synchronization of impulsively coupled systems - Semantic Scholar
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International Journal of Bifurcation and Chaos, Vol. 18, No. 5 (2008) 1539–1549 c World Scientific Publishing Company
SYNCHRONIZATION OF IMPULSIVELY COUPLED SYSTEMS XIUPING HAN, JUN-AN LU∗ and XIAOQUN WU School of Mathematics & Statistics, Wuhan University, Wuhan 430072, P. R. China ∗The State Key Laboratory of Software Engineering, Wuhan University, Wuhan 430072, P. R. China ∗[email protected] Received August 29, 2006; Revised June 12, 2007 In the past years, impulsive control for a single system and impulsive synchronization between two systems have been extensively studied. However, investigation on impulsive control and synchronization of complex networks has just started. In these studies, a network is continuously coupled, and then is synchronized by using impulsive control strategy. In this paper, a new and different coupled model is proposed, where the systems are coupled only at discrete instants through impulsive connections. Several criteria for synchronizing such kind of impulsively coupled complex dynamical systems are established. Two examples are also worked through for illustrating the main results. Keywords: Impulsive differential system; impulsively coupled systems; synchronization.
1. Introduction Impulsive phenomena widely exist in the real world. The mathematical model of an impulsive phenomenon is called an impulsive differential equation [Bainov & Simeonov, 1989; Lakshmikantham et al., 1989]. The most distinct character of impulsive differential systems is that it takes into account the effects of the instantaneous phenomena, such as population-growth models and maneuvers of spacecraft [Liu, 1994; Liu & Willms, 1996], dynamical nerve cell networks [Gu et al., 1992], etc. Studies show that impulsive differential models have been widely applied in many fields, such as space techniques, information science, control system, communication security [Khadra et al., 2003], life science [Fu et al., 2005], and so on. Impulsive control strategy is very effective, robust and with low cost. The control input is ∗
implemented by the “sudden jumps” of some state variables at some instants. It needs only small control impulses and has rapid responsive velocity. In the past decades, impulsive control has been widely studied in stabilizing and synchronizing chaotic systems [Stojanovski et al., 1996; Yang & Chua, 1997; Yang, 2001; Xie et al., 2000; Li et al.., 2001; Sun & Zhang, 2003; Chen et al., 2004]. Guan et al. investigated the synchronization between two nonlinear systems by using the hybrid impulsive and switching control strategy [Guan et al., 2006]. Synchronization of subsystems in a coupled dynamical network is very useful in practical applications, thus it has received a great deal of interest recently. For example, Sun et al. obtained some impulsive synchronization criteria for coupled chaotic systems via unidirectional coupling [Sun et al., 2004], Li and Chen concluded that complex
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dynamical networks can be regarded as a big coupled system [Li & Chen, 2006], Liu et al. discussed the robust impulsive synchronization of uncertain dynamical networks [Liu et al., 2005], Zhou et al. discussed the impulsive control of networks [Zhou et al., 2005], and so on. Previous studies on impulsive control and synchronization for coupled systems were based on continuous coupling. In practice, there are some systems, instead of being coupled with continuous connections, are only coupled by impulsive connections at instants, such as the species-food model in biology, the information transfer and exchange in ants, and the model of integrated circuit. Unlike other existing studies, the model proposed in this paper is coupled by impulsive connections. The coupled systems can reach synchronization by exchanging energy at instants. Sufficient conditions on synchronization of the impulsively coupled systems are presented. Numerical examples are also provided to verify the theoretical results and the effectiveness of the proposed synchronization scheme.
2. Description of the Model Consider a network consisting of N identical nodes, in which each node is an m-dimensional dynamical system. The state equations are x˙ 1 = f (x1 ) = Ax1 + ϕ(x1 ) x˙ = f (x2 ) = Ax2 + ϕ(x2 ) 2 .. (1) . x˙ N −1 = f (xN −1 ) = AxN −1 + ϕ(xN −1 ) x˙ N = f (xN ) = AxN + ϕ(xN ) where xi , ϕ(xi ) ∈ Rm , i = 1, 2, . . . , N and A ∈ Rm×m . Let X = (xT1 , xT2 , . . . , xTN )T and (ϕT (x1 ), ϕT (x2 ), . . . , ϕT (xN ))T = F (X, t), then X˙ = (IN ⊗ A)X+ F (X, t), where IN ⊗A denotes the Kronecker product of matrices IN and A. The impulsive coupling between systems are defined as follows. There is a connection (or impulsive coupling) between systems xi and xj if there is energy exchange at the impulsive instants tk (k = 1, 2, . . .), where t1 < t2 < · · · < tk < · · · , limk→∞ tk = ∞. Let G = (gij )N ×N denote the coupling matrix. Then gij = gji = 1 if there is a connection between node i and j (i = j); otherwise, gij = gji = 0. In this model, it is required that the coupling coefficients satisfy gii = − N j=1,j=i gij .
Then the above defined model of impulsively coupled systems can be described by x˙ i = Axi + ϕ(xi ), t = tk N xi = Bk gij xj , t = tk , k = 1, 2, . . . (2) j=1 + xi (t0 ) = xi0 , i = 1, 2, . . . , N where Bk is an m × m impulsive matrix. Thus (2) can be written as follows. ˙ X = (IN ⊗ A)X + F (X, t), t = tk X = (G ⊗ Bk )X, t = tk + X(t0 ) = X0 . Assume that s = (1/N ) N i=1 xi is the synchronization state of the impulsively coupled systems. Then s˙ = As + (ϕ(x1 ) + ϕ(x2 ) + · · · + ϕ(xN )/N ) = As + ϕ. Let e1 = x1 − s, e2 = x2 − s, . . . , eN = xN − s, ei ∈ Rm , and e = (eT1 , eT2 , . . . , eTN )T . Then, one has the following dynamical error equations without coupling e˙ 1 = Ae1 + ϕ(x1 ) − ϕ = Ae1 + ψ(x1 , s) e˙ 2 = Ae2 + ϕ(x2 ) − ϕ = Ae2 + ψ(x2 , s) (3) .. . e˙ N = Ae1 + ϕ(xN ) − ϕ = AeN + ψ(xN , s) The dynamical error dynamics of impulsively coupled systems (2) is e˙ = (IN ⊗ A)e + H(e, t), t = tk e = (G ⊗ Bk )e, t = tk , (4) + k = 1, 2, . . . e(t0 ) = e0 , where H(e, t) = (ψ T (x1 , s), ψ T (x2 , s), . . . , ψT (xN , s))T . The objective is to find a condition satisfied by matrices Bk , G and the impulsive interval τk , k = 1, 2, . . . , such that the solutions of (2) synchronize with each other, in the sense that lim xi (t) − xj (t) = 0,
t→∞
(5)
for all i, j = 1, 2, . . . , N. That is to say, if the trivial solution of system (4) is stable, the impulsively coupled systems (2) synchronize in the sense of (5).
3. Main Results Since N G is a real symmetric matrix satisfying j=1 gij = 0 for i = 1, 2, . . . , N, we can let
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0 = λ1 > λ2 ≥ λ3 ≥ · · · ≥ λN denote the eigenvalues of G. Thus there exists a matrix U = (u1 , u2 , . . . , uN ) such that G = U ΛU T , where U T U = IN , and Λ = diag(λ1 , λ2 , . . . λN ). The eigenvector corresponding √ √to λ1 = 0 is u1 = √ ((1/ N ), (1/ N ), . . . , (1/ N ))T . Let U ⊗ Im = C, then we have G ⊗ Bk = (U ⊗ Im )(Λ ⊗ Bk )(U T ⊗ Im ) = C(Λ ⊗ Bk )C T and C T (IN ⊗ A)C = IN ⊗ A. Assumption 1. Assume that there exists a nonnegative real constant L, such that ϕ(xi ) − ϕ(xj ) ≤ Lxi − xj for i, j = 1, 2, . . . , N . Here L is called Lipschitzian constant. We have the following theorem: Theorem 1. Let λmax (P ) denote the maximal eigenvalue of P, and
βk = max λmax [(I + λi BkT )(I + λi Bk )], 1≤i≤N
L max η = λmax (AT + A) + N 2≤i≤N N N
× 2(N − 1)|upi upl | l=2 p=1
+
N
(|upi ujl | + |upl uji |)
j=1,j=p
(i) If η < 0 (η is a constant) and there exists a constant α(0 ≤ α < −η), such that ln βk − α(tk − tk−1 ) ≤ 0,
k = 1, 2, . . .
