- Email: [email protected]
Local phase synchronization and clustering for the delayed phase-coupled oscillators with plastic coupling
JID:YJMAA
AID:20537 /FLA
Doctopic: Applied Mathematics
[m3L; v1.181; Prn:11/07/2016; 17:25] P.1 (1-10)
J. Math. Anal. Appl. ••• (••••) •••–•••
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Local phase synchronization and clustering for the delayed phase-coupled oscillators with plastic coupling Yicheng Liu a,∗ , Jianhong Wu b a
College of Science, National University of Defense Technology, Changsha, 410073, PR China Laboratory for Industrial and Applied Mathematics (LIAM), Department of Mathematics and Statistics, York University, Toronto, Ontario M3J 1P3, Canada b
a r t i c l e
i n f o
Article history: Received 22 December 2015 Available online xxxx Submitted by J. Shi Keywords: Frequency synchronization Phase synchronization Kuramoto model Clusters Phase-coupled oscillators
a b s t r a c t In this paper, we investigate the local synchronization problem for delayed phasecoupled oscillators with plastic coupling (also referred as a generalized Kuramoto model). By linearizing the phase-coupled oscillators system at a special solution, an approximation phase-coupled system with delay effects is deduced. Moreover, we find the multiplicity of semisimple zero eigenvalue for coupling strength coefficients matrix and the rank of Vandermonde matrix associated initial values are two sensitive factors. The former is closely related to the maximum number of clusters phase of linear coupled system, and the latter determines clusters phase synchronization. As results, both frequency synchronization criteria and phase synchronization criteria are established in this literature. Also, the final synchronization frequency and phase are formulated in terms of coupling strength, initial values and time delay. © 2016 Elsevier Inc. All rights reserved.
1. Introduction The synchronization problem of phase-coupled oscillators has been widely studied in different disciplines ranging from physics [5,20], networks [4,10,19] and engineering [7,16]. Synchronization of Kuramoto oscillators (a classical phase-coupled system), for example, has been extensively analyzed under various assumptions on homogeneous oscillators and network topology. In particularly, Luca [18] conducted an interesting study on synchronization and stability for systems of homogeneous oscillators with plastic coupling, but the study focused on complete graph topology. Recently, Gushchin–Mallada–Tang [8] discussed a system of phase-coupled oscillators with plastic coupling governed by the following two classes of equations: θ˙i (t) = ωi +
kij · fij (θj (t) − θi (t)), 1 ≤ i ≤ n,
j∈Ni
* Corresponding author. E-mail addresses: [email protected] (Y. Liu), [email protected] (J. Wu). http://dx.doi.org/10.1016/j.jmaa.2016.06.049 0022-247X/© 2016 Elsevier Inc. All rights reserved.
(1)
JID:YJMAA 2
AID:20537 /FLA
Doctopic: Applied Mathematics
[m3L; v1.181; Prn:11/07/2016; 17:25] P.2 (1-10)
Y. Liu, J. Wu / J. Math. Anal. Appl. ••• (••••) •••–•••
k˙ ij (t) = sij (αij Fij (θj (t) − θi (t)) − kij ), (i, j) ∈ E,
(2)
where θi is the phase and ωi the intrinsic frequency of oscillator i, E is the set of edges, Ni is a set of oscillators connected to oscillator i (i.e. the set of its neighbors), kij and fij which are 2π-periodic continx uously differentiable functions. Fij (x) = − 0 fij (t)dt + C with the integration constant C chosen so that π Fij (t)dt = 0. Positive constants αij determine minimum and maximum values of the coupling strengths, 0 and positive constants sij define rates of change of the coupling strengths. This system is coupled by two equations, the first defines behavior of an oscillator, and the second determines dynamics of the coupling strengths. In the special case where kij is constant and fij (θj − θi ) = sin(θj − θi ), system (1) becomes the Kuramoto model [14]. If fij (θj − θi ) = sin(θj − θi ) and Fij (θj − θi ) = cos(θj − θi ), then the system (1)–(2) becomes a generalized Kuramoto model [22]. It is called a homogeneous oscillators system if all intrinsic frequencies of oscillators are equal, i.e. there exists a constant ω such that ω1 = ω2 = · · · = ωn = ω. The synchronization of phase-coupled oscillators may appear at two different levels: frequency synchronization and phase synchronization. The synchronization, including frequency synchronization and phase synchronization, has been widely investigated in recent literatures, see [1,4,9–11,15,16,21] and references therein. Our objective in this work is to develop local phase synchronization and clustering criteria for phasecoupled homogeneous oscillators system with non-constant coupling, distributed delay and general network topology. The distributed delays are incorporated into (1)–(2), so we obtain
θ˙i (t) = ω +
0 kij f (
j∈Ni
μ(s)θj (t + s)ds − θi (t)),
(3)
−τ
0 k˙ ij (t) = sij (αij F (
μ(s)θj (t + s)ds − θi (t)) − kij ),
(4)
−τ
where τ is the maximal coupling delay and the distribution is normalized so that to specify a solution of (3)–(4), we will need to prescribe the initial conditions
0 −τ
μ(s)ds = 1. Naturally,
θi (t) = γi (t) for all t ∈ [−τ, 0], 1 ≤ i ≤ n, where each γi is a continuous function. For more discussions why time delay should be introduced, we refer to [2,5,16,17,19] and references therein. In what follows, we say that system (3)–(4) achieves frequency synchronization if limt→∞ θ˙1 (t) = · · · = limt→∞ θ˙n (t) = Ω and limt→∞ k˙ ij (t) = 0 for all i, j, where Ω is a common synchronization frequency. We say that system (3)–(4) reaches phase synchronization if it achieves frequency synchronization, and limt→∞ (θ1 (t) − Ωt) = limt→∞ (θ2 (t) − Ωt) = · · · = limt→∞ (θn (t) − Ωt) = ϕ (a constant). We say that a system reaches p clusters phase synchronization if it achieves frequency synchronization and for some set of initial values there exist some constants ϕj ∈ R and sets Pj ⊂ {1, 2, · · · , n} satisfying Pj ∩ Pi = ∅ (empty set), ϕj = ϕi (whenever i = j) and ∪j Pj = {1, 2, · · · , n}, such that limt→∞ (θi (t) − Ωt) = ϕj for all i ∈ Pj , j = 1, 2, · · · , p. The numbers ϕj are then called the clustering phases. The structure of the article is as follows. In the next section we linearize the delayed phase-coupled oscillators system and normalize the adjoint matrix. In Section 3, we provide the normal zero-one vectors decomposition in the nullspace. In section 4, a necessary and sufficient condition for clusters phase synchronization is obtained. In this section, both frequency synchronization criteria and phase synchronization criteria are also described. The final synchronization frequency and synchronization phase are formulated in terms of coupling strength, initial values and time delay.
JID:YJMAA
AID:20537 /FLA
Doctopic: Applied Mathematics
[m3L; v1.181; Prn:11/07/2016; 17:25] P.3 (1-10)
Y. Liu, J. Wu / J. Math. Anal. Appl. ••• (••••) •••–•••
3
2. Normalization and linearization Firstly, we assume that the coupling strength coefficients αij satisfy the row sum invariance (here, αij = 0 for (i, j) ∈ / E). That is, there is a constant α so α =
n
αij for all i.
