Synchronous Chaos in Coupled Map Lattices with General Connectivity Topology∗ Jonq Juang† and Yu-Hao Liang†
Abstract. The purpose of the paper is to address the synchronous chaos in coupled map lattices with general connectivity topology. Our main results contain the following. First, the master stability functions also hold for general connectivity topology with coupling through a nonlinear function that needs to be exactly the individual chaotic map. Second, the synchronization curve, composed of pieces of transverse Lyapunov exponent curves, is constructed. Third, necessary and sufficient conditions on coupling strength for yielding the synchronous chaos of the system are given. Moreover, the coupling strength dc giving the fastest convergence rate of the initial values toward the synchronous state
is explicitly obtained. It is also proved that such dc is independent of the choice of the individual
map. Finally, our results here can be applied to address questions of wavelength bifurcations and size instability.
Key words. stable synchronization, Lyapunov exponents, wavelength bifurcation, coupled map lattices AMS subject classifications. 34C15, 37N35
DOI. 10.1137/070705179
1. Introduction. A particularly interesting form of dynamical behavior occurs in networks of coupled systems or oscillators when all of the individual systems or oscillators acquire identical chaotic behavior. Such behavior of a network models many systems of interest in physics, biology, and engineering. A central dynamical question is: When is such synchronous behavior stable, especially in regard to coupling strengths in the network? Much progress in this direction has been made in lattices of coupled chaotic systems. Indeed, many results [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14] give analytical criteria for determining the range of coupling strength to acquire locally or even globally stable synchronization. On the other hand, to the best of our knowledge, there are no general results for global synchronization in coupled map lattices (CMLs). There are, however, globally synchronous results for some special cases (see, e.g., [15]). As to the study of local synchronization in CMLs, the notion of master stability functions (MSFs) that allows one to isolate the contribution of the network structure in terms of the eigenvalues of the coupling matrix was introduced in [8], [16], [17], [18], [19] to determine the possible range of coupling strength. This function then defines a region of stably synchronous state in terms of the coupling strength and the eigenvalues of the coupling matrix. Most of the work done in finding such a region of stability of the synchronous state is numerical. In a few certain cases, such as when the coupling matrix is symmetric, the MSFs can be further reduced to a number of inequalities [20], [21], [22], [23] (1.1) hmax+ ln|1 + dλi| < 0, i = 2, . . . , m.
∗Received by the editors October 12, 2007; accepted for publication (in revised form) by B. Ermentrout April 1,
2008; published electronically July 23, 2008.
http://www.siam.org/journals/siads/7-3/70517.html
†Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan ([email protected]. tw,[email protected]).
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Here hmax is the largest Lyapunov exponent of the individual map, λi are the nonzero
eigen-values of the m× m coupling matrix, and d is the coupling strength. The Gershgorin disk theory is then applied to obtain some sufficient conditions [23] on the coupling strength for lo-cal synchronization. The reason for the huge gap between the theory developed in the lattices of coupled chaotic systems and that of CMLs lies mostly in the fact that it is more natural to have a nonlinear coupling between oscillators in the CMLs. This is because a nonlinear coupling within suitable range of the coupling strength tends to yield an invariant region for the corresponding CMLs while linear coupling cannot. It should be noted that there is no such problem for the lattices of coupled chaotic systems. It should also be mentioned that all the analytical results of the lattices of the coupled chaotic systems stated above are linearly coupled.
