asymptotic synchronization in lattices of coupled ... - CiteSeerX
International Journal of Bifurcation and Chaos, Vol. 10, No. 12 (2000) 2717–2728 c World Scientific Publishing Company
ASYMPTOTIC SYNCHRONIZATION IN LATTICES OF COUPLED NONIDENTICAL LORENZ EQUATIONS CHUANG-HSIUNG CHIU∗ Center for General Education, Southern Taiwan University of Technology, Tainan, 710, Taiwan, R.O.C. WEN-WEI LIN† Department of Mathematics, National Tsing Hua University, Hsinchu, 30043, Taiwan, R.O.C. CHEN-CHANG PENG‡ Department of Mathematics, National Tsing Hua University, Hsinchu, 30043, Taiwan, R.O.C. Received November 5, 1999; Revised February 8, 2000 In this paper we study coupled nonidentical Lorenz equations with three different boundary conditions. Coupling rules and boundary conditions play essential roles in the qualitative analysis of solutions of coupled systems. By using Lyapunov stability theory, a sufficient condition is obtained for the global stability of trivial equilibrium of coupled system with Dirichlet condition. Then we restrict our attention on the dynamics of coupled nonidentical Lorenz equations with Neumann/periodic boundary condition and prove that the asymptotic synchronization occurs provided the coupling strengths are sufficiently large. That is, the difference between any two components of solution is bounded by the quantity O(ε/ max{c1 , c2 , c3 }) as t → ∞, where ε is the maximal deviation of parameters of nonidentical Lorenz equations, and c1 , c2 and c3 are the specified coupling strengths.
1. Introduction Synchronization of coupled chaotic systems has received considerable attention recently. Relevant examples of synchronization are coupled chaotic oscillators [Afraimovich et al., 1997; Afraimovich & Lin, 1998; Chiu et al., 1998; Ermentrout, 1985; Fujisaka & Yamada, 1983; Hale, 1997; Heagy et al., 1994; Mirollo & Strogatz, 1990] and coupled chaotic circuits [Chua et al., 1993; Goldsztein & Strogatz, 1995]. A typical study of synchronization is the coupled identical chaotic systems. This sort of synchronization was studied by Heagy et al. [1994]. In
this considering systems, synchronization appears as a similar behavior of the corresponding variables of the uncoupled subsystems as they evolve in time. In practice, the systems exhibiting identical behaviors are difficult to convert into practical applications because we can never construct two absolutely identical real physical systems. Therefore, synchronization of coupled identical systems has limitation in applications. For this reason, the focus of this paper is on the asymptotic synchronization of coupled nonidentical chaotic systems, where the parameters in the uncoupled subsystems are not required to be equivalent but still close enough. For a
∗
Email: [email protected] Email: [email protected] ‡ Email: [email protected] †
2717
2718 C.-H. Chiu et al.
rigorous description of the asymptotic synchronization, we refer to [Afraimovich et al., 1997; Hale, 1997]. In the case of coupled nonidentical chaotic systems, the diagonal of the phase space is not an invariant set but become a rather complicated geometrical object that the attractor approach the diagonal as the asymptotic synchronization occurs. By the notion of asymptotic synchronization introduced in [Afraimovich et al., 1997], this paper is devoted to analyze asymptotically synchronized behavior of an n × n squared lattice with coupled nonidentical Lorenz equations. For convenience, we designate a point (i1 , i2 ) on a 2-D lattice in a 1-D order by (i1 , i2 ) ↔ ν = i1 + n(i2 − 1), 1 ≤ i1 , i2 ≤ n . 2
Any vector in Rn of the form u = [u(1,1) , . . . , u(n,1) |, . . . , |u(1,n) , . . . , u(n,n) ]T
The operator ∆ ≡ ∆D with Dirichlet boundary condition is a matrix of the form ∆D = I ⊗ ∆d + ∆d ⊗ I with ∆d
−2 1
=
1 −2 .. .
1 ..
.
..
.
..
.
..
.
, 1
−2
1
where the symbol ⊗ denotes the Kronecker product [Alexander, 1981]. It is well-known that the operator ∆D is self-adjoint with mutually orthogonal eigenvectors and λM = max{λν } ≡ −8 sin2 (π/ 2(n + 1)) < 0, for all λν ∈ σ (∆D ). Here σ (∆D ) denotes the spectrum of the matrix ∆D . (2) Neumann boundary conditions: u(0,i2 ) = u(1,i2 ) , u(n+1,i2 ) = u(n,i2 ) , u(i1 ,0) = u(i1 ,1) , u(i1 ,n+1) = u(i1 ,n) , for 1 ≤ i1 , i2 ≤ n .
can be identified by
The operator ∆ ≡ ∆N with Neumann boundary condition is a matrix of the form
u = [uν |ν = 1, 2, . . . , n ] with uν ≡ u(i1 , i2 ) . 2
The dynamics at each node (i1 , i2 ) ↔ ν ≡ i1 + n(i2 − 1) in the squared lattice is represented by x˙ ν = σν (yν − xν ) + c1 (∆x)ν ,
y˙ν = γν xν − yν − xν zν + c2 (∆y)ν ,
z˙ = −b z + x y + c (∆z) , ν ν ν ν ν 3 ν
(1)
where the parameters σν , γν and bν are all positive, and the coupling strength c1 , c2 and c3 are non-negative. The vectors x, y and z here are in 2 Rn with components xν , yν and zν , respectively. The operator ∆ is the discretized Laplacian operator given by (∆u)ν = u(i1 +1,i2 ) + u(i1 ,i2 +1) + u(i1 −1,i2 ) + u(i1 ,i2 −1) − 4u(i1 ,i2 ) , for ν ↔ (i1 , i2 ), 1 ≤ i1 , i2 ≤ n. This type of coupling corresponds to the symmetric nearest neighbor coupling on a 2-D lattice. On the boundary of lattice we impose three various boundary conditions. (1) Dirichlet boundary conditions: u(0,i2 ) = u(i1 ,0) = u(n+1,i2 ) = u(i1 ,n+1) = 0, for 1 ≤ i1 , i2 ≤ n .
∆N = I ⊗ ∆n + ∆n ⊗ I with ∆n
−1 1
=
1 −2 .. .
1 ..
.
..
.
..
.
. 1
−2 1 −1
The matrix ∆N has a simple zero eigenvalue associated with eigenvector e ≡ [1, 1, . . . , 1]T ∈ 2 Rn , and λM = max{λν 6= 0 | λν ∈ σ(∆N )} ≡ −8 sin2 (π/2n). (3) Periodic boundary conditions: u(0,i2 ) = u(n,i2 ) , u(n+1,i2 ) = u(1,i2 ) , u(i1 ,0) = u(i1 ,n) , u(i1 ,n+1) = u(i1 ,1) , for 1 ≤ i1 , i2 ≤ n . The operator ∆ ≡ ∆P with periodic boundary condition is a matrix of the form ∆P = I ⊗ ∆p + ∆p ⊗ I with ∆p
−2 1 1 −2 .. . =
1
1 1 .. . .. .
