Attraction/repulsion functions in a new class of chaotic systems

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1 Physics Letters A ) Attraction/repulsion functions in a new class of chaotic systems Zhisheng Duan, Jin-Zhi Wang, Lin Huang State Key Laboratory for Turbulence and Complex Systems and Department of Mechanics and Engineering Science, Peking University, Beijing , PR China Received 31 March 2004; received in revised form 24 November 2004; accepted 25 November 2004 Available online 23 December 2004 Communicated by A.P. Fordy Abstract In this Letter, a new kind of chaotic system is introduced by combining a three-dimensional linear system with attraction/repulsion functions developed in the context of swarm aggregations. Rich chaotic oscillating phenomena appear in this new system. Some simple extension to multi-input and multi-output systems is also given. The frequency-domain condition for the property of dichotomy is presented for a more general system. Based on the frequency-domain condition given here, some parameter domains can be determined for the nonexistence of chaotic attractors or limit cycles in the system given in this Letter Elsevier B.V. All rights reserved. Keywords: Attraction/repulsion function; Chaos; Frequency-domain method; Dichotomy 1. Introduction The study of chaotic attractors [1 6] has attracted a lot of researchers since Lorenz presented a simple threedimensional chaotic system [2]. During the last four decades, several classical systems for generating chaotic attractors were presented such as Lorenz system, Chua s circuit, etc. Since chaos is gradually found useful in high-tech and industrial engineering applications, the study of chaos control and anti-control has attracted more and more interest. The wide variety of research results show that the study of chaos has a rich and great prospect. In addition, based on the concept of dichotomy, the nonexistence of chaotic attractor was analyzed for Chua s circuit in [7]. Frequency-domain method developed in [7 14] cannot only be used to analyze absolute stability for the traditional Lur e system, but also be applied to the investigation of global properties, the existence of cycles and homoclinical orbits for nonlinear systems with multiple equilibria. Especially, some global properties This work is supported by the National Science Foundation of China under grant , , * Corresponding author. address: duanzs@pku.edu.cn Z. Duan) /$ see front matter 2004 Elsevier B.V. All rights reserved. doi: /j.physleta

2 140 Z. Duan et al. / Physics Letters A ) such as dichotomy, Lagrange stability, gradient-like behavior and Bakaev stability were presented in [7]. And the corresponding frequency-domain criteria were established. These results are important for studying nonlinear systems with multiple equilibria. Since swarming behavior has been observed for a long time in certain living beings such as flocks of birds, schools of fish, herds of animals, modelling of swarming behavior and stability analysis, swarm intelligence have attracted more and more interest of mathematical biologists and physicists [15 21]. Now the general understanding is that the swarming behavior is a result of an interplay between a long range attraction and a short range repulsion between the individuals. Recently swarming behavior was studied in [17 19] by using a class of attraction/repulsion function. In multi-agent systems, the dynamical behavior can become extremely complex and chaotic oscillating phenomena often appear in the presence of interactions of agents, delayed and imperfect knowledge among different agents [22,23]. Chaos control in multi-agent systems was studied in [22]. And it is well known that the interactions of variables in simple systems can give rise to complex chaotic dynamics [24,25]. So it is also interesting to study the possible chaotic phenomena in swarm models. For simplicity, we first construct simple chaotic systems by using attraction/repulsion functions given in [17] for the study of swarm aggregations. It is interesting to see if attraction/repulsion functions can play the same roles as the piece-wise linear PWL) nonlinearity of Chua s circuit has played in chaotic systems [6]. Note that this Letter is the first step to study the possible chaotic phenomena related to the attraction/repulsion functions, the relation of the results here to the study of swarms and chaos study in swarm models are further topics. The rest of this Letter is organized as follows. Some simple preliminaries for frequency-domain method are given in Section 2. A simple three-dimensional system is constructed and some interesting oscillating phenomena are shown by computer simulation in Section 3. Section 4 gives some extension to multi-input and multi-output systems. And some potential relation between the model in this Letter and the swarm model in [17] is addressed. In Section 5, frequency-domain condition of dichotomy is established for a more general system. It can be used to test the nonexistence of chaotic attractors or limit cycles. An example is given to illustrate the results. The last section concludes the Letter. Throughout this Letter, A<0 means that A is a Hermitian and negative definite matrix. The superscript means transpose for real matrices or conjugate transpose for complex matrices. Re{Y } means 1 2 Y + Y ) for any real or complex square matrix Y. 2. Preliminaries First we introduce the definition of dichotomy. Consider the following system, ẋ = ft,x), 1) where f : R + R n R n is continuous and locally Lipschitz continuous in the second argument. Suppose that every solution xt; t 0,x 0 ) of system 1) with t 0 0, and xt 0 ) = x 0 R n can be continued to [t 0, + ). Definition 1. Eq. 1) is said to be dichotomous if every bounded solution is convergent to a certain equilibrium of 1). Obviously, the property of dichotomy can guarantee that there are not chaotic attractors or limit cycles in nonlinear systems. A large class of nonlinear systems can be separated into two parts: linear part and feedback nonlinear part. This kind of systems can be depicted as in Fig. 1. Gs) denotes the transfer function of the linear part, ϕ denotes the feedback nonlinear function. The following well-known Yakubovich Kalman theorem provides a foundation for getting the frequency-domain conditions for some global properties of the nonlinear systems depicted as above. For example, circle criterion and Popov criterion