(5)
j=1
Then we introduce the normalized coupling strength coefficients matrix A = (aij )n×n , where aij =
αij for (i, j) ∈ E; aij = 0 for (i, j) ∈ / E. α
n A is called the normalized weighted adjacency matrix of oscillators network. It is clear that j=1 aij = 1 for all i. Then L = I − A is a normal Laplacian matrix. It is easy to see that zero is an eigenvalue of L and 1 = (1, 1, · · · , 1)T belongs to the corresponding nullspace. In what follows, we assume that zero is a semisimple eigenvalue of L with the multiplicity n0 . We need to consider a special solution of system (3)–(4) as follows θi (t) = Ωt and k˙ ij (t) = 0 for all i, j,
(6)
where Ω denotes the common synchronization frequency, which is one of the roots of the following algebra equation Ω = ω + αF (Ω¯ τ )f (Ω¯ τ ). The static coupling strength k˜ij is given by k˜ij = αij F (Ω¯ τ ), where τ¯ = Introducing the phase variable ϕi (t) of oscillator i by
(7) 0 −τ
sμ(s)ds < 0.
ϕi (t) = θi (t) − Ωt for all i, and linearizing oscillators system (3)–(4) around the special solution (6), we obtain the asymptotically phase-variable system
ϕ˙ i (t) = ε
n j=1
0 aij (
μ(s)ϕj (t + s)ds − ϕi (t))
(8)
−τ
for all i, where ε = αF (Ω¯ τ ) · f (Ω¯ τ ). Consequently, we observed that the delayed phase-coupled homogeneous oscillators system (3)–(4) reaches locally frequency synchronization that is determined by the convergence of system (8). Note also that when system (8) reaches clustering synchronization, the oscillators system (3)–(4) reaches locally clustering phase synchronization. 3. Structure analysis of the linearization As will be clear soon, if 1 can be spitted into k zero-one vectors (zero-one vector means each component of the vector is either 1 or 0), then the phase state set is spitted into possibly k classes. Therefore, we define a number K as follows:
JID:YJMAA 4
AID:20537 /FLA
Doctopic: Applied Mathematics
[m3L; v1.181; Prn:11/07/2016; 17:25] P.4 (1-10)
Y. Liu, J. Wu / J. Math. Anal. Appl. ••• (••••) •••–•••
K = max{k : there are k nonzero zero-one vectors a1 , · · · , ak such that Lai = 0 and a1 + · · · + ak = 1}. Obviously, K is well-defined since L1 = 0. Lemma 3.1. Assume zero is a semisimple eigenvalue of the Laplacian L with multiplicity n0 . Then K = n0 . Proof. Without loss of generality, we assume the zero-one vector vectors a1 , · · · , aK are of the following forms (if necessary, we exchange the rows of matrix L and relabel the subscript of xi ): a1 = (1, · · · , 1, 0, · · · , 0), where the number of components 1 is t1 ; a2 = (0, · · · , 0, 1, · · · , 1, 0, · · · , 0), where t2 components 1 following t1 components 0; ··· ,··· aK = (0, · · · , 0, 1, · · · , 1), where the number of components 1 is tK . Also, we see that ti is the minimum number of components 1 in ai and t1 + t2 + · · · + tK = n. In what follows, a set {a1 , · · · , aK } is called a normal zero-one vector basis if a1 , · · · , aK are of above forms. In this case, noting that Lai = 0(i = 1, · · · , K) and aij ≥ 0, we see that L is a block diagonal matrix, that is, L = diag(D1 , D2 , · · · , DK ). Let f (λ) = det(λI −Di ). Then f (0) = 0. Noting Di is M -matrix and ti is the minimum number, it is easy to see that f (0) = 0. Thus 0 is a simple eigenvalue of matrix Di (i = 1, 2, · · · , K). Hence 0 is an eigenvalue of matrix L with multiplicity K. On the other hand, the multiplicity of semisimple zero eigenvalue is n0 , we conclude K = n0 . 2 Remark 3.1. From the above argument, we see that if zero is a semisimple eigenvalue of the Laplacian L with multiplicity n0 , then L is a block diagonal matrix (if necessary, we exchange the rows of matrix L and renumber the subscript). Lemma 3.2. Assume zero is a semisimple eigenvalue of the Laplacian L with multiplicity n0 . Then there exists a unique family normal zero-one vectors a1 , · · · , an0 such that Lai = 0 and a1 + a2 + · · · + an0 = 1. Proof. From Lemma 3.1, we choose a family of normal zero-one vectors a1 , · · · , an0 such that Lai = 0 and a1 + a2 + · · · + an0 = 1. Thus the existence is established. Next, we show the uniqueness. If there is another family of normal zero-one vectors b1, · · · , bn0 such that Lbi = 0 and b1 + b2 + · · · + bn0 = 1. Without loss of generality, we assume both clusters a1 and b1 include the first oscillator, and b1 has the form b1 = (1, · · · , 1, 0, · · · , 0), where the number of components 1 is s1 . Case s1 < t1 : In this case, we have L(a1 −b1 ) = 0. Also, {b1 , a1 −b1 , a2 , · · · , an0 } is linearly independent. This means that the dimension of the nullspace {v : Lv = 0} is no less than n0 + 1. But its dimension is n0 , a contradiction.
JID:YJMAA
AID:20537 /FLA
Doctopic: Applied Mathematics
[m3L; v1.181; Prn:11/07/2016; 17:25] P.5 (1-10)
Y. Liu, J. Wu / J. Math. Anal. Appl. ••• (••••) •••–•••
5
Case s1 > t1 : Assume t1 < s1 < t2 (if s1 is larger than t2 , then the argument is similar), then L(a1 + a2 − b1 ) = 0. Also, {a1 , a1 + a2 − b1 , b2 , · · · , bn0 } is linearly independent. So, the dimension of the nullspace is also no less than n0 + 1. It is also impossible due to the fact that the dimension of nullspace is n0 . Based on the above arguments, we claim that a1 = b1 . Similarly, we can prove ai = bi , i = 2, · · · , n0 . Thus the uniqueness is verified. This completes the proof. 2 4. Synchronization and phase clustering In this section, we study the synchronization and clustering problems of linear system (8). The goal of this section is to establish the phase synchronization and clustering criteria of system (8). The following standard notations and results about Laplacian matrices can be found in [3,6]. Since all off-diagonal entries of the Laplacian matrix I −A are nonpositive, the real parts of its eigenvalues are nonnegative. Let {λ1 , λ2 , · · · , λn } denote the set of eigenvalues of L. Then, 0 ≤ Re(λi ), i = 1, 2, · · · , n. Furthermore, by an application of the Gershgorin’s circle theorem [13], we have |1 − λi | ≤ 1. Noting that zero is a semi-simple eigenvalue of L with multiplicity n0 . Then there is a nonsingular matrix Q such that
0 Z
On0 L=Q 0
Q−1 ,
where On0 is a n0 -dimensional zero matrix (1 ≤ n0 ≤ n) and Z is a matrix with spectral ρ(Z) ∈ (0, 2]. Let {v1 , · · · , vn } be the columns of Q, which is a basis of Rn . Set {u1 , · · · , un } be the dual basis corresponding to {v1 , · · · , vn }. That is, ui are linearly independent vectors such that < ui , vj > = δji . It is easy to see that n ui are left eigenvectors of L, ui L = 0ui = 0 for i = 1, 2, · · · , n0 . Furthermore, if x = i=1 αi vi ∈ Rn , then αi =< ui , x >. Let x = (ϕ1 , ϕ2 , · · · , ϕn )T . Then system (8) is transformed to the vector version 0 ˙ x(t) = −εx(t) + εA
μ(s)x(t + s)ds.
