# Partial Synchronizability Characterized by Principal Quasi-Submatrices Corresponding to Clusters

Authors: Gang Zhang; Zhongjun Ma; Yi Wang; Jianbao Zhang
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

A partial synchronization problem in an oscillator network is considered. The concept on a principal quasi-submatrix corresponding to the topology of a cluster is proposed for the first time to study partial synchronization. It is shown that partial synchronization can be realized under the condition depending on the principal quasi-submatrix, but not distinctly depending on the intercluster couplings. Obviously, the dimension of any principal quasi-submatrix is usually far less than the one of the network topology matrix. Therefore, our criterion provides us a novel index of partial synchronizability, which reduces the network size when the network is composed of a great mount of nodes. Numerical simulations are carried out to confirm the validity of the method.


---

## Body

## 1. Introduction

Since the pioneering work of Pecora and Carroll [1], extensive researches on chaos synchronization have been carried out due to their potential applications in various disciplines such as physics, engineering, and biology [2–5]. For example, wireless sensor networks are widely researched due to the important applications in the real scenarios. One of the biggest challenges in investigating wireless sensor networks is how to obtain higher synchronization accuracy with minimal overhead [6]. And many other network-induced phenomena have also been discussed extensively in engineering [7].Up to date, there have been many synchronization types being proposed and discussed, including complete synchronization [8, 9], partial synchronization [10], and inner and outer synchronization [11]. Generally speaking, there exists an intimate relationship between the phenomena of synchronization and invariant manifolds of coupled systems [9–12]. Thus, when we carry out researches on partial synchronization of coupled systems, it is always supposed that the corresponding full or partial synchronization manifolds are invariant manifolds. The established tools for the stability of invariant synchronization manifolds mainly consist of local linearization method and Lyapunov function method. One of the prominent results of the former is the master stability function method [9]. The method has obtained two significant factors determining the local stability of the synchronous state, that is, the maximum Lyapunov exponent of the node dynamics and the eigenvalues of the topology matrix. One of the prominent results of the latter is the study of the global attractiveness of the synchronization manifold. A crucial requirement for the method is the condition of the node dynamics satisfyingf
∈
QUAD
(
Δ
,
P
,
Ω
)[10]. In some sense, the condition means that the system can synchronize when the coupling is made sufficiently large. Recently, some new conditions have been obtained for synchronization of networks without Lipschitz condition or QUAD condition [12].The type of synchronization concerned in this paper is partial synchronization. It means that the coupled oscillators split into subgroups called clusters, and all the oscillators in the same cluster behave in the same fashion. Many relative studies have been carried out [13–18]. By using pinning control strategy, partial synchronization of coupled stochastic delayed neural networks was discussed in [14]. Similar control strategy is also proposed to select controlled communities by analyzing the information of each community such as in-degrees and out-degrees [15]. Afterwards, some simple intermittent pinning controls and centralized adaptive intermittent controls are proposed [16]. However, a suitable control law must be presented in order to use pinning control strategy. Some other researches focused on partial synchronization induced by the coupling configuration. An arbitrarily selected partial synchronization manifold was constructed for a network with cooperative and competitive couplings [17]. Recently, a sufficient and necessary condition for partial synchronization manifolds being invariant manifolds was obtained in networks coupled linearly and symmetrically [10]. More significantly, some sufficient conditions for the global attractiveness of the partial synchronization manifold were derived by decomposing the whole space into a direct sum of the synchronization manifold and the transverse space. The results are meaningful and interesting. Based on the method in [10], partial synchronization bifurcations [19] were analyzed for a globally coupled network with a parameter, which topology is not complex. Nevertheless, all the eigenvalues of the topology matrix are essential for that method, which needs a large quantity of computation when the network size is very large.In this paper, a novel criterion on partial synchronization is proposed through the analysis of principal quasi-submatrices corresponding to the clusters. Previous researches have obtained several criteria on partial synchronization [10]. However, these criteria depend heavily on the topology matrix of the whole network. For many complex networks in real world, it is tedious to obtain the eigenvalues and eigenvectors of the topology matrix of the whole network. Therefore, this paper aims to propose a novel criterion, which is not distinctly dependent on the intercluster couplings and the topology matrix of the whole network. Instead, it is sufficient for partial synchronization to verify that the conditions on the intracluster couplings are satisfied. Therefore, it will be advantageous to discuss partial synchronization in the complex network with a large number of nodes because the dimension of any principal quasi-submatrix is usually far less than the one of the network topology matrix. At first, the tedious work of solving the eigenvectors corresponding to the second-largest eigenvalue is avoided. Secondly, partial synchronization is studied by the inner topologies of the individual clusters. Obviously, the network size reduction provides convenience for the study of partial synchronization in networks with great mounts of oscillators. In order to confirm its effectiveness, some numerical simulations are carried out to study partial synchronization in a star-global network. The numerical simulations are in good agreement with the theoretical analysis.The present paper is built up as follows. Some concerned concepts and a lemma for the invariance of the partial synchronization manifold are introduced in Section2. Principal quasi-submatrices corresponding to the clusters are proposed, and our main results are then established in Section 3. Numerical examples are presented to confirm the effectiveness of the results in Section 4. Finally, a brief discussion of the obtained results is given in Section 5.


