DYNAMICS AND CHAOS CONTROL OF NONLINEAR SYSTEMS WITH ATTRACTION/REPULSION FUNCTION

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1 International Journal of Bifurcation and Chaos, Vol. 18, No. 8 (2008) c World Scientific Publishing Company DYNAMICS AND CHAOS CONTROL OF NONLINEAR SYSTEMS WITH ATTRACTION/REPULSION FUNCTION XIAN LIU, JINZHI WANG, ZHISHENG DUAN and LIN HUANG State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, Peking University, Beijing , P. R. China caroline@pku.edu.cn Received December 1, 2006; Revised September 27, 2007 In this paper, a more general third-order chaotic system with attraction/repulsion function is introduced on the basis of [Duan et al., 2005]. A gallery of chaotic attractors, bifurcation diagrams and Lyapunov exponent spectra are presented to show the interesting phenomena of the given system. Based on the absolute stability theory and linear matrix inequality (LMI), a simple method of chaos control for the system is proposed and a stabilizing controller is derived such that chaos oscillations of the system disappear and all chaotic trajectories of it are led to certain equilibrium. Numerical simulations are provided to illustrate the efficiency of the proposed method. Keywords: Attraction/repulsion function; chaos oscillation; absolute stability; stabilizing controller; LMI. 1. Introduction The modeling of swarming behavior [Breder, 1954; Okubo, 1986; Warburton & Lazarus, 1991; Gazi & Passino, 2003, 2004b] has been an active research topic over the past decades since swarming behavior is observed in many biological systems, for examples, flocks of birds, schools of fishes, colonies of bacteria, and it possesses certain advantages like avoiding predators and increasing the chance of finding food. Generally swarming behavior is understood as a result of an interplay between a long range attraction and a short range repulsion between the individuals. A class of attraction/repulsion functions [Gazi & Passino, 2003, 2004a] are developed to characterize swarming behavior. The operational principles from such biological systems are extended to the fields of autonomous multiagent systems [Balch & Arkin, 1998; Egerstedt & Hu, 2001; Ogren et al., 2002; Olfati & Murray, 2002]. It is shown that the dynamical behavior in multiagent systems can become extremely complex, nonlinear oscillation and chaos can often appear [Erwin, 1989; Hogg & Huberman, 1991] due to the existence of imperfect knowledge among different agents. The interactions of variables in simple systems can also lead to complex chaotic phenomena. Therefore, it is interesting to explore the possible chaotic phenomena in swarm models and control chaos if chaos oscillations are unexpected. Duan [Duan et al., 2005] used an attraction/repulsion function to obtain a simple third-order chaotic model, which can be Author for correspondence 2345