(9)
−τ
Let x(t) be a solution of equation (9). Considering the decomposition x(t) = αi (t) =< ui , x(t) >, we have
n i=1
αi (t)vi , where
0 α˙ i (t) = −εαi (t) + ε(1 − λi )
μ(s)αi (t + s)ds
(10)
−τ
for all i. Its characteristic equation is 0 z = −ε + ε(1 − λi )
μ(s)ezs ds.
(11)
−τ
Lemma 4.1. [2] Let ε > 0. If λi = 0, then (11) has a simple root at zero and all other roots have negative real parts. If λi = 0, then all roots have negative real parts. Following Lemma 4.1, we see that lim αi (t) = 0 for all i > n0 .
t→∞
JID:YJMAA
AID:20537 /FLA
Doctopic: Applied Mathematics
[m3L; v1.181; Prn:11/07/2016; 17:25] P.6 (1-10)
Y. Liu, J. Wu / J. Math. Anal. Appl. ••• (••••) •••–•••
6
Hence, lim (x(t) − α1 (t)v1 − · · · − αn0 (t)vn0 ) = 0.
t→∞
In order to calculate the limits of α1 (t), · · · , αn0 (t) as t → ∞, using a similar argument in [2], we consider the initial functional space C([−τ, 0], R), which is the Banach space of real-valued continuous functions on [−τ, 0] equipped with the supremum norm. Let C ∗ = C([−τ, 0], R), and define the bilinear form < ψ, φ >c for φ ∈ C and ψ ∈ C ∗ by 0 0 μ(r)ψ(r − t)φ(t)dtdr.
< ψ, φ >c = ψ(0)φ(0) + ε −τ r
Since λi = 0 for i = 1, 2, · · · , n0 , the characteristic equation (11) has a simple root at zero, thus its characteristic subspace is an one-dimensional space, say, C0 . Denote C0∗ by the dual one-dimensional space. 1 Let constant function Φ(θ) = 1 be the basis of C0 , and then its dual basis in C0∗ is Ψ(θ) = 1−ε¯ τ with < Ψ, Φ >c = 1. Also let at (θ) = αi (t + θ) for all θ ∈ [−τ, 0]. It follows from the theory of functional differential equations [12] that the space C can be decomposed using the invariant subspace C0 and its complement. Thus at = a0t + bt , where a0t ∈ C0 . Following Lemma 4.1, we see that the characteristic equation has a simple root at zero and all other roots have negative real parts. Thus limt→∞ bt (θ) = 0 uniformly for θ ∈ [−τ, 0]. Noting that a0t is a constant and determined by the initial condition, we conclude that it is given as the constant function a0t =< Ψ, a0 >c Φ. Thus, by direct calculation, we have
lim αi (t) =
t→∞
a0t
0 0
1 [αi (0) + ε = 1 − ε¯ τ
μ(θ)αi (r)drdθ]
−τ θ
1 < ui , x(0) + ε = 1 − ε¯ τ
0 0 μ(θ)x(r)drdθ >
−τ θ
for all i = 1, · · · , n0 . We introduce several projection coordinate vectors in the nullspace. Following Lemma 3.2, we conclude that there exists a unique family of normal zero-one vectors a1 , · · · , an0 such that Lai = 0 and a1 + a2 + · · · + an0 = 1. Basing on these vectors, we introduce c = (< u1 , 1 >, · · · , < un0 , 1 >)T , β = (β1 , β2 , · · · , βn0 )T , βi =
1 < ui , x(0) + ε 1 − ε¯ τ
0 0 μ(θ)x(r)drdθ >,
−τ θ
ci = (< u1 , ai >, · · · , < un0 , ai >)T , 1 ≤ i ≤ n0 , U = [v1 , v2 , · · · , vn0 ]. Let
JID:YJMAA
AID:20537 /FLA
Doctopic: Applied Mathematics
[m3L; v1.181; Prn:11/07/2016; 17:25] P.7 (1-10)
Y. Liu, J. Wu / J. Math. Anal. Appl. ••• (••••) •••–•••
⎡
1 ⎢ α ⎢ 1 V (α1 , α2 , · · · , αn0 ) = ⎢ ⎣ ··· α1n0 −1
1 α2 ··· α2n0 −1
··· ··· ··· ···
7
⎤ 1 αn0 ⎥ ⎥ ⎥ ··· ⎦ αnn00 −1
denote the n0 × n0 Vandermonde matrix, and αi =
1 < U β, ai >, 1 ≤ i ≤ n0 . ti
(12)
Theorem 4.1. Assume coupling strength coefficients satisfy (5), F (Ω¯ τ ) · f (Ω¯ τ ) > 0 and zero is a semisimple eigenvalue of the Laplacian L with multiplicity n0 . Then the coupling system (3)–(4) locally reaches m clusters phase synchronization if and only if Rank(V (α1 , α2 , · · · , αn0 )) = m (1 ≤ m ≤ n0 ). Proof. Since F (Ω¯ τ ) · f (Ω¯ τ ) > 0, then ε > 0 and Lemma 4.1 holds. Noting {a1 , a2 , · · · , an0 } is linearly independent, and then it is also a basis of nullspace Span{v1 , v2 , · · · , vn0 }. On the other hand, we see that lim x(t) =
t→+∞
n0
βi vi = U β ∈ Span{a1 , a2 , · · · , an0 }.
i=1
Thus there are some constants αi , such that U β = α1 a1 + α2 a2 + · · · + αn0 an0 . Also, since < ai , aj > = 0 for i = j and < ai , ai > = ti , we have αi = t1i < U β, ai >. Thus the linear system (8) reaches m clusters if and only if α1 , α2 , · · · , αn0 has m pairwise-different numbers. Meanwhile, following the formulation of Vandermonde determinant
Det(V (α1 , α2 , · · · , αn0 )) =
(αi − αj ),
1≤i
we see that the rank of Vandermonde matrix is m if and only if there are m pairwise-different numbers in α1 , α2 , · · · , αn0 . Let φ1 , φ2 , · · · , φm be pairwise-different numbers. Then lim (θi (t) − Ωt) = φj when i ∈ Pj ,
t→∞
where the sets Pj ⊂ {1, 2, · · · , n} satisfy Pj ∩ Pi = ∅, (whenever i = j) and ∪j Pj = {1, 2, · · · , n}. This implies that the coupling system (3)–(4) locally reaches m clusters phase synchronization if and only if Rank(V (α1 , α2 , · · · , αn0 )) = m. This completes the proof. 2 τ )·f (Ω¯ τ) > 0 Corollary 4.1 (Synchronization criteria). Assume coupling strength coefficients satisfy (5), F (Ω¯ and zero is a semisimple eigenvalue of the Laplacian L with multiplicity n0 . The coupling system (3)–(4) locally achieves synchronization if and only if Rank(V (α1 , α2 , · · · , αn0 )) = 1. More precisely, we have 1). If n0 = 1, the coupling system (3)–(4) locally achieves synchronization, and the synchronization frequency is Ω, the synchronization phase is 1 < u1 , x(0) + ε 1 − ε¯ τ
0 0
−τ θ
μ(θ)x(r)drdθ >;
JID:YJMAA
AID:20537 /FLA
Doctopic: Applied Mathematics
[m3L; v1.181; Prn:11/07/2016; 17:25] P.8 (1-10)
Y. Liu, J. Wu / J. Math. Anal. Appl. ••• (••••) •••–•••
8
2). If n0 > 1, the coupling system (3)–(4) locally reaches a conditional synchronization. In other words, the coupling system (3)–(4) locally reaches synchronization if and only if the initial values satisfy < u1 , x(0) + ε
0 0
−τ θ < u1 , 1
j
=
< u , x(0) + ε
0 0
−τ θ < uj , 1
μ(θ)x(r)drdθ > > μ(θ)x(r)drdθ >
(13)
>
for j = 1, · · · , n0 . Finally, the synchronization frequency is Ω and the synchronization phase is given by the constant < u1 , x(0) + ε
α1 =
<
0 0
μ(θ)x(r)drdθ >
−τ θ u1 , 1 > ·(1
− ε¯ τ)
.
