frequency domain method for the dichotomy of ... - Semantic Scholar
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International Journal of Bifurcation and Chaos, Vol. 15, No. 8 (2005) 2485–2505 c World Scientific Publishing Company
FREQUENCY DOMAIN METHOD FOR THE DICHOTOMY OF MODIFIED CHUA’S EQUATIONS ZHISHENG DUAN, JIN-ZHI WANG and LIN HUANG State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, Peking University, Beijing 100871, P. R. China [email protected] Received April 2, 2004; Revised August 11, 2004
On condition of dichotomy, it is pointed out that in Lorenz and a kind of R¨ ossler-like system chaotic attractors or limit cycles will disappear if nonlinearity of the product of two variables is replaced by some single variable nonlinearity, for example, nonlinearity of Chua’s circuit. Furthermore, an extended Chua’s circuit with two nonlinear functions is presented. By computer simulation it is shown that oscillating phenomena in the extended Chua’s circuit are richer than the single Chua’s circuit. The corresponding extension for smooth Chua’s equations is also considered. The effects of input and output coupling are analyzed for the extended Chua’s circuit. Keywords: Frequency domain method; dichotomy; R¨ ossler-like system; Lorenz system; extended Chua’s circuit.
1. Introduction The study of Chaotic attractors has attracted a lot of researchers since Lorenz presented a simple three-dimensional chaotic system, see [Celikovski & Chen, 2002; Chua, 1994; Lorenz, 1963; Madan, 1993; Shil’nikov, 1993] and references therein. During the last four decades two large classes of chaotic systems were studied extensively, one is with nonlinearity of the product of two variables such as Lorenz and R¨ ossler systems [Chen & Ueta, 1999; L¨ u et al., 2002; R¨ ossler, 1979], another is with single variable nonlinear functions such as Chua’s circuits with piecewise linear (PWL) nonlinearity [Chua et al., 1986; Imai et al., 2002; Kapitaniak et al., 1994] and smooth Chua’s equations with cubic nonlinearity studied in [Huang et al., 1996; Khibnik et al., 1993; Liao & Chen, 1998; Tsuneda, 2005]. In addition, R¨ossler equation with nonlinearity of the product of two variables was generalized to a class of R¨ osslerlike equations in which only single variable nonlinearity is involved [Thomas, 1999]. Based on the
concept of feedback circuits, remarkable labyrinth chaos was shown in [Thomas, 1999]. So far, different methods for chaos analysis and control have been established [Chen & Dong, 1998; Madan, 1993]. The various literature references show that the study of chaos is a universal problem. In the study of chaos, it is generally hard to present the existence of chaotic attractors in theory [Stewart, 2000]. Based on the harmonic balance principle, a practical approach for predicting chaos dynamics in nonlinear systems was presented in [Genesio & Tesi, 1992]. This method is practically applicable, but not completely exact. Therefore, it is interesting to know the nonexistence of chaotic attractors. Recently, by using the frequency domain method developed in [Huang, 2003; Leonov et al., 1996] the frequency domain conditions of dichotomy were established to analyze the nonexistence of chaotic attractors or limit cycles in Chua’s circuit or coupled Chua’s circuit [Duan et al., 2004a; Leonov et al., 1996]. And the study of dichotomy
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is an important aspect in view of the existence and nonexistence of complex behavior in dynamical systems [Chua, 1998]. This paper is devoted to studying the effects of input and output coupling in an extended Chua’s circuit with two nonlinear functions. Chaotic systems with more than one nonlinear function were studied and remarkable chaotic phenomena were shown in [Thomas, 1999; Yalcin et al., 2002]. According to the location of the equilibrium points in state space, interesting scroll grid attractors were presented in [Yalcin et al., 2002]. Here, we also pay attention to the effect of nonlinear coupling by establishing the condition of dichotomy. The rest of this paper is organized as follows. In Sec. 2, the basic concept and results in frequency domain method are introduced briefly. In Sec. 3, nonlinearity of the product of two variables and single variable nonlinearity are compared in a R¨ ossler-like system and the Lorenz system. It is pointed out that in Lorenz system and R¨ossler-like system chaotic attractors or limit cycles do not exist if the nonlinearity is replaced by nonlinearity of Chua’s circuit. This also indicates that the frequency domain method here cannot be used to deal with the nonlinearity of the products of different variables. In Sec. 4, an extended Chua’s circuit with two nonlinear functions is presented and some interesting chaotic oscillating phenomena are shown. In Sec. 5, smooth Chua’s equations with cubic nonlinearity are considered similarly. In Sec. 6, the effects of input and output coupling are analyzed in two aspects, one is to avoid chaos, another is to generate new oscillating phenomena. Section 7 concludes the paper. Throughout this paper, 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 .
said to be dichotomous if every bounded solution is convergent to a certain equilibrium of (1). Obviously, if a nonlinear system is dichotomous, then the existence of chaotic attractors or limit cycles is impossible. By using the Yakubovich–Kalman theorem [Leonov et al., 1996], the following system was studied in [Duan et al., 2004a], dy = Ay + Bϕ(x), dt
(2)
where A ∈ Rn×n , B ∈ Rn×m , n > m, y = (y1 , . . . , yn )∗ , x = (x1 , . . . , xm )∗ = (y1 , . . . , ym )∗ , ϕ(x) = (ϕ1 (x1 ), . . . , ϕm (xm ))∗ . Notice that x is a part of y, obviously Chua’s circuits with PWL nonlinearity, smooth Chua’s equations with cubic nonlinearity and R¨ ossler-like systems studied in [Thomas, 1999] can be viewed as special cases of (2). Viewing dx/dt as the output and ϕ(x) as the input, then system (2) can be viewed as a multiinput and multi-output (MIMO) system. Let C be the matrix composed of the first m rows of A, and R be the matrix composed of the first m rows of B. Then the transfer function from ϕ(x) to x˙ is K(s) = C(sI − A)−1 B + R. A simple frequency domain condition of dichotomy for system (2) was established in [Duan et al., 2004a]. Lemma 1. Suppose that A has no pure imaginary
eigenvalues, (A, B) is controllable and (A, C) is observable. If (2) has isolated equilibria and there exist diagonal matrices = diag(1 , . . . , m ) > 0 and κ = diag(κ1 , . . . , κm ) such that the following frequency domain inequality holds: Re{κK(iw) + K ∗ (iw)K(iw)} ≤ 0,
2. Frequency Domain Method for Systems with Single Variable Nonlinearity
∀ w ∈ R, (3)
then system (2) is dichotomous.
In this section, we briefly review a basic definition and two results in frequency domain method. Consider the following system, x˙ = f (t, x),
Definition 1 [Leonov et al., 1996]. Equation (1) is
(1)
where f : R+ × Rn → Rn is continuous and locally Lipschitz continuous in the second argument. Suppose that every solution x(t; t0 , x0 ) of system (1) with t0 ≥ 0, and x(t0 ) = x0 ∈ Rn can be continued to [t0 , +∞).
This lemma can be used to analyze the nonexistence of chaotic attractors or limit cycles. One just needs to test the frequency domain inequality (3). Generally, it is convenient to test this inequality by solving matrix inequality. So we often express the condition (3) in the form of linear matrix inequality (LMI) by KYP Lemma in [Anderson, 1967; Rantzer, 1996]. Lemma 2. If (A, B) is controllable, then (3) holds
if, and only if, there exists P = P ∗ such that
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C ∗ C
1 ∗ C κ + C ∗ R 2 R∗ R + Re{κR}
1 κC + R∗ C 2 P A + A∗ P + B ∗P
PB 0
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we can view y˙ 3 as the output of (4). Let C = (0 0 −5.7), then y˙3 = Cy + y1 y3 + 0.2. But now one can see that (A, C) is not observable. In order to meet the observability condition, we can adjust system (4) a little bit as (we will see the meaning of observability in the forthcoming discussion)
≤ 0.
