2.3 Global synchronization with time-varying coupling
2.3.2 Synchronization Criteria
In the section, we turn our attention back to the dynamics of (2.29a), and analyze the stability of the origin of the system. Let ¯y, ¯yc, and ¯yu be defined as in (2.13). Then, like (2.16a) , Equation (2.29a) can be rewritten as in the form
˙¯yc condi-tions for coupling matrices G and D.
(i) λ = 0 is a simple eigenvalue of G(t), ∀ t ≥ 0 and
e= √1m(1, 1, . . . , 1)T1×m is its corresponding eigenvector; (2.42a) (ii) There is some λ > 0 such that µ2( ¯G(t)) ≤ −λ, ∀t ≥ 0. (2.42b) (iii) Coupling matrix D is of the form
D= We are now in a position to state our first main theorem in the time-varying coupled system.
Theorem 2.3.3. Let coupling matrices G(t) and D satisfy (2.42). Suppose that ¯F, given in (2.29a) or (2.41), satisfies (2.14a), (2.14c), and that (2.14d), and system (2.29a) is bounded dissipative with respect to α. Then lim
t→∞y(t) = 0 for any initial¯ value provided that the coupling strength d satisfies the following inequality
d > b1 Applying the matrix measure inequality (2.1) and hypotheses (2.14a), (2.42b) on ¯yc, for any t ≥ t0, we have that
whenever t ≥ t0,1for some t0,1> t0. Similarly, applying inequality (2.1) and hypotheses (2.14c), (2.14d) on ¯yu1,
whenever t ≥ t1,1 for some t1,1 > t0,1. Inductively, we have k ¯yuj(t)k ≤ α
dcjδj+1, (2.44c)
whenever t ≥ tj,1, for all j = 2, . . . , l. Here cj = bγ2qPj−1
i=0c2i. Letting t1 = tl,1 and summing up (2.44a) to (2.44c), we get
k ¯y(t)k ≤ α is strictly less than 1, and k ¯y(t)k contracts to zero as time progresses. Since δ > 1 can be made arbitrarily close to 1. Consequently, if d is chosen as assumed, then h can still be made to be less than 1. The assertion of the theorem now follows.
In the following, we drive another set of hypotheses to replace that listed in Theorem 2.3.3 to get the easily checkable criteria for the synchronization in coupled system (2.26).
Since r(u, v, t) could depend on all components of u and v, such a decomposition in (2.45) can always be achieved.
Proposition 2.3.4. Suppose fi(·, t), i = 1, . . . , k are uniformly Lipschitz, i.e., there exists a positive constant r > 0 such that
|fi(u, t) − fi(v, t)| ≤ rku − vk (2.46)
for all i = 1, . . . , k. Then the inequality in (2.14a) is satisfied with b1 = r√
The proof of the proposition is completed.
We next turn our attention to the structure of the vector field of the uncoupled parts.
Proposition 2.3.5. (i) Suppose matrix Q(v, t) can be written as the block diagonal form
Q(v, t) = diag(Q1(v, t), . . . , Ql(v, t)),
where the size of matrices Qj(v, t) are kj× kj, ∀j = 1, . . . l and indexes l, kj are given as in (2.14c). Moreover, there is some γ > 0 such that
µ2(Qj(v, t)) ≤ −γ. (2.48a)
Here γ is independent of v, t. Then the inequality in (2.14c) is fulfilled.
(ii) Denoted by s1 = k and sj = k +Pj−1 to obtain F . Using the fact that the row sums of E are all zeros, we have that for 1 ≤ i ≤ m − 1, sj+ 1 ≤ l ≤ sj+ kj,
To save notations, ∀i = 1 . . . , kj, we denote by [rsj+i(xl, x1, t)]ml=2 the vector (rsj+i(x2, x1, t), rsj+i(x3, x1, t), . . . , rsj+i(xm, x1, t))T.
Applying (2.45) and (2.48a)- (2.48c), we shall be able to rewrite h as
−γ. Upon using the similar techniques as those in establishing the inequality in (2.47), we conclude that (2.48a)- (2.48c) hold as asserted.
Now, we are in the position to impose the hypotheses for synchronization. We remark that although these conditions are quite like those given in (2.22) and (2.24), to be self-contained and clear, we list corresponding the hypotheses herein.
(H1) System (1.1a) is bounded dissipative with respect to α.