..
.
..
.
1
. 1
−2
Asymptotic Synchronization in Lattices of Coupled Nonidentical Lorenz Equations 2719
In this case, the matrix ∆P has a simple zero eigenvalue associated with eigenvector e = 2 [1, 1, . . . , 1]T ∈ Rn , and λM = max{λν 6= 0|λν ∈ σ(∆P )} ≡ −8 sin2 πn . From classical definition [Pecora & Carroll, 1990] the coupled system of (1) is said to be synchronized if the difference between any two components of solutions (x(t), y(t), z(t)) approaches zero as t → ∞. Of course, if the system is synchronized, the attractor of (1) must lie on the spatially homogeneous diagonal and this can happen only if all subsystems are identical, i.e. the parameters σν , γν and bν in (1) are all equal. If each subsystem in (1) is nonidentical, then it is impossible to obtain synchronization in the above sense. In this case, we introduce a parameter ε which characterizes the variation of uncoupled subsystems and assume that |σν − σµ | ≤ ε , |γν − γµ | ≤ ε , |bν − bµ | ≤ ε , (2) for all 1 ≤ µ, ν ≤ n2 . The coupled system (1) may possibly be asymptotically synchronized [Hale, 1997]. That is, the difference between any two components of solutions (x(t), y(t), z(t)) is bounded by the quantity O(ε/ max{c1 , c2 , c3 }) as t → ∞. Consequently, the attractor of (1) approaches to the spatially homogeneous diagonal as the coupling strengths become infinity and the variation ε tends to zero. This paper is devoted to analyze the asymptotic synchronization behavior of coupled nonidentical Lorenz equations of (1) with Neumann and periodic boundary conditions, respectively. We first show that the coupled system (1) has pointwise dissipativeness property [Afraimovich et al., 1997; Hale, 1988]. A system is said to be pointwise dissipative if there is a bounded set B so that for any solution U (t, q0 ) through q0 at t = t0 , there is a time t1 ≡ t1 (q0 , B) such that U (t, q0 ) ∈ B, for t ≥ t1 . For the coupled system (1) we can construct a specific Lyapunov function to establish the pointwise dissipativeness property, which is one of the reasons that we restrict ourselves to the study of coupled Lorenz equations. Another reason is that the occurrence of chaotic behavior is over a wide range of parameters [Sparrow, 1982]. Our main theorem asserts that the coupled system (1) with Neumann or periodic boundary condition is asymptotic synchronized under the assumption (2) whenever the coupling strengths are sufficiently large. We prove that the differences of any two
components of (x(t), y(t), z(t)) satisfy lim sup {|xµ (t) − xν (t)|, |yµ (t) − yν (t)|, t→∞
|zµ (t) − zν (t)|} ≤ O
ε max{c1 , c2 , c3 }
, (3)
provided c1 , c2 and c3 are sufficiently large. Note that the full coupling in x-, y- and z-states in (1) (i.e. c1 , c2 , c3 > 0) is essential to establish the asymptotically synchronized regimes for nonidentical subsystems. In our numerical experience (Sec. 5 below), we observe that if any one of the coupling strengths, say c3 , is chosen to be zero, then the asymptotically synchronized behavior as in (3): kzµ (t) − zν (t)k ≤ O(ε/ max{c1 , c2 }) cannot be expected, for 1 ≤ µ, ν ≤ n2 . According to this observation, in this paper we only consider the full state coupling as in (1). The rest of this paper is organized as follows. Section 2 begins with results on the stability for the coupled system (1) with Dirichlet boundary condition. A sufficient condition is obtained for the global stability of trivial equilibrium. In Sec. 3 we study the pointwise dissipativeness of the coupled system (1) with Neumann/periodic boundary condition. In Sec. 4 we prove the asymptotic synchronization for the coupled nonidentical subsystems as in (1) with Neumann/periodic boundary condition. In Sec. 5 we show some numerical results to illustrate the asymptotically synchronized behavior of (1). A concluding remark is contained in the last section.
2. Globally Asymptotic Behavior with Dirichlet Condition The coupled system (1) with Dirichlet boundary condition can be regarded as 1-D vector equations with the individual dynamical system on the νth position x˙ ν = σν (yν − xν ) + c1 (∆D x)ν ,
y˙ = γ x − y − x z + c (∆ y)ν ,
ν ν ν ν ν ν 2 D z˙ = −b z + x y + c (∆ z) . ν ν ν ν ν 3 D ν
(4)
In this section we shall prove globally asymptotic behavior of the coupled system (4) with Dirichlet boundary condition. Define
2720 C.-H. Chiu et al.
λm = max{λν |λν ∈ σ(∆D )},
to λν ∈ σ(∆D ). Now, let
ν
λ0 = min{λν |λν ∈ σ(∆D )},
2
ν
|x| =
γm = max{γν }, σm = max{σν }, σ0 = min{σν }. ν
ν
ν
Let (x(t), y(t), z(t)) be the solution of the coupled system (4) with Dirichlet boundary condition. If Theorem 2.1
λ2m
(σm + γm )2 − 4 c1 λ2m λ0
× c2
!
!
!
V (x, y , z) =
2
ν
+
yν2
+
zν2
n X ν=1
X
n X
λν Xν2 ,
(7)
2
|yν |(∆|y|)ν =
ν
λν Yν2 ,
ν=1
X
2
n X
|xν yν | =
ν
Xν Yν .
ν=1
Xh
−σν x2ν +(σν + γν )xν yν −yν2
From (6) and (7) and after some calculation, we have
ν
i
+ c1 xν (∆D x)ν +c2 yν (∆D y)ν +
Xh
−bν zν2 +c3 zν (∆D z)ν
V˙ ≤
i
Xh
−σ0 Xν2 + (σm + γm )Xν Yν − Yν2
ν
ν
Xh
−σ0 |xν |2 +(σm +γm )|xν yν |−|yν |2
˜ ν |(∆D |x|)ν +c2 λ|y ˜ ν |(∆D |y|)ν + c1 λ|x +
+ c1
ν
Xh
i
i
−bν zν2 +c3 zν (∆D z)ν ,
+
(6)
˜ = λm /λ0 , |x| = [|xν ||ν = 1, 2, . . . , where λ and 2 |y| = [|yν ||ν = 1, 2, . . . , n ]. The last inequality is derived by xν (∆x)ν ≤ λm
X
x2ν
˜ λ0 ≤λ
ν
˜ ≤λ
X
X
!
|xν |
i
ν
=
c1
ν
+ c2 +
X
λm λν − σ0 Xν2 + (σm + γm )Xν Yν λ0
λm λν − 1 Yν2 λ0
[−bν zν2 + c3 zν (∆D z)ν ] .