Based on Lemma 2 one can test the frequency domain condition (3) by solving LMI and discuss feedback controller design problems by the popular LMI method in control theory.
dy = A1 y + B1 (y1 y3 + 0.2), dt where
0 A1 = 1 0.001
Remark 1. The method to avoid chaos by study-
ing the property of dichotomy is different from the chaos control method in [Chen & Dong, 1998]. The external control strategy to avoid chaos sometimes is 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. Based on the harmonic balance principle, a practical approach for predicting chaos dynamics in nonlinear systems was presented in [Genesio & Tesi, 1992]. This method is practically applicable, but not completely exact. The frequency domain condition of dichotomy here can be proved to be strictly correct by the work of Leonov [Leonov et al., 1996]. With Lemma 1, in what follows we first discuss some differences between two classes of nonlinearity in chaotic systems.
3. Comparison Between Two Classes of Nonlinearity
(5)
−1 −1 0.2 0 , 0 −5.7
B1 = B, B and y are given as above. Let C1 = (0.001 0 − 5.7). Computer simulations show that system (5) has a chaotic attractor similar to the R¨ossler attractor, see Fig. 1. Here we call system (5) as a R¨ ossler-like system.
3.2. The Lorenz system The following Lorenz system was studied extensively [Chen & Ueta, 1999; Lorenz, 1963; L¨ u et al., 2002]. Chen’s attractor and L¨ u’s attractor were found in Lorenz-type systems, dy = A2 y + B2 (y1 y3 , y2 y3 )∗ , dt where
−10 28 A2 = 0
10 −1 0
−
0 0 , 8
(6)
0 0 B2 = −1 0 , 0 1
3
see Fig. 2 for the typical Lorenz attractor.
3.1. A R¨ ossler-like system According to [R¨ ossler, 1979], there is a typical chaotic attractor in the following R¨ ossler system dy = Ay + B(y1 y3 + 0.2), dt where
0 −1 −1 0 , A = 1 0.2 0 0 −5.7
0 B = 0 , 1
(4) y1 y = y2. y3
The above R¨ossler system was revisited and generalized to a class of R¨ ossler-like systems [Thomas, 1999]. Here we consider another kind of R¨ossler-like system. Corresponding to system (2),
3.3. PWL nonlinearity of Chua’s circuit In the R¨ ossler-like system and Lorenz system above, one can see some typical chaotic phenomena. But what will happen if we change the nonlinear functions in (5) and (6) into nonlinearity of Chua’s circuit? Are there chaotic phenomena or limit cycles again? In what follows, we discuss such kind of problems. First according to (5), the following system is obtained by substituting the nonlinear function y1 y3 + 0.2 into PWL function, dy = A1 y + B1 f (y3 ), dt
(7)
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15
5
−5
−15 0
40
80
120
160
200
(a)
25
15
5
10 5
10
0 −5 0
−10 −15 −20
−10
(b) Fig. 1.
The solution of (5) with initial value y(0) = [−1, 0, 0]∗ . (a) y1 (t), y2 (t), y3 (t), and (b) y1 , y2 , y3 space.
where A1 , B1 are given as in (5), f (y3 ) = G2 y3 + 0.5(G1 − G2 )(|y3 + 1) − |y3 − 1|), G1 and G2 are parameters to be chosen. Let C1 = (0.001 0 −5.7), D1 = 1, then the transfer function from f (y3 ) to y˙ 3 in system (7) is K1 (s) = C1 (sI − A1 )−1 B1 + D1 = (s3 − 0.2s2 + s/s3 + 5.5s2 − 0.139s + 5.6998). Obviously (A1 , B1 ) and (A1 , C1 ) are controllable and observable, respectively. And by testing the matrix inequality in Lemma 2, we know that K1 (s) satisfies the frequency domain condition in Lemma 1.
Therefore, for any parameters G1 and G2 if (7) has isolated equilibria, then (7) is dichotomous. That is, there exist no chaotic attractors or limit cycles. For example, take G1 = −0.67, G2 = 1.27, see Fig. 3 for the solution of (7) at the same initial value as given in Fig. 1. From Fig. 3, one can see that the solution is dramatically unbounded. Furthermore, by Lemma 1, one knows that when PWL function f (y3 ) in (7) is substituted by any piecewise continuous nonlinear function g(y3 ),
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30
10
−10
−30 0
10
20
30
40
(a)
50
30
10
20
20 15 10
0
5 −20
−10
−5
0
(b) Fig. 2.
The solution of (6) with initial value y(0) = [10, 1, 10]∗ . (a) y1 (t), y2 (t), y3 (t), and (b) y1 , y2 , y3 space.
the corresponding system is dichotomous if it has isolated equilibria. Similarly, if we substitute the nonlinear functions y1 y3 , y2 y3 in the Lorenz system (6) by nonlinearity of Chua’s circuit, then we can get the following system dy = A2 y + B2 (f1 (y2 ), f2 (y3 ))∗ , (8) dt where A2 and B2 are given as in (6), f1 (y2 ) = G12 y2 + 0.5(G11 − G12 )(|y2 + 1) − |y2 − 1|), f2 (y3 ) = G22 y3 + 0.5(G21 − G22 )(|y3 + 1) − |y3 − 1|).
Let C2 be the matrix composed of the last two rows of A2 , D2 be the matrix composed of the last two rows of B2 . Then the transfer function from (f1 (y2 ), f2 (y3 ))∗ to (y˙ 2 , y˙3 )∗ in system (8) is K2 (s) = C2 (sI − A2 )−1 B2 + D2 −s2 − 10s 0 s2 + 11s − 270 = s 0 8 s+ 3
.
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200
0
−200
−400 0
20
40
60
(a)
0.1
0.05
0
−0.05 400 400
200 200
0 0
−200
−200 −400
−400
(b) Fig. 3.
The solution of (7) with initial value y(0) = [−1, 0, 0]∗ . (a) y1 (t), y2 (t), y3 (t), and (b) y1 , y2 , y3 space.
Obviously (A2 , B2 ) and (A2 , C2 ) are controllable and observable, respectively. And by testing the matrix inequality in Lemma 2, we know that K2 (s) satisfies the frequency domain condition in Lemma 1. Therefore, for any parameters G11 , G12 , G21 and G22 if (8) has isolated equilibria, then (8) is dichotomous. Remark 2. From above, one can see that chaotic
phenomena disappear in R¨ ossler-like and Lorenz systems after substituting nonlinearity of the product of different variables by PWL nonlinearity. Of
course, after this substitution, the system equilibria and the interactions of variables are changed largely. This also indicates the complexity and importance of the interactions of different variables in chaotic systems [Thomas, 1999; Chua, 1998]. In addition, we can see that the frequency domain method here cannot be used to deal with nonlinearity of the products of different variables. For simplicity, here we only discuss PWL nonlinearity of Chua’s circuit, some other nonlinear functions in [Huang et al., 1996; Thomas, 1999; Yalcin et al., 2002] can be considered similarly.
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4. An Extended Chua’s Circuit with Two Nonlinear Functions Chua’s circuits were studied extensively [Madan, 1993]. Generally, Chua’s circuits can be viewed as single-input and single-output feedback nonlinear systems. Coupled Chua’s circuits studied in [Duan et al., 2004a; Imai et al., 2002] can be viewed as multi-input and multi-output feedback nonlinear systems. Chua’s circuits and coupled Chua’s circuits can all be viewed as Lur’e systems [Suykens et al., 1997; Suykens et al., 1998]. The effects of input and output coupling were studied in [Duan et al., 2004a] for coupled Chua’s circuit. Some linearly
(a)
Fig. 4.
The extended Chua’s circuit.
4
2
0
−2
−4
−6 0
250
500
(b)
0.4
0
−0.4 4 2
4 0 −2
0 −4 −6
Fig. 5. space.
−4
The solution of (9) with initial value v(0) = (1, −1, 0.4)∗ (G21 = 0, G22 = 0). (a) v1 (t), v2 (t), i3 (t), and (b) v1 , v2 , i3
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interconnected systems were studied in [Duan et al., 2004b]. Generally, the dimension of the coupled system in [Duan et al., 2004a] is 6. In order to study the effects of input and output coupling in lower dimensional systems, we give an extended Chua’s circuit with two nonlinear functions, see Fig. 4. Chaotic systems with more than one nonlinear function were studied and remarkable chaotic phenomena were shown in [Thomas, 1999; Yalcin et al., 2002]. Here, we also pay attention to the effect of nonlinear coupling by establishing the condition of dichotomy. The differential equation of the extended Chua’s circuit in Fig. 4 is v˙ = Av + B(f1 (v1 ), f2 (v2 ))∗ , (a)
where
−1 C1 R1 1 A= C2 R1 0 −1 C1 B= 0 0
(9)
1 C1 R1
0
1 , C2 −R0 L1
−1 C2 R1 −1 L1
v1 v = v2 , i3
0
−1 , C2 0
4
2
0
−2
−4
−6 0
250
500
(b) 0.5
−0.5
0.5
−0.5 5 0
4 −5 5
0 0 −5
−4
Fig. 6. The solution of (9) with initial value v(0) = (1, −1, 0.4)∗ (G21 = 0.1, G22 = −0.01). (a) v1 (t), v2 (t), i3 (t), and (b) v1 , v2 , i3 space.
