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Impulsive control of Rössler systems
Physics Letters A 306 (2003) 306–312 www.elsevier.com/locate/pla
Impulsive control of Rössler systems Jitao Sun ∗ , Yinping Zhang Department of Applied Mathematics, Tongji University, 200092, PR China Received 10 August 2002; received in revised form 10 August 2002; accepted 28 October 2002 Communicated by A.R. Bishop
Abstract In this Letter, several new theorems on the stability of impulsive control systems are presented. These theorems are then used to find the conditions under which the Rössler systems can be asymptotically controlled to the equilibrium point by using impulsive control. Given the parameters of the Rössler system and the impulsive control law. We also present a theory of impulsive synchronization of two Rössler systems. Moreover, a larger upper bound of impulsive intervals for the stabilization and synchronization can be obtained. 2002 Elsevier Science B.V. All rights reserved. Keywords: Impulsive stabilization; Chaos; Rössler system; Impulsive synchronization
1. Introduction Since the seminal paper of Ott, Grebogi, and Yorke (OGY) [2], several methods for control and stabilization of chaotic motions have recently been presented [3–6]. In view of the rich dynamics of chaotic systems, there exists a large variety of approaches for controlling such systems. Some of these approaches include adaptive control [4,5], error-feedback control [7], time-delay feedback control [7], OGY method [2], predictive Poincaré control [8], occasional proportional feedback control [9], and impulsive control [6,11–21]. In fact, the predictive Poincaré control and the occasional proportional feedback control are two impulsive control schemes with varying impulse intervals. Impulsive control is attractive because it allows the stabilization of a chaotic system using only small control impulses, and it offers a direct method for modulating digital information onto a chaotic carrier signal for spread spectrum applications. However, due to a lack of effective tools for analyzing impulsive differential equations [1], most impulse control schemes had been designed mainly by trialand-error. The study of the stability of an impulsive differential equation is much more difficult than that of its “corresponding” differential equation [10]. Yang [11] has presented the theoretical result of impulsive control of the Rössler system to periodic motions and a sufficient condition for existence of periodic trajectory of impulsively controlled Rössler system has been given. * Corresponding author.
E-mail address: [email protected] (J. Sun). 0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 1 4 9 9 - 8
J. Sun, Y. Zhang / Physics Letters A 306 (2003) 306–312
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In this Letter, we investigate the stability of impulsively controlled Rössler systems with varying impulse intervals. First, the results of the stability of the trivial solution for a kind of impulsive differential equation are used to study the conditions under which an impulsive control of Rössler system is asymptotically stable. An estimate of the upper bound of the impulsive interval is also presented. Then, an impulsive control theory is used to study the impulsive synchronization of two chaotic systems. We first show that the impulsive synchronization problem is an impulsive control problem. Then a theorem is given for guaranteeing the asymptotic stability of impulsive synchronization. Since only the synchronization impulses are sent to the driven system is an impulsive synchronization scheme, the information redundancy in the transmitted signal is reduced. In this sense, even low-dimensional chaotic systems can provide high security. The organization of this Letter is as follows. In Section 2, a theory on the stability of impulsive differential equations is given. In Section 3, a stability criterion for impulsive control of Rössler system is presented. In Section 4, the theory of impulsive synchronization of Rössler systems is presented. In Section 5, some concluding remarks are given.
2. Supporting results A impulsive differential system with impulses at fixed times is described by ˙ = f t, X(t) , X(t) t = τk , − ∆ + X(t) = X t − X t = U (k, X), t = τk , k = 1, 2, . . . ,
(1)
where f : R+ × R n → R n is continuous; U : R n × R n → R n is continuous; X ∈ R n is the state variable; 0 < τ1 < τ2 < · · · < τk < τk+1 < · · · , τk → ∞ as k → ∞. Definition 1 [1]. Let V : R+ × R n → R+ , then V is said to belong to class V0 if (1) V is continuous in (τk−1 , τk ] × R n and for each X ∈ R n , k = 1, 2, . . . , lim(t,Y )→(τ + ,X) V (t, Y ) = V (τk+ , X) k exists; (2) V is locally Lipschitzian in X. Definition 2 [1]. For (t, X) ∈ (τi−1 , τi ] × R n we define D + V (t, X) = lim sup ∆
h→0+
1 V t + h, X + hf (t, X) − V (t, X) . h
Definition 3 [1]. Comparison system: Let V ∈ V0 and assume that t= τk , D + V (t, X) g t, V (t, X) , V t, X + U (k, X) Ψk V (t, X) , t = τk , where g : R+ × R+ → R is continuous and Ψk : R+ → R+ is non-decreasing. Then the following system: ω), t = τk , ω˙ = g(t, + ω τk = Ψk ω(τk ) , + ω t0 = ω0 0 is the comparison system of (1). To support our analysis in later sections, the following author’s results are presented.