The purpose of this paper is to give the best possible results for the local synchronization of the CMLs. Indeed, we first prove that (1.1) holds true for general connectivity topology with a limitation that the nonlinear coupling needs to be the individual chaotic map as well. Second, the synchronization curve, composed of pieces of transverse Lyapunov exponent curves, is derived. With the help of the synchronization curve, we give necessary and sufficient conditions on yielding synchronization of the CMLs. Such conditions then lead to the identification of the optimal coupling strength interval for acquiring synchronization of the CMLs. The optimal interval is to be termed the synchronization interval of the CMLs. Moreover, the coupling strength dc, called the center of the synchronization interval and giving the fastest
convergence rate of the initial values toward the synchronous state, can be identified. Such dcis
independent of the choice of the individual map. Like the applications, our work here can also be used to analytically quantify how the small-world scheme improves the synchronizability of the network [24], [25], [26], [27]. Furthermore, our results here can be applied to address questions of wavelength bifurcations [28], [29], [30], [31], [32] and size instability [32]. For CMLs or coupled chaotic systems, the following four scenarios are possible as the coupling varies: (i) no synchronization; (ii) the presence of short wavelength bifurcations (SWBs); (iii) the presence of intermediate wavelength bifurcations (IWBs); and (iv) the presence of long wavelength bifurcations (LWBs). Our main results give the following. First, if the coupling matrix has only real eigenvalues, then only (i) and (ii) are possible. Second, if the coupling matrix has complex eigenvalues, then all four scenarios are possible. Third, the critical values for which wavelength bifurcations occur as well as the exact number of oscillators capable of sustaining stably synchronous chaos can be explicitly computed. Finally, the minimum coupling value where all wavelength modes become de-excited enough to induce the stability of the synchronous state is also explicitly given.
We conclude this introductory section by mentioning the organization of the paper. The main results are contained in section 2. Three types of coupling matrices are provided in section 3 as illustrations and applications to our main results. Some concluding remarks about future research are addressed in section 4.
2. Main results. Consider a network of CMLs consisting of m oscillators. The equations of the motion then read
(2.1) xi(n + 1) = f (xi(n)) + d m k=1 gikh(xk(n)) , i = 1, . . . , m.
Here f :Rl→ Rl, l≥ 1, represents the individual chaotic map, and h : Rl→ Rlis an arbitrary nonlinear function describing how each oscillator’s variables are used in the coupling. The quantities gij are the coupling weights between the oscillators i and j. To consider the notion
of synchronization, we assume that mk=1gik = 0 for each i, and 0 is the simple eigenvalue of
the coupling matrix G = (gij). The quantity d represents the coupling strength of the CMLs
(2.1). To have an invariant region for CMLs (2.1), one usually chooses h as f . Such nonlinear coupling between oscillators is what makes (2.1) harder to treat analytically. In vector-matrix form with h = f , (2.1) becomes
(2.2) x(n + 1) = F (x(n)) + d(G⊗ I)F (x(n)),
where ⊗ denotes the Kronecker product, x(n) = (x1(n), . . . , xm(n))T, and F (x(n)) =
(f (x1(n)), . . . , f (xm(n)))T.
To study the stability of the synchronous state{xi = s∀i} of CML (2.2), we consider the
variational equation of (2.2):
ξ(n + 1) = DF (s(n))ξ(n) + d(G⊗ I)DF (s(n))ξ(n)
= [I⊗ Df(s(n)) + d (G ⊗ I) (I ⊗ Df (s(n)))] ξ(n), (2.3a)
where ξ = (ξ1, . . . , ξm) and each ξiis the perturbation to the ith oscillator. Let J = P−1GP ,
where J = [0]⊕ J1⊕ · · · ⊕ Jp is the real Jordan canonical form of G. Applying the change of
variables η = (P−1⊗ I)ξ, we get
η(n + 1) = [(I + dJ )⊗ Df(s(n))] η(n),
or, equivalently, in block diagonal form,
ηi(n + 1) = [(I + dJi)n⊗ Dfn(s(1))] ηi(1)
=: Ai(n)ηi(1).
(2.3b)
Let σ(A) denote the spectrum of A. Then σ(Ai(n)A∗i(n)) equals
σ([(I + dJi)n⊗ Dfn(s(1))] [(I + dJi∗)n⊗ (Dfn(s(1)))∗])
= σ([(I + dJi)n(I + dJi∗)n]⊗ [Dfn(s(1))· (Dfn(s(1)))∗])
= σ((I + dJi)n(I + dJi∗)n)· σ(Dfn(s(1))· (Dfn(s(1)))∗)
= σ((I + d ¯Ji)n(I + d ¯Ji∗)n)· σ(Dfn(s(1))· (Dfn(s(1)))∗),
where ¯J = [0]⊕ ¯J1⊕ · · · ⊕ ¯Jp is the Jordan canonical form of G. Consequently, the Lyapunov
exponents of (2.2) are
hj+ lim n→∞
lnλk,i
n .