then the trivial solution of system (4) is globally exponentially stable. That is, the impulsively coupled systems (2) synchronize in the sense of (5). (ii) If η ≥ 0 (η is a constant) and there exists a constant α ≥ 1, such that ln(αβk ) + η(tk+1 − tk ) ≤ 0,
k = 1, 2, . . .
then α = 1 implies that the trivial solution of system (4) is stable and α > 1 implies that the trivial solution of system (4) is globally asymptotically stable. That is, the impulsively coupled systems (2) synchronize in the sense of (5). Proof. Introduce a linear transformation
y1 y2 Y = .. = C T e, . yN
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along with (4), then we get y˙ 1 y˙ 2 ˙ = Y .. = C T e˙ . y˙N = C T (I ⊗ A)CY + C TH(e, t), t = tk N λ1 Bk .. Y = (Λ ⊗ Bk )Y = Y, . λN Bk t = tk Y (t+ 0 ) = Y0 (6) where yi ∈ Rm and yi = (uTi ⊗I)e = (u1i e1 +u2i e2 + · · · + uN i eN ) for i = 1, 2, . . . , N. That is, y˙i = Ayi + (uTi ⊗ Im )H(e, t) = Ayi + (u1i ψ(x1 , s) + u2i ψ(x2 , s) + · · · + uN i ψ(xN , s)), t = tk yi = λi Bk yi , t = tk , k = 1, 2, . . . + i = 1, 2, . . . , N. yi (t0 ) = yi0 , (7) √ Then y1 = (uT1 ⊗ I)e = (e1 + e2 + · · · + eN / N ) = 0 corresponds to the synchronizing manifold. From Assumption 1 and the transformation, m u y one has ep = N l=2 pl l ∈ R , (p = 1, 2, . . . , N ). Thus N N N
upl yl ≤ upl yl ≤ |upl |yl ep = l=2
l=2
l=2
and ϕ(xp ) − ϕ = ψ(xp , s) N
L (N − 1)ep + ej ≤ N j=1,j=p
≤
N
L (N − 1) |upl |yl N l=2
+
N N
j=1,j=p
l=2
|ujl |yl ,
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p = 1, 2, . . . , N. We can also obtain that u1i ψ(x1 , s) + u2i ψ(x2 , s) + · · · + uN i ψ(xN , s) ≤
N
upi ψ(xp , s)
p=1
N N N N
L |upl |yl + |ujl |yl ≤ upi (N − 1) N p=1
l=2
j=1,j=p
l=2
N N N N
L (N − 1) = |upi upl |yl + |upi ujl |yl N p=1 l=2
l=2 j=1,j=p
N N N
L
|upi ujl |yl (N − 1)|upi upl |yl + = N p=1
l=2
j=1,j=p
Therefore, the stability of the trivial solution of system (7) implies the stability of the trivial solution of system (4) for i ≥ 2. T For system (7), construct the following Lyapunov function, V (y2 , . . . , yN ) = N i=2 yi yi . Let N
N N
L 2(N − 1)|upi upl | + max (|upi ujl | + |upl uji |) η = λmax (AT + A) + N 2≤i≤N p=1 l=2
j=1,j=p
Thus the time derivative of V along the trajectory of system (7) is V˙ (y2 (t), . . . , yN (t)) =
N
(y˙ iT yi + yiT y˙i )
i=2
=
N
((Ayi + (u1i ψ(x1 , s) + u2i ψ(x2 , s) + · · · + uN i ψ(xN , s)))Ti yi
i=2
+ yiT (Ayi + (u1i ψ(x1 , s) + u2i ψ(x2 , s) + · · · + uN i ψ(xN , s)))) =
N
i=2
yiT (AT + A)yi + 2
≤ λmax (AT + A)V + 2
N
i=2
N
i=2
yiT(u1i ψ(x1 , s) + u2i ψ(x2 , s) + · · · + uN i ψ(xN , s))
yiT(u1i ψ(x1 , s) + u2i ψ(x2 , s) + · · · + uN i ψ(xN , s))
N N N
N
L (N − 1)|upi upl | + yiT |upi ujl |yl ≤ λmax (AT + A)V + 2 N i=2
l=2 p=1
j=1,j=p
N N
N N
L (N − 1)|upi upl | + |upi ujl | 2yiT yl = λmax (AT + A)V + N p=1 i=2 l=2
j=1,j=p
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N N N N
L (N − 1)|upi upl | + ≤ λmax (AT + A)V + |upi ujl | (yiTyi + ylTyl ) N p=1 i=2
l=2
j=1,j=p
N
N N N
L (N − 1)|upi upl | + |upi ujl | yiTyi = λmax (AT + A)V + N
i=2 l=2
p=1
j=1,j=p
N N N N
L + |upi ujl | ylTyl (N − 1)|upi upl | + N p=1
i=2 l=2
j=1,j=p
N N
N N
L (N − 1)|upi upl | + = λmax (AT + A)V + |upi ujl | yiTyi N p=1 i=2 l=2
j=1,j=p
N N N N
L + |upi ujl | ylTyl (N − 1)|upi upl | + N l=2 i=2
p=1
j=1,j=p
N N
N N
L (N − 1)|upi upl | + = λmax (AT + A)V + |upi ujl | yiTyi N i=2 l=2
p=1
j=1,j=p
N N N N
L + |upl uji | yiTyi (N − 1)|upi upl | + N p=1 i=2 l=2
j=1,j=p
N N N
N
L 2(N − 1)|upi upl | + = λmax (AT + A)V + (|upi ujl | + |upl uji |) yiTyi N
i=2
l=2 p=1
≤ λmax (AT + A)V +
L max N 2≤i≤N
= ηV (y2 (t), . . . , yN (t)),
N N
j=1,j=p
(2(N − 1)|upi upl | +
l=2 p=1
t ∈ (tk−1 , tk ],
t ∈ (tk−1 , tk ],
k = 1, 2, . . . . (9)
On the other hand, it follows from system (7) that + V (y2 (t+ k ), . . . , yN (tk ))
=
N
i=2
+ T y(t+ k )i y(tk )i
N
= ((I + λi Bk )y(tk )i )T(I + λi Bk )y(tk )i i=2
(|upi ujl | + |upl uji |) V
k = 1, 2, . . . ,
=
+ ≤ V (y2 (t+ k−1 ), . . . , yN (tk−1 )) exp(η(t − tk−1 )),
j=1,j=p
which implies that V (y2 (t), . . . , yN (t))
N
(8) N
y(tk )Ti (I + λi Bk )T (I + λi Bk )y(tk )i
i=2
≤ βk V (y2 (tk ), . . . , yN (tk ))
(10)
The following results come from (9) and (10). For t ∈ (t0 , t1 ], V (y2 (t), . . . , yN (t)) + ≤ V (y2 (t+ 0 ), . . . , yN (t0 )) exp(η(t − t0 )),
which leads to V (y2 (t1 ), . . . , yN (t1 )) + ≤ V (y2 (t+ 0 ), . . . , yN (t0 )) exp(η(t1 − t0 )),
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and
× β1 β2 · · · βk exp(η(tk+1 − t0 ))
+ V (y2 (t+ 1 ), . . . , yN (t1 ))
+ ≤ V (y2 (t+ 0 ), . . . , yN (t0 ))β1 exp(η(t2 − t1 ))
≤ β1 V (y2 (t1 ), . . . , yN (t1 )) + ≤ β1 V (y2 (t+ 0 ), . . . , yN (t0 )) exp(η(t1 − t0 )).
In general, for t ∈ (tk , tk+1 ], (k = 0, 1, 2, . . .) + ≤ V (y2 (t+ 0 ), . . . , yN (t0 ))β1 β2 · · · βk exp(η(t−t0 )).
Therefore, we have the following results.