(2)
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Theorem 4 [21]. Assume that the following three conditions: (1) V : R+ × R n → R+ , V ∈ V0 , K(t)D + V (t, X) + D + K(t)V (t, X) g(t, K(t)V (t, X)), t = τk , where g is continuous in (τk−1 , τk ] × R n for each x ∈ R n , k = 1, 2, . . . , lim(t,y)→(τ +,x) g(t, y) = g(τk+ , x) exists. K(t) k
m > 0, limt →τ − K(t) = K(τk ), limt →τ + K(t) exists, k = 1, 2, . . . , D + K(t) = limh→0+ sup h1 [K(t + h) − k k K(t)]; (2) K(τk + 0)V (τk + 0, X + U (k, X)) Ψk (K(τk )V (τk , X)), k = 1, 2, . . . ; (3) V (t, 0) = 0 and α(X) V (t, X) on R+ × R n , where α(·) ∈ ℵ (class of continuous strictly increasing functions α : R+ → R+ such that α(0) = 0) are satisfied. Then, the global asymptotic stability of the trivial solution ω = 0 of comparison system implies global asymptotic stability of the trivial solution of impulsive system (1). ˙ Theorem 5 [21]. Let g(t, ω) = λ(t)ω, λ ∈ C 1 [R+ , R+ ], Ψi (ω) = di ω, di 0 for all i. Then, the origin of system (1) is global asymptotically stable if the conditions of Theorem 4 and the following conditions hold: (1) λ(t) is non-decreasing, limt →τ − λ(t) = λ(τk ), limt →τ + λ(t) = λ(τk+ ) exists, for all k = 1, 2, . . . ; k
k
(2) supi {di exp(λ(τi+1 ) − λ(τi+ ))} = ε0 < ∞; + + (3) there exists an r > 1 such that λ(τ2k+3 ) + λ(τ2k+2 ) + ln(rd2k+2d2k+1 ) λ(τ2k+2 ) + λ(τ2k+1 ) holds for all + d2k+2d2k+1 = 0, k = 1, 2, . . . , or there exists an r > 1 such that λ(τk+1 ) + ln(rdk ) λ(τk ) for all k; (4) V (t, 0) = 0 and there exists α(·) in class ℵ such that α(X) V (t, X). 3. Stabilization of Rössler systems using impulsive control In this section, we study the impulsive control of Rössler systems [22] by applying the theory presented in the previous section. The dimensionless form of a Rössler system is given by [22] x˙ = −y − z,
y˙ = x + αy, (3) z˙ = z(x − β) + α, where α, β ∈ R+ . System (3) is chaotic if α = 0.2, β = 5.7. Let (x ∗ , y ∗ , z∗ ) is an equilibrium point of system (3), X1 = x − x ∗ , X2 = y − y ∗ , X3 = z − z∗ , and T X = (X1 , X2 , X3 ), then we can rewrite the Rössler equation into the form X˙ = AX + Φ(X), where
(4)
0 −1 −1 0 A= 1 α 0 . , Φ(X) = 0 z∗ 0 x ∗ − β X1 X3 The impulsive control of a Rössler system is then given by X˙ = AX + Φ(X), t = τi , i = 1, 2, . . . , X|t =τi = BX,
(5)
(6)
where {τi : i = 1, 2, . . . , ∞} are varying but satisfy ∆1 = sup {τ2j +1 − τ2j } < ∞,
(7)
∆2 = sup {τ2j − τ2j −1 } < ∞
(8)
1j <∞ 1j <∞
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and for a given constant ε > 0, τ2j +1 − τ2j ε(τ2j − τ2j −1 ),
∀j ∈ {1, 2, . . . , ∞}. (9)