Here hj are the Lyapunov exponents of the individual system f , and λk,i are the eigenvalues
of (I + dJλi)
n(I + dJ∗ λi)
n, where J
λi is a Jordan block of matrix G and λi is an eigenvalue of
G. Let the size of matrix Jλi be ki× ki, and let N = Jλi− λiI. It should be noted that for sufficiently large n, (I + dJλi) n = ((1 + dλ i)I + dN )n= (1 + dλi)n(I + αN )n = (1 + dλi)n I + ki−1 j=1 n j αjNj =: (1 + dλi)nTi,
where α = d/(1 + dλi). Clearly, the order of the magnitude of each entry of TiTi∗ is at most
O(n2ki−2). We conclude, via the Gershgorin disk theorem, that all eigenvalues of T
iTi∗ are of
the order O(n2ki−2). Consequently, the Lyapunov exponents of (2.2) are
(2.4) hj + ln|1 + dλi|.
We summarize the above as follows.
Theorem 2.1. Let G = (gij) be the coupling matrix satisfying that all its row sums are zero
and zero is a simple eigenvalue. Then the synchronous state of CML (2.2) is (locally) stable provided that
(2.5) hmax+ ln|1 + dλi| < 0, i = 2, . . . , m,
where hmax is the largest Lyapunov exponent of the individual map f and λi ∈ σ(G) − {0},
i = 2, . . . , m. That is to say, if d satisfies the inequalities in (2.5), then for any initial values of (2.2) that are sufficiently close to the synchronous state {xi = s∀i}, we have that each of
the oscillators xi(n) tends to the same state as n goes to infinity. Otherwise, CML (2.2) will
not acquire local synchronization.
Remark. (i) The decoupling form (2.3b) of variational equation (2.3a) was first observed and proposed by Pecora and Carroll [8]. (ii) If the identity matrix I in (2.2) is replaced by a diagonal matrix D with some but not all diagonal elements being zero, then the corresponding system (2.2) is called a partial-state coupling. The partial-state coupling also finds applications in various fields. For instance, in self-pulsating laser diode equations (see, e.g., [33]), only the photon density can be coupled with the electron density of the active region. Moreover, in the case of coupled chaotic systems, the systems that are partial-state coupled may exhibit different dynamic behavior. For instance, it is well known (see, e.g., [7]) that for the coupled Lorentz systems, only if the x-component or y-component is coupled will the resulting system achieve synchronization.
We shall assume from here on that the real parts of the eigenvalues of G are nonpositive. To find the range of the coupling d so that (2.5) is fulfilled, we need to solve the following min max problem:
min
d∈R2max≤i≤m|1 + dλi| = mind>02≤i≤mmax1
|1 + dλi| =: min d>02≤i≤mmax1 ri(d) =: min d>0r(d). (2.6)
Here m1 is the number of eigenvalues lying in upper complex plane or on the real axis. The curves ri(d) are termed the ith mode of the transverse Lyapunov exponent curves. The
equalities above are due to the facts that|1 + dλi| = |1 + d¯λi|, the real parts of the eigenvalues
of G are nonpositive, and (2.5) is violated whenever d≤ 0. Without loss of generality, we may assume that those distinct nonzero eigenvalues are λi, i = 2, . . . , m1, with 0 < |λ2| ≤ · · · ≤
|λm1|. The coupling value d := dc solving the min max problem (2.6) is the optimal choice of
the coupling in the sense that it gives the fastest convergence rate of the initial values toward the synchronous state. To understand how r(d) is formed, we need to know the ordering of
ri(d). For d > 0, direct computations yield
ri(d) = |λi|2d2+ 2 Re(λi)d + 1 1 2 = |λi|2 d−Re(−λi) |λi|2 2 +|λi| 2− Re2(λ i) |λi|2 1 2 =: |λi|2(d− ci)2+ tan2θi 1 2 . (2.7a)
Moreover, ri(d)≥ rj(d) if and only if
(2.7b) Re(λi)≥ Re(λj) if|λi| = |λj|, and (2.7c) (|λi|2− |λj|2)(d− dij)≥ 0 if |λi| = |λj|, where (2.7d) dij = 2(Re(−λi)− Re(−λj)) |λi|2− |λj|2 .
