curve. Case II. Fixed k =,a =., γ =. Similarly to Case I, from condition (), d is positive when parameters α and β satisfy α >.β/.96, and d is positive when α and γ 4. Lagrange stable region Fig.. α γ plane. α

15 A Modified Chua s Circuit with an Attraction-Repulsion Function 879 β Lagrange stable region Fig.. α β plane..747e β have the same sign. Through the same coordinates transformation as in Case I, d =represents a hyperbola on the α β plane with asymptotes β = 8. and α = 9.7. And, for the variable β, d 4 = can be seen as a cluster of parabolas with respect to α. In Fig., the Lagrange stable region of system (9) is shown in the range surrounded by the blue color curve. Compared with the Lagrange stability analysis on the smooth Chua s circuit [Duan et al., 6], the analysis of the Lagrange stable region for (9) is harder because the parameters are coupled together in ().. Attraction-Repulsion Function Versus Cubic Function It is well known that Chua s circuit with a cubic function can generate a large variety of chaos and bifurcation phenomena, and is structurally the simplest but dynamically the most complex, hence more frequently being used in analysis and design of chaotic circuits [Altman, 99a, 99b; Huang et al., 996; Zhong, 994]. In this section, some computer simulations on the modified Chua s circuit with the attractionrepulsion function are presented, in comparison α with those on the modified Chua s circuit with the cubic function. Consider the dimensionless Chua s circuit used in [Tsuneda, ]: ẋ = ˆk ˆα(y x ˆf(x)), ẏ = ˆk(x y + z), ż = ˆk( ˆβy ˆγz), () where ˆf(x) =âx + ˆbx and ˆk = ±. Let ã = ka, b = kb, γ = βγ. Then, system (9) can be rewritten as ẋ = kα(x x f(x )), ẋ = k(x x + x ), ẋ = k( βx γx ), (4) where f(x) =ãx + bx exp(cx )and k = ±. Although systems () and (4) are similar in form, they have different nonlinear parts. Since c is a fixed negative constant, ˆf(x) and f(x) have two variable parameters, respectively. Because bx exp(cx ) is bounded, the linear part ãx is the primary term of f(x) asx tends to infinity. However, for ˆf(x), the nonlinear part âx is the primary term as x tends to infinity. So, the speed of ˆf(x) isfaster than f(x) as they tend to infinity. In the following, some comparisons are presented on the solution orbits of the two Chua s circuits. In Table of [Tsuneda, ], sets of parameter values for system () were given. Here, select several sets of the given parameter values (as shown in Table ) in simulations (see Figs. 4 7). For Chua s circuit with the attraction-repulsion function, some periodic and chaotic orbits are obtained by using the same parameters ˆα, ˆβ and ˆγ as used in the circuit with the cubic function, by adjusting the parameters ã and b. Here, by choosing k = (see Table ), one can observe some similarities and some differences between the dynamics of the two Chua s circuits. Some corresponding computer simulations are shown in Fig. 8. Table. No. ˆα = α ˆβ = β ˆγ = γ â =ã ˆb = b ˆk = k Figure C Fig. 4 C Fig. C Fig. 6 C Fig. 7

16 88 R. Li et al. 4 cubic attraction repulsion. cubic attraction repulsion. f(y) f(y).. 4 y (a) Two functions.... y (a) Two functions Y Y Z 4 X 4 Z X (b) Cubic function (b) Cubic function X() X().6.8 (c) Attraction-repulsion (c) Attraction-repulsion Fig. 4. Nonlinear functions and solutions of system C-. Fig.. Nonlinear functions and solutions of system C-8.

17 A Modified Chua s Circuit with an Attraction-Repulsion Function 88. cubic attraction repulsion. cubic attraction repulsion.. f(y) f(y).... y.... y (a) Two functions (a) Two functions Y Y Z 4 X. Z X (b) Cubic function (b) Cubic function X() X(). (c) Attraction-repulsion (c) Attraction-repulsion Fig. 6. Nonlinear functions and solutions of system C-. Fig. 7. Nonlinear functions and solutions of system C-6.

18 88 R. Li et al. Table. No. α β γ ã b c Figure C Fig. 8(A-) C Fig. 8(A-) C Fig. 8(A-) C Fig. 8(A-4) C Fig. 8(B-) C Fig. 8(B-) C Fig. 8(C-) C Fig. 8(C-) C Fig. 8(D-) C Fig. 8(D-).... X(). X().. (A-) (A-) X()... X().. (A-) Fig. 8. Chua s circuit with an attraction-repulsion function. (A-4)

19 A Modified Chua s Circuit with an Attraction-Repulsion Function X() 4 4 X() (B-) (B-) X() X(). (C-) (C-) 4 4 X() X() (D-) Fig. 8. (Continued ) (D-) 6. Generalization to a Multi-Scroll Circuit It is well known that Chua s circuit with different nonlinear functions, such as multi-breakpoint piecewise-linear function, sine function and hysteresis function, can generate complex multi-scroll chaotic attractors [Lü & Chen, 6] and references therein. In order to study the complex dynamics of the modified Chua s circuit (9), in this section the attention is focused on the generation of multiscroll attractors [Wang, 7]. For this purpose, the attraction-repulsion function in system (9) is

20 884 R. Li et al f(x) (a) Function f(x) x (a) Function f(x) y y x (b) Projection on (x, y)-plane..... x (b) Projection on (x, y)-plane (c) Attractor in (y, x, z)-space Fig. 9. The -scroll attractor generated by system (9) with function (6) and initial value x() = (.,.,.). (c) Attractor in (y, x, z)-space Fig.. The 4-scroll attractor generated by system (9) with function (6) and initial value x() = (.,.,.).