Remark 1. Conditions (7) and (8) imply that the number of switchings is infinite, while conditions (9) implies that impulsive intervals may not be equidistant. Note that there exists a positive number M for chaotic system (6) such that |X1 (t)| M and |X3 (t)| M for all t. Then, the following stability result for the impulsively Rössler system (6) can be obtained. Theorem 6. Let q be the largest eigenvalue of (AT + A), d = ρ 2 (I + B) and ρ(A) denote the spectral radius of A, the origin of the impulsively controlled Rössler system (6) is asymptotically stable if there exists a ξ > 1 and a differentiable at t = τi and non-increasing function K(t) m which satisfies −
+ ) K(τ2i+ )K(τ2i−1 1 K (t) q + 2M ln , K(t) (1 + ε)∆2 K(τ2i+1 )K(τ2i )ξ d 2
(10)
−
K(τi+ ) 1 K (t) q + 2M ln . K(t) max{∆1 , ∆2 } K(τi+1 )ξ d
(11)
or
Proof. Let us construct the Lyapunov function V (t, X) = XT X. For t = τi , we have K(t)D + V (t, X) + K (t)V = K(t) XT AX + XT AT X + XT Φ(X) + Φ T (X)X + K (t)V
K (t) K(t)V (t, X). K(t) qXT X + 2X1 X32 + K (t)V q + 2M + K(t) When t = τi , we have K(τi + 0)V (τi + 0, X + BX) K(τi )(X + BX)T (X + BX) = K(τi )XT (I + B)T (I + B)X dK(τi )V (τi , X). We can get the following comparison system K (t ) ω˙ = q + 2M + K(t ) ω, + ω τi = dω(τi ), ω t0+ = ω0 0. We now consider the conditions in Theorem 5. Since K(τi+1 ) sup d exp (q + 2M)(τi+1 − τi ) + ln d exp (q + 2M) max(∆1 , ∆2 ) < ∞ + K(τi ) i then condition (2) in Theorem 5 is satisfied. Furthermore K(τ2i+1 ) K(τ2i ) + ln + + K(τ2i ) K(τ2i−1 ) K(τ2i+1 ) K(τ2i ) = (q + 2M)(τ2i+1 − τ2i + τ2i − τ2i−1 ) + ln + ln + + K(τ2i ) K(τ2i−1 )
(q + 2M)(τ2i+1 − τ2i−1 ) + ln
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(q + 2M)(∆1 + ∆2 ) + ln
K(τ2i+1 ) K(τ2i ) + ln + K(τ2i+ ) K(τ2i−1 )
(q + 2M)(1 + ε)∆2 + ln
K(τ2i+1 ) K(τ2i ) + ln + + K(τ2i ) K(τ2i−1 )
− ln ξ d 2 or (q + 2M)(τi+1 − τi ) + ln
K(τi+1 ) K(τi+1 ) − ln(ξ d), + (q + 2M) max{∆1 , ∆2 } + ln K(τi ) K(τi+ )
where the last inequality holds from (10) and (11). Thus, condition (3) in Theorem 5 is also satisfied. Therefore, it follows from Theorem 5 that the origin of system (6) is asymptotically stable. ✷ Remark 2. We do not require that B is symmetric. Moreover, we do not require that I + B 1. Thus, our result can be used for a wide class of non-linear systems. Remark 3. Condition (10) implies that V (t, X) is only required to be non-increasing along an odd subsequence of switchings, instead of the whole sequence of switchings when K(τ2i+ ) = K(τ2i ). Thus, our result is less conservative. Example 1. We choose that ε = 0.5 and for all j = 1, 2, . . . τ2j +1 − τ2j = ∆1 = −
ln(ξ d) , 1.5(q + 2M)
τ2j − τ2j −1 = ∆2 = −
2 ln(ξ d) . 1.5(q + 2M)
(12)
We know that the origin of system (6) with above parameters is asymptotically stable from Theorem 6.