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f1 and f2 are two nonlinear functions defined as
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0.0136073, R0 = 0.02969968/0.0136073, G11 = 0.1690817, G12 = −0.4767822, G21 = 0, G22 = 0, see Fig. 5. As one can imagine, the solutions of (9) will change largely with different parameters G21 , G22 of the second nonlinearity, see Figs. 6–9. From Figs. 6–9, one can see more oscillating phenomena in system (9) by choosing the function f2 . So the oscillating phenomena of the extended Chua’s circuit are much richer than the single Chua’s circuit. One can increase the
f1 (x) = G12 x + 0.5(G11 − G12 )(|x + 1| − |x − 1|), f2 (x) = G22 x + 0.5(G21 − G22 )(|x + 1| − |x − 1|). Obviously, if G21 = 0, G22 = 0, the extended Chua’s circuit is the canonical Chua’s circuit studied in [Madan, 1993]. According to the traditional references on Chua’s circuit, there is a chaotic attractor in system (9) with parameters C1 = −1/1.301814, C2 = 1, R1 = 1, L1 = −1/ 4
2
0
−2
−4
−6 0
250
500
(a)
0
−1 1
0
−1 5 0
5 −5 5
0 0 −5
−5
(b) Fig. 7. The solution of (9) with initial value v(0) = (1, −1, 0.4)∗ (G21 = 0.31, G22 = −0.25). (a) v1 (t), v2 (t), i3 (t), and (b) v1 , v2 , i3 space.
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15
0
−15 0
250
500
(a)
3
0
−3 10 5 0
5
−5 −10
−5
−15 −20
−15
(b) Fig. 8. The solution of (9) with initial value v(0) = (1, −1, 0.4)∗ (G21 = 2.98, G22 = −0.35). (a) v1 (t), v2 (t), i3 (t), and (b) v1 , v2 , i3 space.
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0
−4
−8 0
200
400
(a)
0.8
0.4
0 4 2
6 4
0
2 −2 −4 −6
−8
−6
−4
−2
0
(b) Fig. 9. The solution of (9) with initial value v(0) = (1, −1, 0.4)∗ (G21 = −0.26, G22 = −0.01). (a) v1 (t), v2 (t), i3 (t), and (b) v1 , v2 , i3 space.
complexity of behavior in simple systems systematically [Yalcin et al., 2002]. Of course, the extension of nonlinearity here can be used to consider other chaotic systems such as smooth Chua’s equations [Khibnik, 1993] and R¨ ossler-like systems [Thomas, 1999].
5. Smooth Chua’s Equations In contrast to PWL nonlinearity in Chua’s circuit, smooth Chua’s circuits with cubic nonlinearity were
also studied thoroughly [Khibnik et al., 1993; Huang et al., 1996; Liao & Chen, 1998; Tsuneda, 2005]. The choice of a cubic nonlinearity has several advantages over a piecewise linear one. It does not require absolute-valued functions and it is smooth, which is desirable from a mathematical perspective. Moreover, almost all phenomena found in the PWL version also exist in cubic version. In what follows, we extend Chua’s circuit with cubic nonlinearity as above and see some other interesting oscillating phenomena.
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−α B = 0 , 0
The following smooth Chua’s circuit was studied in [Huang et al., 1996] v˙ = Av + Bx3 ,
(10)
According to [Huang et al., 1996], there is a chaotic attractor in system (10) with parameters α = 10, β = 16, c = −0.143, see Fig. 10. We extend system (10) by adding another nonlinearity as follows
where
−αc A= 1 0
α −1 −β
x v = y z
0 1 , 0
v˙ = A1 v + B1 [x3 , y 3 ]∗ ,
0.6
0.2
−0.2
−0.6
−1
0
20
40
60
80
100
(a)
1
0
−1 0.2 0.5 0 0 −0.2
−0.5
(b) Fig. 10.
The solution of (10) with initial value v(0) = (0.1, 0.1, 0.1)∗ . (a) x(t), y(t), z(t), and (b) x, y, z space.
(11)
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where
We can change the solutions of (10) largely by choosing d and f , see Figs. 11–13. Under the parameters α = 10, β = 16, d = 0.5, f = 0.1 if we change c = −0.143 into c = −0.26, one can see the solution in Fig. 14. From above, one can see that the solutions of smooth Chua’s equations can be changed largely by introducing another new nonlinearity. It is interesting to observe a systematic increase or decrease in the complexity of behavior in simple systems.
−αc α 0 −1 + f 1 , A1 = 1 0 −β 0 −α 0 x B1 = 0 d , v = y , 0 0 z d and f are new parameters to be chosen.
(a)
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3
1
−1
−3
−5 0
(b)
10
20
30
40
50
5
0
−5 −2 2
−2 2 0
0 −2
Fig. 11. space.
−2
The solution of (11) with initial value v(0) = (0.1, 0.1, 0.1)∗ (d = −0.2, f = 0.278). (a) x(t), y(t), z(t), and (b) x, y, z
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0.2
−0.2
−0.6
−1
0
50
100
(a)
1
0
−1 0.2 1 0 0 −0.2
−1
(b) Fig. 12. space.
The solution of (11) with initial value v(0) = (0.1, 0.1, 0.1)∗ (d = 3.5, f = 0.1). (a) x(t), y(t), z(t), and (b) x, y, z
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0.6
0.2
−0.2
−0.6
−1
0
50
100
(a)
1
0
−1 0.2 1 0
0.5 0 −0.2
−0.5
(b) Fig. 13. space.
The solution of (11) with initial value v(0) = (0.1, 0.1, 0.1)∗ (d = 16, f = −0.1). (a) x(t), y(t), z(t), and (b) x, y, z
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0.5
−0.5
−1.5 0
50
100
(a)
1.5
0.5
−0.5
−1.5 0.2 1 0 0 −0.2
−1
(b) Fig. 14. The solution of (11) with initial value v(0) = (0.1, 0.1, 0.1)∗ (d = 0.5, f = 0.1, c = −0.26). (a) x(t), y(t), z(t), and (b) x, y, z space.
6. Input and Output Coupling In this section, in order to increase or decrease the complexity of behavior in simple systems, we study the effects of a class of input and output coupling. Viewing (v˙ 1 , v˙ 2 )∗ as the output, (f1 (v1 ), f2 (v2 ))∗ as
the input, system (9) can be viewed as an MIMO system. After some input and output coupling, a new system can be generated, v˙ = Av + B(f1 (α11 v1 + α12 v2 ), f2 (α21 v1 + α22 v2 ))∗ ,
(12)
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where A, B, f1 , f2 and v are given as in (9). Let „
M =
α11
α12
α21
α22
«
,z=M
„
v1 v2
«
, C, R be the matri-
ces composed of the first two rows of A and B, respectively. Here, we call M as a coupling matrix. From (12), one can get z˙ = M Cv + M R(f1 (α11 v1 + α12 v2 ), f2 (α21 v1 + α22 v2 ))∗ .
(13)
))∗
to z˙ The transfer function from (f1 (z1 ), f2 (z2 in systems (12) and (13) is M K(s) = M (C(sI − A)−1 B + R). By Lemma 1, one can get the following result for systems (12) and (13). Theorem 1. Suppose that A has no pure imaginary eigenvalues, (A, B) is controllable and (A, C) is observable. If (12) has isolated equilibria and there exist diagonal matrices = diag(1 , . . . , m ) > 0, and κ = diag(κ1 , . . . , κm ) such that the following frequency domain inequality holds:
Re{κM K(iw) + K ∗ (iw)M ∗ M K(iw)} ≤ 0, ∀ w ∈ R, (14) then system (12) is dichotomous.