V (y2 (t), . . . , yN (t)) + ≤ V (y2 (t+ 0 ), . . . , yN (t0 ))β1 β2 · · · βk exp(η(t − t0 )) + = V (y2 (t+ 0 ), . . . , yN (t0 ))β1 β2 · · · βk × exp(−α(t − t0 )) exp((η + α)(t − t0 )) + ≤ V (y2 (t+ 0 ), . . . , yN (t0 ))β1 β2 · · · βk
× exp(−α(tk − t0 )) exp((η + α)(t − t0 )) exp(−α(t1 − t0 ))β2
× exp(−α(t2 − t1 )) · · · βk × exp(−α(tk − tk−1 )) exp((η + α)(t − t0 ))
which implies that the conclusion (ii) of Theorem 1 holds. This completes the proof. synchronization of the impulsively coupled systems. The result has its physical significance. It implies that the impulsive interval and the impulsive energy are related to the eigenvalues and eigenvectors of coupling matrix G. On the other hand, they are also affected by the eigenvalue of A and the Lipschitzian constant L for ϕ(x), which are respectively determined by the linear and the nonlinear parts of the dynamics. To achieve synchronization, we have to increase the impulsive frequency if we decrease the impulsive energy. In practice, for convenience, the gain matrices Bk are always selected as a constant matrix and the impulsive intervals τk = tk − tk−1 (k = 1, 2, . . .) are set to be a positive constant. Thus we have the following corollary. Corollary 1. Assume τk = τ > 0 and matrices Bk = B (k = 1, 2, . . .)
namely, V (y2 (t), . . . , yN (t)) ≤V
+ (y2 (t+ 0 ), . . . , yN (t0 )) exp((η
+ α)(t − t0 )),
(i) If η < 0 (η is a constant) and there exists a constant α(0 ≤ α < −η), such that ln β − ατ ≤ 0,
t ≥ t0 . We can conclude that the trivial solution of system (7) is globally exponentially stable from the theories in [Bainov & Simeonov, 1989; Yang, 2001]. That is, yi → 0, i = 1, 2, . . . , N as t → ∞. Consequently, limt→∞ ei (t) = 0, i = 1, 2, . . . , N , i.e. limt→∞ xi (t) − xj (t) = 0, i, j = 1, 2, . . . , N. (ii) If η ≥ 0 and there exists a constant α ≥ 1, such that ln(αβk ) + η(tk+1 − tk ) ≤ 0, k = 1, 2, . . . . For t ∈ (tk , tk+1 ], we have V (y2 (t), . . . , yN (t)) ≤V
1 exp(η(t1 − t0 )), αk
Remark. Theorem 1 gives sufficient conditions for
(i) If η < 0 and there exists a constant α(0 ≤ α < −η) such that ln βk − α(tk − tk−1 ) ≤ 0, k = 1, 2, . . . , we obtain that for t ∈ (tk , tk+1 ],
=V
× exp(η(t1 − t0 )) + ≤ V (y2 (t+ 0 ), . . . , yN (t0 ))
V (y2 (t), . . . , yN (t))
+ (y2 (t+ 0 ), . . . , yN (t0 ))β1
× β2 exp(η(t3 − t2 )) · · · βk exp(η(tk+1 − tk ))
+ (y2 (t+ 0 ), . . . , yN (t0 ))
× β1 β2 · · · βk exp(η(t − t0 ))
then the trivial solution of system (4) is globally exponentially stable. (ii) If η ≥ 0 (η is a constant) and there exists a constant α ≥ 1, such that ln(αβ) + ητ ≤ 0, then α = 1 implies that the trivial solution of system (4) is stable and α > 1 implies that the trivial solution of system (4) is globally asymptotically stable. That is, impulsively coupled systems (2) synchronize in the sense of (5). Taking N = 3 and
−2 1 1 1 G = 1 −2 1 1 −2
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as an example, we can derive another sufficient condition for synchronization. What is more, we have wider impulsive interval. The coupled model is thus described by x˙ 1 = f (x1 ) = Ax1 + ϕ(x1 ) x1 = Bk (x2 + x3 − 2x1 ) x˙ = f (x ) = Ax + ϕ(x ) 2 2 2 2 (11) x2 = Bk (x1 + x3 − 2x2 ) x˙ 3 = f (x3 ) = Ax3 + ϕ(x3 ) x3 = Bk (x1 + x2 − 2x3 ) Let e1 = x1 −x2 , e2 = x2 −x3 , e3 = x3 −x1 , then we have the following dynamical error dynamics: e˙ 1 = Ae1 + ϕ(x1 ) − ϕ(x2 ) e1 = Bk (e2 + e3 − 2e1 ) = −3Bk e1 e˙ = Ae + ϕ(x ) − ϕ(x ) 2 2 2 3 (12) = B (e + e − 2e e 2 1 3 2 ) = −3Bk e2 k e˙ 3 = Ae3 + ϕ(x3 ) − ϕ(x1 ) e3 = Bk (e1 + e2 − 2e3 ) = −3Bk e3 And we have the following theorem: Let λmax (P ) denote the maximal eigenvalue of P, and βk = λmax [(I − 3BkT )(I − 3Bk )]. Theorem 2.
(i) If λmax (AT + A) + 2L = η < 0 (η is a constant) and there exists a constant α(0 ≤ α < −η), such that ln βk − α(tk − tk−1 ) ≤ 0,
k = 1, 2, . . .
then the trivial solution of system (12) is globally exponentially stable. That is, the impulsively coupled systems (11) synchronize in the sense of (5). (ii) If λmax (AT + A) + 2L = η ≥ 0 (η is a constant) and there exists a constant α ≥ 1, such that ln(αβk ) + η(tk+1 − tk ) ≤ 0,
k = 1, 2, . . .
then α = 1 implies that the trivial solution of system (12) is stable and α > 1 implies that the trivial solution of system (12) is globally asymptotically stable. That is, the impulsively coupled systems (11) synchronize in the sense of (5). Choose a Lyapunov function V (e1 , e2 , e3 ) = + eT2 e2 + eT3 e3 .
Proof.
eT1 e1
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For t ∈ (tk−1 , tk ], k = 1, 2, . . . , the time derivative of V along the trajectory of system (12) is V˙ (e1 (t), e2 (t), e3 (t)) = e˙T1 e1 + e˙T2 e2 + e˙ T3 e3 + eT1 e˙1 + eT2 e˙ 2 + eT3 e˙ 3 = (Ae1 + ϕ(x1 ) − ϕ(x2 ))Te1 + (Ae2 + ϕ(x2 ) − ϕ(x3 ))Te2 + (Ae3 + ϕ(x3 ) − ϕ(x1 ))Te3 + eT1 (Ae1 + ϕ(x1 ) − ϕ(x2 )) + eT2 (Ae2 + ϕ(x2 ) − ϕ(x3 )) + eT3 (Ae3 + ϕ(x3 ) − ϕ(x1 )) = eT1 (A + AT )e1 + eT2 (A + AT )e2 + eT3 (A +AT )e3 + 2(ϕ(x1 ) − ϕ(x2 ))Te1 + 2(ϕ(x2 ) − ϕ(x3 ))Te2 + 2(ϕ(x3 ) − ϕ(x1 ))Te3 ≤ (λmax (A + AT ) + 2L)V (e1 (t), e2 (t), e3 (t)), which implies that V (e1 (t), e2 (t), e3 (t)) + + ≤ V (e1 (t+ k−1 ), e2 (tk−1 ), e3 (tk−1 )) × exp(η(t − tk−1 )), t ∈ (tk−1 , tk ], k = 1, 2, . . . . On the other hand, we have V (e(t+ k )) + + T + + T + T = e(t+ k )1 e(tk )1 + e(tk )2 e(tk )2 + e(tk )3 e(tk )3
= ((I − 3Bk )e(tk )1 )T (I − 3Bk )e(tk )1 + ((I − 3Bk )e(tk )2 )T (I − 3Bk )e(tk )2 + ((I − 3Bk )e(tk )3 )T (I − 3Bk )e(tk )3 ≤ λmax ((I − 3Bk )T (I − 3Bk ))V (e(tk )) ≤ βk V (e(tk )). The remaining reasoning is similar to that of Theorem 1, so details are omitted. This completes the proof. The following results easily follow from Theorem 2. Corollary 2. Assume τk = τ > 0 and matrices Bk = B (k = 1, 2, . . .)
(i) If η < 0 and there exists a constant α(0 ≤ α < −η), such that ln β − ατ ≤ 0, then the trivial solution of system (12) is globally exponentially stable.
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ln(αβ) + ητ ≤ 0, then α = 1 implies that the trivial solution of system (12) is stable and α > 1 implies that the trivial solution of system (12) is globally asymptotically stable. That is, the impulsively coupled systems (11) synchronize in the sense of (5).
0.1 x11−x21
(ii) If η ≥ 0 (η is a constant) and there exists a constant α ≥ 1, such that
0 −0.1 −0.2
0
2
4
6
8
10
12
0
2
4
6
8
10
12
0
2
4
6 t/s
8
10
12
0.1 x12−x22
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4. Numerical Simulations To demonstrate the above-derived theoretical results, some typical chaotic systems are used as the dynamical nodes of the impulsively coupled systems. Here, we choose the typical Chua’s circuit system and Lorenz system for illustrations. Example 1. Consider the original Chua’s circuit
system described by Chua et al. [1986]: x˙ = µ(y − x − g(x)), y˙ = x − y + z, z˙ = −νy,
(13)
where µ > 0, ν > 0, g(x) is a piecewise linear function given by g(x) = ax + (1/2)(b − a)(|x + 1| − |x − 1|), with b < a < 0. As is well known, Chua’s circuit is a typical chaotic system, which exhibits very rich complex dynamical behaviors [Chua et al., 1993; Chua et al., 1996], and has wide applications. The system has a double-scroll chaos attractor with µ = 10, ν = 14.97, a = −0.68, and b = −1.27. Rewrite system (13) into x˙ = Ax + ϕ(x),
x13−x23
0.2 0 −0.2
Fig. 1. Synchronization errors between the first and the second systems for the impulsively coupled network with three Chua’s systems.