Proof. Following Theorem 4.1, Rank(V (α1 , α2 , · · · , αn0 )) = 1 if and only if all the phase individuals ϕi belong to the same cluster. Thus the linear system reaches a synchronization. Obviously, there are two cases when Rank(V (α1 , α2 , · · · , αn0 )) = 1. Case n0 = 1: In this case, Rank(V (α1 , α2 , · · · , αn0 )) = 1 holds for arbitrary initial conditions. Also, the zero eigenvalue is simple. Then U = v1 = a1 = 1 and β is a constant given by 1 < u1 , x(0) + ε β= 1 − ε¯ τ
0 0 μ(θ)x(r)drdθ > .
−τ θ
Thus, lim x(t) = α1 1 =
t→∞
1 < U β, a1 > 1 = β1. n
This implies that lim (θi (t) − Ωt) = β for all i.
t→∞
Therefore, the coupling system (3)–(4) locally reaches a synchronization for arbitrary initial conditions, and the synchronization frequency is Ω, the synchronization phase is 1 < u1 , x(0) + ε 1 − ε¯ τ
0 0 μ(θ)x(r)drdθ > .
−τ θ
Case n0 > 1: In this case, Rank(V (α1 , α2 , · · · , αn0 )) = 1 if and only if α1 = αj , j ∈ {2, · · · , n0 }. Thus lim x(t) = U β = α1 a1 + α2 a2 + · · · + αn0 an0 = α1 1
t→∞
= α1
n0
< ui , 1 > vi .
i=1
This implies that lim (θi (t) − Ωt) = α1 for all i.
t→∞
JID:YJMAA
AID:20537 /FLA
Doctopic: Applied Mathematics
[m3L; v1.181; Prn:11/07/2016; 17:25] P.9 (1-10)
Y. Liu, J. Wu / J. Math. Anal. Appl. ••• (••••) •••–•••
9
In this case, the coupling system (3)–(4) locally reaches a conditional synchronization if and only if the initial values satisfy (13). Furthermore, the synchronization frequency is Ω and the synchronization phase is given by the coefficient
α1 =
< u1 , x(0) + ε
0 0 −τ
θ
μ(θ)x(r)drdθ >
(1 − ε¯ τ ) < u1 , 1 >
.
2
Remark 4.1. From Theorem 4.1 and Corollary 4.1, we see that the phase clustering criteria condition is completely determined by the structure of linearization and the initial values. Also, the time delay will affect synchronization frequency Ω and the number of clusters phase synchronization. The final synchronization frequency impacts on the final synchronization phase in a nonlinear way. Remark 4.2. Let f (r) = sin(r), F (r) = cos(r). Then system (3)–(4) becomes a generalized Kuramoto model. In this case, the synchronization frequency Ω is determined by the algebra equation Ω = ω + α cos(Ω¯ τ ) sin(Ω¯ τ ). Thus the synchronization frequency Ω ≈ small.
ω 1−α¯ τ
and F (Ω¯ τ )·f (Ω¯ τ ) = cos2 (Ω¯ τ ) > 0 when delay τ is sufficiently
Remark 4.3. When Rank(V (α1 , α2 , · · · , αn0 )) = m, there are m clusters phase synchronization, the clustering phases are determined by the constants αi , which is calculated by equality (12). Acknowledgments This work was partially supported by National Natural Science Foundation of China (11201481 and 11428101) and Natural Science and Engineering Research Council of Canada. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
P. Ashwin, J.W. Swift, The dynamics of N weakly coupled identical oscillators, J. Nonlinear Sci. 2 (1992) 69–108. F.M. Atay, The consensus problem in networks with transmission delays, Philos. Trans. R. Soc. A 371 (2013) 20120460. F. Bauer, Normalized graph Laplacians for directed graphs, Linear Algebra Appl. 436 (2012) 4193–4222. V.N. Belykh, V.S. Petrov, G.V. Osipov, Dynamics of the finite-dimensional Kuramoto model: global and cluster synchronization, Regul. Chaotic Dyn. 20 (1) (2015) 37–48. P.C. Bressloff, S. Coombes, Travelling waves in chains of pulsecoupled integrate-and-fire oscillators with distributed delays, Phys. D 130 (3) (1999) 232–254. F.R.K. Chung, Spectral Graph Theory, American Mathematical Society, Providence, RI, 1997. D. Florian, M. Chertkov, F. Bullo, Synchronization in complex oscillator networks and smart grids, Proc. Natl. Acad. Sci. 110 (6) (2013) 2005–2010. A. Gushchin, E. Mallada, A. Tang, Synchronization of phase-coupled oscillators with plastic coupling strength, in: Information Theory and Applications Workshop, ITA, 2015. S.Y. Ha, T. Ha, J.H. Kim, On the complete synchronization of the Kuramoto phase model, Phys. D 239 (2010) 1692–1700. S.Y. Ha, H.K. Kim, J. Park, Remarks on the complete synchronization of Kuramoto oscillators, Nonlinearity 28 (2015) 1441. S.Y. Ha, Z. Li, Complete synchronization of Kuramoto oscillators with hierarchical leadership, Commun. Math. Sci. 12 (2014) 485–508. J.K. Hale, Theory of Functional Differential Equations, Springer, Berlin, 1977. R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, UK, 1985. Y. Kuramoto, Self-entrainment of a population of coupled nonlinear oscillators, in: International Symposium on Mathematical Problems in Theoretical Physics, vol. 39, 1975, pp. 420–422. Z. Li, X. Xue, D. Yu, On the Lojasiewicz exponent of Kuramoto model, J. Math. Phys. 56 (2015) 022704. T. Liam, L.Q. English, Synchronization in phase-coupled Kuramoto oscillator networks with axonal delay and synaptic plasticity, Phys. Rev. E 89 (3) (2014) 032906. Y.C. Liu, J.H. Wu, Flocking and asymptotic velocity of the Cucker–Smale model with processing delay, J. Math. Anal. Appl. 415 (1) (2014) 53–61.
JID:YJMAA 10
AID:20537 /FLA
Doctopic: Applied Mathematics
[m3L; v1.181; Prn:11/07/2016; 17:25] P.10 (1-10)
Y. Liu, J. Wu / J. Math. Anal. Appl. ••• (••••) •••–•••
[18] S. Luca, Clustering and synchronization in phase models with state dependent coupling, in: 49th IEEE Conference on Decision and Control, CDC, IEEE, 2010. [19] E. Mallada, A. Tang, Synchronization of weakly coupled oscillators: coupling, delay and topology, J. Phys. A 46 (50) (2013) 505101. [20] S.A. Marvel, H.S. Steven, Invariant submanifold for series arrays of Josephson junctions, Chaos 19 (1) (2009) 013132. [21] V. Novičenko, Delayed feedback control of synchronization in weakly coupled oscillator networks, Phys. Rev. E 92 (2015) 022919. [22] P. Seliger, C.Y. Stephen, S.T. Lev, Plasticity and learning in a network of coupled phase oscillators, Phys. Rev. E 65 (4) (2002) 041906.
