Cluster Synchronization of Linearly Coupled Complex Networks Under Pinning Control

Abstract

In this paper, we focus on the problem of driving a general network to a selected cluster synchronization pattern by means of a pinning control strategy. Sufficient conditions are presented to guarantee the realization of the cluster synchronization pattern for all initial values. We also show the detailed steps on how to construct the coupling matrix and to modify the control strengths. Moreover, the method of adapting the coupling strength is provided to refine the result.

Full text

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 56, NO. 4, APRIL 2009 829
Cluster Synchronization of Linearly Coupled
Complex Networks Under Pinning Control
Wei Wu, Wenjuan Zhou, and Tianping Chen
Abstract—In this paper, we focus on the problem of driving a
general network to a selected cluster synchronization pattern by
means of a pinning control strategy. Sufficient conditions are pre-
sented to guarantee the realization of the cluster synchronization
pattern for all initial values. We also show the detailed steps on
how to construct the coupling matrix and to modify the control
strengths. Moreover, the method of adapting the coupling strength
is provided to refine the result.
Index Terms—Adaptive adjustment, cluster synchronization,
globally asymptotically stable, linearly coupled networks, pinning
control.
I. INTRODUCTION
SINCE THE first observation of synchronization phenom-
enon was made by Huygens [1] in the 17th century, this
phenomenon has been discovered in a wide range of real sys-
tems, such as in biology [2], neural networks [3], physiolog-
ical process [4], and others. Recently, inspired by the pioneering
work of Pecora and Carroll [5], the synchronization of coupled
chaotic systems has received increasing attention in the math-
ematical and physical literature because of its potential appli-
cations in various fields, including secure communication, seis-
mology, parallel image processing, chemical reaction, etc. (see,
e.g., [6]–[9]).
In mathematics, synchronization can be defined as a process
wherein two (or many) dynamical systems adjust a given prop-
erty of their motion to a common behavior as time goes to in-
finity, due to coupling or forcing [10]. For ensembles of cou-
pled chaotic oscillators, many different regimes of synchroniza-
tion have been investigated, including phase synchronization,
lag synchronization, full synchronization, cluster synchroniza-
tion, almost synchronization, and so on (see, e.g., [11]–[20]).
The type of synchronization that is referred to in this paper is
cluster synchronization. It requires that the coupled oscillators
split into subgroups called clusters, such that the oscillators syn-
chronize with one another in the same cluster, but there is no
synchronization among different clusters.
Manuscript received May 20, 2008; revised June 19, 2008 and July 16,
2008. First published October 31, 2008; current version published April 10,
2009. This work was supported by the National Science Foundation of China
under Grants 60774074 and 60574044. This paper was recommended by As-
sociate Editor Z. Duan.
The authors are with the Key Laboratory of Nonlinear Mathematics
Science, School of Mathematical Sciences, Fudan University, Shanghai
200433, China (e-mail: 051018023@fudan.edu.cn; 0118133@fudan.edu.cn;
tchen@fudan.edu.cn).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TCSI.2008.2003373
Linearly coupled ordinary differential equations (LCODEs)
are an important class of dynamical systems with continuous
time and state, as well as discrete space, for describing coupled
oscillators. In general, LCODEs can be described as follows:
(1)
where is the network size,
is the state variable of the
th oscillator, is the continuous time,
is a continuous map,
denotes the coupling configuration of
the network, and is a constant matrix of size . Since
the past few years, cluster synchronization of different kinds
of LCODEs has been widely studied by many researchers
[21]–[25]. In [21], an effective method to determine some
possible states of cluster synchronization and ensure their
stability was presented for a given nearest neighborhood
network with zero-flux or periodic boundary conditions.
Two-dimensional and 3-D lattices of diffusively coupled
chaotic oscillators were investigated in [22] and [23]. In [24],
cluster synchronization in a network of strictly semipassive
systems via diffusive coupling was investigated. In [25], an
effective method, which constructs a coupling scheme with
cooperative and competitive weight couplings, was used to
stabilize arbitrarily selected cluster synchronization manifolds
for connected chaotic networks.