(H2) Functions fi(·, t), i = 1, . . . , k in (1.1a) are uniformly Lipschitz in region B given in (H1). That is, there is a constant r > 0 such that |fi(u, t)−fi(v, t)| ≤ rku−vk, whenever t is sufficiently large, and u, v in B.
(H3) The matrix Q(v, t), which is given as in (2.45), is of block diagonal form, i.e., Q(v, t) = diag(Q1(v, t), · · · , Ql(v, t)). Here the sizes of Qj(v, t), j = 1, . . . , l, are kj× kj. Moreover, there is some γ > 0 such that matrix measures µ2(Qj(v, t)) ≤
−γ, for all j, whenever t is sufficiently large, and v in B.
(H4) Denoted by s1 = k and sj = k +Pj−1
i=1ki, j = 2, . . . , l, where ki and l are defined in (H4). Suppose, for any 1 ≤ j ≤ l, there is a δ > 0 such that
k[r(u, v, t)]ssjj+k+1jk ≤ δk[u − v]s1jk,
for t sufficiently large, and u, v in B. Here [u]ji is defined to be (ui, . . . , uj)T. Remark 2.3.2. Using the similar techniques as developed in the proof of Propositions 2.3.4 and 2.3.5, we may also conclude that the global theorems obtained in [17] may still be valid by using the coordinate transformation developed here in this paper. Con-sequently, the size limit problem of their approach can be removed.
The main criterion for synchronization in the time-varying coupled system is now stated in the following. The proof of the main theorem follows directly from Theorem 2.3.3 and Propositions 2.3.4 and 2.3.5.
Theorem 2.3.4. Let the coupling matrices G(t) and D satisfy (2.42). Suppose hy-potheses (H1), (H2), (H3), and (H4) hold true. Then coupled system (2.26) achieves global synchronization whenever
d > r√
k cond(E1ET)
λ 1 + δ2k ˜Ek2kE1ETk2 γ2
!2l
(2.51) where E, E1, and ˜E are given as in Theorem 2.3.1, (2.30), and (2.48c), respectively.
Remark 2.3.3. The small price to pay by introducing the coordinate transformation E is that the lower bound, given as in the right hand side of (2.51), on the coupling strength d, is size dependent.
Chapter 3
Applications for Model I
3.1 Synchronization in coupled Lorenz and coupled Duffing systems
To see the effectiveness of our main results in Chapter 2, we consider two examples in this chapter. These are coupled Lorenz equations [8,63], and coupled Duffing oscilla-tors [105].
Example 1: We shall begin with Lorenz equations. Let x = (x1, x2, x3)T, f(x, t) = f (x) = (σ(x2− x1), rx1− x2− x1x3, −bx3+ x1x2)T
=: (f1(x), f2(x), f3(x))T .
Here σ = 10, r = 28 and b = 83. In the following cases (a), (b), (c) and (d), G denotes the diffusive coupling with zero flux and D is, respectively,
1 0 0 0 0 0 0 0 0
,
0 0 0 0 1 0 0 0 0
,
0 0 0 0 0 0 0 0 1
,
and
0 0 0 0 1 1 0 0 1
. For the first three cases, it was shown in [5] that such the coupled system (2.6) has the topological product of an absorbing domain
B0 = {x21+ x22+ (x3− r − σ)2 < b2(r + σ)2
4(b − 1) =: β}. (3.1)
Hence, in each case, we will concentrate on the illustration of how our main results may or may not be applied.
(a) Let D = D1 =
For “coupled” nonlinearity f1, we get that
|f1(u) − f1(v)| = σ|(u2− v2) − (u1− v1)| ≤√
2σku − vk.
Hence, condition (2.14a) is satisfied. For “uncoupled” nonlinearities f2 and f3, we see that Writing (3.2a)-(3.2b) in the vector form, we get
f2(u) − f2(v) its estimate depends only on coupled space. Hence, conditions (2.24b), and (2.24c) are satisfied.
As in the case (a), the “coupled” nonlinearity f2 is clearly Lipschitz on the absorbing domain. The difference of “uncoupled” nonlinearities f1 and f3 are given as follows.
f1(u) − f1(v) = [−σ(u1− v1)] + σ(u2− v2),
f3(u) − f3(v) = [−b(u3− v3)] + u1(u2− v2) + v2(u1− v1).