AID:20537 /FLA
Doctopic: Applied Mathematics
[m3L; v1.181; Prn:11/07/2016; 17:25] P.1 (1-10)
J. Math. Anal. Appl. ••• (••••) •••–•••
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Local phase synchronization and clustering for the delayed phase-coupled oscillators with plastic coupling Yicheng Liu a,∗ , Jianhong Wu b a
College of Science, National University of Defense Technology, Changsha, 410073, PR China Laboratory for Industrial and Applied Mathematics (LIAM), Department of Mathematics and Statistics, York University, Toronto, Ontario M3J 1P3, Canada b
a r t i c l e
i n f o
Article history: Received 22 December 2015 Available online xxxx Submitted by J. Shi Keywords: Frequency synchronization Phase synchronization Kuramoto model Clusters Phase-coupled oscillators
a b s t r a c t In this paper, we investigate the local synchronization problem for delayed phasecoupled oscillators with plastic coupling (also referred as a generalized Kuramoto model). By linearizing the phase-coupled oscillators system at a special solution, an approximation phase-coupled system with delay effects is deduced. Moreover, we find the multiplicity of semisimple zero eigenvalue for coupling strength coefficients matrix and the rank of Vandermonde matrix associated initial values are two sensitive factors. The former is closely related to the maximum number of clusters phase of linear coupled system, and the latter determines clusters phase synchronization. As results, both frequency synchronization criteria and phase synchronization criteria are established in this literature. Also, the final synchronization frequency and phase are formulated in terms of coupling strength, initial values and time delay. © 2016 Elsevier Inc. All rights reserved.
1. Introduction The synchronization problem of phase-coupled oscillators has been widely studied in different disciplines ranging from physics [5,20], networks [4,10,19] and engineering [7,16]. Synchronization of Kuramoto oscillators (a classical phase-coupled system), for example, has been extensively analyzed under various assumptions on homogeneous oscillators and network topology. In particularly, Luca [18] conducted an interesting study on synchronization and stability for systems of homogeneous oscillators with plastic coupling, but the study focused on complete graph topology. Recently, Gushchin–Mallada–Tang [8] discussed a system of phase-coupled oscillators with plastic coupling governed by the following two classes of equations: θ˙i (t) = ωi +
kij · fij (θj (t) − θi (t)), 1 ≤ i ≤ n,
j∈Ni
* Corresponding author. E-mail addresses: [email protected] (Y. Liu), [email protected] (J. Wu). http://dx.doi.org/10.1016/j.jmaa.2016.06.049 0022-247X/© 2016 Elsevier Inc. All rights reserved.
(1)
JID:YJMAA 2
AID:20537 /FLA
Doctopic: Applied Mathematics
[m3L; v1.181; Prn:11/07/2016; 17:25] P.2 (1-10)
Y. Liu, J. Wu / J. Math. Anal. Appl. ••• (••••) •••–•••
k˙ ij (t) = sij (αij Fij (θj (t) − θi (t)) − kij ), (i, j) ∈ E,
(2)
where θi is the phase and ωi the intrinsic frequency of oscillator i, E is the set of edges, Ni is a set of oscillators connected to oscillator i (i.e. the set of its neighbors), kij and fij which are 2π-periodic continx uously differentiable functions. Fij (x) = − 0 fij (t)dt + C with the integration constant C chosen so that π Fij (t)dt = 0. Positive constants αij determine minimum and maximum values of the coupling strengths, 0 and positive constants sij define rates of change of the coupling strengths. This system is coupled by two equations, the first defines behavior of an oscillator, and the second determines dynamics of the coupling strengths. In the special case where kij is constant and fij (θj − θi ) = sin(θj − θi ), system (1) becomes the Kuramoto model [14]. If fij (θj − θi ) = sin(θj − θi ) and Fij (θj − θi ) = cos(θj − θi ), then the system (1)–(2) becomes a generalized Kuramoto model [22]. It is called a homogeneous oscillators system if all intrinsic frequencies of oscillators are equal, i.e. there exists a constant ω such that ω1 = ω2 = · · · = ωn = ω. The synchronization of phase-coupled oscillators may appear at two different levels: frequency synchronization and phase synchronization. The synchronization, including frequency synchronization and phase synchronization, has been widely investigated in recent literatures, see [1,4,9–11,15,16,21] and references therein. Our objective in this work is to develop local phase synchronization and clustering criteria for phasecoupled homogeneous oscillators system with non-constant coupling, distributed delay and general network topology. The distributed delays are incorporated into (1)–(2), so we obtain
θ˙i (t) = ω +
0 kij f (
j∈Ni
μ(s)θj (t + s)ds − θi (t)),
(3)
−τ
0 k˙ ij (t) = sij (αij F (
μ(s)θj (t + s)ds − θi (t)) − kij ),
(4)
−τ
where τ is the maximal coupling delay and the distribution is normalized so that to specify a solution of (3)–(4), we will need to prescribe the initial conditions
0 −τ
μ(s)ds = 1. Naturally,
θi (t) = γi (t) for all t ∈ [−τ, 0], 1 ≤ i ≤ n, where each γi is a continuous function. For more discussions why time delay should be introduced, we refer to [2,5,16,17,19] and references therein. In what follows, we say that system (3)–(4) achieves frequency synchronization if limt→∞ θ˙1 (t) = · · · = limt→∞ θ˙n (t) = Ω and limt→∞ k˙ ij (t) = 0 for all i, j, where Ω is a common synchronization frequency. We say that system (3)–(4) reaches phase synchronization if it achieves frequency synchronization, and limt→∞ (θ1 (t) − Ωt) = limt→∞ (θ2 (t) − Ωt) = · · · = limt→∞ (θn (t) − Ωt) = ϕ (a constant). We say that a system reaches p clusters phase synchronization if it achieves frequency synchronization and for some set of initial values there exist some constants ϕj ∈ R and sets Pj ⊂ {1, 2, · · · , n} satisfying Pj ∩ Pi = ∅ (empty set), ϕj = ϕi (whenever i = j) and ∪j Pj = {1, 2, · · · , n}, such that limt→∞ (θi (t) − Ωt) = ϕj for all i ∈ Pj , j = 1, 2, · · · , p. The numbers ϕj are then called the clustering phases. The structure of the article is as follows. In the next section we linearize the delayed phase-coupled oscillators system and normalize the adjoint matrix. In Section 3, we provide the normal zero-one vectors decomposition in the nullspace. In section 4, a necessary and sufficient condition for clusters phase synchronization is obtained. In this section, both frequency synchronization criteria and phase synchronization criteria are also described. The final synchronization frequency and synchronization phase are formulated in terms of coupling strength, initial values and time delay.
JID:YJMAA
AID:20537 /FLA
Doctopic: Applied Mathematics
[m3L; v1.181; Prn:11/07/2016; 17:25] P.3 (1-10)
Y. Liu, J. Wu / J. Math. Anal. Appl. ••• (••••) •••–•••
3
2. Normalization and linearization Firstly, we assume that the coupling strength coefficients αij satisfy the row sum invariance (here, αij = 0 for (i, j) ∈ / E). That is, there is a constant α so α =
n
αij for all i.
