J Control Theory Appl 2008 6 (1) 11–15
DOI 10.1007/s11768-008-7187-7
Controlled synchronization of complex network
with different kinds of nodes
Zhengquan YANG, Zhongxin LIU, Zengqiang CHEN, Zhuzhi YUAN
(
Department of Automation, Nankai University, Tianjin 300071, China)
Abstract: In this paper, a new dynamical network model is introduced, in which the nodes of the network are different.
It is shown that by the designed controllers, the state of the network can exponentially synchronize onto a homogeneous
stationary state. Some criteria are derived and some examples are presented. The numerical simulations coincide with
theoretical analysis.
Keywords: Complex network; Synchronization; Pinning control; Chaos; Exponential stable
1 Introduction
Recently, complex networks have been a subject of con-
siderable interest within the science and technology com-
munities. Among them, synchronization is the most inter-
esting. In fact, synchronization is very popular in nature. For
example, synchronization of coupled oscillators can well
explain many natural phenomena such as spatiotemporal
chaos, auto waves, and spiral waves. In addition, many real-
world problems such as the synchronous phenomena on the
Internet, synchronous transfer of digital or analog signals in
communication networks, or biological neural networks are
closely related to synchronization.
Synchronization in large-scale networks of coupled
chaotic systems has been discussed in the past decade
[1∼5]. In these works, some authors focused on completely
regular networks, such as the continuous-time cellular neu-
ral network (CNN) and the discrete-time coupled map lat-
tice (CML) [2∼5], while others addressed synchroniza-
tion of randomly coupled networks [6]. However, many
real-world networks, such as the WWW, food webs, and
metabolic networks are neither completely regular nor com-
pletely random. Thus, more and more studies have been de-
voted to discussing the synchronization phenomena in com-
plex networks in recent years [7∼18].
Very recently, Wang and Chen presented a uniform model
and investigated its synchronization and control in small-
world and scale-free networks [10∼13]. L
¨
u and Chen in-
troduced a time varying model with the same prototype of
Wang and Chen and studied its synchronization [14∼16].
Although these studies reflect the complexity of the network
structure, they focused on N identical nodes. However, in
many real world systems, such as laser arrays and biolog-
ical systems [19, 20], it is hardly the case that every com-
ponent can be assumed to be identical. As a result, more
and more applications of chaos synchronization in secure
communications make it much more important to synchro-
nize two different chaotic systems in recent years [21]. In
this regard, some works on synchronization of two different
chaotic systems have been done [21, 22].
In the real world, most nodes of the complex dynami-
cal networks are different. So in this paper, a new dynami-
cal network model is introduced, which has different nodes.
We want to stabilize the network onto a homogeneous sta-
tionary state. The rest of the paper is organized as follows.
In Section 2, a model with linear coupling matrix is intro-
duced and its controlled synchronization is studied. In Sec-
tion 3, theoretical results are verified by several simulations.
Finally, in Section 4, we conclude the paper.
2 A model with linear coupling matrix and its
pinning control
Now suppose that, at some time, the network consists of
N differently, linearly and diffusively coupled nodes, with
each node being an n-dimensional dynamical system. For
simplicity, we consider two kinds of nodes and assume the
first l(1 <l<N) nodes are the same. The state equations
of the network are
⎧
⎪
⎪
⎨
⎪
⎪
⎩
˙x
i
= g(x
i
)+c
N
j=1
a
ij
Γ x
j
,i=1, ··· ,l;
˙x
i
= f(x
i
)+c
N
j=1
a
ij
Γ x
j
,i= l+1, ··· ,N,
(1)
where f, g are continuous map, x
i
=(x
i1
,x
i2
, ··· ,x
in
)
T
∈
R
n
are the state variables of node i, the constant c>0
Received 12 September 2007.
This work was supported by the National Natural Science Foundation of China (No.60774088, 60574036), the Program for New Century Excellent
Talents in University of China (NCET), the Specialized Research Fund for the Doctoral Program of Higher Education of China (No.20050055013)
and the Science & Technology Research Key Project of Education Ministry of China (No.107024).