B = diag(0.4, 0.4, 0.4), α = 1.1, β = 0.04. From Corollary 2, we obtain that the impulsive interval for synchronization should be less than 0.0778, while it should be less than 0.0055 from Theorem 1. In simulation, choose the impulsive interval τ = 0.015. Figures 1 and 2 display the synchronization errors of the impulsively coupled network with impulsive interval τ = 0.015 Next, we consider an impulsively coupled network with 20 nodes, the coupling structure is shown in Fig. 3. Here, we set Bk = diag(0.2, 0.2, 0.2), α = 1.001 and then βk = 0.8520. We derive from Theorem 1 that impulsive interval τ should be less
(14) 2
that is, the Lipschitzian constant L = µ|b| = 12.7. First, consider the impulsively coupled network with three Chua’s systems, which are coupled with bidirectional ring structure. Let
21
0 0
2
4
6
8
10
12
0
2
4
6
8
10
12
0
2
4
6 t/s
8
10
12
x22−x32
0.1 0 −0.1
33
2 0
23
λmax (A + AT ) = 14.7316 and ϕ(x) − ϕ(y ) ≤ µ|b|x − y ,
1
−1
−2
x −x
x = (x, y, z)T , ϕ(x) = (−µg(x), 0, 0)T , −µ µ 0 A = 1 −1 1 , 0 −ν 0 −2µ µ + 1 0 −2 1 − ν , A + AT = 1 + µ 0 1−ν 0
x −x
31
where
−4
Fig. 2. Synchronization errors between the second and the third systems for the impulsively coupled network with three Chua’s systems.
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Synchronization of Impulsively Coupled Systems
1547
25
20
r
15
10
5
0
0
10
20
30
40
50
t/s
Fig. 4. The synchronization error r of 20 Chua’s systems without coupling. 5
Fig. 3.
An impulsively coupled network with 20 nodes. 4
as the whole synchronization error. Then r → 0 as t → ∞ implies the synchronization of the impulsively coupled systems in sense of (5). Here, we choose τ = 0.0002. Figure 4 shows r of system (1) for 20 Chua’s systems without coupling, which indicates that there is no synchronization. Figure 5 displays r with 20 impulsively coupled Chua’s systems. Figure 6 shows the first variable of the 20 synchronized Chua’s systems. Figure 7 shows dynamics of the first node of the impulsively coupled systems after they reach synchronization.
3
2 r
than 0.00028315. Denote (x11 − x21 )2 + (x21 − x31 )2 + · · · + (x20,1 − x11 )2 + (x12 − x22 )2 + · · · r= + (x 2 2 20,2 − x12 ) + (x13 − x23 ) + · · · + (x20,3 − x13 )2
1
0
−1 −2
6
When a = 10, b = 8/3, c = 28, Lorenz system has a chaotic attractor. Then we have −10 10 0 0 28 −1 A= 8 0 0 − 3
0.1
0.15 t/s
0.2
0.25
0.3
2.5
5 2
4
i1
x (t),(i=1,...,20)
(15)
0.05
Fig. 5. The synchronization error r of 20 impulsively coupled Chua’s systems.
Example 2. A single Lorenz system is described by
x˙ = a(y − x), y˙ = cx − xz − y, z˙ = xy − bz.
0
3 1.5
0
0.01
0.02
0.03
2 1 0 −1 −2 −3
0
5
10
15
20 t/s
25
30
35
40
Fig. 6. Evolutions of the first variables of the 20 coupled systems; the inset figure is the transient.
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1548
200
0
150
r
z
5
100
−5 1
0 −1
y
Fig. 7.
50 −4
−2
−3
−1 x
0
1
2
3
4
The dynamics of the first node after synchronization.
50
60 50 40 r
Consider the impulsively coupled network with three Lorenz systems and choose B = diag(0.4, 0.4, 0.4), α = 1.1, β = 0.04. From Corollary 2, the impulsive interval for synchronization should be less than 0.0293, and that from Theorem 1 should be less than 0.0153. Figure 8 shows the synchronized states with τ = 0.015.
i1
40
70
We can obtain from [Li et al., 2005] that the bound of system (15) is 39.2462. The nonlinear part of the system is ϕ(x, y, z) = (0, −xz, xy)T . It follows from Theorem 1 that L = 39.2462.
x (t)(i=1,2,3)
30
80
λmax (A + A ) = 28.051.
0 −20 0
0.5
1
1.5
2
0
0.5
1
1.5
2
20 0 −20 −40
30 20 10 0 −10 −20
0
0.02
0.04
0.06
0.08
0.1
t/s
Fig. 10. The entire error r of 20 impulsively coupled Lorenz systems.
20
−40
i2
20
Fig. 9. The entire error r of 20 Lorenz systems without coupling.
T
x (t)(i=1,2,3)
10
t/s
and
For the coupling structure shown by Fig. 3, we derive from Theorem 1 that the impulsive interval τ should be less than 0.000091997 for Bk = diag(0.2, 0.2, 0.2), α = 1.001 and βk = 0.8520. Figure 9 shows the entire error r of system (1) with 20 isolate Lorenz systems for τ = 0.00009. Figure 10 shows the entire error of 20 impulsively coupled Lorenz systems.
100 50
5. Conclusions
0
i3
x (t)(i=1,2,3)
0
−50
0
0.5
1 t/s
1.5
2
Fig. 8. Synchronization of the network with three impulsively coupled Lorenz systems.
Unlike other existing works, in this paper, we have discussed the synchronization of multi-systems, which are coupled by impulsive connections only at discrete instants. We have derived sufficient conditions for synchronizing coupled systems with a given
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coupling structure. We have also obtained sufficient conditions for synchronizing three systems which are impulsively coupled with the bidirectional ring structure. The typical Chua’s circuit and the Lorenz system are taken as illustrative examples to confirm the theoretical results. These results can be applied to other systems.
Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant No. 60574045, 70771084), the National Basic Research 973 Program of China (Grant No. 2007CB310805), and China Tianyuan Youth Funding of Mathematics (Grant No. 201161509). Furthermore, the authors thank Mr. Chen Liang for the valuable suggestions.
References Bainov, D. D. & Simeonov, P. S. [1989] Systems with Impulse Effect: Stability, Theory and Applications (Halsted Press, NY). Chen, S. H., Yang, Q. & Wang, C. P. [2004] “Impulsive control and synchronization of unified chaotic system,” Chaos Solit. Fract. 20, 751–758. Chua, L. O., Komuro, M. & Matsumoto, T. [1986] “The double scroll family, Part I: Rigorous proof of chaos,” IEEE Trans. Circuits Syst.-I 33, 1073–1097. Chua, L. O., Itoh, M., Kocarev, L. & Eckert, L. [1993] “Chaos synchronization in Chua’s circuit,” J. Circuits Syst. Comput. 3, 93–108. Chua, L. O., Yang, T., Zhong, G. Q. & Wu, C. W. [1996] “Adaptive synchronization of Chua’s oscillators,” Int. J. Bifurcation and Chaos 6, 189–201. Fu, X. L., Yan, B. Q. & Liu, Y. S. [2005] Introduction of Impulsive Differential System (Academic Press of China, Beijing). Gu, F. J., Li, X. J. & Ruan, J. [1992] “Dynamical neural network model,” Acta Biophys. Sin. 8, 12–32. Guan, Z. H., Hill, D. J. & Yao, J. [2006] “A hybrid impulsive and switching control strategy synchronization of nonlinear systems and application to Chua’s chaotic circuit,” Int. J. Bifurcation and Chaos 16, 229–238. Khadra, A., Liu, X. Z. & Shen, X. M. [2003] “Application of impulsive synchronization to communication security,” IEEE Trans. Circuits. Syst.-I 50, 341–351.