C ∗ M ∗ M C 1 M C + R∗ M ∗ M C 2
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If the coupling matrix M can be chosen deliberately, one can get a simplified result for system (12) by reducing the matrix variable κ. Theorem 2. Suppose that A has no pure imaginary eigenvalues, (A, B) is controllable and (A, C) is observable. If there exist diagonal matrix = diag(1 , . . . , m ) > 0 and any matrix M such that the following frequency domain inequality holds:
Re{M K(iw)} + K ∗ (iw)M ∗ M K(iw) ≤ 0, ∀ w ∈ R,
(15)
and system (12) has isolated equilibria, then system (12) is dichotomous. Furthermore, by KYP Lemma in [Rantzer, 1996], one can also express the condition (15) in the form of linear matrix inequality (LMI). Theorem 3.
If (A, B) is controllable, then (15) holds if, and only if, there exists P = P ∗ such that 1 ∗ ∗ C M + C ∗ M ∗ M R
∗ 2 + P A + A P P B ≤ 0. 0 B ∗P 1 1 R∗ M ∗ M R + M R + R∗ M ∗ 2 2
By using Schur complement, rewrite the matrix inequality in Theorem 3 as 0 P A + A∗ P P B 0 0 B ∗P 0 0 −−1 ∗ C
1 ∗ ∗ R I I + M 0 2 0 ∗ ∗ C
1 I I ≤ 0. (16) + R ∗ M ∗ 0 2 0 In terms of the solvability condition of nonstrict matrix inequality [Helmersson, 1995], if C is with full row rank, (16) holds if, and only if, there exists a scalar β > 0 such that 0 P A + A∗ P P B 0 0 B ∗P 0 0 −−1
C∗ − β R∗ (C 0
and
R
0) ≤ 0,
0 P A + A∗ P P B 0 0 B ∗P 0 0 −−1 0
1 1 I I ≤ 0. −β I 0 2 2
(17)
(18)
I From the frequency domain inequalities (14) and (15), one knows that M = 0 is a trivial solution. When A has no pure imaginary eigenvalues, one can always take M = 0 such that system (12) is dichotomous. According to the method in [Helmersson, 1995], one can parameterize Remark 3.
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1200
1000 0.7
800
0.65 0.6
600
0.55 0.5
400 0.45 0.4
200 0.35 150 1200
100
0
1000 50
800 600 0
−200 0
0.2
0.4
0.6
400 −50
0.8
(a)
200 0
(b)
Fig. 15. The solution of (12) with initial value v(0) = (1, −1, 0.4)∗ (G21 = −0.26, G22 = −0.01). (a) v1 (t), v2 (t), i3 (t), and (b) v1 , v2 , i3 space.
all solutions M to the matrix inequality (16) by solving the inequalities (17) and (18). So one can also consider the invariance of the property of dichotomy by the method here. By choosing the coupling matrix M , one can change the behavior of solutions of system (12) systematically.
From Fig. 9, one knows that system (9) is not But dichotomous for G 21 = −0.26, G22 = −0.01.
we can get M =
−12.522192
21.620156
−2.880473
13.298164
by solv-
ing the LMI in (16) such that the frequency domain inequality in Theorem 2 holds. So system (12) is
10
6
2
−2
−6
−10 0
100
200
300
400
500
(a) Fig. 16. The solution of (12) with initial value v(0) = (1, −1, 0.4)∗ (G21 = −2.98, G22 = −0.35). (a) v1 (t), v2 (t), i3 (t), and (b) v1 , v2 , i3 space.
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0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 10 10
5 5
0 0
−5
−5 −10
−10
(b) Fig. 16.
(Continued )
dichotomous when it has isolated equilibria for any piecewise continuous nonlinear functions (not only for nonlinearity of Chua’s circuit). See Fig. 15 for the solution of (12) with the same initial value as given above.
From above we can see that oscillating solutions can be eliminated by choosing a suitable input and output coupling. On the other hand, we can also see some interesting oscillating phenomena under the input and output coupling
15
5
−5
−15 0
100
200
300
400
500
(a) Fig. 17. The solution of (12) with initial value v(0) = (1, −1, 0.4)∗ (G21 = −2.98, G22 = −0.35). (a) v1 (t), v2 (t), i3 (t), and (b) v1 , v2 , i3 space.
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1
−1
−3 15 10
15
5
10 0
5 −5 −10 −15
−15
−10
−5
0
(b) Fig. 17.
above. For example, take M =
1
0
0.5
1
, we
can see the oscillating solution in Fig. 16 for system (12).
1 −0.68 , we can see the oscilTake M = 1.4
0.35
(Continued )
is dichotomous. The smooth Chua’s equations can be studied similarly. One can also see some new oscillating phenomena with the coupling here. It would be an interesting topic to increase or decrease the complexity of behavior in simple systems systematically.
lating solution in Fig. 17 for system (12). In this section, only nonlinearity of Chua’s circuit is considered. Obviously, other nonlinear systems such as smooth Chua’s equations [Huang et al., 1996; Khibnik et al., 1993; Liao & Chen, 1998; Tsuneda, 2005] and R¨ossler-like systems [Thomas, 1999] can be studied similarly.
Remark 4.
Acknowledgments The authors would like to thank the reviewers for their detailed and helpful comments which help improve the presentation of this paper. This work is supported by the National Science Foundation of China under grant 60204007, 60334030, 10472001.
7. Conclusion
References
The frequency domain method established in [Duan et al., 2004a; Leonov et al., 1996] is used to discuss a class of feedback nonlinear systems. It is seen that a kind of R¨ ossler-like system and the Lorenz system do not have chaotic attractors or limit cycles if the nonlinearity is replaced by some single variable nonlinearity. This also indicates that the frequency domain method here cannot be used to deal with nonlinearity of the products of different variables. Furthermore, an extended Chua’s circuit is presented. Although computer simulations show that much richer oscillating phenomena exist in the extended Chua’s circuit given here than in the single Chua’s circuit, the corresponding system after a suitable input and output coupling
Anderson, B. D. O. [1967] “A system theory criterion for positive real matrices,” SIAM J. Contr. Optim. 5, 171–192. Celikovsky, S. & Chen, G. [2002] “On a generalized Lorenz canonical form of chaotic systems,” Int. J. Bifurcation and Chaos 12, 1789–1812. Chen, G. & Dong, X. [1998] From Chaos to Order: Methodologies, Perspectives and Applications (World Scientific, Singapore). Chen, G. & Ueta, T. [1999] “Yet another chaotic attractor,” Int. J. Bifurcation and Chaos 9, 1465–1466. Chua, L. O., Komuro, M. & Matsumoto, T. [1986] “The double scroll family, Parts I and II,” IEEE Trans. Circuits Syst. 33, 1073–1118. Chua, L. O. [1994] “Chua’s circuit: 10 years later,” Int. J. Cir. Th. Appl. 22, 279–306.