However, approaches to realizing cluster synchronization by
means of linear coupling configurations meet some problems
due essentially to the existence of embedding of invariant syn-
chronization manifolds. For the definition of an invariant mani-
fold for a general system of ordinary differential equations, see
[21, Def. 1]. Previous studies show that, for the linearly coupled
network (1), it may occur that, beginning with different initial
values, the system can attain different clustering patterns, par-
ticularly when there exists an embedding of invariant synchro-
nization manifolds (for details, see [21]–[26] and the references
cited therein). On the other hand, considering that a network can
be formed by an arbitrary number of oscillators, the number of
different existing invariant synchronization manifolds may be
large [24]. Therefore, it is difficult to simply use linear coupling
to achieve a desired cluster synchronization pattern that is in-
dependent of initial values. In fact, similar problems would also
arise for some other coupling schemes, including nonlinear cou-
pling, delayed coupling, and time-varying coupling.
In this paper, we will give an alternative method to gain the
point of realizing cluster synchronization. In [27], the authors
investigated pinning control for linearly coupled network and
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830 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 56, NO. 4, APRIL 2009
suggested that one can pin the coupled network to its equi-
librium by introducing fewer locally negative feedback con-
trollers (although the proof should be revised). In [28], Chen
et al. provided sufficient conditions guaranteeing the synchro-
nization of all the oscillators in the network to a homogenous
solution with a single pinning controller. In this paper, contin-
uing with these previous works, we study the pinning control
problem of linearly coupled networks. For an arbitrarily selected
cluster synchronization pattern, we introduce a single controller
for each cluster and derive sufficient conditions such that the
given cluster synchronization pattern is achieved for any initial
values.
The rest of this paper is organized as follows. In Section II, we
show how to transform the cluster synchronization problem into
a pinning control problem and give the network model discussed
in this paper. In Section III, sufficient conditions are presented
to ensure the pinning control performance. In Section IV, we
provide a method to adapt the coupling strengths. Numerical
examples are given in Section V. We conclude this paper in
Section VI.
II. FORMULATION OF THE PROBLEM
First, we give the mathematical definition of cluster
synchronization.
Definition 1: Let be a partition of the set
into nonempty subsets, i.e., and
.For , let denote
the subscript of the subset in which the number is, i.e.,
. A network with identical oscillators is said to realize
-cluster synchronization with the partition
if, for any initial values, the state variables of the oscilla-
tors satisfy for and
for .
Now, we state the problem under consideration.
When control is introduced, the linearly coupled network (1)
becomes
(2)
where is a subset of denoting the controlled os-
cillator set and denotes the control on the oscillator .
Suppose that we want the network (2) to achieve the -cluster
synchronization with the partition ,
,
, where , ,
and . For the convenience of later use, let the
row vector be denoted by and the partition
be denoted by .
Inspired by the pinning control method investigated in
[28]–[30], in this paper, we use the following approach to
realize this given cluster synchronization pattern: 1) Select
special solutions of the homogenous system
such that
for ; 2) for the coupled system (2), we let be
the identity matrix and the controlled oscillator set be
, and we use linear
negative feedback controllers
where , , are the control strengths, i.e., we con-
sider the following coupled system:
(3)
where ; 3) find suffi-
cient conditions for ensuring the pinning control performance
which is defined by
holding for any initial values. Clearly, if such conditions are
found, the -cluster synchronization with the partition has
been transformed into a pinning problem, i.e., how to make
by pinning control if , where
, , are the particular solutions of the uncou-
pled system .
Finally, we define a class of functions and several classes of
matrices, which will play important roles in determining the de-
sired conditions.
Definition 2: Function class : Let
be a diagonal matrix and
be a positive-definite diagonal matrix.
denotes a class of continuous functions
satisfying
(4)
for some , all , and all .