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Lakshmikantham, V., Bainov, D. & Simeonov, P. [1989] Theory of Impulsive Differential Equations (World Scientific, Singapore). Li, D. M., Lu, J. A., Wu, X. Q. & Chen, G. R. [2005] “Estimating the bounds for the Lorenz family of chaotic systems,” Chaos Solit. Fract. 23, 529–534. Li, Z. & Chen, G. R. [2006] “Global synchronization and asymptotic stability of complex dynamical networks,” IEEE Trans. Circuits Syst.-II 53, 28–33. Li, Z. G., Wen, C. Y. & Soh, Y. C. [2001] “Analysis and design of impulsive control systems,” IEEE Trans. Autom. Contr. 46, 894–903. Liu, B., Liu, X. Z., Chen, G. R. & Wang, H. Y. [2005] “Robust impulsive synchronization of uncertain dynamical networks,” IEEE Trans. Circuits Syst.-I 52, 1431–1440. Liu, X. Z. [1994] “Stability results for impulsive differential systems with applications to population growth models,” Dyn. Stab. Syst. 9, 163–174. Liu, X. Z. & Willms, A. R. [1996] “Impulsive controllability of linear dynamical systems with applications to maneuvers of spacecraft,” Math. Prob. Eng. 2, 277– 299. Stojanovski, T., Kocarev, L. & Parlitz, U. [1996] “Driving and synchronizing by chaotic impulses,” Phys. Rev. E 43, 782–785. Sun, J. T. & Zhang, Y. P. [2003] “Impulsive control of R¨ossler systems,” Phys. Lett. A 306, 306–312. Sun, J. T., Zhang, Y. P., Qiao, F. & Wu, Q. D. [2004] “Some impulsive synchronization criterions for coupled chaotic systems via unidirectional linear error feedback approach,” Chaos Solit. Fract. 19, 1049– 1055. Xie, W. X., Wen, C. Y. & Li, Z. G. [2000] “Impulsive control for the stabilization and synchronization of Lorenz systems,” Phys. Lett. A 275, 67–72. Yang, T. & Chua, L. O. [1997] “Impulsive stabilization for control and synchronization of chaotic systems: Theory and application to secure communication,” IEEE Trans. Circuits Syst.-I 44, 976–988. Yang, T. [2001] Impulsive Control Theory (SpringerVerlag, Berlin). Zhou, J., Chen, T. P. & Gao, Y. H. [2005] “Synchronization dynamics of complex networks with impulsive control,” Second National Forum on Complex Dynamical Networks (Beijing), pp. 226–230.
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International Journal of Bifurcation and Chaos, Vol. 18, No. 5 (2008) 1539–1549 c World Scientific Publishing Company
SYNCHRONIZATION OF IMPULSIVELY COUPLED SYSTEMS XIUPING HAN, JUN-AN LU∗ and XIAOQUN WU School of Mathematics & Statistics, Wuhan University, Wuhan 430072, P. R. China ∗The State Key Laboratory of Software Engineering, Wuhan University, Wuhan 430072, P. R. China ∗[email protected] Received August 29, 2006; Revised June 12, 2007 In the past years, impulsive control for a single system and impulsive synchronization between two systems have been extensively studied. However, investigation on impulsive control and synchronization of complex networks has just started. In these studies, a network is continuously coupled, and then is synchronized by using impulsive control strategy. In this paper, a new and different coupled model is proposed, where the systems are coupled only at discrete instants through impulsive connections. Several criteria for synchronizing such kind of impulsively coupled complex dynamical systems are established. Two examples are also worked through for illustrating the main results. Keywords: Impulsive differential system; impulsively coupled systems; synchronization.
1. Introduction Impulsive phenomena widely exist in the real world. The mathematical model of an impulsive phenomenon is called an impulsive differential equation [Bainov & Simeonov, 1989; Lakshmikantham et al., 1989]. The most distinct character of impulsive differential systems is that it takes into account the effects of the instantaneous phenomena, such as population-growth models and maneuvers of spacecraft [Liu, 1994; Liu & Willms, 1996], dynamical nerve cell networks [Gu et al., 1992], etc. Studies show that impulsive differential models have been widely applied in many fields, such as space techniques, information science, control system, communication security [Khadra et al., 2003], life science [Fu et al., 2005], and so on. Impulsive control strategy is very effective, robust and with low cost. The control input is ∗
implemented by the “sudden jumps” of some state variables at some instants. It needs only small control impulses and has rapid responsive velocity. In the past decades, impulsive control has been widely studied in stabilizing and synchronizing chaotic systems [Stojanovski et al., 1996; Yang & Chua, 1997; Yang, 2001; Xie et al., 2000; Li et al.., 2001; Sun & Zhang, 2003; Chen et al., 2004]. Guan et al. investigated the synchronization between two nonlinear systems by using the hybrid impulsive and switching control strategy [Guan et al., 2006]. Synchronization of subsystems in a coupled dynamical network is very useful in practical applications, thus it has received a great deal of interest recently. For example, Sun et al. obtained some impulsive synchronization criteria for coupled chaotic systems via unidirectional coupling [Sun et al., 2004], Li and Chen concluded that complex
Author for correspondence 1539
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dynamical networks can be regarded as a big coupled system [Li & Chen, 2006], Liu et al. discussed the robust impulsive synchronization of uncertain dynamical networks [Liu et al., 2005], Zhou et al. discussed the impulsive control of networks [Zhou et al., 2005], and so on. Previous studies on impulsive control and synchronization for coupled systems were based on continuous coupling. In practice, there are some systems, instead of being coupled with continuous connections, are only coupled by impulsive connections at instants, such as the species-food model in biology, the information transfer and exchange in ants, and the model of integrated circuit. Unlike other existing studies, the model proposed in this paper is coupled by impulsive connections. The coupled systems can reach synchronization by exchanging energy at instants. Sufficient conditions on synchronization of the impulsively coupled systems are presented. Numerical examples are also provided to verify the theoretical results and the effectiveness of the proposed synchronization scheme.
2. Description of the Model Consider a network consisting of N identical nodes, in which each node is an m-dimensional dynamical system. The state equations are x˙ 1 = f (x1 ) = Ax1 + ϕ(x1 ) x˙ = f (x2 ) = Ax2 + ϕ(x2 ) 2 .. (1) . x˙ N −1 = f (xN −1 ) = AxN −1 + ϕ(xN −1 ) x˙ N = f (xN ) = AxN + ϕ(xN ) where xi , ϕ(xi ) ∈ Rm , i = 1, 2, . . . , N and A ∈ Rm×m . Let X = (xT1 , xT2 , . . . , xTN )T and (ϕT (x1 ), ϕT (x2 ), . . . , ϕT (xN ))T = F (X, t), then X˙ = (IN ⊗ A)X+ F (X, t), where IN ⊗A denotes the Kronecker product of matrices IN and A. The impulsive coupling between systems are defined as follows. There is a connection (or impulsive coupling) between systems xi and xj if there is energy exchange at the impulsive instants tk (k = 1, 2, . . .), where t1 < t2 < · · · < tk < · · · , limk→∞ tk = ∞. Let G = (gij )N ×N denote the coupling matrix. Then gij = gji = 1 if there is a connection between node i and j (i = j); otherwise, gij = gji = 0. In this model, it is required that the coupling coefficients satisfy gii = − N j=1,j=i gij .
Then the above defined model of impulsively coupled systems can be described by x˙ i = Axi + ϕ(xi ), t = tk N xi = Bk gij xj , t = tk , k = 1, 2, . . . (2) j=1 + xi (t0 ) = xi0 , i = 1, 2, . . . , N where Bk is an m × m impulsive matrix. Thus (2) can be written as follows. ˙ X = (IN ⊗ A)X + F (X, t), t = tk X = (G ⊗ Bk )X, t = tk + X(t0 ) = X0 . Assume that s = (1/N ) N i=1 xi is the synchronization state of the impulsively coupled systems. Then s˙ = As + (ϕ(x1 ) + ϕ(x2 ) + · · · + ϕ(xN )/N ) = As + ϕ. Let e1 = x1 − s, e2 = x2 − s, . . . , eN = xN − s, ei ∈ Rm , and e = (eT1 , eT2 , . . . , eTN )T . Then, one has the following dynamical error equations without coupling e˙ 1 = Ae1 + ϕ(x1 ) − ϕ = Ae1 + ψ(x1 , s) e˙ 2 = Ae2 + ϕ(x2 ) − ϕ = Ae2 + ψ(x2 , s) (3) .. . e˙ N = Ae1 + ϕ(xN ) − ϕ = AeN + ψ(xN , s) The dynamical error dynamics of impulsively coupled systems (2) is e˙ = (IN ⊗ A)e + H(e, t), t = tk e = (G ⊗ Bk )e, t = tk , (4) + k = 1, 2, . . . e(t0 ) = e0 , where H(e, t) = (ψ T (x1 , s), ψ T (x2 , s), . . . , ψT (xN , s))T . The objective is to find a condition satisfied by matrices Bk , G and the impulsive interval τk , k = 1, 2, . . . , such that the solutions of (2) synchronize with each other, in the sense that lim xi (t) − xj (t) = 0,
t→∞
(5)
for all i, j = 1, 2, . . . , N. That is to say, if the trivial solution of system (4) is stable, the impulsively coupled systems (2) synchronize in the sense of (5).