21 A Modified Chua s Circuit with an Attraction-Repulsion Function 88 f(x) y x (a) Function f(x) x (b) Projection on (x, y)-plane further modified, as f(x) =f (x) f (x) + f n (x) f n (x)+, () where f i (x) =a i x + b i xe c ix, i =,,...,n.obviously, function () is obtained by overlapping some different attraction-repulsion functions. And it may be simplified as follows: f(x) =ax + b xe c x b xe c x + b n xe c n x b n xe cnx. (6) By choosing appropriate parameters, the number of equilibria of function (6) can be increased. Substituting function (6) into system (9), corresponding multi-scroll attractors may be obtained. For example, by letting parameters n =,a =., b =.8, b =., c =., c =, α=, β =,γ =andk =, a -scroll attractor is generated in system (9) with function (6), as showninfig.9. Moreover, by choosing parameters n =,a =.4, b =,b =,b =,c =, c =, c =, α =,β =,γ = and k =, a 4-scroll attractor is obtained, as shown in Fig.. As one more example, when choosing parameters n =4,a=.7, b =.6, b =.8, b =, b 4 =,c =., c =., c =.4, c 4 =.8, α = 7,β =,γ = and k =, a -scroll attractor is generated, as shown in Fig.. (c) Attractor in (y, x, z)-space Fig.. The -scroll attractor generated by system (9) with function (6) and initial value x() = (.,.,.). 7. Circuit Design and Experimental Observations In this section, a simplified physical circuit is built to visualize the realization of the new system. Since the attraction-repulsion function is an exponential function, as is well known, a truncation of the exponential function is necessary in building a real circuit for its realization. So, an approximate circuitry was built, as shown in Fig., for the purpose of visualization but not accurate circuitry design. The attraction-repulsion function of system () is thus replaced by its approximation f(x) =.x +.x, and the generated one-scroll chaotic orbits and double-scroll chaotic orbits are shown in Fig., obtained by tuning the adjustable resistor.

22 886 R. Li et al. Fig.. Implementation of the chaotic circuit (), where all the active components are supplied by ± volts. Projection on the x x plane Projection on the x x plane Fig.. Experimental observations of system ().

23 A Modified Chua s Circuit with an Attraction-Repulsion Function 887 Projection on the x x plane Fig.. (Continued ) Projection on the x x plane 8. Conclusions In this paper, a new modified Chua s circuit with an attraction-repulsion function has been presented. Basic dynamical properties have been analyzed, including pitchfork bifurcation, Hopf bifurcation and various chaotic behaviors of the circuit, under Lagrange stability conditions. Some chaotic behaviors have also been verified by a simplified electronic circuit. This paper confirms that the attractionrepulsion function can play a similar role as the piecewise-linear function and the cubic function in realizing Chua s circuits. Acknowledgments The authors wish to thank Prof. Wenbo Liu for her assistance in circuit implementation of the new system. References Altman, E. J. [99a] Bifurcation analysis of Chua s circuit with applications for low-level visual sensing, J. Circuits Syst. Comput., 6 9. Altman, E. J. [99b] Normal form analysis of Chua s circuit with applications for trajectory recognition, IEEE Trans. Circuits Syst.-II 4, Chua, L. O., Komuro, M. & Matsumoto, T. [986] The double scroll family, IEEE Trans. Circuits Syst., 7 8. Chua, L. O. [a] The genesis of Chua s circuit, Archiv fur Elektronik und Ubertragungstechnik 46, 7. Chua, L. O. [b] A zoo of strange attractors from the canonical Chua s circuits, Proc. th Midwest Symp. Circuit and Systems, Vol., pp Duan, Z. S., Wang, J. Z. & Huang, L. [] A attraction/repulsion functions in a new chaotic system, Phys. Lett. A, Duan, Z. S., Wang, J. Z. & Huang, L. [7] A generalization of smooth Chua s equations under Lagrange stability, Int. J. Bifurcation and Chaos 7, Gazi, V. & Passino, K. M. [a] Stability analysis of swarms in an environment with an attractant/repellent profile, Proc. American Control Conf. (Anchorage, Alaska), pp Gazi, V. & Passino, K. M. [b] A class of attraction/repulsion functions for stable swarm aggregations, Proc. Conf. Decision Contr. (Las Vegas, Nevada), pp Huang, A., Pivka, L. & Franz, M. [996] Chua s equation with cubic nonlinearity, Int. J. Bifurcation and Chaos, 7. Kennedy, P. & Chua, L. O. [99] Hysteresis in electronic circuits: A circuit theorists perspective, Int. J. Circuit Th. Appl. 9, 47. Lü, J. H. & Chen, G. [6] Generating multiscroll chaotic attractors: Theories, methods and applications, Int. J. Bifurcation and Chaos 6, Okubo, A. [986] Dynamical aspects of animal grouping:swarms,schools,flocks,andherds, Adv. Biophys., 94. Tang, K. S., Man, K. F., Zhong, G. Q. & Chen, G. [] Some new circuit design for chaos generation, Chaos in Circuits and Systems, eds. Chen, G. & Ueta, T., pp

24 888 R. Li et al. Tsuneda, A. [] A gallery of attractors from smooth Chua s equation, Int. J. Bifurcation and Chaos, 49. Wang, B. [7] The study of chaotic systems based on an attraction-repulsion function, Master Thesis, Library of Peking University, Beijing, China (in Chinese). Zhong, G. Q. [994] Implimenttation of Chua s circuit with a cubic nonlinearity, IEEE Trans. Circuits Syst.-I 4, 6 94.