ν
2
ν
From the assumption (5)
!
|xν |(∆|x|)ν
λm λm λν Xν2 + c2 λν Yν2 λ0 λ0
−bν zν2 + c3 zν (∆D z)ν
n2 ]
ν
Xh
X
ν
X
ν=1
|xν |(∆|x|)ν =
where 1 ≤ ν ≤ The derivative of V along the trajectory of the coupled system (4) is
≤
Yν2 ,
2
n2 .
V˙ (x, y, z) =
|yν | =
ν
,
Xν2 ,
2
n X
2
ν
(5)
Construct the Lyapunov function x2ν
|xν | =
ν=1
X
X
X1
2
n X
2
ν
< 0,
Yν hν .
ν=1
X
!
−1
Xν h , |y| =
n X
Then it is easily shown that
− σ0
λ0
2
ν
ν=1
then limt→∞ (x(t), y(t), z(t)) = (0, 0, 0). Proof.
n X
.
ν
Let {hν |ν = 1, 2, . . . , n2 } be the mutually orthonormal eigenvectors, respectively, corresponding
(σm + γm ) − 4 c1 2
× c2
λm λm − σ 0 λ0
λm λm − 1 < 0 λ0
Asymptotic Synchronization in Lattices of Coupled Nonidentical Lorenz Equations 2721
follows that V˙ < 0. This completes the proof of Theorem 2.1.
where
As in (5) of Theorem 2.1, if c2 = c3 = 0 and c1 satisfies
Corollary 2.2.
(σm + γm ) < −4 c1 2
λm λm − σ 0 λ0
The derivative of V along the trajectory of the coupled system (1) becomes
,
V˙ (x, y, z)
then limt→∞ (x(t), y(t), z(t)) = (0, 0, 0).
=
As in (5) of Theorem 2.1 if c1 = c3 = 0 and c2 satisfies
Corollary 2.3.
(σm + γm ) < −4σ0 c2 2
1 X 1 X σ and γ = γν . ν n2 ν n2 ν
σ=
[−σν x2ν + (σν − σ + γν − γ)xν yν − yν2 ]
ν
λm λm − 1 λ0
X
+
,
X
[−bν zν2 + (σ + γ)bν zν ]
ν
X
then limt→∞ (x(t), y(t), z(t)) = (0, 0, 0).
+
Since λ0 is negative, the condition (5) is easy to satisfy, provided the coupling strengths c1 and c2 in (4) are chosen sufficiently large. Note that the globally asymptotic behavior for (4) holds depending only on c1 and c2 without c3 .
+ c3 zν (∆z)ν ] − c3 (σ + γ)
[c1 xν (∆x)ν + c2 yν (∆y)ν
ν
X
(∆z)ν .
(9)
ν
From definition of ∆ ≡ ∆N or ∆P in Sec. 1 it is easily shown that 2
n X
3. Pointwise Dissipativeness In this section we shall prove the pointwise dissipativeness property [Afraimovich et al., 1997; Hale, 1988] for the coupled system (1) with Neumann/periodic boundary condition. Let U (t, (x0 , y0 , z0 )) be a solution of system (1) through (x0 , y0 , z0 ) at t = t0 . As in Sec. 1 the coupled system (1) is said to be pointwise dissipa2 tive if there is a bounded set B ⊆ R3n such 2 that for any point (x0 , y0 , z0 ) ⊆ R3n there is a time t1 ≡ t1 (c1 , c2 , c3 , x0 , y0 , z0 , B) so that U (t, (x0 , y0 , z0 )) ∈ B for t ≥ t1 . Recall that a parameter ε which characterizes the variation of the uncoupled subsystems in (1) satisfying |σµ − σν | ≤ ε, |γµ − γν | ≤ ε, |bµ − bν | ≤ ε ,
(8)
for 1 ≤ µ, ν ≤ n2 . If ε2 − σ0 < 0, where σ0 = minν {σν } , then the coupled system (1) with Neumann/periodic boundary condition is pointwise dissipative.
Proof.
Construct the following Lyapunov function
V (x, y, z) =
" X x2 ν ν
From (10) the last term in (9) becomes zero. From the variation inequalities (8) we have −θx2ν + (σν − σ + γν − γ)xν yν −
ε2 2 y ≤ 0, θ ν
where θ is chosen by assumption satisfying ε2 < θ < σ0 . Since X
[c1 xν (∆x)ν + c2 yν (∆y)ν + c3 zν (∆z)ν ] ≤ 0 ,
ν
from (9) we then have X
"
(−σν +
θ)x2ν
ν
σ+γ × zν − 2
ε2 + −1 + θ
2
+ bν
!
σ+γ 2
yν2 − bν 2 #
Let M be a positive constant satisfying
#
y2 z2 + ν + ν − (σ + γ)zν , 2 2 2
(10)
ν=1
V˙ ≤
Theorem 3.1.
(∆z)ν = eT ∆z = 0 .
M>
X ν
bν
σ+γ 2
2
.
2722 C.-H. Chiu et al.
are positive constants M ∗ and c∗ such that for any (c1 , c2 , c3 ) ∈ W (c∗ ) it holds
and let D be the set defined by (
D=
(x, y, z) ∈ R
3n2
X h (σν − θ)x2ν ν
+ 1−
ε2 θ
!
yν2 + bν zν −
σ+γ 2
2 #
.
2
VK = {(x, y , z) ∈ R3n |V (x, y, z) ≤ K} .
(x, y, z) ∈ R
X h (σν − θ)x2ν
ε2 + 1− θ
yν2
+ bν
σ+γ zν − 2
lim sup |zµ (t) − zν (t)| ≤
M ∗ε , c3
for 1 ≤ µ, ν ≤ n2 .
σ=
ν
!
M ∗ε , c2
t→∞
Proof.
Choose K0 sufficiently large, such that 3n2
lim sup |yµ (t) − yν (t)| ≤ t→∞
This shows that V˙ < 0 for all (x, y, z) ∈ D. Given any K > 0, define
(
M ∗ε , c1
t→∞
)
>M
lim sup |xµ (t) − xν (t)| ≤
2 #
Introduce the constants 1 X 1 X 1 X σν , γ = 2 γν and b = 2 bν . 2 n ν n ν n
Set δν1 = (σν − σ)yν − (σν − σ)xν , δν2 = (γν − γ)xν , δν3 = −(bν − b)zν ,
≤ M } ⊆ VK 0 . 2
Then V˙ < 0, for all (x, y, z) ∈ R3n \VK0 . Therefore, any solution of the coupled system (1) starting with (x0 , y0 , z0 ) will eventually enter and stay in the bounded region VK0 . This completes the proof of Theorem 3.1. Note that the time t1 ≡ t1 (c1 , c2 , c3 , x0 , y0 , z0 , Vk0 ) at which the trajectory (x(t), y(t), z(t)), t ≥ t1 enters and stays in Vk0 , is dependent pointwisely on the initial point (x0 , y0 , z0 ). An uniformly bounded dissipativeness cannot be obtained here.