## 2. Preliminaries

In this section, we introduce some basic concepts of invariant synchronization manifolds and a related lemma, which are required throughout the paper.In past years, many studies [9, 10, 12] of synchronization phenomena focused on oscillator networks coupled linearly and symmetrically. The system can be described by the following ordinary differential equations:
(1)
x
˙
i
=
f
(
x
i
,
t
)
+
ɛ
∑
j
=
1
m
‍
a
i
j
Γ
x
j
,
i
=
1
,
…
,
m
,
wherex
i
=
(
x
i
1
,
…
,
x
i
n
)
⊤is then-dimensional state variable of theith oscillator,m
>
1is the network size,t
∈
[
0
,
+
∞
) is a continuous time,f
:
R
n
×
[
0
,
+
∞
)
→
R
nis a continuous map,ɛ
>
0is the coupling strength, andΓ
=
diag
⁡
{
γ
1
,
…
,
γ
n
}is a nonnegative matrix determining the interaction of variables. The coupling weight matrixA
=
(
a
i
j
)
m
×
mis assumed to satisfy thata
i
j
=
a
j
i
≥
0, fori
≠
j, and∑
j
=
1
m
‍
a
i
j
=
sfori
=
1
,
…
,
m.In order to study partial synchronization of the system (1), the set of nodes{
1
,
…
,
m
}is divided intodnonempty subsets (clusters). LetG
=
{
G
1
,
…
,
G
d
}be the partition, and denoteK
=
(
K
1
,
K
2
,
…
,
K
d
), whereK
k
≥
1is the cardinal number of the clusterG
k,k
=
1
,
…
,
d. Without loss of generality, suppose thatG
1
=
{
1
,
…
,
K
1
}
,
…
,
G
d
=
{
∑
p
=
1
d
-
1
‍
K
p
+
1
,
…
,
m
}
.Based on the partitionG, we rewrite the coupling matrixAas a block matrix,
(2)
A
=
[
A
11
A
12
⋯
A
1
d
A
21
A
22
⋯
A
2
d
⋯
⋯
⋯
⋯
A
d
1
A
d
2
⋯
A
d
d
]
,
where A
k
l
∈
R
K
k
×
K
l
,
k
,
l
=
1
,
…
,
d.We will discuss sufficient conditions for theK
knodes in the clusterG
kto synchronize with each other,k
=
1
,
…
,
d. Before that, the concepts of partial synchronization manifolds and transverse spaces introduced in the following [10]:
(3)
M
K
=
{
∑
(
x
1
⊤
,
…
,
x
m
⊤
)
⊤
∣
(
x
∑
p
=
0
k
-
1
K
p
+
1
⊤
,
…
,
x
∑
p
=
0
k
K
p
⊤
)
⊤
∈
M
K
k
,
k
=
1
,
…
,
d
∑
}
,
where K
0
=
0,
(4)
M
K
k
=
{
∑
(
x
∑
p
=
0
k
-
1
K
p
+
1
⊤
,
…
,
x
∑
p
=
0
k
K
p
⊤
)
⊤
∈
R
n
K
k
∣
x
∑
p
=
0
k
-
1
K
p
+
1
=
⋯
=
x
∑
p
=
0
k
K
p
∑
}
,
are called a partial synchronization manifold. We also call
(5)
L
K
=
{
∑
(
x
1
⊤
,
…
,
x
m
⊤
)
⊤
∣
(
x
∑
p
=
0
k
-
1
K
p
+
1
⊤
,
…
,
x
∑
p
=
0
k
K
p
⊤
)
⊤
∈
L
K
k
,
k
=
1
,
…
,
d
∑
}
,
where
(6)
L
K
k
=
{
∑
i
=
1
K
k
‍
(
x
∑
p
=
0
k
-
1
K
p
+
1
⊤
,
…
,
x
∑
p
=
0
k
K
p
⊤
)
⊤
∈
R
n
K
k
∣
∑
i
=
1
K
k
‍
x
∑
p
=
0
k
-
1
K
p
+
i
=
0
}
,
a transverse space forM
K.In casen
=
1, the four sets mentioned above are denoted by M
K, M
K
k, L
K, and L
K
k, respectively. In cased
=
1, the synchronization manifoldM
Kis called a full synchronization manifold. For simplicity, we denote the full synchronization manifold M
K by M, and the corresponding transverse spaceL
K by L. Sometimes, the manifoldM
Kis also denoted byM
K
(
G
)to emphasize its partitionG.Definition 1.
The synchronization manifoldM
Kis said to be globally attractive for the system (1), or the system (1) is said to realize partial synchronization with the partitionG
=
{
G
1
,
G
2
,
…
,
G
d
}if, for any initial condition(
x
1
⊤
(
0
)
,
x
2
⊤
(
0
)
,
…
,
x
m
⊤
(
0
)
)
⊤
∈
R
m
n,
(7)
lim
⁡
t
→
+
∞
∑
k
=
1
d
‍
∑
i
∈
G
k
‍
∥
x
i
(
t
)
-
x
∑
p
=
0
k
-
1
K
p
+
1
(
t
)
∥
=
0
,
where∥
·
∥denotes 2-norm of vectors. In case ofd
=
1, the system (1) is said to realize full synchronization, if the full synchronization manifold M is globally attractive.Synchronization manifolds are always supposed to be invariant in order to discuss the attractiveness. Therefore, it is necessary to recall the definition of an invariant manifold [20] for the ordinary differential equations,
(8)
x
˙
=
X
(
x
,
t
)
,
x
∈
R
N
,
X
:
R
N
×
[
0
,
+
∞
)
⟶
R
N
.
Denote M as the manifold of codimensionpdefined by a vector equationH
(
x
)
=
0, H
=
(
h
1
,
h
2
,
…
,
h
p
), 1
≤
p
≤
N
-
1. M is called an invariant manifold of the system (8) if
(9)
(
grad
H
·
X
)
|
H
=
0
≡
0
,
which implies that the vector field (8) is tangent to M. For more details of the existence of invariant manifolds, one can refer to the papers by Golubitsky and coworkers [21, 22]. The following lemma gives a sufficient and necessary condition for a partial synchronization manifold being an invariant manifold.Lemma 2 (see [10]).
LetK
=
(
K
1
,
K
2
,
…
,
K
d
). The synchronization manifold M
K is an invariant manifold of the system (1) if and only if the coupling matrixAhas the form (2) with everyA
i
jbeing equal row sum.Remark 3.
Based on Lemma2, we can find all invariant synchronization manifolds for a given coupling matrix. As we know, each diagonal block reveals the intracluster information communication, and each nondiagonal block represents the information communication among different clusters. By the condition that everyA
i
jis equal row sum, we mean that every node in the same cluster receive an equal amount of information communication from every other cluster.