3 Z. Duan et al. / Physics Letters A ) Fig. 1. Feedback nonlinear system. for absolute stability can be implied by Yakubovich Kalman theorem [7]. LetA and B be complex matrices with orders n n and n m, respectively, and Gx, ξ) = x Gx + 2Re { x Dξ } + ξ Γξ be a Hermitian form of x C n and ξ C m. G = G,Γ = Γ and D are complex matrices with orders n n, m m and n m, respectively. Lemma 1 [7,8]. Suppose that A, B) is controllable. Then there exists a matrix H = H satisfying the inequality 2Re { x HAx+ Bξ) } + Gx, ξ) 0, if and only if x C n,ξ C m G iwi A) 1 Bξ,ξ ) 0, for all ξ C m and all w R with detiwi A) 0.IncaseA and B are real matrices, H is real as well. For a given linear system, the following well-known Kalman Yakubovich Popov KYP) lemma establishes the relationship between the frequency domain method and the time domain method. It has been recognized as one of the most basic tools in systems theory. In fact, KYP lemma can be implied by Lemma 1. Lemma 2 [13,14]. Given A R n n, B R n m, M = M R n+m) n+m), with detiwi A) 0 for w R and A, B) is controllable, the following two statements are equivalent: i) iwi A) 1 ) B M iwi A) 1 ) B 0, w R. I I ii) there is a real symmetric matrix P such that PA+ A M + ) P PB B 0. P 0 The corresponding equivalence for strict inequalities holds even if A, B) is not controllable. 3. A new chaotic system By using the nonlinear function given in [17], we consider the following system formed by a linear system and an attraction/repulsion function, { dx = Ax + bf y), y = c 0 x, 2)

4 142 Z. Duan et al. / Physics Letters A ) Fig. 2. Functions f and g. where A = ) ) ) 0 1 u 0 2 v 0.6 0, b= 0, c0 = 1, x = ) x 2, x 3 u and v are parameters to be determined, fy)= y1 20 exp 5y 2 )), y can be viewed as an output of the linear system ẋ = Ax. Please refer to Fig. 2 for the function f. First we discuss the equilibria of system 2). Obviously, any equilibrium x eq = x 1eq,x 2eq,x 3eq ) satisfies 0 1 u v y eq = c 0 x eq. ) x1eq ) ) 0 x 2eq = 0 x 3eq y eq 1 20 exp 5y 2 eq ) ), By the first two equations of 3), one knows that x 1eq = 0.6ux 3eq /v, x 2eq = ux 3eq,andy eq = u u/v)x 3eq. Therefore, 3) holds if and only if the following equation about x 3eq holds, 0.12u/v 3)x 3eq u u/v)x 3eq 1 20 exp 5u u/v) 2 x 2 3eq)) = 0. According to the discussion above, any solution x 3eq of 4) can determine an equilibrium x eq of 2). For example, when u = 1, v = 2.5, Eq. 4) becomes 2.952x 3eq 0.82x 3eq 1 20 exp 3.362x 2 3eq )) = 0. By simple numerical computation, one knows that Eq. 5) has three solutions. Let the left part of 5) be gx 3eq ), please refer to Fig. 2 for the function g. According to the equilibrium analysis above, we know that there are three equilibria in system 2) generally. In what follows, we show some interesting oscillating phenomena in system 2) by computer simulation. First take u = 1, v = 2.5, see Fig. 3 for the solution of 2) at the given initial value. If we change parameters u and v, the solutions of 2) will change dramatically, see Figs. 4 7 for different oscillating solutions of 2) with different parameters at the same initial value given above. From the figures above, one can see rich oscillating phenomena in system 2). It is an interesting topic in chaotic systems to increase the complexity of behavior systematically [24,25]. Here we can change the behavior of system 2) by adding another nonlinear function. x1 3) 4) 5)