4. Asymptotic Synchronization In this section, we shall prove the asymptotic synchronization of the coupled system (1) with Neumann/periodic boundary condition. Let ε be the variation constant given in (8). Define the unbounded region for coupling strengths in parameter space by
for 1 ≤ ν ≤ n2 . Then the equations in (1) can be reformulated in the forms x˙ ν = σ(yν − xν ) + c1 (∆x)ν + δν1 ,
y˙ = γx − y − x z + c (∆y) + δν2 ,
ν ν ν ν ν 2 ν z˙ = −bz + x y + c (∆z) + δ , ν ν ν ν 3 ν ν3
(12)
where ∆ ≡ ∆N or ∆P . Rewrite the system (12) into the vector form x˙ = σ(y − x) + c1 ∆x + δ1 ,
y˙ = γx − y − f (x, z) + c ∆y + δ2 ,
2 z˙ = −bz + g(x, y) + c ∆z + δ , 3 3
(13)
where (f (x, z))ν = xν zν , (g(x, y))ν = xν yν and δi = [δνi |ν = 1, 2, . . . , n2 ] for i = 1, 2, 3. For Neumann/periodic boundary condition, the matrix ∆ as in (13) has a simple zero eigenvalue associated 2 with the eigenvector e = [1, 1, . . . , 1]T ∈ Rn ×1 . Let
1 −1 1
C=
W (c∗ ) = {(c1 , c2 , c3 )|c1 ≥ c∗ , c2 ≥ c∗ , c3 ≥ c∗ } . Let (x(t), y(t), z(t)) be a solution of the coupled system (1) with Neumann/periodic 2 boundary condition. Denote x(t) = [xν (t)]nν=1 , 2 2 y(t) = [yν (t)]nν=1 and z(t) = [zν (t)]nν=1 . Then there
(11)
−1 ..
.
1
Theorem 4.1.
and
C E= 1···1
−1
(n2 −1)×n2
n2 ×n2
.
Asymptotic Synchronization in Lattices of Coupled Nonidentical Lorenz Equations 2723
It is easy to check "
E∆E
−1
˜ ∆ = 0
0 0
c3 are all non-negative. Thus, there is a positive number M0 such that
#
lim kx(t)k ≤ M0 , lim ky(t)k ≤ M0 ,
,
t→∞
lim kz(t)k ≤ M0 , lim kξ(t)k ≤ M0 ,
t→∞
˜ is diagonalizable with λ0 ≡ max{λν |λν ∈ where ∆ ˜ σ(∆)} < 0. Multiplying the matrix E to the system (13) from the left, we get
We now introduce transformations with new variables ξ, ξs , η, ηs , ζ and ζs such that Ex =
!
,
η ηs
Ey =
!
,
ζ ζs
Ez =
lim kη(t)k ≤ M0 , lim kζ(t)k ≤ M0 .
t→∞
ξ(t) = et(−σ+c1 ∆) ξ(τ ) Z τ
ξs =
2
xν , ηs =
ν=1
n X
˜ e(t−s)(−σ+c1 ∆) (ση(s) + δ˜1 )ds .
!
˜
ket(−σ+c1 ∆) k ≤ Ke(−σ+c1 λ0 )t . .
From (17) we have kξ(t)k ≤ Kkξ(τ )ke(−σ+c1 λ0 )t
2
yν and ζs =
ν=1
n X
+ zν .
ν=1
The variables ξ, η and ζ are vectors in R(n satisfying
Consequently, 2 −1)×1
t→∞
for ν = 1, 2, . . . , n2 − 1. Therefore, by Mean Value Theorem, we have ˙ ˜ ˜ ξ = σ(η − ξ) + c1 ∆ξ + δ1 ,
˜ + δ˜ , η˙ = γξ − η − Γ ζ − Γ ξ + c ∆η
lim kξ(t)k ≤ ≤
(ξ)ν = ξν = xν − xν+1 , (η)ν = ην = yν − yν+1 , (ζ)ν = ζν = zν − zν+1 ,
1 3 2 2 ζ˙ = −bζ + Γ η + Γ ξ + c ∆ζ ˜ ˜ + δ3 , 1 2 3
(17)
˜ < 0, there exists a Since λ0 ≡ max{λν |λν ∈ σ(∆)} positive number K such that
(15)
2
t
+
where ξs , ηs and ζs are scalars satisfying n X
t→∞
˜
+c2 E∆E −1 (Ey) + Eδ2 , E z˙ = −bEz + Eg(x, y) + c3 E∆E −1 (Ez) + Eδ3 . (14)
ξ ξs
t→∞
Applying the variation of constant formula to the first equation of (16) we get
−1 E x˙ = σE(y − x) + c1 E∆E (Ex) + Eδ1 , E y˙ = γEx − Ey − Ef (x, z)
t→∞
K σ − c1 λ0 K −c1 λ0
where Γ1 = diag{x1 , x2 , . . . , xn2 −1 }, Γ2 = diag{y2 , y3 , . . . , yn2 }, Γ3 = diag{z2 , z3 , . . . , zn2 } and δ˜i = [δ1,i − δ2,i , δ2,i − δ3,i , . . . , δn2 −1,i − δn2 ,i ] for i = 1, 2, 3. T
From Theorem 3.1, it follows that the coupled system (1) with Neumann/periodic boundary condition is pointwise dissipative whenever c1 , c2 and
(18)
lim kση(t) + δ˜1 k
t→∞
lim kση(t)k + kδ˜1 k .
t→∞
(19) Similarly, the vectors η(t) and ζ(t), respectively, in the second and third equations of system (16), can be estimated by
(16)
Kkση(t) + δ˜1 k . σ − c1 λ0
lim kη(t)k ≤
t→∞
K −c2 λ0
lim (kγξk
t→∞
+ kΓ1 ζk + kΓ3 ξk + kδ˜2 k) ,
lim kζ(t)k ≤
t→∞
K −c3 λ0
lim (kΓ1 ηk
t→∞
+ kΓ2 ξk + kδ˜3 k) .