4. Synchronization of Rössler systems using impulsive control In this section, we study the impulsive synchronization of two Rössler systems. One of the Rössler systems is called the driving system and the other is called the driven system. In an impulsive synchronization configuration, the driving system is given by (4). The driven system is given by ˙ = AX + Φ(X), X
(13)
T = (x, where X ˜ y, ˜ z˜ ) is the state variables of the driven system. At discrete instants, τi , i = 1, 2, . . . , the state variable of the driving system are transmitted to the driven system and then the state variables of driven system are subject to jumps as these instants. In this sense, the driven system is described by the impulsive differential equation ˙ = AX + Φ(X), X t = τi , (14) ∆X|t =τi = −Be, i = 1, 2, . . . , 1 , X2 − X 2 , X3 − X 3 ) is the synchronization error. If where B is a 3 × 3 matrix, and eT = (ex , ey , ez ) = (X1 − X we define
0 = Φ(X) − Φ(X) = Ψ (X, X) 0 1 X 3 X1 X3 − X
J. Sun, Y. Zhang / Physics Letters A 306 (2003) 306–312
then the error system of the impulsive synchronization is given by e˙ = Ae + Ψ (X, X), t = τi , i = 1, 2, . . . . ∆e|t =τi = Be,
311
(15)
We obtain the following theorem to guarantee that our impulsive synchronization is asymptotically stable. Theorem 7. Let q be the largest eigenvalue of (A + AT ), d = ρ 2 (I + B) and ρ(A) denote the spectral radius of A, the origin of the impulsive synchronization of two Rössler systems is asymptotically stable if there exists a ξ > 1 and a differentiable at t = τk and non-increasing function K(t) m > 0 which satisfies −
+ K(τ2i+ )K(τ2i−1 ) K (t) 1 q + 3M ln K(t) (1 + ε)∆2 K(τ2i+1 )K(τ2i )ξ d 2
−
K(τi+ ) K (t) 1 q + 3M ln . K(t) max{∆1 , ∆2 } K(τi+1 )ξ d
or
Similarly, Proof. Observe that the error system in (15) is almost the same as the system in (6) except for Ψ (X, X). T let us construct the Lyapunov function V (t, e) = e e. For t = τi , we have K(t)D + V (t, e) + K (t)V = K(t) eT Ae + eT AT e + eT Ψ (e) + Ψ T (e)e + K (t)V 1 X 3 ) + K (t)V K(t) qeT e + 2ez (X1 X3 − X 1 X 3 + X1 X 3 − X 3 ) + K (t)V = K(t) qeT e + 2ez (X1 X3 − X1 X
T K (t) T K(t)V (t, e). K(t) qe e + 3Me e + K (t)V = q + 3M + K(t)
Hence, condition (1) of Theorem 4 is satisfied with g(t, ω) = (q + 3M + KK(t(t)) )ω. The rest of this proof is the same as that of Theorem 6.
5. Conclusions This Letter has studied the issue on the stabilization and synchronization of Rössler system via an impulsive control with varying impulsive intervals. Through our approach, some less conservative conditions were derived in that the Lyapunov function is only required to be non-increasing along a subsequence of switchings, instead of the whole sequence of switchings. Moreover, a larger upper bound of impulsive intervals for stabilization and synchronization of Rössler system can be obtained.