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Chua, L. O. [1998] CNN: A Paradigm for Complexity, World Scientific Series on Nonlinear Science, Series A, Vol. 31. Duan, Z. S., Wang, J. Z. & Huang, L. [2004a] “Multi-input and multi-output nonlinear systems: Interconnected Chua’s circuit,” Int. J. Bifurcation and Chaos 14, 3065–3081. Duan, Z. S., Huang, L., Wang, L. & Wang, J. Z. [2004b] “Some applications of small gain theorem to interconnected systems,” Syst. Contr. Lett. 52, 263–273. Genesio, A. & Tesi, A. [1992] “Harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems,” Automatica 28, 531–548. Helmersson, A. [1995] Methods for Robust Gain Scheduling (Linkoping, Sweden). Huang, A., Pivka, L. & Franz, M. [1996] “Chua’s equation with cubic nonlinearity,” Int. J. Bifurcation and Chaos 12, 2175–2222. Huang, L. [2003] Fundamental Theory of Stability and Robustness (Scientific Publishing House, Beijing), (in Chinese). Imai, T., Konishi, K., Kokame, H. & Hirata, K. [2002] “An experimental suppression of spatial instability in one-way open coupled Chua’s circuits,” Int. J. Bifurcation and Chaos 12, 2897–2905. Kapitaniak, T., Chua, L. O. & Zhong, G. Q. [1994] “Experimental hyperchaos in coupled Chua’s circuits,” IEEE Trans. Circuits Syst.-I 41, 499–503. Khibnik, A. I., Roose, D. & Chua, L. O. [1993] “On periodic orbits and homoclinic bifurcations in Chua’s circuit with a smooth nonlinearity,” Int. J. Bifurcation and Chaos 3, 363–384. Leonov, G. A., Ponomarenko, D. V. & Smirnova, V. B. [1996] Frequency Domain Method for Nonlinear Analysis: Theory and Applications (World Scientific, Singapore). Liao, T.-L. & Chen, F.-W. [1998] “Control of Chua’s circuit with a cubic nonlinearity via nonlinear
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linearization technique,” Circuits Syst. Sign. Process. 17, 719–731. Lorenz, E. N. [1963] “Deterministic nonperiodic flows,” J. Atmos. Sci. 20, 130–141. L¨ u, J., Chen, G., Cheng, D. & Celikovsky, S. [2002] “Bridge the gap between the Lorenz system and the Chen system,” Int. J. Bifurcation and Chaos 12, 2917–2926. Madan, R. [1993] Chua’s Circuit: A Paradigm for Chaos (World Scientific, Singapore). Rantzer, A. [1996] “On the Kalman–Yakubovich–Popov lemma,” Syst. Contr. Lett. 28, 7–10. Rossler, O. E. [1979] “An equation for hyperchaos,” Phys. Lett. A71, 155–157. Shil’nikov, L. P. [1993] “Chua’s circuits: Rigorous results and future problems,” IEEE Trans. Circuits Syst.-I 40, 784–786. Stewart, I. [2000] “The Lorenz attractor exists,” Nature 406, 948–949. Suykens, J. A. K., Vandewalle, J. & Chua, L. O. [1997] “Nonlinear H∞ synchronization of chaotic Lur’e systems,” Int. J. Bifurcation and Chaos 7, 1323–1335. Suykens, J. A. K., Yang, T. & Chua, L. O. [1998] “Impulsive synchronization of chaotic Lur’e systems by measurement feedback,” Int. J. Bifurcation and Chaos 8, 1371–1381. Thomas, R. [1999] “Deterministic chaos seen in terms of feedback circuits: Analysis, synthesis, ‘Labyrinth chaos’,” Int. J. Bifurcation and Chaos 9, 1889– 1905. Tsuneda, A. [2005] “A gallery of attractors from smooth Chua’s equation,” Int. J. Bifurcation and Chaos 15, 1–49. Yalcin, M., Suykens, J. A. K., Vandewalle, J. & Ozoguz, S. [2002] “Families of scroll grid attractors,” Int. J. Bifurcation and Chaos 12, 23–41.
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FREQUENCY DOMAIN METHOD FOR THE DICHOTOMY OF MODIFIED CHUA’S EQUATIONS ZHISHENG DUAN, JIN-ZHI WANG and LIN HUANG State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, Peking University, Beijing 100871, P. R. China [email protected] Received April 2, 2004; Revised August 11, 2004
On condition of dichotomy, it is pointed out that in Lorenz and a kind of R¨ ossler-like system chaotic attractors or limit cycles will disappear if nonlinearity of the product of two variables is replaced by some single variable nonlinearity, for example, nonlinearity of Chua’s circuit. Furthermore, an extended Chua’s circuit with two nonlinear functions is presented. By computer simulation it is shown that oscillating phenomena in the extended Chua’s circuit are richer than the single Chua’s circuit. The corresponding extension for smooth Chua’s equations is also considered. The effects of input and output coupling are analyzed for the extended Chua’s circuit. Keywords: Frequency domain method; dichotomy; R¨ ossler-like system; Lorenz system; extended Chua’s circuit.
1. Introduction The study of Chaotic attractors has attracted a lot of researchers since Lorenz presented a simple three-dimensional chaotic system, see [Celikovski & Chen, 2002; Chua, 1994; Lorenz, 1963; Madan, 1993; Shil’nikov, 1993] and references therein. During the last four decades two large classes of chaotic systems were studied extensively, one is with nonlinearity of the product of two variables such as Lorenz and R¨ ossler systems [Chen & Ueta, 1999; L¨ u et al., 2002; R¨ ossler, 1979], another is with single variable nonlinear functions such as Chua’s circuits with piecewise linear (PWL) nonlinearity [Chua et al., 1986; Imai et al., 2002; Kapitaniak et al., 1994] and smooth Chua’s equations with cubic nonlinearity studied in [Huang et al., 1996; Khibnik et al., 1993; Liao & Chen, 1998; Tsuneda, 2005]. In addition, R¨ossler equation with nonlinearity of the product of two variables was generalized to a class of R¨ osslerlike equations in which only single variable nonlinearity is involved [Thomas, 1999]. Based on the
concept of feedback circuits, remarkable labyrinth chaos was shown in [Thomas, 1999]. So far, different methods for chaos analysis and control have been established [Chen & Dong, 1998; Madan, 1993]. The various literature references show that the study of chaos is a universal problem. In the study of chaos, it is generally hard to present the existence of chaotic attractors in theory [Stewart, 2000]. Based on the harmonic balance principle, a practical approach for predicting chaos dynamics in nonlinear systems was presented in [Genesio & Tesi, 1992]. This method is practically applicable, but not completely exact. Therefore, it is interesting to know the nonexistence of chaotic attractors. Recently, by using the frequency domain method developed in [Huang, 2003; Leonov et al., 1996] the frequency domain conditions of dichotomy were established to analyze the nonexistence of chaotic attractors or limit cycles in Chua’s circuit or coupled Chua’s circuit [Duan et al., 2004a; Leonov et al., 1996]. And the study of dichotomy
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is an important aspect in view of the existence and nonexistence of complex behavior in dynamical systems [Chua, 1998]. This paper is devoted to studying the effects of input and output coupling in an extended Chua’s circuit with two nonlinear functions. Chaotic systems with more than one nonlinear function were studied and remarkable chaotic phenomena were shown in [Thomas, 1999; Yalcin et al., 2002]. According to the location of the equilibrium points in state space, interesting scroll grid attractors were presented in [Yalcin et al., 2002]. Here, we also pay attention to the effect of nonlinear coupling by establishing the condition of dichotomy. The rest of this paper is organized as follows. In Sec. 2, the basic concept and results in frequency domain method are introduced briefly. In Sec. 3, nonlinearity of the product of two variables and single variable nonlinearity are compared in a R¨ ossler-like system and the Lorenz system. It is pointed out that in Lorenz system and R¨ossler-like system chaotic attractors or limit cycles do not exist if the nonlinearity is replaced by nonlinearity of Chua’s circuit. This also indicates that the frequency domain method here cannot be used to deal with the nonlinearity of the products of different variables. In Sec. 4, an extended Chua’s circuit with two nonlinear functions is presented and some interesting chaotic oscillating phenomena are shown. In Sec. 5, smooth Chua’s equations with cubic nonlinearity are considered similarly. In Sec. 6, the effects of input and output coupling are analyzed in two aspects, one is to avoid chaos, another is to generate new oscillating phenomena. Section 7 concludes the paper. Throughout this paper, 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 .
said to be dichotomous if every bounded solution is convergent to a certain equilibrium of (1). Obviously, if a nonlinear system is dichotomous, then the existence of chaotic attractors or limit cycles is impossible. By using the Yakubovich–Kalman theorem [Leonov et al., 1996], the following system was studied in [Duan et al., 2004a], dy = Ay + Bϕ(x), dt
(2)
where A ∈ Rn×n , B ∈ Rn×m , n > m, y = (y1 , . . . , yn )∗ , x = (x1 , . . . , xm )∗ = (y1 , . . . , ym )∗ , ϕ(x) = (ϕ1 (x1 ), . . . , ϕm (xm ))∗ . Notice that x is a part of y, obviously Chua’s circuits with PWL nonlinearity, smooth Chua’s equations with cubic nonlinearity and R¨ ossler-like systems studied in [Thomas, 1999] can be viewed as special cases of (2). Viewing dx/dt as the output and ϕ(x) as the input, then system (2) can be viewed as a multiinput and multi-output (MIMO) system. Let C be the matrix composed of the first m rows of A, and R be the matrix composed of the first m rows of B. Then the transfer function from ϕ(x) to x˙ is K(s) = C(sI − A)−1 B + R. A simple frequency domain condition of dichotomy for system (2) was established in [Duan et al., 2004a]. Lemma 1. Suppose that A has no pure imaginary
eigenvalues, (A, B) is controllable and (A, C) is observable. If (2) has isolated equilibria and there exist diagonal matrices = diag(1 , . . . , m ) > 0 and κ = diag(κ1 , . . . , κm ) such that the following frequency domain inequality holds: Re{κK(iw) + K ∗ (iw)K(iw)} ≤ 0,
2. Frequency Domain Method for Systems with Single Variable Nonlinearity
∀ w ∈ R, (3)
then system (2) is dichotomous.