Let Ai = {j : 2 ≤ j ≤ m1 and |λi| = |λj|}. Then maxj∈Ai|1 + dλj| = |1 + dλk|, where k is chosen so that Re(λk) ≥ Re(λj) ∀j ∈ Ai. This gives that within each of the index set Ai,
their corresponding quantities|1 + dλi| are well ordered for any d > 0. Consequently, to solve
(2.6), we may assume, without loss of generality, that 0 <|λ2| < · · · < |λm1| from here on.
Using the terminology in [32], we see that the numbers 2 and m1 correspond to the longest
and shortest wavelength modes, respectively. The numbers in between 2 and m1 are to be
called intermediate wavelength modes. Since dij = dji for any i and j we consider only dij
with i > j. Our reduction process is now complete.
The following procedures are proposed to determine the “actual” node points of r(d) from the candidate set {dij : i > j}.
(A) Set k0 = 0, and k1 = max{l | Re(λi)≤ Re(λl) ∀i = 2 . . . , m1}. Let k2 be the largest
index so that 0 < dk2k1 ≤ dkk1 ∀k1 < k≤ m1.
(B) Let k3 be the largest index so that dk2k1 < dk3k2 ≤ dkk2 ∀k2 < k. The process can be
continued until kp = m1 for some p≤ m1.
The next result shows that {ki}pi=1 is the set of “actual” node points of r(d).
Theorem 2.2. Let G be given as in Theorem 2.1. Assume that the real parts of the eigen-values of G are nonpositive. Then r(d) = rki(d) whenever dkiki−1 ≤ d ≤ dki+1ki, i = 1, . . . , p.
Here dk1,k0 = 0 and dkp+1kp =∞.
Proof. Denote Ij = [dkjkj−1, dkj+1kj]. It then follows from (2.7c) that if i > j and dij > 0,
then ri(d) > rj(d) whenever d > dij and ri(d) < rj(d) whenever 0 < d < dij. We then
conclude that
(i) the ordering of ri(d) and rj(d) remains the same until both curves meet;
(2.8a)
(ii) if ri(d∗) > rj(d∗) for some d∗ > 0 with i > j, then ri(d) > rj(d)∀d ≥ d∗.
(2.8b)
Using the first inequality in (2.7a), we have that r(d) = rk1(d) for 1 > d ≥ 0. Here 1 is sufficiently small. It then follows from (2.8a), (2.8b), and procedure (A) that r(d) = rk1(d)
on I1. Upon using (2.8a), we conclude that r(d) = rk2(d) for d ∈ (dk2k1, dk2k1 + 2). Here
2 is sufficiently small. Similarly, r(d) = rk2(d) on I2. We omit the proof of the remaining assertions of the theorem due to the similarity.
Note that not all cki given in (2.7a) could be critical points of r(d). In fact, the critical
points of r(d) may not even come from the set {cki}. We next identify the “actual” critical points of r(d). Our next main result shows that r(d) has exactly one critical point.
Theorem 2.3. The curve r(d) has a unique critical point dc that solves the min max problem
(2.6). Moreover, the optimal range of coupling d to sustain stably synchronous chaos of (2.2) is (dl, dr). Here dland dr, dl< dr, are the intersection points (if any exist) of the straight line
y = e−hmax and the curve y = r(d). Consequently, CML (2.2) acquires local synchronization
if and only if d∈ (dl, dr).
Proof. We break up the proof of the theorem into the following three steps. Step I. We first claim that the number of cki lying in the interior
◦
Ii of Ii is at most one.
Indeed, suppose there exist cka ∈
◦
Ia and ckb ∈
◦
Ib with cka < ckb. Then the following hold
true: (i) rka(ckb) > rka(cka). (ii) rka(cka) > rkb(cka). (iii) rkb(cka) > rkb(ckb). Inequalities (i)
and (iii) hold true since cka and ckb are, respectively, the minimum points of rka(d) and rkb(d).
The fact that rka(d) lies above all other curves on Ialeads to inequality (ii). Combining these
inequalities, we have that rka(ckb) > rkb(ckb), which is in contradiction to the fact that rkb is
the maximum curve on Ikb.
Step II. We next show that if cki ∈
◦
Ii, then r(d) is decreasing on (0, cki) and increasing
on (cki,∞). Indeed, for d ∈ Ii+1, r(d) = rki+1(d) > rki(d) > rki(dki+1ki) = rki+1(dki+1ki).