(5)
j=1
Then we introduce the normalized coupling strength coefficients matrix A = (aij )n×n , where aij =
αij for (i, j) ∈ E; aij = 0 for (i, j) ∈ / E. α
n A is called the normalized weighted adjacency matrix of oscillators network. It is clear that j=1 aij = 1 for all i. Then L = I − A is a normal Laplacian matrix. It is easy to see that zero is an eigenvalue of L and 1 = (1, 1, · · · , 1)T belongs to the corresponding nullspace. In what follows, we assume that zero is a semisimple eigenvalue of L with the multiplicity n0 . We need to consider a special solution of system (3)–(4) as follows θi (t) = Ωt and k˙ ij (t) = 0 for all i, j,
(6)
where Ω denotes the common synchronization frequency, which is one of the roots of the following algebra equation Ω = ω + αF (Ω¯ τ )f (Ω¯ τ ). The static coupling strength k˜ij is given by k˜ij = αij F (Ω¯ τ ), where τ¯ = Introducing the phase variable ϕi (t) of oscillator i by
(7) 0 −τ
sμ(s)ds < 0.
ϕi (t) = θi (t) − Ωt for all i, and linearizing oscillators system (3)–(4) around the special solution (6), we obtain the asymptotically phase-variable system
ϕ˙ i (t) = ε
n j=1
0 aij (
μ(s)ϕj (t + s)ds − ϕi (t))
(8)
−τ
for all i, where ε = αF (Ω¯ τ ) · f (Ω¯ τ ). Consequently, we observed that the delayed phase-coupled homogeneous oscillators system (3)–(4) reaches locally frequency synchronization that is determined by the convergence of system (8). Note also that when system (8) reaches clustering synchronization, the oscillators system (3)–(4) reaches locally clustering phase synchronization. 3. Structure analysis of the linearization As will be clear soon, if 1 can be spitted into k zero-one vectors (zero-one vector means each component of the vector is either 1 or 0), then the phase state set is spitted into possibly k classes. Therefore, we define a number K as follows:
JID:YJMAA 4
AID:20537 /FLA
Doctopic: Applied Mathematics
[m3L; v1.181; Prn:11/07/2016; 17:25] P.4 (1-10)
Y. Liu, J. Wu / J. Math. Anal. Appl. ••• (••••) •••–•••
K = max{k : there are k nonzero zero-one vectors a1 , · · · , ak such that Lai = 0 and a1 + · · · + ak = 1}. Obviously, K is well-defined since L1 = 0. Lemma 3.1. Assume zero is a semisimple eigenvalue of the Laplacian L with multiplicity n0 . Then K = n0 . Proof. Without loss of generality, we assume the zero-one vector vectors a1 , · · · , aK are of the following forms (if necessary, we exchange the rows of matrix L and relabel the subscript of xi ): a1 = (1, · · · , 1, 0, · · · , 0), where the number of components 1 is t1 ; a2 = (0, · · · , 0, 1, · · · , 1, 0, · · · , 0), where t2 components 1 following t1 components 0; ··· ,··· aK = (0, · · · , 0, 1, · · · , 1), where the number of components 1 is tK . Also, we see that ti is the minimum number of components 1 in ai and t1 + t2 + · · · + tK = n. In what follows, a set {a1 , · · · , aK } is called a normal zero-one vector basis if a1 , · · · , aK are of above forms. In this case, noting that Lai = 0(i = 1, · · · , K) and aij ≥ 0, we see that L is a block diagonal matrix, that is, L = diag(D1 , D2 , · · · , DK ). Let f (λ) = det(λI −Di ). Then f (0) = 0. Noting Di is M -matrix and ti is the minimum number, it is easy to see that f (0) = 0. Thus 0 is a simple eigenvalue of matrix Di (i = 1, 2, · · · , K). Hence 0 is an eigenvalue of matrix L with multiplicity K. On the other hand, the multiplicity of semisimple zero eigenvalue is n0 , we conclude K = n0 . 2 Remark 3.1. From the above argument, we see that if zero is a semisimple eigenvalue of the Laplacian L with multiplicity n0 , then L is a block diagonal matrix (if necessary, we exchange the rows of matrix L and renumber the subscript). Lemma 3.2. Assume zero is a semisimple eigenvalue of the Laplacian L with multiplicity n0 . Then there exists a unique family normal zero-one vectors a1 , · · · , an0 such that Lai = 0 and a1 + a2 + · · · + an0 = 1. Proof. From Lemma 3.1, we choose a family of normal zero-one vectors a1 , · · · , an0 such that Lai = 0 and a1 + a2 + · · · + an0 = 1. Thus the existence is established. Next, we show the uniqueness. If there is another family of normal zero-one vectors b1, · · · , bn0 such that Lbi = 0 and b1 + b2 + · · · + bn0 = 1. Without loss of generality, we assume both clusters a1 and b1 include the first oscillator, and b1 has the form b1 = (1, · · · , 1, 0, · · · , 0), where the number of components 1 is s1 . Case s1 < t1 : In this case, we have L(a1 −b1 ) = 0. Also, {b1 , a1 −b1 , a2 , · · · , an0 } is linearly independent. This means that the dimension of the nullspace {v : Lv = 0} is no less than n0 + 1. But its dimension is n0 , a contradiction.
JID:YJMAA
AID:20537 /FLA
Doctopic: Applied Mathematics
[m3L; v1.181; Prn:11/07/2016; 17:25] P.5 (1-10)
Y. Liu, J. Wu / J. Math. Anal. Appl. ••• (••••) •••–•••
5
Case s1 > t1 : Assume t1 < s1 < t2 (if s1 is larger than t2 , then the argument is similar), then L(a1 + a2 − b1 ) = 0. Also, {a1 , a1 + a2 − b1 , b2 , · · · , bn0 } is linearly independent. So, the dimension of the nullspace is also no less than n0 + 1. It is also impossible due to the fact that the dimension of nullspace is n0 . Based on the above arguments, we claim that a1 = b1 . Similarly, we can prove ai = bi , i = 2, · · · , n0 . Thus the uniqueness is verified. This completes the proof. 2 4. Synchronization and phase clustering In this section, we study the synchronization and clustering problems of linear system (8). The goal of this section is to establish the phase synchronization and clustering criteria of system (8). The following standard notations and results about Laplacian matrices can be found in [3,6]. Since all off-diagonal entries of the Laplacian matrix I −A are nonpositive, the real parts of its eigenvalues are nonnegative. Let {λ1 , λ2 , · · · , λn } denote the set of eigenvalues of L. Then, 0 ≤ Re(λi ), i = 1, 2, · · · , n. Furthermore, by an application of the Gershgorin’s circle theorem [13], we have |1 − λi | ≤ 1. Noting that zero is a semi-simple eigenvalue of L with multiplicity n0 . Then there is a nonsingular matrix Q such that
0 Z
On0 L=Q 0
Q−1 ,
where On0 is a n0 -dimensional zero matrix (1 ≤ n0 ≤ n) and Z is a matrix with spectral ρ(Z) ∈ (0, 2]. Let {v1 , · · · , vn } be the columns of Q, which is a basis of Rn . Set {u1 , · · · , un } be the dual basis corresponding to {v1 , · · · , vn }. That is, ui are linearly independent vectors such that < ui , vj > = δji . It is easy to see that n ui are left eigenvectors of L, ui L = 0ui = 0 for i = 1, 2, · · · , n0 . Furthermore, if x = i=1 αi vi ∈ Rn , then αi =< ui , x >. Let x = (ϕ1 , ϕ2 , · · · , ϕn )T . Then system (8) is transformed to the vector version 0 ˙ x(t) = −εx(t) + εA
μ(s)x(t + s)ds.