Definition 3: Suppose that .If
( ), ( ), and is
irreducible, then we say that .
Definition 4: For an symmetric matrix
(5)
with , , if each block is a
zero-row-sum matrix, then we say that . Further-
more, if , , then we say that .
III. SUFFICIENT CONDITIONS FOR ENSURING THE PINNING
CONTROL PERFORMANCE
In this section, we derive sufficient conditions under which,
for any initial values, the solutions of the system (3) satisfy
.
At first, basing on the Lyapunov function method and matrix
theorem, we prove the following theorem.
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WU et al.: CLUSTER SYNCHRONIZATION OF LINEARLY COUPLED COMPLEX NETWORKS UNDER PINNING CONTROL 831
Theorem 1: Suppose that the coupling matrix in (3) sat-
isfies . Let be a diagonal
matrix and be a positive-definite diag-
onal matrix such that .If
(6)
where is the identity matrix and
is the diagonal matrix with
entries ,
and for , then, for any initial values,
the solutions of the system (3) satisfy
.
Proof: Denote , . Since
, which means that holds for all
and , it is clear that
Therefore, for ,wehave
and for ,wehave
That is to say, satisfy the following differ-
ential equations:
.
Define a Lyapunov function as
Denote ,
. Calculating the derivative of [noticing
that ], we have
(7)
Combining (7) and the condition (6), we get
(8)
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832 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 56, NO. 4, APRIL 2009
From (8), we can conclude that, for any initial values,
the solutions of the system (3) satisfy
.
Remark 1: In fact, under the assumption of the coupling ma-
trix , the cluster synchronization manifold
is an in-
variant manifold of the coupled system (1). In addition, it is
clear that when , each block in block form
(5) has nonzero elements or is a zero matrix. For ,if
with and , then the coupling be-
tween oscillators and is called an inhibitory coupling, which
can be regarded as a mechanism to desynchronize and ;if
, then there is no coupling between oscillators
in the th cluster and oscillators in the th cluster. Of course, for
a nontrivial coupling configuration, it is required that, for each
, there exists a such that .
From Theorem 1, it can be seen that, in order to realize the
pinning control performance, we should construct a coupling
matrix and select control strengths
such that the matrix inequalities (6) hold. In the following, we
will give an efficient construction method to ensure the matrix
inequalities (6).
Before giving the method, we introduce some notations and
lemmas.
For a matrix , denote
. For a matrix ,
denote the maximal eigenvalue of by .
Lemma 1 ([28Proposition 1]): Suppose that and
with are two matrices of size
. Then, the matrix is negative definite.
Lemma 2: For a matrix ,
holds for all and .
Now, we give the method by proving the following theorem.
Theorem 2: Suppose that are positive con-
stants and that
where , . Denote
, . We pick the cou-
pling matrix
(9)
and the control strengths
(10)
where is a positive constant. For a diagonal matrix
, if the coupling strength
satisfies
(11)
where
(12)
then we have , , where is
defined as in Theorem 1.
Remark 2: In fact, the construction method can be divided
into three steps. First, arbitrarily choose a matrix
and positive constants . Second, construct the cou-
pling matrix as (9), and select the control strengths
as (10). Since , it is clear that . Finally,
pick a coupling strength satisfying (11).
Proof: From the definitions of and ,wehave
.
For a vector , we let
. Then, for all and all
, by Lemma 2, we have
According to Lemma 1, is negative definite, which
implies that . Thus, under the condition of
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WU et al.: CLUSTER SYNCHRONIZATION OF LINEARLY COUPLED COMPLEX NETWORKS UNDER PINNING CONTROL 833
, we have that holds
for all and all .
IV. ADAPTIVE ADJUSTMENT OF THE COUPLING STRENGTH
In the previous section, we proposed a construction method
and proved that the pinning control performance can be realized
if the coupling strength is large enough. However, in practice,
it is not allowed that the coupling strength is arbitrarily large.