If l = 1 is chosen, then (2.24c) is violated. For in the case, the norm estimate in the right hand side of (2.24c) can only depend on u2− v2. Now, if we choose l = 2 and pick the space of the first diagonal block being the one associated with the nonlinearity f1, then Qu,v,1 = (−σ) and r1 = σ(u2 − v2). Consequently, (2.24b) and (2.24c) are satisfied. Moreover, we have Qu,v,2 = (−b) and r2 = u1(u2− v2) + v2(u1− v1), which depends only on the coupled space and the first uncoupled space. Thus, r2 satisfies (2.24c).
(c) For illustration, we also consider D = D3 =
In this case, the un-coupled nonlinearities of f1 and f2 both contain the terms x2 and x1. The only fea-sible choice to break the uncoupled space is not to do any breaking. Consequently, Qu,v,1 =
−σ σ
r − u3(t) −1
. For such Qu,v,1, its matrix measure can not stay neg-ative for all time. An indicated, see e.g., [63], synchronization fails for this type of partial coupling.
To apply Theorem 2.2.3, we first note that for D =
D5 =
the corresponding coupled system is indeed globally synchronized, and hence, so is the coupled system with D = D4. Note that bounded dissipation of the coupled system can be verified similarly as in [63].
(e) The work that are most related to ours are those in [4,8]. While their estimates for dmin seems to be sharper than ours, which we shall illustrate in case (f), their connec-tivity topology requires that off-diagonal entries be nonnegative. We only assume our connectivity topology satisfies (2.7). Consider for instant the following matrix:
G=
Such G has some negative off-diagonal entries and satisfy (2.7a) and (2.7b). In fact, the eigenvalues of G are 0, −1 ± √
5i, and −6. Clearly, applying our results, we see immediately that the coupled system (2.6) with D = Di, i = 1, 2, 4, is globally synchronized. Numerical results (see Figure 3.1.) indeed confirm synchronization of such connectivity topology. We remark that by constructing the Lyapunov function as given in [63], one would be able to show bounded dissipation of the coupled system with this particular connectivity topology.
0 10 20 30 40 50 60 70 80
-20 0 20 40
0 20 40 60 80 100
-40 -20 0 20
0 20 40 60 80 100
-50 0 50
x-differencey-differencez-difference
t t t
Figure 3.1: The difference of each component of two coupled oscillators in case (e).
(f) In this part, we shall compute the lower bound for global synchronization for case (a) by using our method, those obtained in [8] and MSF, respectively. To compute dmin, given in (2.15), we note that ¯G= E2GE2T(E2E2T)−1 = E2(E2TE2)E2T(E2E2T)−1
= E2E2T. Here E2 is given as in (2.30). Since ¯G is symmetric, c and ǫ, given as in (2.16c), can be chosen to be 1, and 0, respectively. Consequently,
dmin =
√2σp
1 + β + 2σ√ β + σ2
4 sin2(2mπ ) . (3.3)
Here 4 sin2(2mπ ) = |λ1|. Applying Theorem 2.2.3, we see that the coupled system is globally synchronized provided that the coupling strength d is greater than dmin. For
m = 4, dmin ≈ 1189. In [8], the bound ¯dmin for threshold of global synchronization is Using the MSF-criteria, we numerically (see Figure 3.2.) compute the maximum Lyapunov exponent of the variational equations with respect to the parameter α. We have in this example that if
α = dλ1 < −7.778, (3.4)
then its maximum Lyapunov exponent is negative. Here λ1 = −4 sin2 π8 is the largest nonzero eigenvalues of G. Hence if d > −7.778λ
1 ≈ 13.3, then local synchronization of the coupled system (2.6) can be realized.
−10 −15
Figure 3.2: The vertical axis denotes the maximum Lyapunov exponent of the varia-tional equations. While the horizontal axis represents α = dλ.
(g) Let coupling matrix G(t) be time dependent as given in (2.33), (2.35), (2.37) or (2.39), and the coupling matrix D = Di, i = 1, 2. Then by Theorem 2.3.4 and above arguments, we can also have coupled system is globally synchronized whenever the coupling strength d is sufficient large. Fig. 3.3 illustrates the phenomenon of synchronization with G(t) of the form (2.37), d1(t) = 32 − sin(t), and D = D1.
0 5 -10
0 10
0 5
-50 0 50
0 5
-50 0 50
x-differencey-differencez-difference
t t t
Figure 3.3: The difference of components of the first two coupled oscillators. Here the x-component partial-state coupling is considered. and the G(t) is given as in (2.37) with d1(t) = 32 − sin(t) and m = 8.