(9)
−τ
Let x(t) be a solution of equation (9). Considering the decomposition x(t) = αi (t) =< ui , x(t) >, we have
n i=1
αi (t)vi , where
0 α˙ i (t) = −εαi (t) + ε(1 − λi )
μ(s)αi (t + s)ds
(10)
−τ
for all i. Its characteristic equation is 0 z = −ε + ε(1 − λi )
μ(s)ezs ds.
(11)
−τ
Lemma 4.1. [2] Let ε > 0. If λi = 0, then (11) has a simple root at zero and all other roots have negative real parts. If λi = 0, then all roots have negative real parts. Following Lemma 4.1, we see that lim αi (t) = 0 for all i > n0 .
t→∞
JID:YJMAA
AID:20537 /FLA
Doctopic: Applied Mathematics
[m3L; v1.181; Prn:11/07/2016; 17:25] P.6 (1-10)
Y. Liu, J. Wu / J. Math. Anal. Appl. ••• (••••) •••–•••
6
Hence, lim (x(t) − α1 (t)v1 − · · · − αn0 (t)vn0 ) = 0.
t→∞
In order to calculate the limits of α1 (t), · · · , αn0 (t) as t → ∞, using a similar argument in [2], we consider the initial functional space C([−τ, 0], R), which is the Banach space of real-valued continuous functions on [−τ, 0] equipped with the supremum norm. Let C ∗ = C([−τ, 0], R), and define the bilinear form < ψ, φ >c for φ ∈ C and ψ ∈ C ∗ by 0 0 μ(r)ψ(r − t)φ(t)dtdr.
< ψ, φ >c = ψ(0)φ(0) + ε −τ r
Since λi = 0 for i = 1, 2, · · · , n0 , the characteristic equation (11) has a simple root at zero, thus its characteristic subspace is an one-dimensional space, say, C0 . Denote C0∗ by the dual one-dimensional space. 1 Let constant function Φ(θ) = 1 be the basis of C0 , and then its dual basis in C0∗ is Ψ(θ) = 1−ε¯ τ with < Ψ, Φ >c = 1. Also let at (θ) = αi (t + θ) for all θ ∈ [−τ, 0]. It follows from the theory of functional differential equations [12] that the space C can be decomposed using the invariant subspace C0 and its complement. Thus at = a0t + bt , where a0t ∈ C0 . Following Lemma 4.1, we see that the characteristic equation has a simple root at zero and all other roots have negative real parts. Thus limt→∞ bt (θ) = 0 uniformly for θ ∈ [−τ, 0]. Noting that a0t is a constant and determined by the initial condition, we conclude that it is given as the constant function a0t =< Ψ, a0 >c Φ. Thus, by direct calculation, we have
lim αi (t) =
t→∞
a0t
0 0
1 [αi (0) + ε = 1 − ε¯ τ
μ(θ)αi (r)drdθ]
−τ θ
1 < ui , x(0) + ε = 1 − ε¯ τ
0 0 μ(θ)x(r)drdθ >
−τ θ
for all i = 1, · · · , n0 . We introduce several projection coordinate vectors in the nullspace. Following Lemma 3.2, we conclude that there exists a unique family of normal zero-one vectors a1 , · · · , an0 such that Lai = 0 and a1 + a2 + · · · + an0 = 1. Basing on these vectors, we introduce c = (< u1 , 1 >, · · · , < un0 , 1 >)T , β = (β1 , β2 , · · · , βn0 )T , βi =
1 < ui , x(0) + ε 1 − ε¯ τ
0 0 μ(θ)x(r)drdθ >,
−τ θ
ci = (< u1 , ai >, · · · , < un0 , ai >)T , 1 ≤ i ≤ n0 , U = [v1 , v2 , · · · , vn0 ]. Let
JID:YJMAA
AID:20537 /FLA
Doctopic: Applied Mathematics
[m3L; v1.181; Prn:11/07/2016; 17:25] P.7 (1-10)
Y. Liu, J. Wu / J. Math. Anal. Appl. ••• (••••) •••–•••
⎡
1 ⎢ α ⎢ 1 V (α1 , α2 , · · · , αn0 ) = ⎢ ⎣ ··· α1n0 −1
1 α2 ··· α2n0 −1
··· ··· ··· ···
7
⎤ 1 αn0 ⎥ ⎥ ⎥ ··· ⎦ αnn00 −1
denote the n0 × n0 Vandermonde matrix, and αi =
1 < U β, ai >, 1 ≤ i ≤ n0 . ti
(12)
Theorem 4.1. Assume coupling strength coefficients satisfy (5), F (Ω¯ τ ) · f (Ω¯ τ ) > 0 and zero is a semisimple eigenvalue of the Laplacian L with multiplicity n0 . Then the coupling system (3)–(4) locally reaches m clusters phase synchronization if and only if Rank(V (α1 , α2 , · · · , αn0 )) = m (1 ≤ m ≤ n0 ). Proof. Since F (Ω¯ τ ) · f (Ω¯ τ ) > 0, then ε > 0 and Lemma 4.1 holds. Noting {a1 , a2 , · · · , an0 } is linearly independent, and then it is also a basis of nullspace Span{v1 , v2 , · · · , vn0 }. On the other hand, we see that lim x(t) =
t→+∞
n0
βi vi = U β ∈ Span{a1 , a2 , · · · , an0 }.
i=1
Thus there are some constants αi , such that U β = α1 a1 + α2 a2 + · · · + αn0 an0 . Also, since < ai , aj > = 0 for i = j and < ai , ai > = ti , we have αi = t1i < U β, ai >. Thus the linear system (8) reaches m clusters if and only if α1 , α2 , · · · , αn0 has m pairwise-different numbers. Meanwhile, following the formulation of Vandermonde determinant
Det(V (α1 , α2 , · · · , αn0 )) =
(αi − αj ),
1≤i
we see that the rank of Vandermonde matrix is m if and only if there are m pairwise-different numbers in α1 , α2 , · · · , αn0 . Let φ1 , φ2 , · · · , φm be pairwise-different numbers. Then lim (θi (t) − Ωt) = φj when i ∈ Pj ,
t→∞
where the sets Pj ⊂ {1, 2, · · · , n} satisfy Pj ∩ Pi = ∅, (whenever i = j) and ∪j Pj = {1, 2, · · · , n}. This implies that the coupling system (3)–(4) locally reaches m clusters phase synchronization if and only if Rank(V (α1 , α2 , · · · , αn0 )) = m. This completes the proof. 2 τ )·f (Ω¯ τ) > 0 Corollary 4.1 (Synchronization criteria). Assume coupling strength coefficients satisfy (5), F (Ω¯ and zero is a semisimple eigenvalue of the Laplacian L with multiplicity n0 . The coupling system (3)–(4) locally achieves synchronization if and only if Rank(V (α1 , α2 , · · · , αn0 )) = 1. More precisely, we have 1). If n0 = 1, the coupling system (3)–(4) locally achieves synchronization, and the synchronization frequency is Ω, the synchronization phase is 1 < u1 , x(0) + ε 1 − ε¯ τ
0 0
−τ θ
μ(θ)x(r)drdθ >;
JID:YJMAA
AID:20537 /FLA
Doctopic: Applied Mathematics
[m3L; v1.181; Prn:11/07/2016; 17:25] P.8 (1-10)
Y. Liu, J. Wu / J. Math. Anal. Appl. ••• (••••) •••–•••
8
2). If n0 > 1, the coupling system (3)–(4) locally reaches a conditional synchronization. In other words, the coupling system (3)–(4) locally reaches synchronization if and only if the initial values satisfy < u1 , x(0) + ε
0 0
−τ θ < u1 , 1
j
=
< u , x(0) + ε
0 0
−τ θ < uj , 1
μ(θ)x(r)drdθ > > μ(θ)x(r)drdθ >
(13)
>
for j = 1, · · · , n0 . Finally, the synchronization frequency is Ω and the synchronization phase is given by the constant < u1 , x(0) + ε
α1 =
<
0 0
μ(θ)x(r)drdθ >
−τ θ u1 , 1 > ·(1
− ε¯ τ)
.