2 2346 X. Liu et al. viewed as a simplified and abstracted system of swarm model studied in [Gazi & Passino, 2003]. Some parameter domains are determined for the nonexistence of chaotic attractors in the system by using the dichotomous criterion [Leonov et al., 1996]. In this paper, on the basis of [Duan et al., 2005], a more general third-order chaotic system with attraction/repulsion function is introduced. It contains many kinds of systems including the model given in [Duan et al., 2005], canonical Chua s circuit with the piecewise-linear nonlinearity substituted by the attraction/repulsion function and other more complex systems. Fifteen groups of chaotic attractors together with corresponding bifurcation diagrams and Lyapunov exponent spectra are presented to show the interesting phenomena of this system and emphasize the importance of the attraction/repulsion function. Nevertheless, chaos oscillation in many systems is not expected and should be avoided or eliminated. Motivated by this reason, a simple method of chaos control for the system is proposed by using the results of absolute stability theory. A stabilizing controller is derived in terms of LMI such that chaos oscillations of the system disappear and all chaotic its trajectories are led to certain equilibrium. The rest of this paper is organized as follows. In Sec. 2, a simple third-order chaotic system is introduced and fifteen groups of chaotic attractors together with corresponding bifurcation diagrams and Lyapunov exponent spectra of the system are presented. The frequency-domain condition for absolute stability of a class of nonlinear systems is turned into LMI and the method of controller design is proposed in Sec. 3. In Sec. 4, based on the results of Sec. 3, a stabilizing controller is derived such that chaos oscillations of the system disappear and all its trajectories are led to certain equilibrium. Numerical simulations are also given to illustrate the effectiveness of the proposed method. In Sec. 5, concluding remarks are given. The following standard notations are used throughout this paper. The superscripts T and represent transpose and conjugate transpose, respectively. A>0(A 0) denotes A is a Hermitian and positive definite (positive semi-definite) matrix. A < 0 (A 0) denotes A is a Hermitian and negative definite (negative semi-definite) matrix. 2. Chaotic Attractors of Nonlinear Systems with Attraction/ Repulsion Function Let us consider a third-order nonlinear system with attraction/repulsion function given in [Gazi & Passino, 2003] ẋ 1 = mx 1 + qx 2 ẋ 2 = nx 1 + sx 2 + px 3 (1) ẋ 3 = c 1 x 2 + c 2 x 3 + rf (y) where x 1, x 2 and x 3 are states, f(y) = y(a 1 a 2 exp( a 3 y 2 )) denotes the attraction/repulsion function, y is a function of x 1, x 2 and x 3 expressed as y = d 1 x 1 + d 2 x 2 + d 3 x 3, m, q, n, s, p, c 1, c 2, r, a 1, a 2, a 3, d 1, d 2 and d 3 are parameters. The attraction/repulsion function with a 1 = 1, a 2 = 20 and a 3 = 5 is shown in Fig. 1. It should be pointed out that f(y) is consistent with inter-individual attraction and repulsion in biological systems if y represents the distances between the individuals. In this section, the dynamics of (1) is explored by using numerical simulations. It is observed that system (1) can exhibit a wide variety of chaotic phenomena. Remark 1. The following model was investigated in [Duan et al., 2005] ż 1 = z 2 uz 3 ż 2 = vz 1 0.6z 2 (2) ż 3 = 0.2z 1 3z 3 f(y 1 ) f(y) y Fig. 1. Attraction/repulsion function f( ).

3 Dynamics and Chaos Control of Nonlinear Systems with Attraction/Repulsion Function 2347 where u and v are parameters, y 1 =2z 1 z z 3 and f(y 1 )=y 1 (1 20 exp( 5y1 2)). Let x 1 = z 2, x 2 = z 1, x 3 = z 3 and y 2 = x 1 +2x x 3.The system (2) can be rewritten as ẋ 1 = 0.6x 1 vx 2 ẋ 2 = x 1 ux 3 (3) ẋ 3 = 0.2x 2 3x 3 f(y 2 ) It is obvious that (3) is a particular case of (1). Remark 2. Consider Chua s circuit [Matsumoto, 1984; Chua, 1994; Tsuneda, 2005], as shown in Fig. 2. As is well known, this circuit is described by the following state equations C 1 v 1 = v 2 v 1 R g(v 1) C 2 v 2 = v 1 v 2 R + i (4) L L i L = v 2 R 1 i L where v 1 and v 2 are the voltages across the capacitors C 1 and C 2, i L denotes the current through the inductances L, R 1 and R are linear resistors, RC 2 > 0, the term g(v 1 ) represents the current through the nonlinear resistor N R which is a piecewise-linear function expressed as g(v 1 ) = G 2 v 1 +(1/2)(G 1 G 2 )[ v 1 + E v 1 E ]. Let x 1 = i L R/E, x 2 = v 2 /E, x 3 = v 1 /E, σ = C 2 /C 1, β = R 2 C 2 /L, α = RR 1 C 2 /L, m 0 = RG 1, m 1 = RG 2 and k = 1/RC 2. Then (4) is transformed into ẋ 1 = k(αx 1 + βx 2 ) ẋ 2 = k(x 1 x 2 + x 3 ) (5) ẋ 3 = kσ(x 2 x 3 g(x 3 )) where g(x 3 )=m 1 x 3 +(1/2)(m 0 m 1 )( x 3 +1 x 3 1 ). Assume that if RC 2 > 0, k =1ischosen. It is not difficult to find that Chua s circuit can be considered as a special case of system (1). In fact, taking n = p = s =1,c 1 = c 2 = r, d 1 = d 2 = 0, d 3 =1andy = x 3, then (1) is simplified as ẋ 1 = mx 1 + qx 2 ẋ 2 = x 1 x 2 + x 3 (6) ẋ 3 = c 1 (x 2 x 3 f(x 3 )) where f(x 3 )= x 3 (a 1 a 2 exp( a 3 x 2 3 )). One can see that (6) has the same form as (5), which indicates that (6) can be implemented by the circuit showninfig.2withg(v 1 ) substituted by an attraction/repulsion function. In order to show the interesting phenomena exhibited in (1) and underline the importance of the attraction/repulsion function, fifteen groups of chaotic attractors together with corresponding bifurcation diagrams and Lyapunov exponent spectra are presented in Figs respectively. The parameter values are given in Table 1. The bifurcation diagram is derived by fixing all the parameter values except p which denotes the variation of the state x 1 with p. Remark 3. It is shown from Figs that (1) can generate a wide variety of chaotic phenomena. The small variances of parameters may result in large variances of the dynamics of (1), see Figs. 10 and 15, 13 and 14. Therefore, the attraction/repulsion function plays an important role in (1). Remark 4. System (6), which is viewed as Chua s circuit with the piecewise-linear function substituted by the attraction/repulsion function, can exhibit chaotic phenomena, see Figs. 3 and 4. Thus we deduce that the attraction/repulsion function Fig. 2. Chua s circuit.