## 3. Main Results

Noticing the sufficient and necessary condition in Lemma2, we assume that every submatrixA
k
l
∈
R
K
k
×
K
lis an equal row sum matrix with row sums
k
l,k
,
l
=
1
,
…
,
d. SinceA
sis an equal row sum matrix, that is, ∑
j
=
1
m
‍
a
i
j
=
∑
k
=
1
d
‍
s
i
^
k
=
s, where the denotationi
^representsk, for alli
∈
G
k, we define the matrices A
¯
k
k
=
(
a
¯
i
j
)
K
k
×
K
k as
(10)
A
¯
k
k
=
A
k
k
+
(
s
-
s
k
k
)
E
K
k
=
A
k
k
+
∑
l
=
1
,
l
≠
k
d
‍
s
k
l
E
K
k
,
whereE
K
k
∈
R
K
k
×
K
kis the identity matrix,k
=
1
,
…
,
d. It is easy to conclude from (10) that ∑
j
∈
G
i
^
‍
a
¯
i
j
=
s and
(11)
a
¯
i
j
=
{
a
i
j
,
i
≠
j
;
a
i
j
+
∑
l
=
1
,
l
≠
i
^
d
‍
s
i
^
l
,
i
=
j
.
Noticing thatA
k
kis a principal submatrix ofA, we call the matrixA
¯
k
ka principal quasi-submatrix corresponding to the clusterG
k,k
=
1
,
…
,
d.Now, we are now in a position to carry out the following theorem with the help of Lyapunov function method.Theorem 4.
LetK
=
(
K
1
,
K
2
,
…
,
K
d
),P
=
diag
⁡
(
p
1
,
…
,
p
n
)be a positive-definite diagonal matrix, and letΔ
=
diag
⁡
(
δ
1
,
…
,
δ
n
)be a diagonal matrix. Supposeδ
j
≤
0ifj
∉
J, whereJ
=
{
j
:
1
≤
j
≤
n
,
γ
j
≠
0
}. Then under the following three conditions. (i)
Every submatrixA
k
lin the block matrix (2) has equal row sums
k
l, and every principal submatrixA
k
kis irreducible.
(ii)
There exists a constantϵ
>
0such that, for anyu
,
v
∈
R
n and  all t
≥
0,
(12)
(
u
-
v
)
⊤
P
{
[
f
(
u
,
t
)
-
f
(
v
,
t
)
]
-
Δ
(
u
-
v
)
}
≤
-
ϵ
(
u
-
v
)
⊤
(
u
-
v
)
.
(iii)
For allj
=
1
,
…
,
n, k
=
1
,
…
,
d, the matricesɛ
γ
j
A
¯
k
k
+
δ
j
E
K
kare negative semidefinite in the transverse spaceL
K
k, that is,
(13)
z
⊤
(
ɛ
γ
j
A
¯
k
k
+
δ
j
E
K
k
)
z
≤
0
,
z
∈
L
K
k
,or, in particular,
(14)
ɛ
γ
j
λ
k
(
2
)
+
δ
j
≤
0
,
where λ
k
(
2
) is the second-largest eigenvalue ofA
¯
k
k.
The synchronization manifoldM
Kis globally attractive for the system (1).For convenience of the proof, the following notations are introduced.(15)
x
¯
k
(
t
)
=
1
K
k
∑
i
∈
G
k
‍
x
i
(
t
)
,
x
¯
(
t
)
=
[
x
¯
1
^
⊤
(
t
)
,
…
,
x
¯
m
^
⊤
(
t
)
]
⊤
,
δ
x
i
(
t
)
=
x
i
(
t
)
-
x
¯
i
^
(
t
)
,
δ
x
(
t
)
=
[
δ
x
1
⊤
(
t
)
,
…
,
δ
x
m
⊤
(
t
)
]
⊤
,
δ
x
~
k
s
(
t
)
=
[
δ
x
∑
p
=
0
k
-
1
K
p
+
1
s
(
t
)
,
…
,
δ
x
∑
p
=
0
k
K
p
s
(
t
)
]
⊤
,
δ
x
~
k
(
t
)
=
[
δ
x
~
k
1
⊤
(
t
)
,
…
,
x
~
k
n
⊤
(
t
)
]
⊤
,
wherek
=
1
,
…
,
d
,
i
=
1
,
…
,
m
,
s
=
1
,
…
,
n.Then any vectorx
=
(
x
1
⊤
,
…
,
x
m
⊤
)
⊤
∈
R
m
ncan be decomposed into
(16)
x
=
x
¯
⊕
δ
x
,
x
¯
∈
M
K
,
δ
x
∈
L
K
.
Therefore, the attractiveness of the invariant synchronization manifoldM
Kis equivalent toδ
x
→
0whent
→
+
∞; that is, the dynamical flow in the transverse subspaceL
Kconverges to zero.Proof.
Noticing the condition (i), one gets(17)
∑
j
=
1
m
‍
a
i
j
Γ
x
j
(
t
)
=
∑
l
=
1
d
‍
∑
j
∈
G
l
‍
a
i
j
Γ
[
δ
x
j
(
t
)
+
x
¯
l
(
t
)
]
=
∑
l
=
1
m
‍
a
i
j
Γ
δ
x
j
(
t
)
+
∑
l
=
1
d
‍
s
i
^
l
Γ
x
¯
l
(
t
)
.
Therefore,
(18)
d
δ
x
i
(
t
)
d
t
=
d
x
i
(
t
)
d
t
-
1
K
i
^
∑
p
∈
G
i
^
‍
d
x
p
(
t
)
d
t
=
f
(
x
i
(
t
)
,
t
)
-
f
(
x
¯
i
^
(
t
)
,
t
)
+
ɛ
∑
j
=
1
m
‍
a
i
j
Γ
δ
x
j
(
t
)
+
J
i
^
,
where
(19)
J
i
^
=
f
(
x
¯
i
^
(
t
)
,
t
)
-
1
K
i
^
∑
p
∈
G
i
^
‍
[
f
(
x
p
(
t
)
,
t
)
+
ɛ
∑
q
=
1
m
‍
a
p
q
Γ
x
q
(
t
)
]
+
ɛ
∑
l
=
1
d
‍
s
i
^
l
Γ
x
¯
l
(
t
)
.
In order to utilize theQUAD
(
Δ
,
P
,
R
n
) condition, a Lyapunov function is defined as follows:
(20)
V
(
δ
x
(
t
)
)
=
1
2
∑
j
=
1
m
‍
δ
x
i
⊤
(
t