For synchronization, it was pointed out in [31] that the theoret-
ical value of the coupling strength is usually much larger than
needed in practice. Therefore, the following question arose in
[31]: Can we find relatively small coupling strength? Similarly,
in the pinning process, it is also important to make the coupling
strength as small as applicable.
A simple way to achieve this is to adapt the coupling strength.
For that purpose, we give the following theorem.
Theorem 3: Suppose that are positive constants
and that
Pick the coupling matrix
(13)
and the control strengths
(14)
where is the adaptive coupling strength. Let
be a diagonal matrix and
be a positive-definite diagonal matrix
such that . Then, the coupled system
(15)
where and , can achieve the pinning control
performance.
Proof: Define a Lyapunov function as
where positive constants and will be decided later.
Denote , where ,
, are defined as in Theorem 2. Calculating the deriva-
tive of , similarly, as in Theorem 1, we have
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834 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 56, NO. 4, APRIL 2009
In the following, we try to prove that, by picking proper and
, the time-varying matrix
is negative definite for all and all .
Similarly, as in Theorem 2, for any vector , denote
.
Then, for any , all , and all ,wehave
(16)
Since and , , are negative-definite,
we can pick and such that for
and for
. Then, by (16), is
negative definite for and all .
Therefore
which implies that and , i.e.,
, where is a nonnegative constant.
Theorem 3 is proved.
V. N UMERICAL SIMULATIONS
In this section, we give several numerical examples to verify
the theorems given in the previous sections.
We consider a network composed of identical os-
cillators. The uncoupled subsystem is a
3-D neural network or Chua’s circuit.
The 3-D neural network that we consider is described by
(17)
with
, and , where
. As indicated by [32],
system (17) has a double-scrolling chaotic attractor. In
[33], it was obtained that, for the 3-D neural network (17),
.
Chua’s circuit is described in dimensionless form by
(18)
where , , and
. In [34], it was obtained that, for Chua’s circuit (18),
.
In this simulation, we want to realize a three-cluster
synchronization with the partition ,
, . According to the
pinning control approach with an adaptive coupling strength,
we consider the coupled system (15) with .
We let , , and be the solutions of the ho-
mogenous system with initial values
, , and
; ; be a matrix
chosen randomly in ; ; and . The
following quantity is used to investigate the process of cluster
synchronization:
Solving (15) numerically by the fourth-order Runge–Kutta
method with a fixed step size, we obtain Figs. 1 and 2.
Fig. 1(a) shows how and evolve in pinning 3-D
neural networks with initial values chosen randomly in the in-
terval .
Fig. 1(b) shows how and evolve in pinning 3-D
neural networks with initial values ,
.
Fig. 2(a) shows how and evolve in pinning Chua’s
circuits with initial values chosen randomly in the interval
.
Fig. 2(b) shows how and evolve in pinning
Chua’s circuits with initial values ,
.
Remark 3: In [25], a linear coupling scheme with cooperative
and competitive weight couplings was proposed to stabilize ar-
bitrarily selected cluster synchronization patterns in connected
chaotic networks. However, for networks with this linear cou-
pling configuration, there must be some invariant submanifolds
in the selected cluster synchronization manifold, such as the full
synchronization manifold. It implies that the selected cluster
synchronization patterns cannot be realized via this coupling
scheme if initial values are chosen in these invariant subman-
ifolds. In our paper, it was proved that, by introducing pinning
control, we can realize an arbitrarily selected cluster synchro-
nization pattern for any initial values. In Figs. 1(b) and 2(b), one
can see that the three-cluster synchronization with the partition
, ,
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WU et al.: CLUSTER SYNCHRONIZATION OF LINEARLY COUPLED COMPLEX NETWORKS UNDER PINNING CONTROL 835
Fig. 1. Time evolution of and in pinning 3-D neural networks.
(a) Initial values are chosen randomly in . (b) Initial values
, .
can be realized even if initial values are chosen in the full syn-
chronization manifold.