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5 Dynamics and Chaos Control of Nonlinear Systems with Attraction/Repulsion Function 2349 Fig. 4.

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19 Dynamics and Chaos Control of Nonlinear Systems with Attraction/Repulsion Function 2363 Table 1. Parameter values of (1). No. of Fig. m q n s p c 1 c 2 r d 1 d 2 d 3 Fig Fig Fig Fig Fig Fig Fig Fig Fig Fig Fig Fig Fig Fig Fig in (6) may play the same role as the piecewiselinear nonlinearity in Chua s circuit, which causes the appearance of many interesting nonlinear phenomena such as chaos and bifurcation. 3. Stability Analysis and Control of a Class of Nonlinear Systems Consider the following system ẋ = Ax + bϕ(σ), σ = c T x, (7) where A R n n is a constant matrix, b R n and c R n are constant vectors. Suppose that ϕ(σ) :R R is continuously differentiable on R and satisfies the following inequalities 0 ϕ(σ)σ µσ 2, <µ 1 ϕ (σ) µ 2 < +, (8) where µ 1 0 and µ 2 µ. The transfer function of linear part from ϕ(σ) to σ is χ(p) = c T (A pi) 1 b. Lemma 1 [Leonov et al., 1996]. Let (A, b) be controllable and A be Hurwitzian. Suppose that there exist numbers ν and τ 0 such that µ 1 +Re{(1 + iων)χ(iω)+τω 2 (µ 1 χ(iω)+1) (µ 2 χ(iω)+1)} 0, ω R. (9) Then (7 ) is absolutely stable. The following KYP lemma establishes the equivalence between a frequency-domain inequality and a time-domain inequality. Lemma 2 [Rantzer, 1996]. Given A R n n,b R n m and symmetric matrix Ω R (n+m) (n+m), with det(iωi A) 0 for ω R and the pair (A, B) is controllable, the following two statements are equivalent (i) [ ] [ ] (iωi A) 1 B (iωi A) 1 B Ω 0, I I ω R. (10) (ii) There exists a matrix P = P T such that [ ] A T P + PA PB B T +Ω 0. (11) P 0 The equivalence for strict inequalities holds even if (A, B) is not controllable. Further, if A is Hurwitzian and the top-left submatrix of Ω is positive semi-definite, then (11) implies P 0. Theorem 1. Let (A, b) be controllable and A be Hurwitzian. Suppose that there exist numbers ν, τ 0 and a matrix P = P T such that [ ] A T P + PA αa T cc T A Pb+ E b T P + E T < 0. (12) h Then (7 ) is absolutely stable, where α = τµ 1 µ 2, E = αc T ba T c +(1/2)c +(ν/2)a T c + βa T A T c, β = τ(µ 1 + µ 2 )/2, h = αb T cc T b + c T b + 2βc T Ab µ 1.