Example 2: Now, we consider the coupled Duffing oscillators, where the individual system considered is defined by
˙x1 = −αx1− x32+ a cos wt (3.5a)
˙x2 = x1, (3.5b)
where α and a are positive constants. Letting x = (x1, x2)T, we have
f(x, t) = (f1(x, t), f2(x)) = (−αx1− x32+ a cos wt, x1), (3.6a) Assume coupling matrices D and G are, respectively,
D(c) =
1 c 0 0
(3.6b)
and
where ǫ > 0 and r are scalar diffusive and gradient coupling parameters, respectively.
In this way, the coupled Duffing systems can be written as
˙˜x1 = −α ˜x1− ˜x32+ g(t) + dcG(ǫ, r) ˜x2+ dG(ǫ, r) ˜x1 (3.7a)
˙˜x2 = ˜x1. (3.7b)
Here ˜x32 = (x31,2, . . . , x3m,2)T, and g(t) = a cos(wt) (1, · · · , 1)T. Note that f2(u) − f2(v) = 0(u2− v2) + (u1− v1)
and so the matrix measure of the corresponding Qu,v,1 is zero. To apply our theorem, we need to make the following coordinate change.
Letting y2 = x2 and y1 = qx1+ px2, we see that (3.5) becomes
and the corresponding coupled system (3.7) becomes
˙˜y1 = (p
in the case of c > 0, (3.9) becomes
˙˜y1 = dG(ǫ, r) ˜y1+ (c − α − 1
d) ˜y1+ (α − c + 1
d) ˜y2− ˜y32+ g(t) + G(ǫ, r) ˜y2
=: dG(ǫ, r) ˜y1+ ˜Fc( ˜y, t)
˙˜y2 = − 1
c −1dy˜2+ ˜y1.
The purpose of the coordinate transformation is two-fold. First, to make the dynamics of the linear part on the uncoupled space stable. In this case, the coefficient of ˜y2
becomes negative when d > 2c. Second, to make sure the parameters in the nonlinear part of coupled space contain no bad influence of d, coupling strength. Otherwise, we may not be able to control its corresponding dynamics by choosing d large.
It is then easy to check that assumptions for Theorem 2.2.1 are all satisfied, and similar arguments can be followed for the case of c = 0. What the remainder is the checking of the bounded disspation of the coupled system.
Consider the following scalar-valued function as the Lyapunov function of the coupled system (3.7)
U( ˜x1, ˜x2) = 1
2 < ˜x1, ˜x1 > + Xm
i=1
x4i,2
4 + c < ˜x2, ˜x1 >, (3.10) Taking the time derivative of U along solutions of the coupled system (3.7), we have
dU
dt =< ˜x1, ˙˜x1 > + Xm
i=1
x3i,2xi,1+ c < ˜x1, ˜x1 > +c < ˜x2, ˙˜x1 >
= (c − α) < ˜x1, ˜x1 > −cα < ˜x2, ˜x1 > −c < ˜x2, ˜x32 > + < ˜x1+ c ˜x2, g(t) >
+ d < ˜x1, G(ǫ, r) ˜x1 > +2dc < ˜x1, G(ǫ, r) ˜x2 > +dc2 < ˜x2, G(ǫ, r) ˜x2 >
= (c − α) < ˜x1, ˜x1 > −cα < ˜x2, ˜x1 > −c < ˜x2, ˜x32 > + < ˜x1+ c ˜x2, g(t) >
+ d ˜xT1, ˜xT2
1 c c c2
⊗ G(ǫ, r)
x˜1
˜ x2
≤ (c − α) < ˜x1, ˜x1 > −cα < ˜x2, ˜x1 > −c < ˜x2, ˜x32 > + < ˜x1+ c ˜x2, g(t) >
Note that the last inequality holds true since
and G(ǫ, r) + G(ǫ, r)T is a nonpositive definite matrix. On the other hand, since
< ˜x2, ˜x32 >=
We are now in a position to show bounded dissipation of the coupled system (3.7).
Proposition 3.1.1.
(i) If c satisfies the inequality
0 < c < min{ 4α
4 + α2m, α} = 4α
4 + α2m. (3.11)
Then there exists a constant c0 so that dUdt < 0 for k ˜x2k21+ k ˜x2k22 ≥ c0. (ii) If c = 0, then the first assertion of the proposition still holds true.