(20)
Let M be a positive number satisfying max{2n2 M0 , σ, γ, b} ≤ M . Then from (11) it holds lim kδ˜i k ≤ M ε, i = 1, 2, 3 .
t→∞
(21)
2724 C.-H. Chiu et al.
Choose c∗ > 0 satisfying −c∗ λ0 ≥ 3KM l with some integer l > 1. Then, for c1 > c∗ and c3 > c∗ , from (19) and (20) follows that lim kξ(t)k
t→∞
1 M 2l−3
+
ASYMPTOTIC SYNCHRONIZATION IN LATTICES OF COUPLED NONIDENTICAL LORENZ EQUATIONS CHUANG-HSIUNG CHIU∗ Center for General Education, Southern Taiwan University of Technology, Tainan, 710, Taiwan, R.O.C. WEN-WEI LIN† Department of Mathematics, National Tsing Hua University, Hsinchu, 30043, Taiwan, R.O.C. CHEN-CHANG PENG‡ Department of Mathematics, National Tsing Hua University, Hsinchu, 30043, Taiwan, R.O.C. Received November 5, 1999; Revised February 8, 2000 In this paper we study coupled nonidentical Lorenz equations with three different boundary conditions. Coupling rules and boundary conditions play essential roles in the qualitative analysis of solutions of coupled systems. By using Lyapunov stability theory, a sufficient condition is obtained for the global stability of trivial equilibrium of coupled system with Dirichlet condition. Then we restrict our attention on the dynamics of coupled nonidentical Lorenz equations with Neumann/periodic boundary condition and prove that the asymptotic synchronization occurs provided the coupling strengths are sufficiently large. That is, the difference between any two components of solution is bounded by the quantity O(ε/ max{c1 , c2 , c3 }) as t → ∞, where ε is the maximal deviation of parameters of nonidentical Lorenz equations, and c1 , c2 and c3 are the specified coupling strengths.
1. Introduction Synchronization of coupled chaotic systems has received considerable attention recently. Relevant examples of synchronization are coupled chaotic oscillators [Afraimovich et al., 1997; Afraimovich & Lin, 1998; Chiu et al., 1998; Ermentrout, 1985; Fujisaka & Yamada, 1983; Hale, 1997; Heagy et al., 1994; Mirollo & Strogatz, 1990] and coupled chaotic circuits [Chua et al., 1993; Goldsztein & Strogatz, 1995]. A typical study of synchronization is the coupled identical chaotic systems. This sort of synchronization was studied by Heagy et al. [1994]. In
this considering systems, synchronization appears as a similar behavior of the corresponding variables of the uncoupled subsystems as they evolve in time. In practice, the systems exhibiting identical behaviors are difficult to convert into practical applications because we can never construct two absolutely identical real physical systems. Therefore, synchronization of coupled identical systems has limitation in applications. For this reason, the focus of this paper is on the asymptotic synchronization of coupled nonidentical chaotic systems, where the parameters in the uncoupled subsystems are not required to be equivalent but still close enough. For a
∗
Email: [email protected] Email: [email protected] ‡ Email: [email protected] †
2717
2718 C.-H. Chiu et al.
rigorous description of the asymptotic synchronization, we refer to [Afraimovich et al., 1997; Hale, 1997]. In the case of coupled nonidentical chaotic systems, the diagonal of the phase space is not an invariant set but become a rather complicated geometrical object that the attractor approach the diagonal as the asymptotic synchronization occurs. By the notion of asymptotic synchronization introduced in [Afraimovich et al., 1997], this paper is devoted to analyze asymptotically synchronized behavior of an n × n squared lattice with coupled nonidentical Lorenz equations. For convenience, we designate a point (i1 , i2 ) on a 2-D lattice in a 1-D order by (i1 , i2 ) ↔ ν = i1 + n(i2 − 1), 1 ≤ i1 , i2 ≤ n . 2
Any vector in Rn of the form u = [u(1,1) , . . . , u(n,1) |, . . . , |u(1,n) , . . . , u(n,n) ]T
The operator ∆ ≡ ∆D with Dirichlet boundary condition is a matrix of the form ∆D = I ⊗ ∆d + ∆d ⊗ I with ∆d
−2 1
=
1 −2 .. .
1 ..
.
..
.
..
.
..
.
, 1
−2
1
where the symbol ⊗ denotes the Kronecker product [Alexander, 1981]. It is well-known that the operator ∆D is self-adjoint with mutually orthogonal eigenvectors and λM = max{λν } ≡ −8 sin2 (π/ 2(n + 1)) < 0, for all λν ∈ σ (∆D ). Here σ (∆D ) denotes the spectrum of the matrix ∆D . (2) Neumann boundary conditions: u(0,i2 ) = u(1,i2 ) , u(n+1,i2 ) = u(n,i2 ) , u(i1 ,0) = u(i1 ,1) , u(i1 ,n+1) = u(i1 ,n) , for 1 ≤ i1 , i2 ≤ n .
can be identified by
The operator ∆ ≡ ∆N with Neumann boundary condition is a matrix of the form
u = [uν |ν = 1, 2, . . . , n ] with uν ≡ u(i1 , i2 ) . 2
The dynamics at each node (i1 , i2 ) ↔ ν ≡ i1 + n(i2 − 1) in the squared lattice is represented by x˙ ν = σν (yν − xν ) + c1 (∆x)ν ,
y˙ν = γν xν − yν − xν zν + c2 (∆y)ν ,
z˙ = −b z + x y + c (∆z) , ν ν ν ν ν 3 ν
(1)
where the parameters σν , γν and bν are all positive, and the coupling strength c1 , c2 and c3 are non-negative. The vectors x, y and z here are in 2 Rn with components xν , yν and zν , respectively. The operator ∆ is the discretized Laplacian operator given by (∆u)ν = u(i1 +1,i2 ) + u(i1 ,i2 +1) + u(i1 −1,i2 ) + u(i1 ,i2 −1) − 4u(i1 ,i2 ) , for ν ↔ (i1 , i2 ), 1 ≤ i1 , i2 ≤ n. This type of coupling corresponds to the symmetric nearest neighbor coupling on a 2-D lattice. On the boundary of lattice we impose three various boundary conditions. (1) Dirichlet boundary conditions: u(0,i2 ) = u(i1 ,0) = u(n+1,i2 ) = u(i1 ,n+1) = 0, for 1 ≤ i1 , i2 ≤ n .
∆N = I ⊗ ∆n + ∆n ⊗ I with ∆n
−1 1
=
1 −2 .. .
1 ..
.
..
.
..
.
. 1
−2 1 −1
The matrix ∆N has a simple zero eigenvalue associated with eigenvector e ≡ [1, 1, . . . , 1]T ∈ 2 Rn , and λM = max{λν 6= 0 | λν ∈ σ(∆N )} ≡ −8 sin2 (π/2n). (3) Periodic boundary conditions: u(0,i2 ) = u(n,i2 ) , u(n+1,i2 ) = u(1,i2 ) , u(i1 ,0) = u(i1 ,n) , u(i1 ,n+1) = u(i1 ,1) , for 1 ≤ i1 , i2 ≤ n . The operator ∆ ≡ ∆P with periodic boundary condition is a matrix of the form ∆P = I ⊗ ∆p + ∆p ⊗ I with ∆p
−2 1 1 −2 .. . =
1
1 1 .. . .. .