20 2364 X. Liu et al. Proof. Inequality (9) can be rewritten as αg (iω)g(iω)+µ (χ(iω)+χ (iω)) + ν 2 (G(iω)+G (iω)) β(m(iω)+m (iω)) + τω 2 0, (13) where G(iω) =c T A(A iωi) 1 b c T b, M(iω) = c T A 2 (A iωi) 1 b c T Ab iωc T b. Without affecting the result we can eliminate τω 2 in (13). Then the obtained frequency-domain inequality can be rewritten as the form of (10) which is equivalent to (12) from Lemma 2. This completes the proof. Corollary 1. Let (A, b) be controllable and A be Hurwitzian. Suppose that there exist a number ν and amatrixp>0such that A T P + PA P b + c 2 + νat c 2 b T P + ct 2 + νct A c T b 1 < 0. (14) 2 µ QA T + AQ + Y T b T 1 + b 1Y b T + ct Q 2 + νct AQ 2 + νct b 1 Y 2 Then (7 ) is absolutely stable. In what follows, the method of controller design is proposed. The purpose is to find control law u = Kx such that the following closed-loop system ẋ = Ax + bϕ(σ)+b 1 u, σ = c T x (15) is absolutely stable, where K R 1 n and b 1 R n. Note that (15) can be formulated as ẋ =(A + b 1 K)x + bϕ(σ), σ = c T x. (16) The control matrix K can be derived by using Corollary 1. Replacing A in (14) with A + b 1 K,the following result is obtained. Theorem 2. Suppose that there exist a number ν, matrices Q>0 and Y such that b+ Qc 2 + νqat c + νy T b T 1 c 2 2 c T b 1 < 0. (17) µ Then K = YQ 1.Furthermore, if (A + b 1 K, b) is controllable, then (16) is absolutely stable. 4. Chaos Control for Nonlinear Systems with Attraction/ Repulsion Function Since chaos oscillations in many systems are unexpected, it is preferred to avoid their appearances. As many generalized Chua s circuits [Suykens et al., 1997], the third-order system included in this paper canalsoberepresentedinlur eform.basedonthe results of Sec. 3, the stabilizing controller is derived such that the trajectories of the system are led to an expected equilibrium. First consider Fig. 2 with the piecewise-linear nonlinearity g(v 1 ) replaced by the attraction/repulsion function f(v 1 )= v 1 (ã 1 ã 2 exp( ã 3 v1 2 )). The state equations are as follows C 1 v 1 = v 2 v 1 R f(v 1) C 2 v 2 = v 1 v 2 R + i (18) L L i L = v 2 R 1 i L where v 1, v 2, i L, C 1, C 2, L, R 1, R and N R denote the same meanings as in Fig. 2. Letting x 1 = i L R, x 2 = v 2, x 3 = v 1, c 1 = 1/RC 1, q = R/L, m = R 1 /L and RC 2 = 1, (18) is transformed into ẋ 1 = mx 1 + qx 2 ẋ 2 = x 1 x 2 + x 3 (19) ẋ 3 = c 1 (x 2 x 3 f(x 3 )) where f(x 3 ) = x 3 (a 1 a 2 exp( a 3 x 2 3 )), a 1 = Rã 1 = 1, a 2 = Rã 2 = 20 and a 3 = ã 3 =5. Observe that (19) has the same form as (6) and it can exhibit chaotic phenomena when the parametersarechosenasthevaluesgiveninfigs.3and4 in Table 1. In order to control chaos oscillations, an external signal i u is inserted. Please refer to Fig. 18. The state equations describing the circuit are as follows C 1 v 1 = v 2 v 1 R f(v 1)+i u C 2 v 2 = v 1 v 2 R + i (20) L L i L = v 2 R 1 i L