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E.R. Hunt, G. Johnson, IEEE Spectrum 32 (1993) 36. A.M. Samoilenko, N.A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995. T. Yang, Impulsive Control Theory, Springer-Verlag, Berlin–Heidelberg, 2001. T. Yang, L.B. Yang, C.M. Yang, Phys. Lett. A 226 (1997) 349. T. Yang, Impulsive Systems and Control: Theory and Application, Nova Science, 2001. T. Yang, L.B. Yang, C.M. Yang, Physica D 110 (1997) 18. T. Yang, C.M. Yang, L.B. Yang, Phys. Lett. A 232 (1997) 356. T. Yang, L.O. Chua, IEEE Trans. Circuits Systems I 44 (10) (1997) 976. T. Yang, IEEE Trans. Automat. Control 44 (5) (1999) 1081. W.X. Xie, C.Y. Wen, Z.G. Li, Phys. Lett. A 275 (2000) 67. Z.G. Li, C.Y. Wen, Y.C. Soh, IEEE Trans. Automat. Control 46 (6) (2001) 894. J.T. Sun, Y.P. Zhang, Q.D. Wu, Phys. Lett. A 298 (2002) 153. J.T. Sun, Y.P. Zhang, Q.D. Wu, IEEE Trans. Automat. Control, in press. O.E. Rössler, Phys. Lett. A 57 (1976) 397.
Impulsive control of Rössler systems Jitao Sun ∗ , Yinping Zhang Department of Applied Mathematics, Tongji University, 200092, PR China Received 10 August 2002; received in revised form 10 August 2002; accepted 28 October 2002 Communicated by A.R. Bishop
Abstract In this Letter, several new theorems on the stability of impulsive control systems are presented. These theorems are then used to find the conditions under which the Rössler systems can be asymptotically controlled to the equilibrium point by using impulsive control. Given the parameters of the Rössler system and the impulsive control law. We also present a theory of impulsive synchronization of two Rössler systems. Moreover, a larger upper bound of impulsive intervals for the stabilization and synchronization can be obtained. 2002 Elsevier Science B.V. All rights reserved. Keywords: Impulsive stabilization; Chaos; Rössler system; Impulsive synchronization
1. Introduction Since the seminal paper of Ott, Grebogi, and Yorke (OGY) [2], several methods for control and stabilization of chaotic motions have recently been presented [3–6]. In view of the rich dynamics of chaotic systems, there exists a large variety of approaches for controlling such systems. Some of these approaches include adaptive control [4,5], error-feedback control [7], time-delay feedback control [7], OGY method [2], predictive Poincaré control [8], occasional proportional feedback control [9], and impulsive control [6,11–21]. In fact, the predictive Poincaré control and the occasional proportional feedback control are two impulsive control schemes with varying impulse intervals. Impulsive control is attractive because it allows the stabilization of a chaotic system using only small control impulses, and it offers a direct method for modulating digital information onto a chaotic carrier signal for spread spectrum applications. However, due to a lack of effective tools for analyzing impulsive differential equations [1], most impulse control schemes had been designed mainly by trialand-error. The study of the stability of an impulsive differential equation is much more difficult than that of its “corresponding” differential equation [10]. Yang [11] has presented the theoretical result of impulsive control of the Rössler system to periodic motions and a sufficient condition for existence of periodic trajectory of impulsively controlled Rössler system has been given. * Corresponding author.
E-mail address: [email protected] (J. Sun). 0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 1 4 9 9 - 8
J. Sun, Y. Zhang / Physics Letters A 306 (2003) 306–312
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In this Letter, we investigate the stability of impulsively controlled Rössler systems with varying impulse intervals. First, the results of the stability of the trivial solution for a kind of impulsive differential equation are used to study the conditions under which an impulsive control of Rössler system is asymptotically stable. An estimate of the upper bound of the impulsive interval is also presented. Then, an impulsive control theory is used to study the impulsive synchronization of two chaotic systems. We first show that the impulsive synchronization problem is an impulsive control problem. Then a theorem is given for guaranteeing the asymptotic stability of impulsive synchronization. Since only the synchronization impulses are sent to the driven system is an impulsive synchronization scheme, the information redundancy in the transmitted signal is reduced. In this sense, even low-dimensional chaotic systems can provide high security. The organization of this Letter is as follows. In Section 2, a theory on the stability of impulsive differential equations is given. In Section 3, a stability criterion for impulsive control of Rössler system is presented. In Section 4, the theory of impulsive synchronization of Rössler systems is presented. In Section 5, some concluding remarks are given.