)
P
δ
x
i
(
t
)
.
One can conclude from∑
i
∈
G
k
‍
δ
x
i
(
t
)
=
0that
(21)
∑
i
=
1
m
‍
δ
x
i
⊤
(
t
)
P
J
i
^
=
∑
k
=
1
d
‍
[
∑
i
∈
G
k
‍
δ
x
i
⊤
(
t
)
]
P
J
i
^
=
0
,
and then,
(22)
d
V
(
δ
x
i
(
t
)
)
d
t
=
∑
i
=
1
m
‍
δ
x
i
⊤
(
t
)
P
[
f
(
x
i
(
t
)
,
t
)
-
f
(
x
¯
i
^
(
t
)
,
t
)
+
ɛ
∑
j
=
1
m
‍
a
i
j
Γ
δ
x
j
(
t
)
]
≤
-
ϵ
∑
i
=
1
m
‍
δ
x
i
⊤
(
t
)
δ
x
i
(
t
)
+
∑
i
=
1
m
‍
δ
x
i
⊤
(
t
)
P
[
ɛ
∑
j
=
1
m
‍
a
i
j
Γ
δ
x
j
(
t
)
+
Δ
δ
x
i
(
t
)
]
.
Denote the second term in the right hand of (22) asSfor convenience; then,
(23)
S
=
∑
k
=
1
d
‍
∑
i
∈
G
k
‍
δ
x
i
⊤
(
t
)
P
[
ɛ
∑
l
=
1
d
‍
∑
j
∈
G
l
‍
a
i
j
Γ
δ
x
j
(
t
)
+
Δ
δ
x
i
(
t
)
]
=
∑
k
=
1
d
‍
∑
i
∈
G
k
‍
δ
x
i
⊤
(
t
)
P
[
ɛ
∑
j
∈
G
k
‍
a
i
j
Γ
δ
x
j
(
t
)
+
Δ
δ
x
i
(
t
)
+
ɛ
∑
l
=
1
,
l
≠
k
d
‍
∑
j
∈
G
l
‍
a
i
j
Γ
δ
x
j
(
t
)
]
.
Sincei
,
j
∈
G
kholds for the first term in the right hand above, replacinga
i
jwitha
¯
i
jaccording to (11) gives rise to that
(24)
S
=
∑
k
=
1
d
‍
∑
i
∈
G
k
‍
δ
x
i
⊤
(
t
)
P
[
ɛ
∑
j
∈
G
k
‍
a
¯
i
j
Γ
δ
x
j
(
t
)
+
Δ
δ
x
i
(
t
)
]
+
ɛ
∑
k
=
1
d
‍
∑
i
∈
G
k
‍
δ
x
i
⊤
(
t
)
P
∑
l
=
1
,
l
≠
i
^
d
‍
[
-
s
i
^
l
Γ
δ
x
i
(
t
)
+
∑
j
∈
G
l
‍
a
i
j
Γ
δ
x
j
(
t
)
]
=
S
1
+
S
2
.
As a result of the condition (13), one obtains that
(25)
S
1
=
∑
k
=
1
d
‍
∑
i
∈
G
k
‍
δ
x
i
⊤
(
t
)
P
[
ɛ
∑
j
∈
G
k
‍
a
¯
i
j
Γ
δ
x
j
(
t
)
+
Δ
δ
x
i
(
t
)
]
=
∑
k
=
1
d
‍
∑
s
=
1
n
‍
p
s
δ
x
~
k
s
⊤
(
t
)
(
ɛ
γ
s
A
¯
k
k
+
δ
s
E
K
k
)
δ
x
~
k
s
(
t
)
≤
0
.
Sinces
i
^
l
=
∑
j
∈
G
l
‍
a
i
j, the sumS
2can be decomposed into
(26)
S
2
=
ɛ
∑
k
=
1
d
‍
∑
i
∈
G
k
‍
δ
x
i
⊤
(
t
)
P
∑
l
=
1
,
l
≠
k
d
‍
∑
j
∈
G
l
‍
a
i
j
Γ
(
δ
x
j
(
t
)
-
δ
x
i
(
t
)
)
=
ɛ
∑
k
=
1
d
-
1
‍
∑
l
=
k
+
1
d
‍
∑
i
∈
G
k
‍
∑
j
∈
G
l
‍
a
i
j
δ
x
i
⊤
(
t
)
P
Γ
(
δ
x
j
(
t
)
-
δ
x
i
(
t
)
)
+
ɛ
∑
l
=
1
d
-
1
‍
∑
k
=
l
+
1
d
‍
∑
i
∈
G
k
‍
∑
j
∈
G
l
‍
a
i
j
δ
x
i
⊤
(
t
)
P
Γ
(
δ
x
j
(
t
)
-
δ
x
i
(
t
)
)
.
Renaming in the second termkbyl,ibyj, and vice versa [23] and utilizing the symmetry of a
i
j, or A
l
k
⊤
=
A
k
l, one gets
(27)
S
2
=
ɛ
∑
k
=
1
d
-
1
‍
∑
l
=
k
+
1
d
‍
∑
i
∈
G
k
‍
∑
j
∈
G
l
‍
a
i
j
δ
x
i
⊤
(
t
)
P
Γ
(
δ
x
j
(
t
)
-
δ
x
i
(
t
)
)
+
ɛ
∑
k
=
1
d
-
1
‍
∑
l
=
k
+
1
d
‍
∑
j
∈
G
l
‍
∑
i
∈
G
k
‍
a
j
i
δ
x
j
⊤
(
t
)
P
Γ
(
δ
x
i
(
t
)
-
δ
x
j
(
t
)
)
=
-
ɛ
∑
k
=
1
d
-
1
‍
∑
l
=
k
+
1
d
‍
∑
i
∈
G
k
‍
∑
j
∈
G
l
‍
a
i
j
(
δ
x
j
(
t
)
-
δ
x
i
(
t
)
)
⊤
P
Γ
×
(
δ
x
j
(
t
)
-
δ
x
i
(
t
)
)
≤
0
.
Therefore, one obtains thatS
≤
0and
(28)
d
V
(
δ
x
i
(
t
)
)
d
t
≤
-
ϵ
∑
i
=
1
m
‍
δ
x
i
⊤
(
t
)
δ
x
i
(
t
)
≤
-
2
ϵ
V
(
δ
x
i
(
t
)
)
max
⁡
i
⁡
p
i
,
which implies that the partial synchronization manifoldM
Kis globally attractive for the system (1).
The remainder of the proof is to show that condition (14) is also sufficient forS
1
≤
0.
It is well known that a symmetric matrixA
¯
k
khas the decompositionA
¯
k
k
=
U
k
Λ
k
U
k
⊤, whereΛ
k
=
diag
⁡
{
λ
k
(
1
)
,
…
,
λ
k
(
K
k
)
}is a real diagonal matrix andU
k
∈
R
K
k
×
K
kis an unitary matrix,that
is
,
U
k
U
k
⊤
=
E
K
k. The diagonal elements ofΛ
kare the eigenvalues ofA
¯
k
ksatisfyings
=
λ
k
(
1
)
>
λ
k
(
2
)
≥
⋯
≥
λ
k
(
K
k
). Theith column ofU
kis the eigenvector ofA
¯
k
kcorresponding to the eigenvalueλ
k
(
i
)
,
i
=
1
,
…
,
K
k. By changes of variablesδ
x
~
k
s
(
t
)
=
U
η
k
s
(
t
), the quadratic form (25) can be diagonalized as follows:
(29)
S
1
=
∑
k
=
1
d
‍
∑
s
=
1
n
‍
p
s
δ
x
~
k
s
⊤
(
t
)
(
ɛ
γ
s
A
¯
k
k
+
δ
s
E
K
k
)
δ
x
~
k
s
(
t
)
=
∑
k
=
1
d
‍
∑
s
=
1
n
‍
p
s