3. Main Results Since N G is a real symmetric matrix satisfying j=1 gij = 0 for i = 1, 2, . . . , N, we can let
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0 = λ1 > λ2 ≥ λ3 ≥ · · · ≥ λN denote the eigenvalues of G. Thus there exists a matrix U = (u1 , u2 , . . . , uN ) such that G = U ΛU T , where U T U = IN , and Λ = diag(λ1 , λ2 , . . . λN ). The eigenvector corresponding √ √to λ1 = 0 is u1 = √ ((1/ N ), (1/ N ), . . . , (1/ N ))T . Let U ⊗ Im = C, then we have G ⊗ Bk = (U ⊗ Im )(Λ ⊗ Bk )(U T ⊗ Im ) = C(Λ ⊗ Bk )C T and C T (IN ⊗ A)C = IN ⊗ A. Assumption 1. Assume that there exists a nonnegative real constant L, such that ϕ(xi ) − ϕ(xj ) ≤ Lxi − xj for i, j = 1, 2, . . . , N . Here L is called Lipschitzian constant. We have the following theorem: Theorem 1. Let λmax (P ) denote the maximal eigenvalue of P, and
βk = max λmax [(I + λi BkT )(I + λi Bk )], 1≤i≤N
L max η = λmax (AT + A) + N 2≤i≤N N N
× 2(N − 1)|upi upl | l=2 p=1
+
N
(|upi ujl | + |upl uji |)
j=1,j=p
(i) If η < 0 (η is a constant) and there exists a constant α(0 ≤ α < −η), such that ln βk − α(tk − tk−1 ) ≤ 0,
k = 1, 2, . . .
then the trivial solution of system (4) is globally exponentially stable. That is, the impulsively coupled systems (2) synchronize in the sense of (5). (ii) If η ≥ 0 (η is a constant) and there exists a constant α ≥ 1, such that ln(αβk ) + η(tk+1 − tk ) ≤ 0,
k = 1, 2, . . .
then α = 1 implies that the trivial solution of system (4) is stable and α > 1 implies that the trivial solution of system (4) is globally asymptotically stable. That is, the impulsively coupled systems (2) synchronize in the sense of (5). Proof. Introduce a linear transformation
y1 y2 Y = .. = C T e, . yN
1541
along with (4), then we get y˙ 1 y˙ 2 ˙ = Y .. = C T e˙ . y˙N = C T (I ⊗ A)CY + C TH(e, t), t = tk N λ1 Bk .. Y = (Λ ⊗ Bk )Y = Y, . λN Bk t = tk Y (t+ 0 ) = Y0 (6) where yi ∈ Rm and yi = (uTi ⊗I)e = (u1i e1 +u2i e2 + · · · + uN i eN ) for i = 1, 2, . . . , N. That is, y˙i = Ayi + (uTi ⊗ Im )H(e, t) = Ayi + (u1i ψ(x1 , s) + u2i ψ(x2 , s) + · · · + uN i ψ(xN , s)), t = tk yi = λi Bk yi , t = tk , k = 1, 2, . . . + i = 1, 2, . . . , N. yi (t0 ) = yi0 , (7) √ Then y1 = (uT1 ⊗ I)e = (e1 + e2 + · · · + eN / N ) = 0 corresponds to the synchronizing manifold. From Assumption 1 and the transformation, m u y one has ep = N l=2 pl l ∈ R , (p = 1, 2, . . . , N ). Thus N N N
upl yl ≤ upl yl ≤ |upl |yl ep = l=2
l=2
l=2
and ϕ(xp ) − ϕ = ψ(xp , s) N
L (N − 1)ep + ej ≤ N j=1,j=p
≤
N
L (N − 1) |upl |yl N l=2
+
N N
j=1,j=p
l=2
|ujl |yl ,
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p = 1, 2, . . . , N. We can also obtain that u1i ψ(x1 , s) + u2i ψ(x2 , s) + · · · + uN i ψ(xN , s) ≤
N
upi ψ(xp , s)
p=1
N N N N
L |upl |yl + |ujl |yl ≤ upi (N − 1) N p=1
l=2
j=1,j=p
l=2
N N N N
L (N − 1) = |upi upl |yl + |upi ujl |yl N p=1 l=2
l=2 j=1,j=p
N N N
L
|upi ujl |yl (N − 1)|upi upl |yl + = N p=1
l=2
j=1,j=p
Therefore, the stability of the trivial solution of system (7) implies the stability of the trivial solution of system (4) for i ≥ 2. T For system (7), construct the following Lyapunov function, V (y2 , . . . , yN ) = N i=2 yi yi . Let N
N N
L 2(N − 1)|upi upl | + max (|upi ujl | + |upl uji |) η = λmax (AT + A) + N 2≤i≤N p=1 l=2
j=1,j=p
Thus the time derivative of V along the trajectory of system (7) is V˙ (y2 (t), . . . , yN (t)) =
N
(y˙ iT yi + yiT y˙i )
i=2
=
N
((Ayi + (u1i ψ(x1 , s) + u2i ψ(x2 , s) + · · · + uN i ψ(xN , s)))Ti yi
i=2
+ yiT (Ayi + (u1i ψ(x1 , s) + u2i ψ(x2 , s) + · · · + uN i ψ(xN , s)))) =
N
i=2
yiT (AT + A)yi + 2
≤ λmax (AT + A)V + 2
N
i=2
N
i=2
yiT(u1i ψ(x1 , s) + u2i ψ(x2 , s) + · · · + uN i ψ(xN , s))
yiT(u1i ψ(x1 , s) + u2i ψ(x2 , s) + · · · + uN i ψ(xN , s))
N N N
N
L (N − 1)|upi upl | + yiT |upi ujl |yl ≤ λmax (AT + A)V + 2 N i=2
l=2 p=1
j=1,j=p
N N
N N
L (N − 1)|upi upl | + |upi ujl | 2yiT yl = λmax (AT + A)V + N p=1 i=2 l=2
j=1,j=p
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1543
N N N N
L (N − 1)|upi upl | + ≤ λmax (AT + A)V + |upi ujl | (yiTyi + ylTyl ) N p=1 i=2
l=2
j=1,j=p
N
N N N
L (N − 1)|upi upl | + |upi ujl | yiTyi = λmax (AT + A)V + N
i=2 l=2
p=1
j=1,j=p
N N N N
L + |upi ujl | ylTyl (N − 1)|upi upl | + N p=1
i=2 l=2
j=1,j=p
N N
N N
L (N − 1)|upi upl | + = λmax (AT + A)V + |upi ujl | yiTyi N p=1 i=2 l=2
j=1,j=p
N N N N
L + |upi ujl | ylTyl (N − 1)|upi upl | + N l=2 i=2
p=1
j=1,j=p
N N
N N
L (N − 1)|upi upl | + = λmax (AT + A)V + |upi ujl | yiTyi N i=2 l=2
p=1
j=1,j=p
N N N N
L + |upl uji | yiTyi (N − 1)|upi upl | + N p=1 i=2 l=2
j=1,j=p
N N N
N
L 2(N − 1)|upi upl | + = λmax (AT + A)V + (|upi ujl | + |upl uji |) yiTyi N
i=2
l=2 p=1
≤ λmax (AT + A)V +
L max N 2≤i≤N
= ηV (y2 (t), . . . , yN (t)),
N N
j=1,j=p
(2(N − 1)|upi upl | +
l=2 p=1
t ∈ (tk−1 , tk ],
t ∈ (tk−1 , tk ],
k = 1, 2, . . . . (9)
On the other hand, it follows from system (7) that + V (y2 (t+ k ), . . . , yN (tk ))
=
N
i=2
+ T y(t+ k )i y(tk )i
N
= ((I + λi Bk )y(tk )i )T(I + λi Bk )y(tk )i i=2
(|upi ujl | + |upl uji |) V
k = 1, 2, . . . ,
=
+ ≤ V (y2 (t+ k−1 ), . . . , yN (tk−1 )) exp(η(t − tk−1 )),
j=1,j=p
which implies that V (y2 (t), . . . , yN (t))
N
(8) N
y(tk )Ti (I + λi Bk )T (I + λi Bk )y(tk )i
i=2
≤ βk V (y2 (tk ), . . . , yN (tk ))
(10)
The following results come from (9) and (10). For t ∈ (t0 , t1 ], V (y2 (t), . . . , yN (t)) + ≤ V (y2 (t+ 0 ), . . . , yN (t0 )) exp(η(t − t0 )),
which leads to V (y2 (t1 ), . . . , yN (t1 )) + ≤ V (y2 (t+ 0 ), . . . , yN (t0 )) exp(η(t1 − t0 )),
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and
× β1 β2 · · · βk exp(η(tk+1 − t0 ))
+ V (y2 (t+ 1 ), . . . , yN (t1 ))
+ ≤ V (y2 (t+ 0 ), . . . , yN (t0 ))β1 exp(η(t2 − t1 ))
≤ β1 V (y2 (t1 ), . . . , yN (t1 )) + ≤ β1 V (y2 (t+ 0 ), . . . , yN (t0 )) exp(η(t1 − t0 )).