2. Supporting results A impulsive differential system with impulses at fixed times is described by ˙ = f t, X(t) , X(t) t = τk , − ∆ + X(t) = X t − X t = U (k, X), t = τk , k = 1, 2, . . . ,
(1)
where f : R+ × R n → R n is continuous; U : R n × R n → R n is continuous; X ∈ R n is the state variable; 0 < τ1 < τ2 < · · · < τk < τk+1 < · · · , τk → ∞ as k → ∞. Definition 1 [1]. Let V : R+ × R n → R+ , then V is said to belong to class V0 if (1) V is continuous in (τk−1 , τk ] × R n and for each X ∈ R n , k = 1, 2, . . . , lim(t,Y )→(τ + ,X) V (t, Y ) = V (τk+ , X) k exists; (2) V is locally Lipschitzian in X. Definition 2 [1]. For (t, X) ∈ (τi−1 , τi ] × R n we define D + V (t, X) = lim sup ∆
h→0+
1 V t + h, X + hf (t, X) − V (t, X) . h
Definition 3 [1]. Comparison system: Let V ∈ V0 and assume that t= τk , D + V (t, X) g t, V (t, X) , V t, X + U (k, X) Ψk V (t, X) , t = τk , where g : R+ × R+ → R is continuous and Ψk : R+ → R+ is non-decreasing. Then the following system: ω), t = τk , ω˙ = g(t, + ω τk = Ψk ω(τk ) , + ω t0 = ω0 0 is the comparison system of (1). To support our analysis in later sections, the following author’s results are presented.
(2)
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Theorem 4 [21]. Assume that the following three conditions: (1) V : R+ × R n → R+ , V ∈ V0 , K(t)D + V (t, X) + D + K(t)V (t, X) g(t, K(t)V (t, X)), t = τk , where g is continuous in (τk−1 , τk ] × R n for each x ∈ R n , k = 1, 2, . . . , lim(t,y)→(τ +,x) g(t, y) = g(τk+ , x) exists. K(t) k
m > 0, limt →τ − K(t) = K(τk ), limt →τ + K(t) exists, k = 1, 2, . . . , D + K(t) = limh→0+ sup h1 [K(t + h) − k k K(t)]; (2) K(τk + 0)V (τk + 0, X + U (k, X)) Ψk (K(τk )V (τk , X)), k = 1, 2, . . . ; (3) V (t, 0) = 0 and α(X) V (t, X) on R+ × R n , where α(·) ∈ ℵ (class of continuous strictly increasing functions α : R+ → R+ such that α(0) = 0) are satisfied. Then, the global asymptotic stability of the trivial solution ω = 0 of comparison system implies global asymptotic stability of the trivial solution of impulsive system (1). ˙ Theorem 5 [21]. Let g(t, ω) = λ(t)ω, λ ∈ C 1 [R+ , R+ ], Ψi (ω) = di ω, di 0 for all i. Then, the origin of system (1) is global asymptotically stable if the conditions of Theorem 4 and the following conditions hold: (1) λ(t) is non-decreasing, limt →τ − λ(t) = λ(τk ), limt →τ + λ(t) = λ(τk+ ) exists, for all k = 1, 2, . . . ; k
k
(2) supi {di exp(λ(τi+1 ) − λ(τi+ ))} = ε0 < ∞; + + (3) there exists an r > 1 such that λ(τ2k+3 ) + λ(τ2k+2 ) + ln(rd2k+2d2k+1 ) λ(τ2k+2 ) + λ(τ2k+1 ) holds for all + d2k+2d2k+1 = 0, k = 1, 2, . . . , or there exists an r > 1 such that λ(τk+1 ) + ln(rdk ) λ(τk ) for all k; (4) V (t, 0) = 0 and there exists α(·) in class ℵ such that α(X) V (t, X). 3. Stabilization of Rössler systems using impulsive control In this section, we study the impulsive control of Rössler systems [22] by applying the theory presented in the previous section. The dimensionless form of a Rössler system is given by [22] x˙ = −y − z,
y˙ = x + αy, (3) z˙ = z(x − β) + α, where α, β ∈ R+ . System (3) is chaotic if α = 0.2, β = 5.7. Let (x ∗ , y ∗ , z∗ ) is an equilibrium point of system (3), X1 = x − x ∗ , X2 = y − y ∗ , X3 = z − z∗ , and T X = (X1 , X2 , X3 ), then we can rewrite the Rössler equation into the form X˙ = AX + Φ(X), where