5 Z. Duan et al. / Physics Letters A ) Fig. 3. The solution of 2) with initial value x0) =[ 0.1, 1.5, 2].x 1 t), x 2 t), x 3 t);x 1,x 2,x 3 space. Fig. 4. The solution of 2) with u = 0.8, v = 2.5. x 1 t), x 2 t), x 3 t);x 1,x 2,x 3 space. Fig. 5. The solution of 2) with u = 0.5, v = 2.5. x 1 t), x 2 t), x 3 t);x 1,x 2,x 3 space.

6 144 Z. Duan et al. / Physics Letters A ) Extension to multi-input and multi-output MIMO) systems Chaotic systems with more than one nonlinear function were studied and remarkable chaotic oscillating phenomena were shown in [24,25]. Here we can also extend system 2) by adding another nonlinear function as follows { dx = Ax + B [ f 1 y 1 ), f 2 y 2 ) ], 6) y = Cx, where A is as given in 2), ) 0 0 ) y1 B = 0 1, y =, C = y f 1 t) = f 2 t) = t 1 20 exp 5t 2)). c1 c 2 ) ) =, One can see that c 1 = c 0, c 0 is given as in 2). y 2 = c 2 x can be viewed as another output of linear system ẋ = Ax, f 2 y 2 ) can be viewed as another input. In this way system 2) is extended to an MIMO system. Of course, one can also see some interesting oscillating phenomena in system 6). For example, with the parameters given above, one can see the oscillating solution as in Fig. 9 with the same initial value given as in the above section. Comparing Fig. 7 with Fig. 8, one can see some differences by adding another new nonlinear function. As one can imagine, we can change solutions of system 2) largely by choosing the function f 2 and the vector c 2. For example, taking f 2 t) = t1 30 exp 10t 2 )), c 2 = ), one can see the oscillating solution as in Fig. 9 with the same initial value given as in the above section. Fig. 6. The solution of 2) with u = 0.8, v = 5. x 1 t), x 2 t), x 3 t);x 1,x 2,x 3 space. Fig. 7. The solution of 2) with u = 1, v = 15. x 1 t), x 2 t), x 3 t);x 1,x 2,x 3 space.

7 Z. Duan et al. / Physics Letters A ) Stability of swarms was studied in [17] by using attraction/repulsion functions. Here we pay attention to the possible chaotic oscillating phenomena in simple systems related to attraction/repulsion functions. In what follows we discuss the potential relation between the model 6) and the swarm model given in [17]. First we see another equivalent model of 6). Take c 3 = 0 0 1) and let P = c1 c2 c3 ). Obviously, P is a nonsingular matrix. Let y = y 1 y 2 y 3 ) = Px, then one can get the following equivalent system of 6) ẏ = PAP 1 y + PB [ f 1 y 1 ), f 2 y 2 ) ]. 7) The swarm model studied in [17] is as follows, M ẋ i = g x i x j ), j=1,j i i = 1,...,M, 8) where x i represents the ith individual, x i can be a n-dimensional vector in [17]. M is the number of individuals, gt) = ta b exp t 2 c )) represents the function of attraction and repulsion between the members. Here we only discuss the case of three individuals, i.e., M = 3, and x i is one-dimensional. Let z 1 = x 1 x 2,z 2 = x 2 x 3,z 3 = x 3 and z = z 1 z 2 z 3 ). Noticing that g t) = gt), obviously system 8) with M = 3 can be Fig. 8. The solution of 6) with u = 1, v = 15. x 1 t), x 2 t), x 3 t);x 1,x 2,x 3 space. Fig. 9. The solution of 6) with u = 1, v = 15, c 2 = ).x 1 t), x 2 t), x 3 t);x 1,x 2,x 3 space.