η
k
s
⊤
(
t
)
(
ɛ
γ
s
Λ
k
+
δ
s
E
K
k
)
η
k
s
(
t
)
.
Noticing that the first column ofU
kis[
1
,
…
,
1
]
⊤, one can conclude from∑
i
∈
G
k
‍
δ
x
i
(
t
)
=
0andη
k
s
(
t
)
=
U
⊤
δ
x
~
k
s
(
t
)thatη
k
1
(
t
)
=
0. Therefore, condition (14) is sufficient forS
1
≤
0.
The proof is completed.Compared with the previous results [10], the conditions in Theorem 4 are not dependent on the intercluster couplings, which are eliminated in the proof through a set of mathematical skills in inequalities (25) and (27). The obtained results greatly reduce the network sizes theoretically. However, another question arises naturally: how to implement the proposed condition in reality? The following remarks might answer these questions.Remark 5.
In case that the network size is not very large and the coupling matrix is given, it is easy to verify all the conditions in Theorem4. In case that the network consists of great mounts of oscillators, it should still be full of challenges to implement though our result reduces the network size greatly. And it might be hard to implement.In order to make clear the implications of Theorem4, several remarks on the conditions are given as follows.Remark 6.
(
1
)Many chaotic oscillators have been proved to satisfy condition (ii), such as Chua circuits [24] and standard Hopfield neural networks [25]. However, many other systems are not the case such as a lattice ofx-coupled Rössler systems, in which the stability of synchronization regime is lost with the increasing of coupling [26].
(
2
)Providing that the trajectories of the uncoupled systemsx
i
=
f
(
x
i
,
t
),i
=
1
,
…
,
mare eventually dissipative, that is, the trajectories will be in the absorbing domain B eventually, it has been proved that each trajectory of the coupled system (1) is also eventually dissipative and will be in the absorbing domainB
×
⋯
×
B
︸
m[17] eventually. Therefore, condition (ii) holds when the timetis large enough. For example, the uncoupled Lorenz system
(30)
u
˙
=
σ
(
v
-
u
)
,
σ
=
10
,
v
˙
=
r
u
-
v
-
u
w
,
r
=
28
,
w
˙
=
-
b
w
+
u
v
,
b
=
8
3
,
is eventually dissipative, where
(31)
B
=
{
(
u
)
2
+
(
v
)
2
+
(
w
-
σ
-
r
)
2
<
b
2
(
σ
+
r
)
2
4
(
b
-
1
)
=
B
}
.
It has been proved thatx-coupled [27] or y-coupled Lorenz systems [28] satisfy condition (ii) when the timetis large enough.
(
3
)
If there exists a k
0
∈
{
1,2
,
…
,
d
} such that K
k
0
=
1, which implies that the subset G
k
0 contains only one element,  which does not synchronize with any other node. Without loss of generality, suppose thatK
k
=
1fork
>
d
¯. In this case, the corresponding principal quasi-submatrixA
¯
k
k
=
s and the constantscan be regarded as the single eigenvalue ofA
¯
k
ksinceA
¯
k
k
v
→
=
s
v
→for anyv
→
∈
R. Therefore, the second-largest eigenvalueλ
k
(
2
)does not exist. But notice the definition of partial synchronization in Section 2, synchronization in the clusterG
kalways occurs, and conditions (13) and (14) should be regarded to hold for anyɛ
>
0,k
>
d
¯.
(
4
)SinceA
k
kis irreducible and diffusive, the largest eigenvalue ofA
k
kis zero, which is simple. Therefore, the largest eigenvalue ofA
¯
k
kiss, and it is simple also. In cases
=
0, it can be seen thatλ
k
(
2
)
<
0and condition (14) is equivalent to
(32)
ɛ
≥
max
⁡
j
∈
J
⁡
{
0
,
δ
j
/
γ
j
}
min
⁡
1
≤
k
≤
d
¯
⁡
|
λ
k
(
2
)
|
.
Condition (14) provides us a novel index of partial synchronizability.