In general, for t ∈ (tk , tk+1 ], (k = 0, 1, 2, . . .) + ≤ V (y2 (t+ 0 ), . . . , yN (t0 ))β1 β2 · · · βk exp(η(t−t0 )).
Therefore, we have the following results.
V (y2 (t), . . . , yN (t)) + ≤ V (y2 (t+ 0 ), . . . , yN (t0 ))β1 β2 · · · βk exp(η(t − t0 )) + = V (y2 (t+ 0 ), . . . , yN (t0 ))β1 β2 · · · βk × exp(−α(t − t0 )) exp((η + α)(t − t0 )) + ≤ V (y2 (t+ 0 ), . . . , yN (t0 ))β1 β2 · · · βk
× exp(−α(tk − t0 )) exp((η + α)(t − t0 )) exp(−α(t1 − t0 ))β2
× exp(−α(t2 − t1 )) · · · βk × exp(−α(tk − tk−1 )) exp((η + α)(t − t0 ))
which implies that the conclusion (ii) of Theorem 1 holds. This completes the proof. synchronization of the impulsively coupled systems. The result has its physical significance. It implies that the impulsive interval and the impulsive energy are related to the eigenvalues and eigenvectors of coupling matrix G. On the other hand, they are also affected by the eigenvalue of A and the Lipschitzian constant L for ϕ(x), which are respectively determined by the linear and the nonlinear parts of the dynamics. To achieve synchronization, we have to increase the impulsive frequency if we decrease the impulsive energy. In practice, for convenience, the gain matrices Bk are always selected as a constant matrix and the impulsive intervals τk = tk − tk−1 (k = 1, 2, . . .) are set to be a positive constant. Thus we have the following corollary. Corollary 1. Assume τk = τ > 0 and matrices Bk = B (k = 1, 2, . . .)
namely, V (y2 (t), . . . , yN (t)) ≤V
+ (y2 (t+ 0 ), . . . , yN (t0 )) exp((η
+ α)(t − t0 )),
(i) If η < 0 (η is a constant) and there exists a constant α(0 ≤ α < −η), such that ln β − ατ ≤ 0,
t ≥ t0 . We can conclude that the trivial solution of system (7) is globally exponentially stable from the theories in [Bainov & Simeonov, 1989; Yang, 2001]. That is, yi → 0, i = 1, 2, . . . , N as t → ∞. Consequently, limt→∞ ei (t) = 0, i = 1, 2, . . . , N , i.e. limt→∞ xi (t) − xj (t) = 0, i, j = 1, 2, . . . , N. (ii) If η ≥ 0 and there exists a constant α ≥ 1, such that ln(αβk ) + η(tk+1 − tk ) ≤ 0, k = 1, 2, . . . . For t ∈ (tk , tk+1 ], we have V (y2 (t), . . . , yN (t)) ≤V
1 exp(η(t1 − t0 )), αk
Remark. Theorem 1 gives sufficient conditions for
(i) If η < 0 and there exists a constant α(0 ≤ α < −η) such that ln βk − α(tk − tk−1 ) ≤ 0, k = 1, 2, . . . , we obtain that for t ∈ (tk , tk+1 ],
=V
× exp(η(t1 − t0 )) + ≤ V (y2 (t+ 0 ), . . . , yN (t0 ))
V (y2 (t), . . . , yN (t))
+ (y2 (t+ 0 ), . . . , yN (t0 ))β1
× β2 exp(η(t3 − t2 )) · · · βk exp(η(tk+1 − tk ))
+ (y2 (t+ 0 ), . . . , yN (t0 ))
× β1 β2 · · · βk exp(η(t − t0 ))
then the trivial solution of system (4) is globally exponentially stable. (ii) If η ≥ 0 (η is a constant) and there exists a constant α ≥ 1, such that ln(αβ) + ητ ≤ 0, then α = 1 implies that the trivial solution of system (4) is stable and α > 1 implies that the trivial solution of system (4) is globally asymptotically stable. That is, impulsively coupled systems (2) synchronize in the sense of (5). Taking N = 3 and
−2 1 1 1 G = 1 −2 1 1 −2
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as an example, we can derive another sufficient condition for synchronization. What is more, we have wider impulsive interval. The coupled model is thus described by x˙ 1 = f (x1 ) = Ax1 + ϕ(x1 ) x1 = Bk (x2 + x3 − 2x1 ) x˙ = f (x ) = Ax + ϕ(x ) 2 2 2 2 (11) x2 = Bk (x1 + x3 − 2x2 ) x˙ 3 = f (x3 ) = Ax3 + ϕ(x3 ) x3 = Bk (x1 + x2 − 2x3 ) Let e1 = x1 −x2 , e2 = x2 −x3 , e3 = x3 −x1 , then we have the following dynamical error dynamics: e˙ 1 = Ae1 + ϕ(x1 ) − ϕ(x2 ) e1 = Bk (e2 + e3 − 2e1 ) = −3Bk e1 e˙ = Ae + ϕ(x ) − ϕ(x ) 2 2 2 3 (12) = B (e + e − 2e e 2 1 3 2 ) = −3Bk e2 k e˙ 3 = Ae3 + ϕ(x3 ) − ϕ(x1 ) e3 = Bk (e1 + e2 − 2e3 ) = −3Bk e3 And we have the following theorem: Let λmax (P ) denote the maximal eigenvalue of P, and βk = λmax [(I − 3BkT )(I − 3Bk )]. Theorem 2.
(i) If λmax (AT + A) + 2L = η < 0 (η is a constant) and there exists a constant α(0 ≤ α < −η), such that ln βk − α(tk − tk−1 ) ≤ 0,
k = 1, 2, . . .
then the trivial solution of system (12) is globally exponentially stable. That is, the impulsively coupled systems (11) synchronize in the sense of (5). (ii) If λmax (AT + A) + 2L = η ≥ 0 (η is a constant) and there exists a constant α ≥ 1, such that ln(αβk ) + η(tk+1 − tk ) ≤ 0,
k = 1, 2, . . .
then α = 1 implies that the trivial solution of system (12) is stable and α > 1 implies that the trivial solution of system (12) is globally asymptotically stable. That is, the impulsively coupled systems (11) synchronize in the sense of (5). Choose a Lyapunov function V (e1 , e2 , e3 ) = + eT2 e2 + eT3 e3 .
Proof.
eT1 e1
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For t ∈ (tk−1 , tk ], k = 1, 2, . . . , the time derivative of V along the trajectory of system (12) is V˙ (e1 (t), e2 (t), e3 (t)) = e˙T1 e1 + e˙T2 e2 + e˙ T3 e3 + eT1 e˙1 + eT2 e˙ 2 + eT3 e˙ 3 = (Ae1 + ϕ(x1 ) − ϕ(x2 ))Te1 + (Ae2 + ϕ(x2 ) − ϕ(x3 ))Te2 + (Ae3 + ϕ(x3 ) − ϕ(x1 ))Te3 + eT1 (Ae1 + ϕ(x1 ) − ϕ(x2 )) + eT2 (Ae2 + ϕ(x2 ) − ϕ(x3 )) + eT3 (Ae3 + ϕ(x3 ) − ϕ(x1 )) = eT1 (A + AT )e1 + eT2 (A + AT )e2 + eT3 (A +AT )e3 + 2(ϕ(x1 ) − ϕ(x2 ))Te1 + 2(ϕ(x2 ) − ϕ(x3 ))Te2 + 2(ϕ(x3 ) − ϕ(x1 ))Te3 ≤ (λmax (A + AT ) + 2L)V (e1 (t), e2 (t), e3 (t)), which implies that V (e1 (t), e2 (t), e3 (t)) + + ≤ V (e1 (t+ k−1 ), e2 (tk−1 ), e3 (tk−1 )) × exp(η(t − tk−1 )), t ∈ (tk−1 , tk ], k = 1, 2, . . . . On the other hand, we have V (e(t+ k )) + + T + + T + T = e(t+ k )1 e(tk )1 + e(tk )2 e(tk )2 + e(tk )3 e(tk )3
= ((I − 3Bk )e(tk )1 )T (I − 3Bk )e(tk )1 + ((I − 3Bk )e(tk )2 )T (I − 3Bk )e(tk )2 + ((I − 3Bk )e(tk )3 )T (I − 3Bk )e(tk )3 ≤ λmax ((I − 3Bk )T (I − 3Bk ))V (e(tk )) ≤ βk V (e(tk )). The remaining reasoning is similar to that of Theorem 1, so details are omitted. This completes the proof. The following results easily follow from Theorem 2. Corollary 2. Assume τk = τ > 0 and matrices Bk = B (k = 1, 2, . . .)