21 Dynamics and Chaos Control of Nonlinear Systems with Attraction/Repulsion Function 2365 Any equilibrium x eq = (x 1eq,x 2eq,x 3eq ) T of (19) satisfies Ax eq + bf (y eq ) = 0, i.e. mx 1eq + qx 2eq =0 x 1eq x 2eq + x 3eq =0 (24) c 1 x 2eq c 1 x 3eq c 1 f(y eq )=0 Fig. 18. The circuit implementation of (18) with external signal i u. By the same transformation as (18), (20) is formulated as ẋ 1 = mx 1 + qx 2 ẋ 2 = x 1 x 2 + x 3 (21) ẋ 3 = c 1 (x 2 x 3 f(x 3 )) + u where u = i u /C 1 canbeconsideredasacontrol signal. Rewrite (21) as where ẋ = Ax + bf (y)+b 1 u, y = c T x, (22) m q 0 0 A = 1 1 1, b = 0, 0 c 1 c 1 c x 1 b 1 = 0, c = 0, x = x 2, 1 1 f(y) = y(1 20 exp( 5y 2 )) satisfies 19y 2 f(y)y y 2 and 19 f (y) 10. Further by simple transformation (22) is turned into where ẋ = Ax + bg(y)+b 1 u, y = c T x, (23) m q 0 A = 1 1 1, 0 c 1 18c 1 b, b 1,c and x are the same as those in (22), g(y) = f(y) +19y satisfies 0 g(y)y 20y 2, 0 g (y) 29. x 3 where y eq = x 3eq. It follows from (24) that x 2eq = (m/q)x 1eq, x 1eq = (q/(q + m))x 3eq and f(x 3eq ) = (q/(q + m))x 3eq. The equilibria are the points of intersection of the straight line f = (q/(q + m))x 3eq and the curve f = x 3eq (1 20 exp( 5x 2 3eq )), which means that the equilibria satisfy the equation qx 3eq +(q + m)x 3eq (1 20 exp( 5x 2 3eq )) = 0. Once the parameter values of the system are known, x 1eq, x 2eq and x 3eq can be solved according to above formulae. In addition, x eq = (x 1eq,x 2eq,x 3eq ) T also satisfies ẋ eq = Ax eq + bg(y eq ), y eq = c T x eq. (25) The aim of chaos control is to design u = K(x x eq ) such that the states of the closed-loop system (21) are led to certain equilibrium x eq. Defining e = x x eq and subtracting (25) from (23), one can obtain ė =(A + b 1 K)e + bη(c T e, x eq ), (26) where η(c T e, x eq )=g(c T (e+x eq )) g(c T x eq )satisfies 0 g(ct (e + x eq )) g(c T x eq ) c T e = η(ct e, x eq ) c T 29, 0 η (c T e, x eq ) 29. e From the above presentation, it is shown that the chaos control problem of (19) is transformed into the absolute stability problem of (26), which can be settled by using Theorem 2. Choose a real value of parameter ν and solve (17) in terms of LMI toolbox in MATLAB. If LMI (17) is feasible for variables Y, Q>0, then the control matrix K = YQ 1 is followed. Otherwise adjust the value of parameter ν until (17) is feasible. Generally ν is not unique. According to the above process, we can derive the control matrix K for system (21) with the parameter values given in Figs. 3 and 4 in Table 1. The corresponding equilibria, control matrix K and value of parameter ν are listed in Table 2. The states of (21) with u = K(x x eq ) are shown in Figs. 19 and 20, respectively.