## 4. Numerical Examples

Consider the system (1) with2
m
y-coupled Lorenz systems
(33)
x
˙
i
=
f
(
x
i
,
t
)
+
ɛ
∑
j
=
1
2
m
‍
a
i
j
(
θ
)
Γ
x
j
,
i
=
1
,
…
,
2
m
,
wheref
(
x
i
,
t
)
,
x
i
=
(
u
i
,
v
i
,
w
i
)
⊤defined by the system (30),Γ
=
diag
⁡
(
0,1
,
0
). LetP
=
diag
⁡
(
B
/
2
(
b
-
ϵ
)
σ
+
ϵ
/
σ
,
1,1
), andΔ
=
diag
⁡
(
0,2
b
-
ϵ
-
1
+
[
B
-
2
(
b
-
ϵ
)
(
σ
-
ϵ
)
]
2
/
2
B
(
b
-
ϵ
)
,
0
), and thefin (27) has been proved to satisfy condition (ii) when the timetis large enough. Through translating time, the network system (27) satisfies condition (ii) in Theorem 4.


### 4.1. A Star-Global Network

Design the following topology matrix of the system (33) as
(34)
A
(
θ
)
=
[
A
11
(
θ
)
θ
E
m
θ
E
m
A
22
(
θ
)
]
2
m
×
2
m
,
whereθ
≥
0is the coupling weight parameter of couplings between the two clusters, and the identity matrixE
mimplies that theith (1
≤
i
≤
m) oscillator in the first cluster is coupled with the(
m
+
i
)th oscillator in the second cluster. As a special case, the submatricesA
11
(
θ
)andA
22
(
θ
)are taken as follows:
(35)
A
11
(
θ
)
=
[
1
-
m
-
θ
1
⋯
1
1
1
-
m
-
θ
⋯
1
⋯
⋯
1
1
⋯
1
-
m
-
θ
]
,
A
22
(
θ
)
=
[
1
-
m
-
θ
1
1
⋯
1
1
-
1
-
θ
0
⋯
0
⋯
⋯
1
0
0
⋯
-
1
-
θ
]
.
Obviously,A
11
(
θ
)andA
22
(
θ
)are the topology matrices of a globally coupled network and a star-coupled network, respectively. Therefore, we call the network a star-global (coupled) network.