(i) If η < 0 and there exists a constant α(0 ≤ α < −η), such that ln β − ατ ≤ 0, then the trivial solution of system (12) is globally exponentially stable.
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ln(αβ) + ητ ≤ 0, then α = 1 implies that the trivial solution of system (12) is stable and α > 1 implies that the trivial solution of system (12) is globally asymptotically stable. That is, the impulsively coupled systems (11) synchronize in the sense of (5).
0.1 x11−x21
(ii) If η ≥ 0 (η is a constant) and there exists a constant α ≥ 1, such that
0 −0.1 −0.2
0
2
4
6
8
10
12
0
2
4
6
8
10
12
0
2
4
6 t/s
8
10
12
0.1 x12−x22
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0 −0.1
4. Numerical Simulations To demonstrate the above-derived theoretical results, some typical chaotic systems are used as the dynamical nodes of the impulsively coupled systems. Here, we choose the typical Chua’s circuit system and Lorenz system for illustrations. Example 1. Consider the original Chua’s circuit
system described by Chua et al. [1986]: x˙ = µ(y − x − g(x)), y˙ = x − y + z, z˙ = −νy,
(13)
where µ > 0, ν > 0, g(x) is a piecewise linear function given by g(x) = ax + (1/2)(b − a)(|x + 1| − |x − 1|), with b < a < 0. As is well known, Chua’s circuit is a typical chaotic system, which exhibits very rich complex dynamical behaviors [Chua et al., 1993; Chua et al., 1996], and has wide applications. The system has a double-scroll chaos attractor with µ = 10, ν = 14.97, a = −0.68, and b = −1.27. Rewrite system (13) into x˙ = Ax + ϕ(x),
x13−x23
0.2 0 −0.2
Fig. 1. Synchronization errors between the first and the second systems for the impulsively coupled network with three Chua’s systems.
B = diag(0.4, 0.4, 0.4), α = 1.1, β = 0.04. From Corollary 2, we obtain that the impulsive interval for synchronization should be less than 0.0778, while it should be less than 0.0055 from Theorem 1. In simulation, choose the impulsive interval τ = 0.015. Figures 1 and 2 display the synchronization errors of the impulsively coupled network with impulsive interval τ = 0.015 Next, we consider an impulsively coupled network with 20 nodes, the coupling structure is shown in Fig. 3. Here, we set Bk = diag(0.2, 0.2, 0.2), α = 1.001 and then βk = 0.8520. We derive from Theorem 1 that impulsive interval τ should be less
(14) 2
that is, the Lipschitzian constant L = µ|b| = 12.7. First, consider the impulsively coupled network with three Chua’s systems, which are coupled with bidirectional ring structure. Let
21
0 0
2
4
6
8
10
12
0
2
4
6
8
10
12
0
2
4
6 t/s
8
10
12
x22−x32
0.1 0 −0.1
33
2 0
23
λmax (A + AT ) = 14.7316 and ϕ(x) − ϕ(y ) ≤ µ|b|x − y ,
1
−1
−2
x −x
x = (x, y, z)T , ϕ(x) = (−µg(x), 0, 0)T , −µ µ 0 A = 1 −1 1 , 0 −ν 0 −2µ µ + 1 0 −2 1 − ν , A + AT = 1 + µ 0 1−ν 0
x −x
31
where
−4
Fig. 2. Synchronization errors between the second and the third systems for the impulsively coupled network with three Chua’s systems.
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25
20
r
15
10
5
0
0
10
20
30
40
50
t/s
Fig. 4. The synchronization error r of 20 Chua’s systems without coupling. 5
Fig. 3.
An impulsively coupled network with 20 nodes. 4
as the whole synchronization error. Then r → 0 as t → ∞ implies the synchronization of the impulsively coupled systems in sense of (5). Here, we choose τ = 0.0002. Figure 4 shows r of system (1) for 20 Chua’s systems without coupling, which indicates that there is no synchronization. Figure 5 displays r with 20 impulsively coupled Chua’s systems. Figure 6 shows the first variable of the 20 synchronized Chua’s systems. Figure 7 shows dynamics of the first node of the impulsively coupled systems after they reach synchronization.
3
2 r
than 0.00028315. Denote (x11 − x21 )2 + (x21 − x31 )2 + · · · + (x20,1 − x11 )2 + (x12 − x22 )2 + · · · r= + (x 2 2 20,2 − x12 ) + (x13 − x23 ) + · · · + (x20,3 − x13 )2
1
0
−1 −2
6
When a = 10, b = 8/3, c = 28, Lorenz system has a chaotic attractor. Then we have −10 10 0 0 28 −1 A= 8 0 0 − 3
0.1
0.15 t/s
0.2
0.25
0.3
2.5
5 2
4
i1
x (t),(i=1,...,20)
(15)
0.05
Fig. 5. The synchronization error r of 20 impulsively coupled Chua’s systems.
Example 2. A single Lorenz system is described by
x˙ = a(y − x), y˙ = cx − xz − y, z˙ = xy − bz.
0
3 1.5
0
0.01
0.02
0.03
2 1 0 −1 −2 −3
0
5
10
15
20 t/s
25
30
35
40
Fig. 6. Evolutions of the first variables of the 20 coupled systems; the inset figure is the transient.
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200
0
150
r
z
5
100
−5 1
0 −1
y
Fig. 7.
50 −4
−2
−3
−1 x
0
1
2
3
4
The dynamics of the first node after synchronization.
50
60 50 40 r
Consider the impulsively coupled network with three Lorenz systems and choose B = diag(0.4, 0.4, 0.4), α = 1.1, β = 0.04. From Corollary 2, the impulsive interval for synchronization should be less than 0.0293, and that from Theorem 1 should be less than 0.0153. Figure 8 shows the synchronized states with τ = 0.015.
i1
40
70
We can obtain from [Li et al., 2005] that the bound of system (15) is 39.2462. The nonlinear part of the system is ϕ(x, y, z) = (0, −xz, xy)T . It follows from Theorem 1 that L = 39.2462.
x (t)(i=1,2,3)
30
80
λmax (A + A ) = 28.051.
0 −20 0
0.5
1
1.5
2
0
0.5
1
1.5
2
20 0 −20 −40
30 20 10 0 −10 −20
0
0.02
0.04
0.06
0.08
0.1
t/s
Fig. 10. The entire error r of 20 impulsively coupled Lorenz systems.
20
−40
i2
20
Fig. 9. The entire error r of 20 Lorenz systems without coupling.
T
x (t)(i=1,2,3)
10
t/s
and
For the coupling structure shown by Fig. 3, we derive from Theorem 1 that the impulsive interval τ should be less than 0.000091997 for Bk = diag(0.2, 0.2, 0.2), α = 1.001 and βk = 0.8520. Figure 9 shows the entire error r of system (1) with 20 isolate Lorenz systems for τ = 0.00009. Figure 10 shows the entire error of 20 impulsively coupled Lorenz systems.
100 50
5. Conclusions
0
i3
x (t)(i=1,2,3)
0
−50
0
0.5
1 t/s
1.5
2
Fig. 8. Synchronization of the network with three impulsively coupled Lorenz systems.
Unlike other existing works, in this paper, we have discussed the synchronization of multi-systems, which are coupled by impulsive connections only at discrete instants. We have derived sufficient conditions for synchronizing coupled systems with a given
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coupling structure. We have also obtained sufficient conditions for synchronizing three systems which are impulsively coupled with the bidirectional ring structure. The typical Chua’s circuit and the Lorenz system are taken as illustrative examples to confirm the theoretical results. These results can be applied to other systems.
Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant No. 60574045, 70771084), the National Basic Research 973 Program of China (Grant No. 2007CB310805), and China Tianyuan Youth Funding of Mathematics (Grant No. 201161509). Furthermore, the authors thank Mr. Chen Liang for the valuable suggestions.
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