Using the conclusion in Step I and the fact that r2k
i(d) is parabolic, we conclude that rki+1(d)
must be increasing on Ii+1. On the other hand, rki−1(d) must be decreasing on Ii−1 since
rki−1(d) > rk(d) > rki(dkiki−1) = rki−1(dkiki−1). The monotonicity of r(d) on each interval Ij,
1≤ j ≤ m1, can be similarly determined.
Step III. Since r(d) is decreasing initially on I1and increasing eventually on Ip, there must
be at least one critical point. If such points do not lie in the set of node points, then r(d) has a unique critical point. Suppose cki ∈/
◦
Ii ∀i = 1, . . . , p. Then r(d) is monotonic on each interval
Ii. Suppose r(d) first changes its monotonicity at dkl+1kl for some l. Then an argument similar
to that given in Step II shows that once r(d) becomes increasing on Il+1, it will stay increasing
the rest of the way. We have just completed the proof of the theorem.
Remark. (i) If the straight line y = e−hmax and the curve y = r(d) do not intersect, then
CML (2.2) will not achieve synchronization for any coupling strength. Suppose drand dlexist.
Then, as soon as d exceeds dr, a certain wavelength mode is excited, which, in turn, causes
the instability of the synchronous state. The illustration in Examples 2 and 3 shows that the excited wavelength mode could be either the shortest wavelength mode, the intermediate wavelength mode, or the longest wavelength mode. In any event, dris the exact critical value
where wavelength bifurcation occurs. On the other hand, dl is the exact critical value where
all wavelength modes become de-excited enough to induce the stability of the synchronous state. (ii) Such r(d) is called the synchronization curve of CML (2.2), and the interval (dl, dr),
if it exists, is termed the synchronization interval of CML (2.2). Clearly, dc ∈ (dl, dr) and
depends only on the connectivity topology.
Theorem 2.4. Suppose the coupling matrix G has nonpositive real eigenvalues. Denote by {λi}m1i=2 the distinct nonzero eigenvalues of G. Then
r(d) =
λ2(d), d∈ [0, dm12] = I1, λm1(d), d∈ (dm12,∞) = I2,
and dc = dm12 = λ2+λ−2
m1. Consequently, depending on the quantity of hmax, either CML (2.2)
achieves no synchronization or SWB occurs as d varies. Furthermore, if dl and dr exist, then
the synchronization interval of the corresponding CMLs is 1−e−λ−hmax
2 ,
1+e−hmax
−λm1
.
Proof. It is easily seen that k1 = 2 and k2 = m1 since dij = λi−2+λj. Thus, r(d) is as
asserted. The proof then follows from the facts that cm1 = −λ1
m1 <
−2
λ2+λm1 < − 1
λ2 = c2
and dc = dm12. Solving equations y = r(d) and y = e−hmax, we have that dl and dr are as
claimed.
3. Illustrations and applications. We illustrate our theorems with the following examples.
Example 1. Let the oscillators be diffusively coupled with periodic boundary conditions. For such G, m1= m, −λm1 = 4 sin2 [
m
2]π
m , and −λ2 = 4 sin
2 π
m.
Let f (x) = 4x(1− x), 0 ≤ x ≤ 1. Then hmax = ln 2, and the corresponding candidates for dl and dr are, respectively, 18sin−2 πm and 38sin−2 [
m
2]π
m . However, dl ≤ dr only if m≤ 5.
Hence, we conclude that the maximum number of oscillators to sustain synchronous chaos is 5.
We next compare our results with those obtained in [23], [34]. Their sufficient conditions on the coupling strength for obtaining stable synchronization are, respectively, given as follows:
1−e−hmax m < d gij < 1+e−hmax m and m k=1, k =i|gki− gji|
+1d+ gii− gji< 1de−hmax ∀i, j with
i = j. However, the first inequality above fails to find any suitable coupling strength provided that G has zero off-diagonal elements. If G is given as above with m≥ 4 and f(x) = 4x(1−x), then the second inequality also fails to find any suitable coupling strength.
Example 2. Consider synchronization in a directed ring of 2K nearest neighbor coupled oscillators [19] with K = 2 and m = 9. Specifically, the coupling matrix G under consideration is a circulant matrix of the form
G = circ(−30, 13, 2, 0, . . . , 0, 5.4, 9.6).