In this section, we briefly review a basic definition and two results in frequency domain method. Consider the following system, x˙ = f (t, x),
Definition 1 [Leonov et al., 1996]. Equation (1) is
(1)
where f : R+ × Rn → Rn is continuous and locally Lipschitz continuous in the second argument. Suppose that every solution x(t; t0 , x0 ) of system (1) with t0 ≥ 0, and x(t0 ) = x0 ∈ Rn can be continued to [t0 , +∞).
This lemma can be used to analyze the nonexistence of chaotic attractors or limit cycles. One just needs to test the frequency domain inequality (3). Generally, it is convenient to test this inequality by solving matrix inequality. So we often express the condition (3) in the form of linear matrix inequality (LMI) by KYP Lemma in [Anderson, 1967; Rantzer, 1996]. Lemma 2. If (A, B) is controllable, then (3) holds
if, and only if, there exists P = P ∗ such that
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C ∗ C
1 ∗ C κ + C ∗ R 2 R∗ R + Re{κR}
1 κC + R∗ C 2 P A + A∗ P + B ∗P
PB 0
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we can view y˙ 3 as the output of (4). Let C = (0 0 −5.7), then y˙3 = Cy + y1 y3 + 0.2. But now one can see that (A, C) is not observable. In order to meet the observability condition, we can adjust system (4) a little bit as (we will see the meaning of observability in the forthcoming discussion)
≤ 0.
Based on Lemma 2 one can test the frequency domain condition (3) by solving LMI and discuss feedback controller design problems by the popular LMI method in control theory.
dy = A1 y + B1 (y1 y3 + 0.2), dt where
0 A1 = 1 0.001
Remark 1. The method to avoid chaos by study-
ing the property of dichotomy is different from the chaos control method in [Chen & Dong, 1998]. The external control strategy to avoid chaos sometimes is 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. Based on the harmonic balance principle, a practical approach for predicting chaos dynamics in nonlinear systems was presented in [Genesio & Tesi, 1992]. This method is practically applicable, but not completely exact. The frequency domain condition of dichotomy here can be proved to be strictly correct by the work of Leonov [Leonov et al., 1996]. With Lemma 1, in what follows we first discuss some differences between two classes of nonlinearity in chaotic systems.
3. Comparison Between Two Classes of Nonlinearity
(5)
−1 −1 0.2 0 , 0 −5.7
B1 = B, B and y are given as above. Let C1 = (0.001 0 − 5.7). Computer simulations show that system (5) has a chaotic attractor similar to the R¨ossler attractor, see Fig. 1. Here we call system (5) as a R¨ ossler-like system.
3.2. The Lorenz system The following Lorenz system was studied extensively [Chen & Ueta, 1999; Lorenz, 1963; L¨ u et al., 2002]. Chen’s attractor and L¨ u’s attractor were found in Lorenz-type systems, dy = A2 y + B2 (y1 y3 , y2 y3 )∗ , dt where
−10 28 A2 = 0
10 −1 0
−
0 0 , 8
(6)
0 0 B2 = −1 0 , 0 1
3
see Fig. 2 for the typical Lorenz attractor.
3.1. A R¨ ossler-like system According to [R¨ ossler, 1979], there is a typical chaotic attractor in the following R¨ ossler system dy = Ay + B(y1 y3 + 0.2), dt where
0 −1 −1 0 , A = 1 0.2 0 0 −5.7
0 B = 0 , 1
(4) y1 y = y2. y3
The above R¨ossler system was revisited and generalized to a class of R¨ ossler-like systems [Thomas, 1999]. Here we consider another kind of R¨ossler-like system. Corresponding to system (2),
3.3. PWL nonlinearity of Chua’s circuit In the R¨ ossler-like system and Lorenz system above, one can see some typical chaotic phenomena. But what will happen if we change the nonlinear functions in (5) and (6) into nonlinearity of Chua’s circuit? Are there chaotic phenomena or limit cycles again? In what follows, we discuss such kind of problems. First according to (5), the following system is obtained by substituting the nonlinear function y1 y3 + 0.2 into PWL function, dy = A1 y + B1 f (y3 ), dt
(7)
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15
5
−5
−15 0
40
80
120
160
200
(a)
25
15
5
10 5
10
0 −5 0
−10 −15 −20
−10
(b) Fig. 1.
The solution of (5) with initial value y(0) = [−1, 0, 0]∗ . (a) y1 (t), y2 (t), y3 (t), and (b) y1 , y2 , y3 space.
where A1 , B1 are given as in (5), f (y3 ) = G2 y3 + 0.5(G1 − G2 )(|y3 + 1) − |y3 − 1|), G1 and G2 are parameters to be chosen. Let C1 = (0.001 0 −5.7), D1 = 1, then the transfer function from f (y3 ) to y˙ 3 in system (7) is K1 (s) = C1 (sI − A1 )−1 B1 + D1 = (s3 − 0.2s2 + s/s3 + 5.5s2 − 0.139s + 5.6998). Obviously (A1 , B1 ) and (A1 , C1 ) are controllable and observable, respectively. And by testing the matrix inequality in Lemma 2, we know that K1 (s) satisfies the frequency domain condition in Lemma 1.
Therefore, for any parameters G1 and G2 if (7) has isolated equilibria, then (7) is dichotomous. That is, there exist no chaotic attractors or limit cycles. For example, take G1 = −0.67, G2 = 1.27, see Fig. 3 for the solution of (7) at the same initial value as given in Fig. 1. From Fig. 3, one can see that the solution is dramatically unbounded. Furthermore, by Lemma 1, one knows that when PWL function f (y3 ) in (7) is substituted by any piecewise continuous nonlinear function g(y3 ),
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50
30
10
−10
−30 0
10
20
30
40
(a)
50
30
10
20
20 15 10
0
5 −20
−10
−5
0
(b) Fig. 2.
The solution of (6) with initial value y(0) = [10, 1, 10]∗ . (a) y1 (t), y2 (t), y3 (t), and (b) y1 , y2 , y3 space.
the corresponding system is dichotomous if it has isolated equilibria. Similarly, if we substitute the nonlinear functions y1 y3 , y2 y3 in the Lorenz system (6) by nonlinearity of Chua’s circuit, then we can get the following system dy = A2 y + B2 (f1 (y2 ), f2 (y3 ))∗ , (8) dt where A2 and B2 are given as in (6), f1 (y2 ) = G12 y2 + 0.5(G11 − G12 )(|y2 + 1) − |y2 − 1|), f2 (y3 ) = G22 y3 + 0.5(G21 − G22 )(|y3 + 1) − |y3 − 1|).
Let C2 be the matrix composed of the last two rows of A2 , D2 be the matrix composed of the last two rows of B2 . Then the transfer function from (f1 (y2 ), f2 (y3 ))∗ to (y˙ 2 , y˙3 )∗ in system (8) is K2 (s) = C2 (sI − A2 )−1 B2 + D2 −s2 − 10s 0 s2 + 11s − 270 = s 0 8 s+ 3
.
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200
0
−200
−400 0
20
40
60
(a)
0.1
0.05
0
−0.05 400 400
200 200
0 0
−200
−200 −400
−400
(b) Fig. 3.
The solution of (7) with initial value y(0) = [−1, 0, 0]∗ . (a) y1 (t), y2 (t), y3 (t), and (b) y1 , y2 , y3 space.