8 146 Z. Duan et al. / Physics Letters A ) rewritten as follows ż = W [ gz 1 ), gz 2 ), gz 1 + z 2 ) ], 9) where W is the corresponding matrix after the variable transformation. Separating a linear part from 9) by changing the parameter a in nonlinear function g, system9) can be written as ż = A 0 z + B 0 [ g1 z 1 ), g 2 z 2 ), g 3 z 1 + z 2 ) ], where g i t) = ta i b exp t 2 c )), i = 1, 2, 3. Comparing system 7) with 10), one can see that there is another nonlinear function g 3 z 1 +z 2 ) in 10).Sosystem7) can be viewed as a simplified and abstracted system of 10).Of course, there is not necessary meaning in swarms for the parameters in 7) or 6)). Note that this Letter is the first step towards constructing chaotic systems by using attraction/repulsion functions, the study of chaotic phenomena in the model 10) or 8) is a topic of further research. We have reasons to believe that the attraction/repulsion functions can play the same roles as the PWL nonlinearity of Chua s circuit has played in chaotic systems [6]. From discussions above, one can see some interesting oscillating phenomena in system 2) or 6). In what follows, we consider the nonexistence of chaotic attractors or limit cycles by the frequency-domainmethod established in [7,26]. 10) 5. Frequency-domain method System 6) studied in the section above can be generalized to the following high order system { dx = Ax + BF y), y = Cx, where A R n n, B R n m, C R m n, x = x 1,...,x n ), y = y 1,...,y m ), Fy)= f 1 y 1 ),...,f m y m ) ), fi y i ) = y i αi β i exp γ i yi 2 )), α i, β i, γ i are scalars. y can be viewed as the output of system ẋ = Ax.Fromy = Cx, one can get ẏ = CAx + CBFy). Viewing ẏ as the output of system 11), then the transfer function from Fy)to ẏ is Kp)= CApI A) 1 B + CB. Obviously, K0) = 0. In addition, suppose that nonlinear functions f i t), i = 1,...,m have bounded derivatives and µ i1 f i t) µ i2. 12) Let µ 1 = diagµ 11,...,µ m1 ), µ 2 = diagµ 12,...,µ m2 ). With the transfer function Kp), by using the method in [7,26] one can get the following result for the property of dichotomy which is an important property for testing the nonexistence of limit cycles or chaotic attractors. Theorem 1. Suppose A has no pure imaginary eigenvalues, A, B) is controllable and A, C) is observable. System 11) has isolated equilibria. If there exist diagonal matrices κ,ɛ, τ with τ 0, ɛ>0 such that the following frequency domain inequality holds: 11) Re { κkiw)+ K iw)ɛkiw) [ µ 1 Kiw) iwi ] [ τ µ2 Kiw) iwi ]} 0, w R, then system 11) is dichotomous. 13)

9 Z. Duan et al. / Physics Letters A ) Proof. The functions f i,i = 1,...,m are differentiable, i.e., df i y i t))/ exist for all t 0. With the following notations ) ) A B 0 A C ) ) xt) Q =, L=, D= 0 0 I B C, zt)=, Fyt)) it is obvious that any solution of system 11) satisfies the system dzt) = Qzt) + Lξt), dyt) = D zt), t 0, where ξt) = df yt))/. Then one can complete the proof by using the method in [7,26]. 14) Remark 1. We should point out that the form of system 11) is different from the system form given in [7]. Especially, K0) = 0insystem11).In[7], K0) is required to be nonsingular generally. Remark 2. From the definition of dichotomy, one knows that the existence of chaotic attractors or limit cycles is impossible in dichotomous systems. Obviously, the method to avoid chaos by studying the property of dichotomy is different from the chaos control method in [4]. The external control strategies to avoid chaos sometimes are unrealistic in real physical systems. The property of dichotomy is a kind of physical property. Therefore, studying the property of dichotomy to know the nonexistence of chaos is more reasonable in physical systems. If the frequency-domain condition in Theorem 1 holds with τ = 0, it can be turned into linear matrix inequality LMI) by Lemma 2. Corollary 1. If A, B) is controllable, the frequency-domain condition in Theorem 1 holds with τ = 0, if and only if there exists P = P such that A C ɛca 1 2 A C κ + A C ɛcb 1 2 κca+ B C ɛca CBɛB C κcb B C κ ) + PA+ A P PB B P 0 ) 0. According to Corollary 1, one can test the frequency-domain condition with τ = 0 in Theorem 1 by solving the LMI 15). And based on LMI method, feedback controller design problems can be discussed for system 11). Remark 3. Obviously, one can see that the frequency-domain condition in Theorem 1 is conservative. For the characteristics of nonlinear functions f i, only the boundedness of derivatives is used. For some other nonlinear functions g i y i ), if the bounds of derivatives of g i are the same with the ones of f i, the corresponding system obtained by substituting f i by g i in system 11) is also dichotomous under the conditions of Theorem 1. One can reduce the conservatism of Theorem 1 to some degree by the following method. Transform system 11) into the following system, { dx = A + αbc)x + B Fy) αy ), y = Cx, where α is any real diagonal matrix. Obviously, system 16) is equivalent to system 11). But one can get a transfer function Gp) different from Kp) when α 0. Taking different parameter matrices α, one can get different transfer functions. And the bounds of the corresponding nonlinear functions are also different. For any parameter matrix α, iftheconditionsoftheorem1aresatisfied forsystem 16), thensystem 11) is dichotomous. Inthisway, one can reduce the conservatism of the frequency-domain inequality in Theorem 1. 15) 16)