..
.
..
.
1
. 1
−2
Asymptotic Synchronization in Lattices of Coupled Nonidentical Lorenz Equations 2719
In this case, the matrix ∆P has a simple zero eigenvalue associated with eigenvector e = 2 [1, 1, . . . , 1]T ∈ Rn , and λM = max{λν 6= 0|λν ∈ σ(∆P )} ≡ −8 sin2 πn . From classical definition [Pecora & Carroll, 1990] the coupled system of (1) is said to be synchronized if the difference between any two components of solutions (x(t), y(t), z(t)) approaches zero as t → ∞. Of course, if the system is synchronized, the attractor of (1) must lie on the spatially homogeneous diagonal and this can happen only if all subsystems are identical, i.e. the parameters σν , γν and bν in (1) are all equal. If each subsystem in (1) is nonidentical, then it is impossible to obtain synchronization in the above sense. In this case, we introduce a parameter ε which characterizes the variation of uncoupled subsystems and assume that |σν − σµ | ≤ ε , |γν − γµ | ≤ ε , |bν − bµ | ≤ ε , (2) for all 1 ≤ µ, ν ≤ n2 . The coupled system (1) may possibly be asymptotically synchronized [Hale, 1997]. That is, the difference between any two components of solutions (x(t), y(t), z(t)) is bounded by the quantity O(ε/ max{c1 , c2 , c3 }) as t → ∞. Consequently, the attractor of (1) approaches to the spatially homogeneous diagonal as the coupling strengths become infinity and the variation ε tends to zero. This paper is devoted to analyze the asymptotic synchronization behavior of coupled nonidentical Lorenz equations of (1) with Neumann and periodic boundary conditions, respectively. We first show that the coupled system (1) has pointwise dissipativeness property [Afraimovich et al., 1997; Hale, 1988]. A system is said to be pointwise dissipative if there is a bounded set B so that for any solution U (t, q0 ) through q0 at t = t0 , there is a time t1 ≡ t1 (q0 , B) such that U (t, q0 ) ∈ B, for t ≥ t1 . For the coupled system (1) we can construct a specific Lyapunov function to establish the pointwise dissipativeness property, which is one of the reasons that we restrict ourselves to the study of coupled Lorenz equations. Another reason is that the occurrence of chaotic behavior is over a wide range of parameters [Sparrow, 1982]. Our main theorem asserts that the coupled system (1) with Neumann or periodic boundary condition is asymptotic synchronized under the assumption (2) whenever the coupling strengths are sufficiently large. We prove that the differences of any two
components of (x(t), y(t), z(t)) satisfy lim sup {|xµ (t) − xν (t)|, |yµ (t) − yν (t)|, t→∞
|zµ (t) − zν (t)|} ≤ O
ε max{c1 , c2 , c3 }
, (3)
provided c1 , c2 and c3 are sufficiently large. Note that the full coupling in x-, y- and z-states in (1) (i.e. c1 , c2 , c3 > 0) is essential to establish the asymptotically synchronized regimes for nonidentical subsystems. In our numerical experience (Sec. 5 below), we observe that if any one of the coupling strengths, say c3 , is chosen to be zero, then the asymptotically synchronized behavior as in (3): kzµ (t) − zν (t)k ≤ O(ε/ max{c1 , c2 }) cannot be expected, for 1 ≤ µ, ν ≤ n2 . According to this observation, in this paper we only consider the full state coupling as in (1). The rest of this paper is organized as follows. Section 2 begins with results on the stability for the coupled system (1) with Dirichlet boundary condition. A sufficient condition is obtained for the global stability of trivial equilibrium. In Sec. 3 we study the pointwise dissipativeness of the coupled system (1) with Neumann/periodic boundary condition. In Sec. 4 we prove the asymptotic synchronization for the coupled nonidentical subsystems as in (1) with Neumann/periodic boundary condition. In Sec. 5 we show some numerical results to illustrate the asymptotically synchronized behavior of (1). A concluding remark is contained in the last section.
2. Globally Asymptotic Behavior with Dirichlet Condition The coupled system (1) with Dirichlet boundary condition can be regarded as 1-D vector equations with the individual dynamical system on the νth position x˙ ν = σν (yν − xν ) + c1 (∆D x)ν ,
y˙ = γ x − y − x z + c (∆ y)ν ,
ν ν ν ν ν ν 2 D z˙ = −b z + x y + c (∆ z) . ν ν ν ν ν 3 D ν
(4)
In this section we shall prove globally asymptotic behavior of the coupled system (4) with Dirichlet boundary condition. Define
2720 C.-H. Chiu et al.
λm = max{λν |λν ∈ σ(∆D )},
to λν ∈ σ(∆D ). Now, let
ν
λ0 = min{λν |λν ∈ σ(∆D )},
2
ν
|x| =
γm = max{γν }, σm = max{σν }, σ0 = min{σν }. ν
ν
ν
Let (x(t), y(t), z(t)) be the solution of the coupled system (4) with Dirichlet boundary condition. If Theorem 2.1
λ2m
(σm + γm )2 − 4 c1 λ2m λ0
× c2
!
!
!
V (x, y , z) =
2
ν
+
yν2
+
zν2
n X ν=1
X
n X
λν Xν2 ,
(7)
2
|yν |(∆|y|)ν =
ν
λν Yν2 ,
ν=1
X
2
n X
|xν yν | =
ν
Xν Yν .
ν=1
Xh
−σν x2ν +(σν + γν )xν yν −yν2
From (6) and (7) and after some calculation, we have
ν
i
+ c1 xν (∆D x)ν +c2 yν (∆D y)ν +
Xh
−bν zν2 +c3 zν (∆D z)ν
V˙ ≤
i
Xh
−σ0 Xν2 + (σm + γm )Xν Yν − Yν2
ν
ν
Xh
−σ0 |xν |2 +(σm +γm )|xν yν |−|yν |2
˜ ν |(∆D |x|)ν +c2 λ|y ˜ ν |(∆D |y|)ν + c1 λ|x +
+ c1
ν
Xh
i
i
−bν zν2 +c3 zν (∆D z)ν ,
+
(6)
˜ = λm /λ0 , |x| = [|xν ||ν = 1, 2, . . . , where λ and 2 |y| = [|yν ||ν = 1, 2, . . . , n ]. The last inequality is derived by xν (∆x)ν ≤ λm
X
x2ν
˜ λ0 ≤λ
ν
˜ ≤λ
X
X
!
|xν |
i
ν
=
c1
ν
+ c2 +
X
λm λν − σ0 Xν2 + (σm + γm )Xν Yν λ0
λm λν − 1 Yν2 λ0
[−bν zν2 + c3 zν (∆D z)ν ] .
