To study synchronization of (2.3), we first make a coordinate change to decom-pose the synchronous subspace. Let A be an m × m matrix of the form
5
A=
where e is given as in (2.4a). It is then easy to see that CCT is invertible and that
A−1= matrix A. Multiplying E to the both side of equation (2.6a), we get
˙˜y =: E ˙˜x = E˜F(˜x, t) + dE(D ⊗ G)E−1y˜
we have that the dynamics of ¯yis satisfied by following equation
˙¯y = d(D ⊗ ¯G)¯y+ ¯F(¯y, t). (3.3)
6
Here ¯Fis obtained from E ˜F(E−1y˜, t) accordingly.
The task of obtaining global synchronization of system (2.3) is now reduced to showing that the origin is globally and asymptotically stable with respect to system (3.3). To this end, the space ¯y is broken into two parts ¯yc, the coupled space, and
¯
yu, the uncoupled space.
¯
. The dynamics on the coupled space
with respect to the linear part is under the influence of ¯G, which is asymptotically stable. The dynamics of the nonlinear part on coupled space can then be controlled by choosing large coupling strength. As a matter of fact, it is easier to obtain synchronization of coupled chaotic systems with a larger coupled space. On the other hand, the uncoupled space has no stable matrix ¯G to play with. Thus, its corresponding vector field ¯Fu(¯y, t) must have a certain structure to make the trajectory stay closer to the origin as time progresses. As we shall explain latter.
Now, assume that ¯Fc(¯y, t) satisfies a dual-Lipschitz condition with a dual-Lipschitz constant b1. That is,
k¯Fc(¯y, t)k ≤ b1k¯yk (3.5a)
whenever ¯y in the ball B(m−1)n(α), and for all time t. Since the estimate in the right-hand side of (3.5a) depends on the whole space ¯y, condition (3.5a) is a mild assumption provided that the coupled system is bounded dissipative. Write ¯Fu(¯y, t) as
(i) The matrix measures µi(Uj(t)) are less than −γ for all t and all j,
dual-Lipschitz condition with a strong dual-Lipschitz constant b2. Specifically, let
¯
, written in accordance with the block structure of U(t). Then we
assume that
Specifically, we break the vector field ¯Fu into (time dependent) linear part U(t)¯yu and nonlinear part ¯Ru(¯y, t). We will further break U(t) into certain block diagonal form if necessary. Note that form (3.5b) can always be achieved since the remainder term ¯Rustill depends on the whole space ¯y. To take control of the dy-namics on the linear part, we assume that the matrix measure of each diagonal block Uj(t) is negative. As to contain corresponding dynamics on the nonlinear part, we assume that (3.5d) holds. Note that though the nonlinear terms Ruj(¯y, t) could possibly depend on the whole space, their norm estimates are required to depend only on the coupled space and uncoupled subspaces with their indexes proceeding j. In this set up, the nonlinear dynamics on uncoupled space can be iteratively controlled by choosing large coupling strength. We also remark that if (3.5c) and (3.5d) are satisfied for l, the number of diagonal blocks, being one, then we do not need to further break U(t). Such further breaking is needed only if (3.5c) and (3.5d) are not satisfied. The proof in the following theorem gives exactly how the above strategy can be realized.
Theorem 3.1. Let G and D be given as in (2.4). Assume that ¯F satisfies (3.5a-d), and system (3.3) is uniformly bounded dissipative with respect to α. Let λ1= max{λj|λj ∈ Re(σ( ¯G))}. If
Proof. Since system (3.3) is uniformly bounded dissipative with respect to α,
Applying the variation of constant formula to (3.7a) on ¯yc, we get
¯
It then follows from (3.5c-d) and (3.8a) that
k¯yu1(t)k ≤ αe−γ(t−t0,1)+α
whenever t ≥ t1,1 for some t1,1 ≥ t0,1. Inductively, we get factor h is strictly less than 1, and k¯y(t)k contracts as time progresses. To complete the proof of the theorem, we note that δ > 1 can be made arbitrary close to 1.
Consequently, if d >
1 + (bγ2)22l
cb1
|ν|, then h can still be made to be less than
1.
Remark 3.1. (i) In case that ¯Gis symmetric, then c and ǫ can be chosen to be one and zero, respectively. (ii) b1and b2could possibly depend on α. (iii) If system (3.3) is only bounded dissipative, then the estimate in (3.6) is still valid. However, in this case, b1 and b2 depend not only on α but also on x0.
Corollary 3.1. Suppose ¯Fand G are given as in Theorem 3.1. Let
D= D¯k×k 0
0 0
!
n×n,
where Re( σ( ¯D) ) > 0. (3.9a) Assume, in addition, that either σ(G) or σ( ¯D) has no complex eigenvalue.