Proof. Following Theorem 4.1, Rank(V (α1 , α2 , · · · , αn0 )) = 1 if and only if all the phase individuals ϕi belong to the same cluster. Thus the linear system reaches a synchronization. Obviously, there are two cases when Rank(V (α1 , α2 , · · · , αn0 )) = 1. Case n0 = 1: In this case, Rank(V (α1 , α2 , · · · , αn0 )) = 1 holds for arbitrary initial conditions. Also, the zero eigenvalue is simple. Then U = v1 = a1 = 1 and β is a constant given by 1 < u1 , x(0) + ε β= 1 − ε¯ τ
0 0 μ(θ)x(r)drdθ > .
−τ θ
Thus, lim x(t) = α1 1 =
t→∞
1 < U β, a1 > 1 = β1. n
This implies that lim (θi (t) − Ωt) = β for all i.
t→∞
Therefore, the coupling system (3)–(4) locally reaches a synchronization for arbitrary initial conditions, and the synchronization frequency is Ω, the synchronization phase is 1 < u1 , x(0) + ε 1 − ε¯ τ
0 0 μ(θ)x(r)drdθ > .
−τ θ
Case n0 > 1: In this case, Rank(V (α1 , α2 , · · · , αn0 )) = 1 if and only if α1 = αj , j ∈ {2, · · · , n0 }. Thus lim x(t) = U β = α1 a1 + α2 a2 + · · · + αn0 an0 = α1 1
t→∞
= α1
n0
< ui , 1 > vi .
i=1
This implies that lim (θi (t) − Ωt) = α1 for all i.
t→∞
JID:YJMAA
AID:20537 /FLA
Doctopic: Applied Mathematics
[m3L; v1.181; Prn:11/07/2016; 17:25] P.9 (1-10)
Y. Liu, J. Wu / J. Math. Anal. Appl. ••• (••••) •••–•••
9
In this case, the coupling system (3)–(4) locally reaches a conditional synchronization if and only if the initial values satisfy (13). Furthermore, the synchronization frequency is Ω and the synchronization phase is given by the coefficient
α1 =
< u1 , x(0) + ε
0 0 −τ
θ
μ(θ)x(r)drdθ >
(1 − ε¯ τ ) < u1 , 1 >
.
2
Remark 4.1. From Theorem 4.1 and Corollary 4.1, we see that the phase clustering criteria condition is completely determined by the structure of linearization and the initial values. Also, the time delay will affect synchronization frequency Ω and the number of clusters phase synchronization. The final synchronization frequency impacts on the final synchronization phase in a nonlinear way. Remark 4.2. Let f (r) = sin(r), F (r) = cos(r). Then system (3)–(4) becomes a generalized Kuramoto model. In this case, the synchronization frequency Ω is determined by the algebra equation Ω = ω + α cos(Ω¯ τ ) sin(Ω¯ τ ). Thus the synchronization frequency Ω ≈ small.
ω 1−α¯ τ
and F (Ω¯ τ )·f (Ω¯ τ ) = cos2 (Ω¯ τ ) > 0 when delay τ is sufficiently
Remark 4.3. When Rank(V (α1 , α2 , · · · , αn0 )) = m, there are m clusters phase synchronization, the clustering phases are determined by the constants αi , which is calculated by equality (12). Acknowledgments This work was partially supported by National Natural Science Foundation of China (11201481 and 11428101) and Natural Science and Engineering Research Council of Canada. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
P. Ashwin, J.W. Swift, The dynamics of N weakly coupled identical oscillators, J. Nonlinear Sci. 2 (1992) 69–108. F.M. Atay, The consensus problem in networks with transmission delays, Philos. Trans. R. Soc. A 371 (2013) 20120460. F. Bauer, Normalized graph Laplacians for directed graphs, Linear Algebra Appl. 436 (2012) 4193–4222. V.N. Belykh, V.S. Petrov, G.V. Osipov, Dynamics of the finite-dimensional Kuramoto model: global and cluster synchronization, Regul. Chaotic Dyn. 20 (1) (2015) 37–48. P.C. Bressloff, S. Coombes, Travelling waves in chains of pulsecoupled integrate-and-fire oscillators with distributed delays, Phys. D 130 (3) (1999) 232–254. F.R.K. Chung, Spectral Graph Theory, American Mathematical Society, Providence, RI, 1997. D. Florian, M. Chertkov, F. Bullo, Synchronization in complex oscillator networks and smart grids, Proc. Natl. Acad. Sci. 110 (6) (2013) 2005–2010. A. Gushchin, E. Mallada, A. Tang, Synchronization of phase-coupled oscillators with plastic coupling strength, in: Information Theory and Applications Workshop, ITA, 2015. S.Y. Ha, T. Ha, J.H. Kim, On the complete synchronization of the Kuramoto phase model, Phys. D 239 (2010) 1692–1700. S.Y. Ha, H.K. Kim, J. Park, Remarks on the complete synchronization of Kuramoto oscillators, Nonlinearity 28 (2015) 1441. S.Y. Ha, Z. Li, Complete synchronization of Kuramoto oscillators with hierarchical leadership, Commun. Math. Sci. 12 (2014) 485–508. J.K. Hale, Theory of Functional Differential Equations, Springer, Berlin, 1977. R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, UK, 1985. Y. Kuramoto, Self-entrainment of a population of coupled nonlinear oscillators, in: International Symposium on Mathematical Problems in Theoretical Physics, vol. 39, 1975, pp. 420–422. Z. Li, X. Xue, D. Yu, On the Lojasiewicz exponent of Kuramoto model, J. Math. Phys. 56 (2015) 022704. T. Liam, L.Q. English, Synchronization in phase-coupled Kuramoto oscillator networks with axonal delay and synaptic plasticity, Phys. Rev. E 89 (3) (2014) 032906. Y.C. Liu, J.H. Wu, Flocking and asymptotic velocity of the Cucker–Smale model with processing delay, J. Math. Anal. Appl. 415 (1) (2014) 53–61.
JID:YJMAA 10
AID:20537 /FLA
Doctopic: Applied Mathematics
[m3L; v1.181; Prn:11/07/2016; 17:25] P.10 (1-10)
Y. Liu, J. Wu / J. Math. Anal. Appl. ••• (••••) •••–•••
[18] S. Luca, Clustering and synchronization in phase models with state dependent coupling, in: 49th IEEE Conference on Decision and Control, CDC, IEEE, 2010. [19] E. Mallada, A. Tang, Synchronization of weakly coupled oscillators: coupling, delay and topology, J. Phys. A 46 (50) (2013) 505101. [20] S.A. Marvel, H.S. Steven, Invariant submanifold for series arrays of Josephson junctions, Chaos 19 (1) (2009) 013132. [21] V. Novičenko, Delayed feedback control of synchronization in weakly coupled oscillator networks, Phys. Rev. E 92 (2015) 022919. [22] P. Seliger, C.Y. Stephen, S.T. Lev, Plasticity and learning in a network of coupled phase oscillators, Phys. Rev. E 65 (4) (2002) 041906.