Proof. Suppose k ˜x2k2 ≥ 1. Then
u(k ˜x1k2, k ˜x2k2) ≤ (c − α)k ˜x1k22+ cαk ˜x2k2k ˜x1k2− c
mk ˜x2k22+√
ma(k ˜x1k2+ ck ˜x2k2)
=: ¯u(k ˜x1k2, k ˜x2k2).
It then follows from (3.11) that the the level curve of ¯u is a bounded closed curve.
We shall call such curve ellipse-like is an elliptic in the plane. Thus, there exists a c1 so that dUdt < 0 whenever k ˜x2k21+ k ˜x2k22 ≥ c1 and k ˜x2k2 ≥ 1. Let k ˜x2k2 < 1 and k ˜x2k21+ k ˜x2k22 ≥ c2. Here c2 is a constant to be determined. Then
u(k ˜x1k2, k ˜x2k2) ≤ (c − α)k ˜x1k22+ (cα +√
ma)k ˜x1k2+√
mac =: h(k ˜x1k2).
Since h(k ˜x1k2) is a parabola-like curve which is open downward, there exists a c3 > 1 such that h(k ˜x1k2) < 0 whenever k ˜x1k2 ≥ c3.Thus, if c2 ≥ c23+1, then u(k ˜x1k2, k ˜x2k2) <
0 whenever k ˜x2k2 < 1 and k ˜x1k22+ k ˜x2k22 ≥ c2. Picking c0 = max{c1, c2}, we have that the assertion of the proposition holds true.
Proposition 3.1.2. Assume (3.11) holds true. Then lim
r→∞U( ˜x1, ˜x2) = ∞, where r = pk ˜x1k2+ k ˜x2k2.
Proof. From (3.10), we have that
U( ˜x1, ˜x2) = 1
2k ˜x1k2+ Xm
i=1
x4i,2
4 + c < ˜x2, ˜x1 >
≥ 1
2k ˜x1k2+ 1
4mk ˜x2k4− ck ˜x2k · k ˜x1k, Let 4m1 b21 > c2. Then suppose k ˜x2k > b1, we have
U( ˜x1, ˜x2) ≥ 1
2k ˜x1k2+ c2k ˜x2k2− ck ˜x2kk ˜x1k =: h1(k ˜x1k, k ˜x2k).
Since the level curve of h1(k ˜x1k, k ˜x2k) is elliptic-like in the plane. Thus, for any given M > 0, there exists a d1 > 0 such that U( ˜x1, ˜x2) > M whenever k ˜x1k2+ k ˜x2k2 ≥ d21
and k ˜x2k > b1. Let k ˜x2k ≤ b1. Then
U( ˜x1, ˜x2) ≥ 1
2k ˜x1k2− cb1k ˜x1k =: h2(k ˜x1k, k ˜x2k),
since h2(k ˜x1k, k ˜x2k) is a parabola-like curve which is open upward in the plane. Thus, for any given M > 0, there exists a d2 > 0 such that U( ˜x1, ˜x2) > M whenever k ˜x1k2+ k ˜x2k2 ≥ d22 and k ˜x2k ≤ b1. Picking δ = max{d1, d2}, we have that U( ˜x1, ˜x2) >
M for all k ˜x1k2+ k ˜x2k2 ≥ δ2. Thus, the assertion of the proposition holds true.
Theorem 3.1.1. The coupled system (3.7) is bounded dissipative if condition (3.11) holds true.
Proof. The proof is direct consequences of Propositions 3.1.1 and 3.1.2.
Thus, summarizing above results and applying Theorem 2.2.1, we get the follow-ing conclusion.
Theorem 3.1.2. Let f , D(c) and G(ǫ, r) be given as in (3.6a), (3.6b) and (3.6c), respectively. Let 0 ≤ c < 4+α4α2m. Then the coupled Duffing system (3.7) is globally synchronized provided that d is chosen sufficiently large.
Proof. It remains only to verify that G(ǫ, r) satisfies assumptions (2.7a) and (2.7b).
Indeed G(ǫ, r) is a circulant matrix (see e.g., [22]), the eigenvalues λk of G(ǫ, r) are
λk = −2ǫ(1 − cos2kπ
m ) − i 2r sin2kπ
m , k = 0, . . . , m − 1.