(4)
0 −1 −1 0 A= 1 α 0 . , Φ(X) = 0 z∗ 0 x ∗ − β X1 X3 The impulsive control of a Rössler system is then given by X˙ = AX + Φ(X), t = τi , i = 1, 2, . . . , X|t =τi = BX,
(5)
(6)
where {τi : i = 1, 2, . . . , ∞} are varying but satisfy ∆1 = sup {τ2j +1 − τ2j } < ∞,
(7)
∆2 = sup {τ2j − τ2j −1 } < ∞
(8)
1j <∞ 1j <∞
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and for a given constant ε > 0, τ2j +1 − τ2j ε(τ2j − τ2j −1 ),
∀j ∈ {1, 2, . . . , ∞}. (9)
Remark 1. Conditions (7) and (8) imply that the number of switchings is infinite, while conditions (9) implies that impulsive intervals may not be equidistant. Note that there exists a positive number M for chaotic system (6) such that |X1 (t)| M and |X3 (t)| M for all t. Then, the following stability result for the impulsively Rössler system (6) can be obtained. Theorem 6. Let q be the largest eigenvalue of (AT + A), d = ρ 2 (I + B) and ρ(A) denote the spectral radius of A, the origin of the impulsively controlled Rössler system (6) is asymptotically stable if there exists a ξ > 1 and a differentiable at t = τi and non-increasing function K(t) m which satisfies −
+ ) K(τ2i+ )K(τ2i−1 1 K (t) q + 2M ln , K(t) (1 + ε)∆2 K(τ2i+1 )K(τ2i )ξ d 2
(10)
−
K(τi+ ) 1 K (t) q + 2M ln . K(t) max{∆1 , ∆2 } K(τi+1 )ξ d
(11)
or
Proof. Let us construct the Lyapunov function V (t, X) = XT X. For t = τi , we have K(t)D + V (t, X) + K (t)V = K(t) XT AX + XT AT X + XT Φ(X) + Φ T (X)X + K (t)V
K (t) K(t)V (t, X). K(t) qXT X + 2X1 X32 + K (t)V q + 2M + K(t) When t = τi , we have K(τi + 0)V (τi + 0, X + BX) K(τi )(X + BX)T (X + BX) = K(τi )XT (I + B)T (I + B)X dK(τi )V (τi , X). We can get the following comparison system K (t ) ω˙ = q + 2M + K(t ) ω, + ω τi = dω(τi ), ω t0+ = ω0 0. We now consider the conditions in Theorem 5. Since K(τi+1 ) sup d exp (q + 2M)(τi+1 − τi ) + ln d exp (q + 2M) max(∆1 , ∆2 ) < ∞ + K(τi ) i then condition (2) in Theorem 5 is satisfied. Furthermore K(τ2i+1 ) K(τ2i ) + ln + + K(τ2i ) K(τ2i−1 ) K(τ2i+1 ) K(τ2i ) = (q + 2M)(τ2i+1 − τ2i + τ2i − τ2i−1 ) + ln + ln + + K(τ2i ) K(τ2i−1 )
(q + 2M)(τ2i+1 − τ2i−1 ) + ln
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(q + 2M)(∆1 + ∆2 ) + ln
K(τ2i+1 ) K(τ2i ) + ln + K(τ2i+ ) K(τ2i−1 )
(q + 2M)(1 + ε)∆2 + ln
K(τ2i+1 ) K(τ2i ) + ln + + K(τ2i ) K(τ2i−1 )
− ln ξ d 2 or (q + 2M)(τi+1 − τi ) + ln
K(τi+1 ) K(τi+1 ) − ln(ξ d), + (q + 2M) max{∆1 , ∆2 } + ln K(τi ) K(τi+ )
where the last inequality holds from (10) and (11). Thus, condition (3) in Theorem 5 is also satisfied. Therefore, it follows from Theorem 5 that the origin of system (6) is asymptotically stable. ✷ Remark 2. We do not require that B is symmetric. Moreover, we do not require that I + B 1. Thus, our result can be used for a wide class of non-linear systems. Remark 3. Condition (10) implies that V (t, X) is only required to be non-increasing along an odd subsequence of switchings, instead of the whole sequence of switchings when K(τ2i+ ) = K(τ2i ). Thus, our result is less conservative. Example 1. We choose that ε = 0.5 and for all j = 1, 2, . . . τ2j +1 − τ2j = ∆1 = −
ln(ξ d) , 1.5(q + 2M)
τ2j − τ2j −1 = ∆2 = −
2 ln(ξ d) . 1.5(q + 2M)
(12)
We know that the origin of system (6) with above parameters is asymptotically stable from Theorem 6.