Obviously (A2 , B2 ) and (A2 , C2 ) are controllable and observable, respectively. And by testing the matrix inequality in Lemma 2, we know that K2 (s) satisfies the frequency domain condition in Lemma 1. Therefore, for any parameters G11 , G12 , G21 and G22 if (8) has isolated equilibria, then (8) is dichotomous. Remark 2. From above, one can see that chaotic
phenomena disappear in R¨ ossler-like and Lorenz systems after substituting nonlinearity of the product of different variables by PWL nonlinearity. Of
course, after this substitution, the system equilibria and the interactions of variables are changed largely. This also indicates the complexity and importance of the interactions of different variables in chaotic systems [Thomas, 1999; Chua, 1998]. In addition, we can see that the frequency domain method here cannot be used to deal with nonlinearity of the products of different variables. For simplicity, here we only discuss PWL nonlinearity of Chua’s circuit, some other nonlinear functions in [Huang et al., 1996; Thomas, 1999; Yalcin et al., 2002] can be considered similarly.
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4. An Extended Chua’s Circuit with Two Nonlinear Functions Chua’s circuits were studied extensively [Madan, 1993]. Generally, Chua’s circuits can be viewed as single-input and single-output feedback nonlinear systems. Coupled Chua’s circuits studied in [Duan et al., 2004a; Imai et al., 2002] can be viewed as multi-input and multi-output feedback nonlinear systems. Chua’s circuits and coupled Chua’s circuits can all be viewed as Lur’e systems [Suykens et al., 1997; Suykens et al., 1998]. The effects of input and output coupling were studied in [Duan et al., 2004a] for coupled Chua’s circuit. Some linearly
(a)
Fig. 4.
The extended Chua’s circuit.
4
2
0
−2
−4
−6 0
250
500
(b)
0.4
0
−0.4 4 2
4 0 −2
0 −4 −6
Fig. 5. space.
−4
The solution of (9) with initial value v(0) = (1, −1, 0.4)∗ (G21 = 0, G22 = 0). (a) v1 (t), v2 (t), i3 (t), and (b) v1 , v2 , i3
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interconnected systems were studied in [Duan et al., 2004b]. Generally, the dimension of the coupled system in [Duan et al., 2004a] is 6. In order to study the effects of input and output coupling in lower dimensional systems, we give an extended Chua’s circuit with two nonlinear functions, see Fig. 4. Chaotic systems with more than one nonlinear function were studied and remarkable chaotic phenomena were shown in [Thomas, 1999; Yalcin et al., 2002]. Here, we also pay attention to the effect of nonlinear coupling by establishing the condition of dichotomy. The differential equation of the extended Chua’s circuit in Fig. 4 is v˙ = Av + B(f1 (v1 ), f2 (v2 ))∗ , (a)
where
−1 C1 R1 1 A= C2 R1 0 −1 C1 B= 0 0
(9)
1 C1 R1
0
1 , C2 −R0 L1
−1 C2 R1 −1 L1
v1 v = v2 , i3
0
−1 , C2 0
4
2
0
−2
−4
−6 0
250
500
(b) 0.5
−0.5
0.5
−0.5 5 0
4 −5 5
0 0 −5
−4
Fig. 6. The solution of (9) with initial value v(0) = (1, −1, 0.4)∗ (G21 = 0.1, G22 = −0.01). (a) v1 (t), v2 (t), i3 (t), and (b) v1 , v2 , i3 space.
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f1 and f2 are two nonlinear functions defined as
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0.0136073, R0 = 0.02969968/0.0136073, G11 = 0.1690817, G12 = −0.4767822, G21 = 0, G22 = 0, see Fig. 5. As one can imagine, the solutions of (9) will change largely with different parameters G21 , G22 of the second nonlinearity, see Figs. 6–9. From Figs. 6–9, one can see more oscillating phenomena in system (9) by choosing the function f2 . So the oscillating phenomena of the extended Chua’s circuit are much richer than the single Chua’s circuit. One can increase the
f1 (x) = G12 x + 0.5(G11 − G12 )(|x + 1| − |x − 1|), f2 (x) = G22 x + 0.5(G21 − G22 )(|x + 1| − |x − 1|). Obviously, if G21 = 0, G22 = 0, the extended Chua’s circuit is the canonical Chua’s circuit studied in [Madan, 1993]. According to the traditional references on Chua’s circuit, there is a chaotic attractor in system (9) with parameters C1 = −1/1.301814, C2 = 1, R1 = 1, L1 = −1/ 4
2
0
−2
−4
−6 0
250
500
(a)
0
−1 1
0
−1 5 0
5 −5 5
0 0 −5
−5
(b) Fig. 7. The solution of (9) with initial value v(0) = (1, −1, 0.4)∗ (G21 = 0.31, G22 = −0.25). (a) v1 (t), v2 (t), i3 (t), and (b) v1 , v2 , i3 space.
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15
0
−15 0
250
500
(a)
3
0
−3 10 5 0
5
−5 −10
−5
−15 −20
−15
(b) Fig. 8. The solution of (9) with initial value v(0) = (1, −1, 0.4)∗ (G21 = 2.98, G22 = −0.35). (a) v1 (t), v2 (t), i3 (t), and (b) v1 , v2 , i3 space.
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4
0
−4
−8 0
200
400
(a)
0.8
0.4
0 4 2
6 4
0
2 −2 −4 −6
−8
−6
−4
−2
0
(b) Fig. 9. The solution of (9) with initial value v(0) = (1, −1, 0.4)∗ (G21 = −0.26, G22 = −0.01). (a) v1 (t), v2 (t), i3 (t), and (b) v1 , v2 , i3 space.
complexity of behavior in simple systems systematically [Yalcin et al., 2002]. Of course, the extension of nonlinearity here can be used to consider other chaotic systems such as smooth Chua’s equations [Khibnik, 1993] and R¨ ossler-like systems [Thomas, 1999].
5. Smooth Chua’s Equations In contrast to PWL nonlinearity in Chua’s circuit, smooth Chua’s circuits with cubic nonlinearity were
also studied thoroughly [Khibnik et al., 1993; Huang et al., 1996; Liao & Chen, 1998; Tsuneda, 2005]. The choice of a cubic nonlinearity has several advantages over a piecewise linear one. It does not require absolute-valued functions and it is smooth, which is desirable from a mathematical perspective. Moreover, almost all phenomena found in the PWL version also exist in cubic version. In what follows, we extend Chua’s circuit with cubic nonlinearity as above and see some other interesting oscillating phenomena.
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−α B = 0 , 0
The following smooth Chua’s circuit was studied in [Huang et al., 1996] v˙ = Av + Bx3 ,
(10)
According to [Huang et al., 1996], there is a chaotic attractor in system (10) with parameters α = 10, β = 16, c = −0.143, see Fig. 10. We extend system (10) by adding another nonlinearity as follows
where
−αc A= 1 0
α −1 −β
x v = y z
0 1 , 0
v˙ = A1 v + B1 [x3 , y 3 ]∗ ,
0.6
0.2
−0.2
−0.6
−1
0
20
40
60
80
100
(a)
1
0
−1 0.2 0.5 0 0 −0.2
−0.5
(b) Fig. 10.
The solution of (10) with initial value v(0) = (0.1, 0.1, 0.1)∗ . (a) x(t), y(t), z(t), and (b) x, y, z space.
(11)
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where
We can change the solutions of (10) largely by choosing d and f , see Figs. 11–13. Under the parameters α = 10, β = 16, d = 0.5, f = 0.1 if we change c = −0.143 into c = −0.26, one can see the solution in Fig. 14. From above, one can see that the solutions of smooth Chua’s equations can be changed largely by introducing another new nonlinearity. It is interesting to observe a systematic increase or decrease in the complexity of behavior in simple systems.
−αc α 0 −1 + f 1 , A1 = 1 0 −β 0 −α 0 x B1 = 0 d , v = y , 0 0 z d and f are new parameters to be chosen.
(a)
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5
3
1
−1
−3
−5 0
(b)
10
20
30
40
50
5
0
−5 −2 2
−2 2 0
0 −2
Fig. 11. space.
−2
The solution of (11) with initial value v(0) = (0.1, 0.1, 0.1)∗ (d = −0.2, f = 0.278). (a) x(t), y(t), z(t), and (b) x, y, z
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0.6
0.2
−0.2
−0.6
−1
0
50
100
(a)
1
0
−1 0.2 1 0 0 −0.2
−1
(b) Fig. 12. space.
The solution of (11) with initial value v(0) = (0.1, 0.1, 0.1)∗ (d = 3.5, f = 0.1). (a) x(t), y(t), z(t), and (b) x, y, z
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0.6
0.2
−0.2
−0.6
−1
0
50
100
(a)
1
0
−1 0.2 1 0
0.5 0 −0.2
−0.5
(b) Fig. 13. space.