Remark 4: In our paper, we consider the following coupled
system with pinning controllers [see (3)]:
(19)
and try to find conditions such that
Thus,
should be a solution of the coupled system
(19). Suppose that each block is an equal-row-sum matrix.
Fig. 2. Time evolution of and in pinning Chua’s circuits.
(a) Initial values are chosen randomly in . (b) Initial values
, .
Let be the row sum of the block . Then, we have that
satisfy the differential equations
(20)
In our paper, are chosen to be solutions of the
chaotic oscillator and satisfy
for . Thus, holds for all
, which means that the row sum is zero for every block
. Therefore, the assumption that ALL blocks ,
, are zero row sum is natural and necessary.
In [35], the authors studied the coupled system
(21)
where is an -dimensional state vector and is an -di-
mensional state vector. In this coupled system, there are two
distinct groups of nodes where the dynamical systems on each
node within a group are the same but are different for nodes in
different groups. Therefore, the coupled system realizes two-
cluster/group synchronization as long as the synchronization
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836 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 56, NO. 4, APRIL 2009
manifold
is stable.
However, for coupled systems composed of identical nodes
(i.e., ), the problem of cluster synchronization becomes
complicated due essentially to the existence of embedding of
invariant synchronization manifolds (we have addressed this
topic in detail). For example, for the following linearly coupled
system:
(22)
even if the cluster synchronization manifold
is globally stable, we cannot say that the system
(22) realizes -cluster synchronization. It is because there may
be some submanifolds of which are invariant (even glob-
ally stable). That is to say, when is globally stable, perhaps
beginning with some (even all) initial values, the system attains
other clustering patterns. Moreover, because of the complex dy-
namics of the system (22), it is difficult to exclude this possi-
bility. Here, cluster synchronization was defined as follows: “If
an invariant manifold is globally stable asymptotically, and
the full synchronization manifold is unstable, then the flow in
defines cluster synchronization of dimension .” (for a
similar definition, also see [21]).
Now, we give a simple example to verify our conclusions.
Consider a network composed of identical oscillators.
The uncoupled oscillator is described by
(23)
where . This system has three equilibria
It is easy to verify that and are stable equilibria and that
is an unstable equilibrium.
In the following, we discuss how to realize a two-cluster syn-
chronization with the partition , .
A. Simulation 1
In this simulation, we consider the linearly coupled system
with pinning controllers discussed in this paper
(24)
where the controlled oscillator set , and the orbits
and are chosen to be two equilibria of the system
(23): and .
First, we pick the coupling matrix
Fig. 3. Time evolution of , , , and with and
.
Fig. 4. Time evolution of , , , and with and
.
which is a matrix in the class , i.e., each block is a zero-
row-sum matrix.
In this case, cluster synchronization can be realized.
Fig. 3 shows how , , , and evolve with
and initial values chosen randomly in the interval
. It can be seen that the desired two-cluster synchro-
nization is realized.
Second, we pick the coupling matrix
Obviously, the block
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WU et al.: CLUSTER SYNCHRONIZATION OF LINEARLY COUPLED COMPLEX NETWORKS UNDER PINNING CONTROL 837
is not a zero-row-sum matrix but an equal-row-sum matrix,
which means that the sum of the couplings from the oscillators
in the first (or second) group to those in the second (or first)
one is a nonzero constant.
In this case, cluster synchronization cannot be ensured.
Fig. 4 shows how , , , and evolve with
and initial values chosen randomly in the interval
. It can be seen that the desired two-cluster synchro-
nization is not realized.
From Simulation 1, we can have the following conclusion.
For the case discussed in this paper of cluster synchronization
with pinning control to some different trajectories of the uncou-
pled system, the assumption that all are zero row sum is nat-
ural and necessary. Yet, under different circumstances, it is also
possible to achieve cluster synchronization without introducing
pinning control; this case will be briefly discussed hereinafter.
B. Simulation 2
In this simulation, we consider the linearly coupled system
without controller
(25)
We pick the coupling matrix
where and are constant (not zero) row sum.