The spectrum of G is{−30+13e2(j−1)π9 i+2e 4(j−1)π 9 i+5.4e 14(j−1)π 9 i+9.6e 16(j−1)π 9 i : j = 1, . . . , 9}.
Here λ2 ≈ −11.4024 + 1.1629i, λ3 ≈ −33.0293 + 2.1855i, λ4 ≈ −45 + 5.8890i, and λ5 ≈ −45.5683 + 3.3483i. Direct computations yield that d42≈ 0.0348 < d52 < d32, d54 ≈ 0.0406, and c5 < c4 < d42< d54< c2 (see Figure 1). Consequently,
r(d) = ⎧ ⎨ ⎩ r2(d), d∈ I1= [0, d42], r4(d), d∈ I2= [d42, d54], r5(d), d∈ I3= [d54,∞],
the node points of r(d) are d42 and d54, and the critical point of r(d) occurs at d42. Let fμ(x) = μx(1− x). For μ = 4, since e−hmax = e− ln 2 = 0.5 < r(d42), the synchronization
interval does not exist. As μ varies from μ∞ ≈ 3.57 to μ = 4, scenarios (i), (ii), and (iii) described in the introductory section can be clearly observed from the figure.
On the other hand, the maximum number of oscillators on such connectivity topology to sustain stably synchronous chaos is 7. The claim above is done by checking the intersection of the equations y = 12 and y = r(d) ∀m ≤ 8.
0 0.08 0.1 0 0.5 1 1.2 c2 c5c4 r (d)2 r (d)5 d42d54 r (d)4
Figure 1. Graph of r(d) in Example2. Here d42≈ 0.0348, d54≈ 0.0406, r(d42)≈ 0.6040, r(d54)≈ 0.8604, c5≈ 0.02182, c4 ≈ 0.02185, and c2≈ 0.0868. The critical point of r(d) is d42.
Example 3. The following example shows that LWB is also possible. Let G be given as
follows: ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ −30 3 12 5 10 10 −30 3 12 5 5 10 −30 3 12 12 5 10 −30 3 3 12 5 10 −30 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠.
The spectrum of G is{0, −35.2639 + 10.7719i =: λ2,−39.7361 + 2.5429i =: λ3, ¯λ2, ¯λ3}. Then the graph of r(d) is demonstrated in Figure 2. Consider f (x) = 4x(1− x). Then the longest wavelength mode becomes excited to induce instability as d is increased beyond dr.
Example 4. To illustrate the accuracy of our theorems, synchronization intervals estab-lished in Theorem 2.4 are compared with those obtained by the computer simulation. In particular, theoretically and numerically predicted synchronization intervals for three
Figure 2. Graph of r(d) in Example 3. Here d32 ≈ 0.0396, c2 ≈ 0.0259, c3 ≈ 0.0251, dl≈ 0.0149, and
dr≈ 0.0369. The critical point of r(d) is c2.
ples above are almost identical. Such comparisons are recorded in Figure3. They are “almost” identical. This simulation is set up so that the differences between the initial values xi are
within 10−5. Synchronization is achieved when their differences are within 10−15.
d (coupling strength) G (coupling matrix) Ex3,m=5 Ex2,m=9 Ex1,m=4 0.01 0.02 0.03 0.04 0 0.25 0.5 0.75 1 0.2 0.25 0.3 0.35 0.4
Figure 3. Three typical synchronization intervals for coupled logistic map with various coupling matrices are shown. Solid (bold) lines are synchronization intervals obtained by computer simulation. Dotted (fine) lines are synchronization intervals predicted by our theorems. All are scaled for clear visualization.
4. Conclusions. We conclude this paper by mentioning the difficulty one might face by applying our methods to more general cases, D = I or h = f, and a possible approach to solving them.
Our main results in this paper are based on the study of the inequalities in (2.5). However, in the case that D = I or h = f, it seems to be a nontrivial matter to find their corresponding inequalities such as (2.5). One possible approach is to find the lower and upper bounds of the Lyapunov exponents of (2.2), where both bounds have expressions similar to those in (2.4).
Acknowledgment. The authors would like to thank the referees for their helpful com-ments.
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