22 2366 X. Liu et al. Table 2. Equilibria, parameter value ν and the control matrix K. Fig. with u Fig. with u = 0 Equilibria ν K Fig. 19 Fig. 3 (0, 0, 0), (±0.6903, ±0.1318, ) 0.2 K 1 =[ ] Fig. 20 Fig. 4 (0, 0, 0), (±0.6909, ±0.1382, ) 0.1 K 2 =[ ] Fig. 21 Fig. 5 (0, 0, 0), ( , ±0.2629, ) 0.18 K 3 =[ ] Fig. 22 Fig. 6 (0, 0, 0), ( 0.831, ±0.554, 0.831) 0.5 K 4 =[ ] Fig. 23 Fig. 7 (0, 0, 0), ( , ±0.2803, 1.09) 0.17 K 5 =[ ] Fig. 24 Fig. 8 (0, 0, 0), ( , , ) 0.44 K 6 =[ ] Fig. 25 Fig. 9 (0, 0, 0), ( , ±0.6981, ±1.7450) 3.0 K 7 =[ ] Fig. 26 Fig. 10 (0, 0, 0), ( , ±0.2481, ±2.8518) 1.1 K 8 =[ ] Fig. 27 Fig. 11 (0, 0, 0), ( , , ±0.2217) 4.95 K 9 =[ ] Fig. 28 Fig. 12 (0, 0, 0), ( , , ±0.1912) 1.15 K 10 =[ ] Fig. 29 Fig. 13 (0, 0, 0), ( , ±1.6251, ±2.7409) 1.3 K 11 =[ ] Fig. 30 Fig. 14 (0, 0, 0), ( 0.837, ±1.712, ±2.3101) 1.2 K 12 =[ ] Fig. 31 Fig. 15 (0, 0, 0), ( , ±0.1927, ±2.9079) 1.0 K 13 =[ ] Fig. 19. States of (21) with u = K 1 (x x eq) (converge to (0, 0, 0) and (0.6903, , ) respectively). Fig. 20. States of (21) with u = K 2 (x x eq) (converge to (0, 0, 0) and (0.6909, , ) respectively).

23 Dynamics and Chaos Control of Nonlinear Systems with Attraction/Repulsion Function 2367 Fig. 21. States of (28) with u = K 3 (x x eq) (converge to (0, 0, 0) and ( , , ) respectively). Fig. 22. States of (28) with u = K 4 (x x eq) (converge to (0, 0, 0) and ( 0.831, 0.554, 0.831) respectively). Now consider (1) with d 1 =1andd 2 = d 3 =0 and s = 0. Then (1) is simplified as ẋ 1 = mx 1 + qx 2 ẋ 2 = nx 1 + px 3 (27) ẋ 3 = c 1 x 2 + c 2 x 3 + rf (y) where y = x 1. In order to control chaos, a controller is added to the state equations. The purpose of chaos control for (27) is also to design a controller u = K(x x eq ) such that the states of the following equation ẋ 1 = mx 1 + qx 2 ẋ 2 = nx 1 + px 3 + u (28) ẋ 3 = c 1 x 2 + c 2 x 3 + rf (x 1 ) are led to certain equilibrium x eq of (27). Equation (28) can be rewritten in the form of (22) with m q A = n 0 p, b = 0, b 1 = 1, 0 c 1 c 2 r 0 1 x 1 c = 0, x = x 2. 0 Further (28) can be expressed in the form of (23) with m q 0 A = n 0 p. 19r c 1 c 2 x 3

24 2368 X. Liu et al. Fig. 23. States of (28) with u = K 5 (x x eq) (converge to (0, 0, 0) and ( , , 1.09) respectively). Fig. 24. States of (28) with u = K 6 (x x eq) (converge to (0, 0, 0) and ( , , ) respectively). Fig. 25. States of (28) with u = K 7 (x x eq) (converge to (0, 0, 0) and ( , , ) respectively).

25 Dynamics and Chaos Control of Nonlinear Systems with Attraction/Repulsion Function 2369 Fig. 26. States of (28) with u = K 8 (x x eq) (converge to (0, 0, 0) and ( , , ) respectively). Fig. 27. States of (28) with u = K 9 (x x eq) (converge to (0, 0, 0) and ( , , ) respectively). Fig. 28. States of (28) with u = K 10 (x x eq) (converge to (0, 0, 0) and ( , , ) respectively).

26 2370 X. Liu et al. Fig. 29. States of (28) with u = K 11 (x x eq) (converge to (0, 0, 0) and ( , , ) respectively). Fig. 30. States of (28) with u = K 12 (x x eq) (converge to (0, 0, 0) and ( 0.837, 1.712, ) respectively). Fig. 31. States of (28) with u = K 13 (x x eq) (converge to (0, 0, 0) and ( , , ) respectively).