### 4.2. Numerical Simulations

As an example, the topology structure of a star-global network withm
=
4 is shown in Figure 1. Due to the specific topological structure of the network, the following partitions satisfy condition (
i
),
(36)
G
1
=
{
1,2
,
3,4
,
5,6
,
7,8
}
,
G
2
=
{
1,2
,
3,4
;
5,6
,
7,8
}
,
G
3
=
{
1,5
;
2,3
,
4,6
,
7,8
}
.Figure 1
Topology structure of a star-global network composed of8oscillators. The solid lines represent the inner couplings of each clusters; the dashed lines represent the outer couplings between different clusters.Remark 7.
According to the criterion in the previous researches [10], we should firstly find the eigenvalues and the corresponding eigenvectors of the8
×
8matrix A
(
θ
). As we know, it should be of a vast amount of calculations. However, based on Theorem 4, it is enough for partial synchronization with the partition G
2
(
G
3
)if we can find the eigenvalues of two 4
×
4
matrices (four
2
×
2matrices). Obviously, Theorem 4 reduces the calculations greatly.The principal quasi-submatrices corresponding to the clusters in the partitionsG
1orG
2, areA
(
θ
)orA
11
(
θ
)
|
θ
=
0andA
22
(
θ
)
|
θ
=
0, respectively. And denote the ones in the partitionG
3asA
¯
11
(
θ
)andA
¯
22
(
θ
). Further analysis gives rise to the eigenvalues sets of the quasi-submatrices,
(37)
σ
(
A
(
θ
)
)
=
{
0
,
-
4
,
-
2
θ
,
-
4
-
2
θ
,
λ
±
,
λ
±
}
,
σ
(
A
11
(
θ
)
|
θ
=
0
)
=
{
0
,
-
4
,
-
4
,
-
4
}
,
σ
(
A
22
(
θ
)
|
θ
=
0
)
=
{
0
,
-
1
,
-
1
,
-
4
}
,
σ
(
A
¯
11
(
θ
)
)
=
{
0
,
-
2
θ
}
,
σ
(
A
¯
22
(
θ
)
)
=
{
0
,
-
2
θ
,
λ
′
±
,
λ
′
±
}
,
whereλ
±
=
-
(
2
θ
+
5
±
9
+
4
θ
2
)
/
2,λ
′
±
=
-
(
2
θ
+
3
±
9
+
4
θ
2
)
/
2. Therefore, partial synchronization with the partitionG
2occurs ifɛ
≥
δ; partial synchronization with the partitionG
3occurs ifɛ
≥
δ
/
|
λ
′
-
(
θ
)
|; and full synchronization with the partitionG
1occurs ifɛ
≥
max
⁡
{
δ
/
2
θ
,
δ
/
|
λ
-
(
θ
)
|
}. These are seen much more clearly in Figure 2.Figure 2
Threshold of the coupling strengthɛand the coupling weightθfor partial synchronization with different partitions.By fixingθ
∈
(
0,0.5
]and increasingɛgradually, partial synchronization with the partitionG
2will firstly occur; then, the one withG
3and full synchronization occurs at the same time. Figure 2 also implies that the threshold for partial synchronization with the partitionG
3is sufficient for full synchronization.In order to validate the effectiveness of Figure2 numerically, the following average cluster errors of the system (33) are defined to measure partial synchronization with partitionsG
1
,
G
2, andG
3, respectively,
(38)
e
1
=
1
8
∑
i
=
1
8
‍
∥
x
i
(
t
0
)
-
x
1
(
t
0
)
∥
,
e
2
=
1
8
∑
i
=
1
4
‍
∥
x
i
(
t
0
)
-
x
1
(
t
0
)
∥
+
1
8
∑
i
=
5
8
‍
∥
x
i
(
t
0
)
-
x
5
(
t
0
)
∥
,
e
3
=
1
8
∥
x
5
(
t
0
)
-
x
1
(
t
0
)
∥
+
1
8
∑
i
=
2
,
i
≠
5
8
‍
∥
x
i
(
t
0
)
-
x
2
(
t
0
)
∥
.Choose the initial conditions(
u
i
(
0
)
,
v
i
(
0
),  and  w
i
(
0
)
)randomly on[
-
1,1
]
×
[
-
1,1
]
×
[
-
1,1
], and pickt
0
=
500. Fix the coupling weight θ at 0.2, and5; the dependence of the cluster errors onɛis shown, respectively, in Figure 3. As can be seen, there is a good agreement between Figures 3 and 2.Dependence of the average cluster errors on the coupling strengthɛ for the system (33) with the topology matrix (34). The errorse
1
(
t
0
)
,
e
2
(
t
0
)
,
and
e
3
(
t
0
)measure the attractiveness of the manifoldsM
K
(
G
1
), M
K
(
G
2
),  and  M
K
(
G
3
), respectively.
(a)
θ
=
0.2
(b)
θ
=
5


## 4.1. A Star-Global Network

Design the following topology matrix of the system (33) as
(34)
A
(
θ
)
=
[
A
11
(
θ
)
θ
E
m
θ
E
m
A
22
(
θ
)
]
2
m
×
2
m
,
whereθ
≥
0is the coupling weight parameter of couplings between the two clusters, and the identity matrixE
mimplies that theith (1
≤
i
≤
m) oscillator in the first cluster is coupled with the(
m
+
i
)th oscillator in the second cluster. As a special case, the submatricesA
11
(
θ
)andA
22
(
θ
)are taken as follows:
(35)
A
11
(
θ
)
=
[
1
-
m
-
θ
1
⋯
1
1
1
-
m
-
θ
⋯
1
⋯
⋯
1
1
⋯
1
-
m
-
θ
]
,
A
22
(
θ
)
=
[
1
-
m
-
θ
1
1
⋯
1
1
-
1
-
θ
0
⋯
0
⋯
⋯
1
0
0
⋯
-
1
-
θ
]
.
Obviously,A
11
(
θ
)andA
22
(
θ
)are the topology matrices of a globally coupled network and a star-coupled network, respectively. Therefore, we call the network a star-global (coupled) network.