Then assertions in Theorem 3.1 still hold true, except dc needs to be replaced by dc= c b1
Proof. Assumption on D is to ensure that (3.7b) is still valid. Other parts of the proof are similar to those in Theorem 3.1 and are thus omitted. We next turn our attention to finding conditions on the nonlinearities fi(u, t), i = 1, . . . , n, u ∈ Rn, so that assumptions (3.5a-d) are satisfied. To this end, we need the following notations. Let ˜xi and ˜xbe given as in (2.5). Define
[˜xi]−=
We then break ˜Fas given in (2.6a) into two parts so that the breaking is in consis-tent with ¯yin (3.4). Specifically, we shall write
F(˜˜ x, t) = F˜c(˜x, t) F˜u(˜x, t)
!
.
(3.11) We are now in the position to state the following propositions.
Proposition 3.1. Suppose that fi(x, t), i = 1, 2, . . . , k satisfy a Lipschitz condition
we conclude that (3.5a) holds.
From the above proposition, we see that the nonlinearities on the corresponding coupled space are only assumed to be Lipchitz. The following proposition is very useful in the sense that by checking how each component fi of the nonlinearity f is formed, one would then be able to conclude whether (3.5c-d) are satisfied.
Proposition 3.2. Let u = (u1, . . . , un)T and v = (v1, . . . , vn)T be vectors in
i = 1, . . . , kp,as
fwp−1+i(u, t) − fwp−1+i(v, t)
=
kp
X
j=1
qwp−1+i,wp−1+j(u, v, t)(uwp−1+j− vwp−1+j) + rwp−1+i(u, v, t).
(3.14a) We further assume that the followings are true.
(i) For p = 1, . . . , l, let Qu,v,p= (qwp−1+i,wp−1+j(u, v, t)), where 1 ≤ i, j ≤ kp. Then µ∗(Vp) < −γ for all p, u, v in Bn(α2) and all time t, where ∗ =
1, 2, ∞. (3.14b)
(ii) Let rp= rwp−1+1(u, v, t), . . . , rwp(u, v, t)T
. We have that
krpk ≤ b2k
u1− v1 ... uwp−1− vwp−1
k (3.14c)
for all p, u, v in Bn(α2) and all time t.
Then (3.5c) and (3.5d) hold true for ∗ = 1, 2, ∞.
Proof. Since ri(u, v, t) depend on the whole space, fi(u, t) − fi(v, t) can always be written as the form in (3.14a). Using (3.14a) and (3.13), we have that the matrices Up(t) in the linear part of ¯Fu(¯y, t) take the form
Up(t) =
m−1
X
w=1
Qxw,xw+1,p(t) ⊗ Dw, (3.15)
where xw are given as in (2.2), and
(Dw)ij=
( 1 i = j = w,
0 otherwise, 1 ≤ i, j ≤ m − 1.
It then follows from (2.7a,b), and (3.15) that µ∗(Up(t)) < −γ for ∗ = 1 or ∞. For
∗ = 2, we have that
m−1
[
w=1
σ{Qxw,xw+1,p(t) + Qxw,xw+1,p(t)T
}
= σ (m−1
X
w=1
Qxw,xw+1,p(t) ⊗ Dw+ Qxw,xw+1,p(t)T
⊗ Dw )
= σ Up(t) + UTp(t) ,
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where σ(A) is the spectrum of A. We remark that the first equality above can be verified by the definition of eigenvalues due to the structure of Up(t). It then follows from (2.7c) that µ2(Up(t)) < −γ. The remainder of the proof is similar to
that of Proposition 3.1, and is thus omitted.
Remark 3.2. The upshot of Proposition 3.2 is that by only checking the “struc-ture” of the vector field f of the single oscillator, one should be able to determine if our main result can be applied. To be precise, we begin with saving notations by setting f as f = f (x, t) = (f1(x, t), . . . , fn(x, t))T. We then check the form of the difference of “uncoupled” part of dynamics. That is, we write fi(u, t) − fi(v, t) in the form of (3.14a) with i = k + 1, . . . , n. If (3.14b, c) can be satisfied, then l = 1 gets the job done. Otherwise, we further break the uncoupled states into a set of smaller pieces to see if the resulting (3.14b, c) are satisfied.
We are now ready to state the main theorems of the paper.
Theorem 3.2. Assume that system (2.3) is (resp., uniformly) bounded dissipative.
Let coupling matrices G and D satisfy (2.4) and the nonlinearities fi(x, t), i = 1, 2, . . . , n, satisfy (3.12) and (3.14). Suppose d is greater than dc, as given in (3.6). Then system (2.3) is (resp., uniformly,) globally synchronized.
Proof. The proof is direct consequences of Propositions 3.1 and 3.2, and Theorem
3.1.
Remark 3.3. From here on, we will refer the assumptions in Theorem 3.2 as synchronization hypotheses.
Theorem 3.3. Coupled system (D, G, F(x, t)), given as in Corollary 3.1, is also (resp., uniformly,) globally synchronized provided that its coupled system is (resp., uniformly) bounded dissipative and that d is greater than dc. Here dc is given in (3.9b).