27 Dynamics and Chaos Control of Nonlinear Systems with Attraction/Repulsion Function 2371 Using the same method of chaos control for (19), the control matrices K for (28) with the parameter values given in Figs in Table 1 are obtained. The corresponding equilibria of (27), control matrix K and value of parameter ν are listed in Table 2. The states of (28) with u = K(x x eq )areshown in Figs respectively. It is shown from Figs that the state variables of (1) with the parameter values given in Table 1 can lead to certain equilibrium efficiently by using the proposed method. In addition, the designed controller has simple form. Remark 5. There are three different equilibria for (1) with the parameter values given in Table 1. Figures present the stabilization of the origin and one nonsingular equilibrium of (19) and (27). The stabilization of another nonsingular equilibrium can be obtained similarly. Remark 6. From Table 2 it is seen that K is identical for the stabilization of the origin and nonsingular equilibria of the same system with the same parameter values. The proposed method can also be applied to chaos control for a more general system (1). 5. Conclusions In this paper, we introduce a more general chaotic system by combining a linear system with the attraction/repulsion function. Fifteen groups of chaotic attractors together with corresponding bifurcation diagrams and Lyapunov exponent spectra are presented by computer simulations to show the interesting phenomena of the system. It is seen that the attraction/repulsion function plays an important role in the system and may play the same role as the piecewise-linear nonlinearity in Chua s circuit. A simple method of chaos control for the system is proposed by using the results of absolute stability theory. From numerical simulations it is shown that chaotic oscillations of the system are well controlled by using the proposed method. Acknowledgments This work is supported by National Science Foundation of China under Grant , , and also supported by Engineering Research Institute of Peking University. References Balch, T. & Arkin, R. C. [1998] Behavior based formation control for multi-robot teams, IEEE Trans. Robot. Autom. 14, Breder, C. M. [1954] Equations descriptive of fish schools and other animal aggregations, Ecology 35, Chua, L. O. [1994] Chuas circuit: An overview ten years later, J. Circuits Syst. Comput. 4, Duan, Z., Wang, J. & Huang, L. [2005] Attraction/ repulsion functions in a new class of chaotic systems, Phys. Lett. A 335, Egerstedt, M. & Hu, X. [2001] Formation constrained multi-agent control, IEEE Trans. Robot. Autom. 17, Erwin, H. R. [1989] Mixing and sensitive dependence on initial conditions in computer systems, Comput. Measur. Group Trans. 65, 3 6. Gazi, V. & Passino, K. M. [2003] Stability analysis of swarms, IEEE Trans. Autom. Contr. 48, Gazi, V. & Passino, K. M. [2004a] A class of attraction/repulsion functions for stable swarm aggregations, Int. J. Contr. 77, Gazi, V. & Passino, K. M. [2004b] Stability analysis of social foraging swarms, IEEE Trans. Syst. Man Cybern. 34, Hogg, T. & Huberman, B. A. [1991] Controlling chaos in distributed systems, IEEE Trans. Syst. Man Cybern. 21, Leonov, G. A., Ponomarenko, D. V. & Smirnova, V. B. [1996] Frequency-Domain Methods for Nonlinear Analysis-Theory and Applications (World Scientific, Singapore), Chap. 1, pp Matsumoto, T. [1984] A chaotic attractor from Chua s circuit, IEEE Trans. Circuits Syst. 31, Ogren, P., Egerstedt, M. & Hu, X. [2002] A control Lyapunov function approach to multi-agent coordination, IEEE Trans. Robot. Autom. 18, Okubo, A. [1986] Dynamical aspects of animal grouping: Swarms, schools, flocks, and herds, Adv. Biophys. 22, Olfati, R. S. & Murray, R. M. [2002] Distributed cooperative control of multiple vehicle formations using structural potential functions, in IFAC World Congress, Spain, pp Rantzer, A. [1996] On the Kalman Yakubovich Popov lemma, Syst. Contr. Lett. 28, Suykens, J. A. K., Huang, A. & Chua, L. O. [1997] A family of n-scroll attractors from a generalized Chua s circuit, Int. J. Electron. Commun. 51, Tsuneda, A. [2005] A gallery of attractors from smooth Chua s equation, Int. J. Bifurcation and Chaos 15, Warburton, K. & Lazarus, J. [1991] Tendency distance models of social cohesion in animal groups, J. Theoret. Biol. 150,