ν
2
ν
From the assumption (5)
!
|xν |(∆|x|)ν
λm λm λν Xν2 + c2 λν Yν2 λ0 λ0
−bν zν2 + c3 zν (∆D z)ν
n2 ]
ν
Xh
X
ν
X
ν=1
|xν |(∆|x|)ν =
where 1 ≤ ν ≤ The derivative of V along the trajectory of the coupled system (4) is
≤
Yν2 ,
2
n2 .
V˙ (x, y, z) =
|yν | =
ν
,
Xν2 ,
2
n X
2
ν
(5)
Construct the Lyapunov function x2ν
|xν | =
ν=1
X
X
X1
2
n X
2
ν
< 0,
Yν hν .
ν=1
X
!
−1
Xν h , |y| =
n X
Then it is easily shown that
− σ0
λ0
2
ν
ν=1
then limt→∞ (x(t), y(t), z(t)) = (0, 0, 0). Proof.
n X
.
ν
Let {hν |ν = 1, 2, . . . , n2 } be the mutually orthonormal eigenvectors, respectively, corresponding
(σm + γm ) − 4 c1 2
× c2
λm λm − σ 0 λ0
λm λm − 1 < 0 λ0
Asymptotic Synchronization in Lattices of Coupled Nonidentical Lorenz Equations 2721
follows that V˙ < 0. This completes the proof of Theorem 2.1.
where
As in (5) of Theorem 2.1, if c2 = c3 = 0 and c1 satisfies
Corollary 2.2.
(σm + γm ) < −4 c1 2
λm λm − σ 0 λ0
The derivative of V along the trajectory of the coupled system (1) becomes
,
V˙ (x, y, z)
then limt→∞ (x(t), y(t), z(t)) = (0, 0, 0).
=
As in (5) of Theorem 2.1 if c1 = c3 = 0 and c2 satisfies
Corollary 2.3.
(σm + γm ) < −4σ0 c2 2
1 X 1 X σ and γ = γν . ν n2 ν n2 ν
σ=
[−σν x2ν + (σν − σ + γν − γ)xν yν − yν2 ]
ν
λm λm − 1 λ0
X
+
,
X
[−bν zν2 + (σ + γ)bν zν ]
ν
X
then limt→∞ (x(t), y(t), z(t)) = (0, 0, 0).
+
Since λ0 is negative, the condition (5) is easy to satisfy, provided the coupling strengths c1 and c2 in (4) are chosen sufficiently large. Note that the globally asymptotic behavior for (4) holds depending only on c1 and c2 without c3 .
+ c3 zν (∆z)ν ] − c3 (σ + γ)
[c1 xν (∆x)ν + c2 yν (∆y)ν
ν
X
(∆z)ν .
(9)
ν
From definition of ∆ ≡ ∆N or ∆P in Sec. 1 it is easily shown that 2
n X
3. Pointwise Dissipativeness In this section we shall prove the pointwise dissipativeness property [Afraimovich et al., 1997; Hale, 1988] for the coupled system (1) with Neumann/periodic boundary condition. Let U (t, (x0 , y0 , z0 )) be a solution of system (1) through (x0 , y0 , z0 ) at t = t0 . As in Sec. 1 the coupled system (1) is said to be pointwise dissipa2 tive if there is a bounded set B ⊆ R3n such 2 that for any point (x0 , y0 , z0 ) ⊆ R3n there is a time t1 ≡ t1 (c1 , c2 , c3 , x0 , y0 , z0 , B) so that U (t, (x0 , y0 , z0 )) ∈ B for t ≥ t1 . Recall that a parameter ε which characterizes the variation of the uncoupled subsystems in (1) satisfying |σµ − σν | ≤ ε, |γµ − γν | ≤ ε, |bµ − bν | ≤ ε ,
(8)
for 1 ≤ µ, ν ≤ n2 . If ε2 − σ0 < 0, where σ0 = minν {σν } , then the coupled system (1) with Neumann/periodic boundary condition is pointwise dissipative.
Proof.
Construct the following Lyapunov function
V (x, y, z) =
" X x2 ν ν
From (10) the last term in (9) becomes zero. From the variation inequalities (8) we have −θx2ν + (σν − σ + γν − γ)xν yν −
ε2 2 y ≤ 0, θ ν
where θ is chosen by assumption satisfying ε2 < θ < σ0 . Since X
[c1 xν (∆x)ν + c2 yν (∆y)ν + c3 zν (∆z)ν ] ≤ 0 ,
ν
from (9) we then have X
"
(−σν +
θ)x2ν
ν
σ+γ × zν − 2
ε2 + −1 + θ
2
+ bν
!
σ+γ 2
yν2 − bν 2 #
Let M be a positive constant satisfying
#
y2 z2 + ν + ν − (σ + γ)zν , 2 2 2
(10)
ν=1
V˙ ≤
Theorem 3.1.
(∆z)ν = eT ∆z = 0 .
M>
X ν
bν
σ+γ 2
2
.
2722 C.-H. Chiu et al.
are positive constants M ∗ and c∗ such that for any (c1 , c2 , c3 ) ∈ W (c∗ ) it holds
and let D be the set defined by (
D=
(x, y, z) ∈ R
3n2
X h (σν − θ)x2ν ν
+ 1−
ε2 θ
!
yν2 + bν zν −
σ+γ 2
2 #
.
2
VK = {(x, y , z) ∈ R3n |V (x, y, z) ≤ K} .
(x, y, z) ∈ R
X h (σν − θ)x2ν
ε2 + 1− θ
yν2
+ bν
σ+γ zν − 2
lim sup |zµ (t) − zν (t)| ≤
M ∗ε , c3
for 1 ≤ µ, ν ≤ n2 .
σ=
ν
!
M ∗ε , c2
t→∞
Proof.
Choose K0 sufficiently large, such that 3n2
lim sup |yµ (t) − yν (t)| ≤ t→∞
This shows that V˙ < 0 for all (x, y, z) ∈ D. Given any K > 0, define
(
M ∗ε , c1
t→∞
)
>M
lim sup |xµ (t) − xν (t)| ≤
2 #
Introduce the constants 1 X 1 X 1 X σν , γ = 2 γν and b = 2 bν . 2 n ν n ν n
Set δν1 = (σν − σ)yν − (σν − σ)xν , δν2 = (γν − γ)xν , δν3 = −(bν − b)zν ,
≤ M } ⊆ VK 0 . 2
Then V˙ < 0, for all (x, y, z) ∈ R3n \VK0 . Therefore, any solution of the coupled system (1) starting with (x0 , y0 , z0 ) will eventually enter and stay in the bounded region VK0 . This completes the proof of Theorem 3.1. Note that the time t1 ≡ t1 (c1 , c2 , c3 , x0 , y0 , z0 , Vk0 ) at which the trajectory (x(t), y(t), z(t)), t ≥ t1 enters and stays in Vk0 , is dependent pointwisely on the initial point (x0 , y0 , z0 ). An uniformly bounded dissipativeness cannot be obtained here.