4. Synchronization of Rössler systems using impulsive control In this section, we study the impulsive synchronization of two Rössler systems. One of the Rössler systems is called the driving system and the other is called the driven system. In an impulsive synchronization configuration, the driving system is given by (4). The driven system is given by ˙ = AX + Φ(X), X
(13)
T = (x, where X ˜ y, ˜ z˜ ) is the state variables of the driven system. At discrete instants, τi , i = 1, 2, . . . , the state variable of the driving system are transmitted to the driven system and then the state variables of driven system are subject to jumps as these instants. In this sense, the driven system is described by the impulsive differential equation ˙ = AX + Φ(X), X t = τi , (14) ∆X|t =τi = −Be, i = 1, 2, . . . , 1 , X2 − X 2 , X3 − X 3 ) is the synchronization error. If where B is a 3 × 3 matrix, and eT = (ex , ey , ez ) = (X1 − X we define
0 = Φ(X) − Φ(X) = Ψ (X, X) 0 1 X 3 X1 X3 − X
J. Sun, Y. Zhang / Physics Letters A 306 (2003) 306–312
then the error system of the impulsive synchronization is given by e˙ = Ae + Ψ (X, X), t = τi , i = 1, 2, . . . . ∆e|t =τi = Be,
311
(15)
We obtain the following theorem to guarantee that our impulsive synchronization is asymptotically stable. Theorem 7. Let q be the largest eigenvalue of (A + AT ), d = ρ 2 (I + B) and ρ(A) denote the spectral radius of A, the origin of the impulsive synchronization of two Rössler systems is asymptotically stable if there exists a ξ > 1 and a differentiable at t = τk and non-increasing function K(t) m > 0 which satisfies −
+ K(τ2i+ )K(τ2i−1 ) K (t) 1 q + 3M ln K(t) (1 + ε)∆2 K(τ2i+1 )K(τ2i )ξ d 2
−
K(τi+ ) K (t) 1 q + 3M ln . K(t) max{∆1 , ∆2 } K(τi+1 )ξ d
or
Similarly, Proof. Observe that the error system in (15) is almost the same as the system in (6) except for Ψ (X, X). T let us construct the Lyapunov function V (t, e) = e e. For t = τi , we have K(t)D + V (t, e) + K (t)V = K(t) eT Ae + eT AT e + eT Ψ (e) + Ψ T (e)e + K (t)V 1 X 3 ) + K (t)V K(t) qeT e + 2ez (X1 X3 − X 1 X 3 + X1 X 3 − X 3 ) + K (t)V = K(t) qeT e + 2ez (X1 X3 − X1 X
T K (t) T K(t)V (t, e). K(t) qe e + 3Me e + K (t)V = q + 3M + K(t)
Hence, condition (1) of Theorem 4 is satisfied with g(t, ω) = (q + 3M + KK(t(t)) )ω. The rest of this proof is the same as that of Theorem 6.
5. Conclusions This Letter has studied the issue on the stabilization and synchronization of Rössler system via an impulsive control with varying impulsive intervals. Through our approach, some less conservative conditions were derived in that the Lyapunov function is only required to be non-increasing along a subsequence of switchings, instead of the whole sequence of switchings. Moreover, a larger upper bound of impulsive intervals for stabilization and synchronization of Rössler system can be obtained.
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