10 148 Z. Duan et al. / Physics Letters A ) Example. Consider system 2) again. The transfer function from fy)to ẏ is Kp)= 0.3p3 + 2u 0.18)p 2 uv + 0.3v 1.2u)p p p 2 + v u)p + 3v 0.12u. And f y) satisfies 19 = µ 1 f y) µ 2 = 10. Testing the frequency-domain inequality in Theorem 1 on system 2), for the parameters u = 1,v= 2.5; u = 0.8, v = 2.5; u = 0.5, v = 2.5; u = 0.8, v = 5; u = 1, v = 15, respectively, the frequency-domain condition is broken. Refer to Section 3, one can see various oscillating solutions corresponding to these parameters. In addition, by using the conditions given in Theorem 1 one can determine some parameter domains for the nonexistence of chaotic attractors or limit cycles in system 2). For example, if v = 2.5isfixed,whenu, 0] we know that system 2) is dichotomous except at two points: u = and one point near u = 0.602where the observability condition is broken. That is, it is impossible for the existence of chaotic attractors or limit cycles at these parameters in system 2). For simplicity, one can consider system 6) similarly by Corollary Conclusions and future works In this Letter, a new chaotic system is constructed by combining a three-dimensional linear system and the attraction/repulsion functions presented in [17] for the study of swarm aggregations. Rich oscillating phenomena are shown by computer simulation. And some extension to multi-input and multi-output systems is also given. In addition, some parameter domains for the nonexistence of chaotic attractors or limit cycles can be determined for the given system by using the frequency-domain condition established here. Attraction/repulsion functions which played important roles in the stability analysis of swarms [17 19] are a class of interesting nonlinear functions with special characteristics. In the field of chaos study, the nonlinearity of Chua s circuit has played important roles for constructing chaotic attractors. It is interesting to see if the attraction/repulsion functions can play the same roles as the piece-wise linear nonlinearity of Chua s circuit has played in chaotic systems. This can give a useful extension of simple chaotic systems. The attraction/repulsion functions used in this Letter are closely related to swarm aggregations [17]. It is known that chaos appears in multi-agent systems [22,23]. The study of possible chaotic phenomena in swarm models given in [17] is an interesting topic which will give a special meaning for the study of swarms. On the other hand, in the study of chaos, it is generally hard to give the existence of chaotic attractors in theory. So it is interesting to know some parameter domains for the nonexistence of bounded oscillating solutions. The frequency domain method established in this Letter for testing the property of dichotomy is generally conservative. More effective methods are needed for this line by exploiting the special characteristics of given systems. We hope the results of this Letter can be helpful for opening a further study on the effects of attraction/repulsion functions in chaotic systems and other related topics. References [1] S. Celikovsky, G. Chen, Int. J. Bifur. Chaos 12 8) 2002) [2] E.N. Lorenz, J. Atmos. Sci ) 130. [3] G. Chen, T. Ueta, Int. J. Bifur. Chaos ) [4] G. Chen, X. Dong, From Chaos to Order: Methodologies, Perspectives and Applications, World Scientific, Singapore, [5] J. Lü, G. Chen, D. Cheng, S. Celikovsky, Int. J. Bifur. Chaos 12 12) 2002) [6] R. Madan, Chua s Circuit: A Paradigm for Chaos, World Scientific, Singapore, [7] G.A. Leonov, D.V. Ponomarenko, V.B. Sminova, Frequency-Domain Method for Nonlinear Analysis: Theory and Applications, World Scientific, Singapore, 1996.

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