The solution of (11) with initial value v(0) = (0.1, 0.1, 0.1)∗ (d = 16, f = −0.1). (a) x(t), y(t), z(t), and (b) x, y, z
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0.5
−0.5
−1.5 0
50
100
(a)
1.5
0.5
−0.5
−1.5 0.2 1 0 0 −0.2
−1
(b) Fig. 14. The solution of (11) with initial value v(0) = (0.1, 0.1, 0.1)∗ (d = 0.5, f = 0.1, c = −0.26). (a) x(t), y(t), z(t), and (b) x, y, z space.
6. Input and Output Coupling In this section, in order to increase or decrease the complexity of behavior in simple systems, we study the effects of a class of input and output coupling. Viewing (v˙ 1 , v˙ 2 )∗ as the output, (f1 (v1 ), f2 (v2 ))∗ as
the input, system (9) can be viewed as an MIMO system. After some input and output coupling, a new system can be generated, v˙ = Av + B(f1 (α11 v1 + α12 v2 ), f2 (α21 v1 + α22 v2 ))∗ ,
(12)
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where A, B, f1 , f2 and v are given as in (9). Let „
M =
α11
α12
α21
α22
«
,z=M
„
v1 v2
«
, C, R be the matri-
ces composed of the first two rows of A and B, respectively. Here, we call M as a coupling matrix. From (12), one can get z˙ = M Cv + M R(f1 (α11 v1 + α12 v2 ), f2 (α21 v1 + α22 v2 ))∗ .
(13)
))∗
to z˙ The transfer function from (f1 (z1 ), f2 (z2 in systems (12) and (13) is M K(s) = M (C(sI − A)−1 B + R). By Lemma 1, one can get the following result for systems (12) and (13). Theorem 1. Suppose that A has no pure imaginary eigenvalues, (A, B) is controllable and (A, C) is observable. If (12) has isolated equilibria and there exist diagonal matrices = diag(1 , . . . , m ) > 0, and κ = diag(κ1 , . . . , κm ) such that the following frequency domain inequality holds:
Re{κM K(iw) + K ∗ (iw)M ∗ M K(iw)} ≤ 0, ∀ w ∈ R, (14) then system (12) is dichotomous.
C ∗ M ∗ M C 1 M C + R∗ M ∗ M C 2
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If the coupling matrix M can be chosen deliberately, one can get a simplified result for system (12) by reducing the matrix variable κ. Theorem 2. Suppose that A has no pure imaginary eigenvalues, (A, B) is controllable and (A, C) is observable. If there exist diagonal matrix = diag(1 , . . . , m ) > 0 and any matrix M such that the following frequency domain inequality holds:
Re{M K(iw)} + K ∗ (iw)M ∗ M K(iw) ≤ 0, ∀ w ∈ R,
(15)
and system (12) has isolated equilibria, then system (12) is dichotomous. Furthermore, by KYP Lemma in [Rantzer, 1996], one can also express the condition (15) in the form of linear matrix inequality (LMI). Theorem 3.
If (A, B) is controllable, then (15) holds if, and only if, there exists P = P ∗ such that 1 ∗ ∗ C M + C ∗ M ∗ M R
∗ 2 + P A + A P P B ≤ 0. 0 B ∗P 1 1 R∗ M ∗ M R + M R + R∗ M ∗ 2 2
By using Schur complement, rewrite the matrix inequality in Theorem 3 as 0 P A + A∗ P P B 0 0 B ∗P 0 0 −−1 ∗ C
1 ∗ ∗ R I I + M 0 2 0 ∗ ∗ C
1 I I ≤ 0. (16) + R ∗ M ∗ 0 2 0 In terms of the solvability condition of nonstrict matrix inequality [Helmersson, 1995], if C is with full row rank, (16) holds if, and only if, there exists a scalar β > 0 such that 0 P A + A∗ P P B 0 0 B ∗P 0 0 −−1
C∗ − β R∗ (C 0
and
R
0) ≤ 0,
0 P A + A∗ P P B 0 0 B ∗P 0 0 −−1 0
1 1 I I ≤ 0. −β I 0 2 2
(17)
(18)
I From the frequency domain inequalities (14) and (15), one knows that M = 0 is a trivial solution. When A has no pure imaginary eigenvalues, one can always take M = 0 such that system (12) is dichotomous. According to the method in [Helmersson, 1995], one can parameterize Remark 3.
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1200
1000 0.7
800
0.65 0.6
600
0.55 0.5
400 0.45 0.4
200 0.35 150 1200
100
0
1000 50
800 600 0
−200 0
0.2
0.4
0.6
400 −50
0.8
(a)
200 0
(b)
Fig. 15. The solution of (12) with initial value v(0) = (1, −1, 0.4)∗ (G21 = −0.26, G22 = −0.01). (a) v1 (t), v2 (t), i3 (t), and (b) v1 , v2 , i3 space.
all solutions M to the matrix inequality (16) by solving the inequalities (17) and (18). So one can also consider the invariance of the property of dichotomy by the method here. By choosing the coupling matrix M , one can change the behavior of solutions of system (12) systematically.
From Fig. 9, one knows that system (9) is not But dichotomous for G 21 = −0.26, G22 = −0.01.
we can get M =
−12.522192
21.620156
−2.880473
13.298164
by solv-
ing the LMI in (16) such that the frequency domain inequality in Theorem 2 holds. So system (12) is
10
6
2
−2
−6
−10 0
100
200
300
400
500
(a) Fig. 16. The solution of (12) with initial value v(0) = (1, −1, 0.4)∗ (G21 = −2.98, G22 = −0.35). (a) v1 (t), v2 (t), i3 (t), and (b) v1 , v2 , i3 space.
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0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 10 10
5 5
0 0
−5
−5 −10
−10
(b) Fig. 16.
(Continued )
dichotomous when it has isolated equilibria for any piecewise continuous nonlinear functions (not only for nonlinearity of Chua’s circuit). See Fig. 15 for the solution of (12) with the same initial value as given above.
From above we can see that oscillating solutions can be eliminated by choosing a suitable input and output coupling. On the other hand, we can also see some interesting oscillating phenomena under the input and output coupling
15
5
−5
−15 0
100
200
300
400
500
(a) Fig. 17. The solution of (12) with initial value v(0) = (1, −1, 0.4)∗ (G21 = −2.98, G22 = −0.35). (a) v1 (t), v2 (t), i3 (t), and (b) v1 , v2 , i3 space.
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1
−1
−3 15 10
15
5
10 0
5 −5 −10 −15
−15
−10
−5
0
(b) Fig. 17.
above. For example, take M =
1
0
0.5
1
, we
can see the oscillating solution in Fig. 16 for system (12).
1 −0.68 , we can see the oscilTake M = 1.4
0.35
(Continued )
is dichotomous. The smooth Chua’s equations can be studied similarly. One can also see some new oscillating phenomena with the coupling here. It would be an interesting topic to increase or decrease the complexity of behavior in simple systems systematically.
lating solution in Fig. 17 for system (12). In this section, only nonlinearity of Chua’s circuit is considered. Obviously, other nonlinear systems such as smooth Chua’s equations [Huang et al., 1996; Khibnik et al., 1993; Liao & Chen, 1998; Tsuneda, 2005] and R¨ossler-like systems [Thomas, 1999] can be studied similarly.
Remark 4.
Acknowledgments The authors would like to thank the reviewers for their detailed and helpful comments which help improve the presentation of this paper. This work is supported by the National Science Foundation of China under grant 60204007, 60334030, 10472001.
7. Conclusion
References
The frequency domain method established in [Duan et al., 2004a; Leonov et al., 1996] is used to discuss a class of feedback nonlinear systems. It is seen that a kind of R¨ ossler-like system and the Lorenz system do not have chaotic attractors or limit cycles if the nonlinearity is replaced by some single variable nonlinearity. This also indicates that the frequency domain method here cannot be used to deal with nonlinearity of the products of different variables. Furthermore, an extended Chua’s circuit is presented. Although computer simulations show that much richer oscillating phenomena exist in the extended Chua’s circuit given here than in the single Chua’s circuit, the corresponding system after a suitable input and output coupling
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