4. Asymptotic Synchronization In this section, we shall prove the asymptotic synchronization of the coupled system (1) with Neumann/periodic boundary condition. Let ε be the variation constant given in (8). Define the unbounded region for coupling strengths in parameter space by
for 1 ≤ ν ≤ n2 . Then the equations in (1) can be reformulated in the forms x˙ ν = σ(yν − xν ) + c1 (∆x)ν + δν1 ,
y˙ = γx − y − x z + c (∆y) + δν2 ,
ν ν ν ν ν 2 ν z˙ = −bz + x y + c (∆z) + δ , ν ν ν ν 3 ν ν3
(12)
where ∆ ≡ ∆N or ∆P . Rewrite the system (12) into the vector form x˙ = σ(y − x) + c1 ∆x + δ1 ,
y˙ = γx − y − f (x, z) + c ∆y + δ2 ,
2 z˙ = −bz + g(x, y) + c ∆z + δ , 3 3
(13)
where (f (x, z))ν = xν zν , (g(x, y))ν = xν yν and δi = [δνi |ν = 1, 2, . . . , n2 ] for i = 1, 2, 3. For Neumann/periodic boundary condition, the matrix ∆ as in (13) has a simple zero eigenvalue associated 2 with the eigenvector e = [1, 1, . . . , 1]T ∈ Rn ×1 . Let
1 −1 1
C=
W (c∗ ) = {(c1 , c2 , c3 )|c1 ≥ c∗ , c2 ≥ c∗ , c3 ≥ c∗ } . Let (x(t), y(t), z(t)) be a solution of the coupled system (1) with Neumann/periodic 2 boundary condition. Denote x(t) = [xν (t)]nν=1 , 2 2 y(t) = [yν (t)]nν=1 and z(t) = [zν (t)]nν=1 . Then there
(11)
−1 ..
.
1
Theorem 4.1.
and
C E= 1···1
−1
(n2 −1)×n2
n2 ×n2
.
Asymptotic Synchronization in Lattices of Coupled Nonidentical Lorenz Equations 2723
It is easy to check "
E∆E
−1
˜ ∆ = 0
0 0
c3 are all non-negative. Thus, there is a positive number M0 such that
#
lim kx(t)k ≤ M0 , lim ky(t)k ≤ M0 ,
,
t→∞
lim kz(t)k ≤ M0 , lim kξ(t)k ≤ M0 ,
t→∞
˜ is diagonalizable with λ0 ≡ max{λν |λν ∈ where ∆ ˜ σ(∆)} < 0. Multiplying the matrix E to the system (13) from the left, we get
We now introduce transformations with new variables ξ, ξs , η, ηs , ζ and ζs such that Ex =
!
,
η ηs
Ey =
!
,
ζ ζs
Ez =
lim kη(t)k ≤ M0 , lim kζ(t)k ≤ M0 .
t→∞
ξ(t) = et(−σ+c1 ∆) ξ(τ ) Z τ
ξs =
2
xν , ηs =
ν=1
n X
˜ e(t−s)(−σ+c1 ∆) (ση(s) + δ˜1 )ds .
!
˜
ket(−σ+c1 ∆) k ≤ Ke(−σ+c1 λ0 )t . .
From (17) we have kξ(t)k ≤ Kkξ(τ )ke(−σ+c1 λ0 )t
2
yν and ζs =
ν=1
n X
+ zν .
ν=1
The variables ξ, η and ζ are vectors in R(n satisfying
Consequently, 2 −1)×1
t→∞
for ν = 1, 2, . . . , n2 − 1. Therefore, by Mean Value Theorem, we have ˙ ˜ ˜ ξ = σ(η − ξ) + c1 ∆ξ + δ1 ,
˜ + δ˜ , η˙ = γξ − η − Γ ζ − Γ ξ + c ∆η
lim kξ(t)k ≤ ≤
(ξ)ν = ξν = xν − xν+1 , (η)ν = ην = yν − yν+1 , (ζ)ν = ζν = zν − zν+1 ,
1 3 2 2 ζ˙ = −bζ + Γ η + Γ ξ + c ∆ζ ˜ ˜ + δ3 , 1 2 3
(17)
˜ < 0, there exists a Since λ0 ≡ max{λν |λν ∈ σ(∆)} positive number K such that
(15)
2
t
+
where ξs , ηs and ζs are scalars satisfying n X
t→∞
˜
+c2 E∆E −1 (Ey) + Eδ2 , E z˙ = −bEz + Eg(x, y) + c3 E∆E −1 (Ez) + Eδ3 . (14)
ξ ξs
t→∞
Applying the variation of constant formula to the first equation of (16) we get
−1 E x˙ = σE(y − x) + c1 E∆E (Ex) + Eδ1 , E y˙ = γEx − Ey − Ef (x, z)
t→∞
K σ − c1 λ0 K −c1 λ0
where Γ1 = diag{x1 , x2 , . . . , xn2 −1 }, Γ2 = diag{y2 , y3 , . . . , yn2 }, Γ3 = diag{z2 , z3 , . . . , zn2 } and δ˜i = [δ1,i − δ2,i , δ2,i − δ3,i , . . . , δn2 −1,i − δn2 ,i ] for i = 1, 2, 3. T
From Theorem 3.1, it follows that the coupled system (1) with Neumann/periodic boundary condition is pointwise dissipative whenever c1 , c2 and
(18)
lim kση(t) + δ˜1 k
t→∞
lim kση(t)k + kδ˜1 k .
t→∞
(19) Similarly, the vectors η(t) and ζ(t), respectively, in the second and third equations of system (16), can be estimated by
(16)
Kkση(t) + δ˜1 k . σ − c1 λ0
lim kη(t)k ≤
t→∞
K −c2 λ0
lim (kγξk
t→∞
+ kΓ1 ζk + kΓ3 ξk + kδ˜2 k) ,
lim kζ(t)k ≤
t→∞
K −c3 λ0
lim (kΓ1 ηk
t→∞
+ kΓ2 ξk + kδ˜3 k) .
(20)
Let M be a positive number satisfying max{2n2 M0 , σ, γ, b} ≤ M . Then from (11) it holds lim kδ˜i k ≤ M ε, i = 1, 2, 3 .
t→∞
(21)
2724 C.-H. Chiu et al.
Choose c∗ > 0 satisfying −c∗ λ0 ≥ 3KM l with some integer l > 1. Then, for c1 > c∗ and c3 > c∗ , from (19) and (20) follows that lim kξ(t)k
t→∞
1 M 2l−3
+