## 4.2. Numerical Simulations

As an example, the topology structure of a star-global network withm
=
4 is shown in Figure 1. Due to the specific topological structure of the network, the following partitions satisfy condition (
i
),
(36)
G
1
=
{
1,2
,
3,4
,
5,6
,
7,8
}
,
G
2
=
{
1,2
,
3,4
;
5,6
,
7,8
}
,
G
3
=
{
1,5
;
2,3
,
4,6
,
7,8
}
.Figure 1
Topology structure of a star-global network composed of8oscillators. The solid lines represent the inner couplings of each clusters; the dashed lines represent the outer couplings between different clusters.Remark 7.
According to the criterion in the previous researches [10], we should firstly find the eigenvalues and the corresponding eigenvectors of the8
×
8matrix A
(
θ
). As we know, it should be of a vast amount of calculations. However, based on Theorem 4, it is enough for partial synchronization with the partition G
2
(
G
3
)if we can find the eigenvalues of two 4
×
4
matrices (four
2
×
2matrices). Obviously, Theorem 4 reduces the calculations greatly.The principal quasi-submatrices corresponding to the clusters in the partitionsG
1orG
2, areA
(
θ
)orA
11
(
θ
)
|
θ
=
0andA
22
(
θ
)
|
θ
=
0, respectively. And denote the ones in the partitionG
3asA
¯
11
(
θ
)andA
¯
22
(
θ
). Further analysis gives rise to the eigenvalues sets of the quasi-submatrices,
(37)
σ
(
A
(
θ
)
)
=
{
0
,
-
4
,
-
2
θ
,
-
4
-
2
θ
,
λ
±
,
λ
±
}
,
σ
(
A
11
(
θ
)
|
θ
=
0
)
=
{
0
,
-
4
,
-
4
,
-
4
}
,
σ
(
A
22
(
θ
)
|
θ
=
0
)
=
{
0
,
-
1
,
-
1
,
-
4
}
,
σ
(
A
¯
11
(
θ
)
)
=
{
0
,
-
2
θ
}
,
σ
(
A
¯
22
(
θ
)
)
=
{
0
,
-
2
θ
,
λ
′
±
,
λ
′
±
}
,
whereλ
±
=
-
(
2
θ
+
5
±
9
+
4
θ
2
)
/
2,λ
′
±
=
-
(
2
θ
+
3
±
9
+
4
θ
2
)
/
2. Therefore, partial synchronization with the partitionG
2occurs ifɛ
≥
δ; partial synchronization with the partitionG
3occurs ifɛ
≥
δ
/
|
λ
′
-
(
θ
)
|; and full synchronization with the partitionG
1occurs ifɛ
≥
max
⁡
{
δ
/
2
θ
,
δ
/
|
λ
-
(
θ
)
|
}. These are seen much more clearly in Figure 2.Figure 2
Threshold of the coupling strengthɛand the coupling weightθfor partial synchronization with different partitions.By fixingθ
∈
(
0,0.5
]and increasingɛgradually, partial synchronization with the partitionG
2will firstly occur; then, the one withG
3and full synchronization occurs at the same time. Figure 2 also implies that the threshold for partial synchronization with the partitionG
3is sufficient for full synchronization.In order to validate the effectiveness of Figure2 numerically, the following average cluster errors of the system (33) are defined to measure partial synchronization with partitionsG
1
,
G
2, andG
3, respectively,
(38)
e
1
=
1
8
∑
i
=
1
8
‍
∥
x
i
(
t
0
)
-
x
1
(
t
0
)
∥
,
e
2
=
1
8
∑
i
=
1
4
‍
∥
x
i
(
t
0
)
-
x
1
(
t
0
)
∥
+
1
8
∑
i
=
5
8
‍
∥
x
i
(
t
0
)
-
x
5
(
t
0
)
∥
,
e
3
=
1
8
∥
x
5
(
t
0
)
-
x
1
(
t
0
)
∥
+
1
8
∑
i
=
2
,
i
≠
5
8
‍
∥
x
i
(
t
0
)
-
x
2
(
t
0
)
∥
.Choose the initial conditions(
u
i
(
0
)
,
v
i
(
0
),  and  w
i
(
0
)
)randomly on[
-
1,1
]
×
[
-
1,1
]
×
[
-
1,1
], and pickt
0
=
500. Fix the coupling weight θ at 0.2, and5; the dependence of the cluster errors onɛis shown, respectively, in Figure 3. As can be seen, there is a good agreement between Figures 3 and 2.Dependence of the average cluster errors on the coupling strengthɛ for the system (33) with the topology matrix (34). The errorse
1
(
t
0
)
,
e
2
(
t
0
)
,
and
e
3
(
t
0
)measure the attractiveness of the manifoldsM
K
(
G
1
), M
K
(
G
2
),  and  M
K
(
G
3
), respectively.
(a)
θ
=
0.2
(b)
θ
=
5


## 5. Conclusions

In summary, this paper introduced a novel index of partial synchronizability of a network. It is shown that partial synchronization can be ensured by the conditions merely on the quasi-submatrices corresponding to the clusters. If a network is composed of a great mount of nodes, the enormous amount of calculation can be reduced by replacing the coupling matrix with several quasi-submatrices. Numerically, different types of partial synchronization occur in a star-global network when the coupling strength is increased, the order of which is forecasted accurately by our result. It should be a meaningful and effective method to study partial synchronization with different partitions.In the past decades, the networked control systems have attracted much attention due to their applications covering a wide range of industries. And the network-induced phenomena under consideration in engineering have been discussed widely, including missing measurements [29], fading measurements [30], and probabilistic sensor delays [31]. Therefore, the obtained approach for partial synchronization in this paper might be applicable to the complex networks with networked induced phenomena. The related studies should be one of the future research topics.



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*Source: 584790-2014-07-16.xml*