In this case, cluster synchronization can be realized but not
ensured.
In fact, the cluster synchronization problem reduces to the
problem of the existence of equilibria and their stability for the
following system:
(26)
It is clear that if is an equilibrium point of the system
(28)
then the equilibrium point , is
the cluster synchronization state.
Pick the initial values of , ,
, and .
Fig. 5 shows how , , , and evolve with
time. It can be seen that the coupled system realizes two-cluster
synchronization.
Instead, if is an equilibrium point of the system
(29)
then (or ) is the full synchro-
nization state.
Fig. 5. Time evolution of , , , and with and
.
Fig. 6. Time evolution of , , , and with and
.
Pick the initial values of , ,
, and .
Fig. 6 shows how , , , and evolve. It can
be seen that the coupled system realizes full synchronization.
It should be noted that, in both cases, the synchronization
state never satisfies .
From Simulation 2, we can have the following conclusion.
By the linearly coupled system without controllers discussed
in [35], where all are constant row sum, the cluster synchro-
nization to the different trajectories of cannot
be ensured if the uncoupled systems are identical, i.e.,
in (21). The assumption that all are zero row sum is natural
and needed.
The general case that are constant row sum is very inter-
esting and complicated. This case is discussed in [36].
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838 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 56, NO. 4, APRIL 2009
VI. CONCLUSION
In this paper, we have discussed the problem of driving lin-
early coupled networks to an arbitrarily selected cluster syn-
chronization pattern via pinning control. We proved that, by in-
troducing a single negative feedback controller for each cluster,
we can pin the coupled system to the assigned cluster synchro-
nization pattern for any initial values. Moreover, the method of
adjusting the coupling strength adaptively is proposed to ensure
more practical applications. Numerical examples are also pro-
vided to verify our theoretical results.
ACKNOWLEDGMENT
The authors would like to thank the anonymous reviewers for
their comments. In particular, one reviewer read this paper very
carefully and raised very important queries that greatly helped
the authors in revising this paper.
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Wei Wu received the B.S. degree in computational
mathematics and the Ph.D. degree in applied math-
ematics from the School of Mathematical Sciences,
Fudan University, Shanghai, China, in 2003 and
2008, respectively.
He is currently with the Software Development
Center, Industrial and Commercial Bank of China.
His research interests include nonlinear dynamical
systems, complex networks, and neural networks.
Authorized licensed use limited to: CityU. Downloaded on April 11, 2009 at 06:43 from IEEE Xplore. Restrictions apply.

WU et al.: CLUSTER SYNCHRONIZATION OF LINEARLY COUPLED COMPLEX NETWORKS UNDER PINNING CONTROL 839
Wenjuan Zhou received the B.S. degree in mathe-
matics and the M.S. degree in applied mathematics
from the School of Mathematical Sciences, Fudan
University, Shanghai, China, in 2005 and 2008,
respectively.
She is currently with the Qingpu Power Supply
Branch, Shanghai Municipal Electric Power Com-
pany. Her research interests include nonlinear
dynamical systems and complex networks.
Tianping Chen graduated as a Postgraduate Student
from the Department of Mathematics, Fudan Univer-
sity, Shanghai, China, in 1965.
He is currently a Professor with the School of
Mathematical Sciences, Fudan University, and with
Key Laboratory of Nonlinear Mathematics Science.
His research interests include complex networks,
neural networks, signal processing, dynamical sys-
tems, harmonic analysis, and approximation theory.
Prof. Chen is a recipient of several awards,
including the 2002 National Natural Sciences Award
of China (second prize), the 1997 Outstanding Paper Award of the IEEE
TRANSACTIONS ON NEURAL NETWORKS, and the 1997 Best Paper Award of the
Japanese Neural Network Society.
Authorized licensed use limited to: CityU. Downloaded on April 11, 2009 at 06:43 from IEEE Xplore. Restrictions apply.