Pinning Group Synchronization in Complex Dynamical Networks with Different Groups of Oscillators

This paper investigates the pinning control schemes and corresponding criteria for group synchronization in a complex dynamical network with different types of chaotic oscillators. We present the linear pinning and adaptive pinning control schemes to make different groups of oscillators synchronize to their own synchronization states, respectively. The globally asymptotically stable criteria for group synchronization are derived, which indicate that the group synchronization can be realized only by pinning a part of nodes in a general network. Finally, some numerical simulations are provided to verify the theoretical results.


Introduction
Nowadays, complex networks widely exist in nature and society, such as the World Wide Web, the Internet, traffic network, communication network, power network, and social network.Over the past three decades, complex networks have received increasing attention from various fields such as biology, physics, chemistry, engineering, and information [1][2][3].
Synchronization is a kind of typical collective behaviors and basic motion in nature.As a typical network dynamics, the synchronization of complex networks has been intensively studied recently.Different synchronization phenomena have been revealed, such as global synchronization, generalized synchronization, partial synchronization, and cluster synchronization [4][5][6][7][8][9][10][11].In particular, the group (cluster) synchronization, which means that nodes in the same group achieve the same synchronization state while nodes in different groups achieve different synchronization states, is considered to be significant in biological science, laser technology, and communication engineering [1,2,12].In [13], the authors investigated the group synchronization in complex networks with two clusters of coupled dynamical systems and revealed the group synchronization motions whenever there are connections or not within a cluster.In [14], the authors investigated the group synchronization in delay-coupled dynamical networks with groups of different and identical local dynamics, respectively.In [15], we presented a complex dynamical network model with  groups of different chaotic oscillators and derived some criteria to guarantee the group synchronization for a linearly or adaptively coupled network, respectively, under the "same input" condition.However, in many cases, the "same input" condition is too conservative to satisfy.If the "same input" condition cannot be satisfied, does the network achieve group synchronization?In this paper, we will propose control schemes to solve this problem.
As we known now, the real-world complex networks normally have a large number of nodes.Therefore, it is usually difficult to control a complex network by adding the controllers to all nodes.To reduce the number of the controllers, a usual approach, which is called the pinning control, is used to control part of nodes.Wang and Chen [16] and Li et al. [17] proposed some effective methods to pin a complex dynamical network to its equilibrium.In [18], Chen et al. proved that a single controller can pin a coupled complex network to a homogenous solution.Zhou et al. presented a simple approximate formula for estimating the detailed number of pinning nodes and the magitude of the coupling strength for a given general complex dynamical network in [19].However, the researchers mentioned above focused on the global synchronization of complex dynamical networks by pinning control.In this work, we will use the pinning control schemes to realize the group synchronization in a general complex dynamical network.
In this paper, based on the model in [15], we further investigate the group synchronization in a complex network containing  groups of different oscillators.Different from [15], this work mainly focuses on the pinning control schemes and the corresponding criteria to realize the group synchronization in a complex dynamical network.In particular, the network has a general structure rather than the "same input" condition.
The rest of the paper is organized as follows.In Section 2, the network model of our research and mathematical preliminaries are introduced.In Section 3, we present the linear feedback and adaptive feedback pinning schemes, respectively.The corresponding criteria for the globally asymptotically stable of the group synchronization manifold are derived.In Section 4, some numerical simulations are provided to verify the theoretical results.Finally, a brief summary is given in Section 5.
Assumption 1 (A1).Suppose that there exists positive constant   such that where z 1 , z 2 are time-varying vectors.
Definition 2. The set is called group synchronous manifold of network (1).
In order to analyze the group synchronization, the following lemmas are needed.
According to Lemma 3, one has λ − max That is, all eigenvalues of Ĝ are negative.Thus the proof is completed.Remark 6.From the derivation process, the conclusion of Corollary 5 is also valid when  1 , . . .,   are in any positions of the diagonal of Δ; that is to say, the result holds without any dependence on the sequence of the diagonal elements of Δ.

Schemes and Criteria
In this section, we present the linear feedback and adaptive feedback pinning schemes, respectively, and derive the corresponding criteria to achieve the group synchronization in network (1).
Remark 9.In this linear feedback pinning scheme, one only needs to pin a few nodes which have external connections; then the network achieves the group synchronization with large enough coupling strength.In particular, it is only relevant to network connectivity, without any limitation to the network structure.For a cluster network, in general, the connection within a cluster is more than the external connection between clusters.When the network has a cluster property, the higher the clustering, the better the control effect.Since the network with higher clustering means that the number of nodes with external connection is less, thus the less nodes are needed to be pinned.Therefore, the higher the degree of clustering, the more obvious the advantages.

Adaptive Feedback Pinning Control for Group Synchronization.
In order to avoid estimating the feedback control gain, in this section, we present an adaptive feedback pining scheme to realize the group synchronization in network (1).For network (1), we add the adaptive feedback pinning controllers as follows: Then the error equation ( 8) can be rewritten as Theorem 10.Suppose that 1 holds.The group synchronous manifold  of the controlled network (1) under the adaptive feedback pinning controllers (18) is globally asymptotically stable if the following inequality holds: where  = (  ) × , D = diag{, . .Its time derivative along the trajectory of ( 19) is The rest is similar to the proof of Theorem 7; one has lim  → +∞ ‖   ‖ = 0 ( = 1, . . .,   ,  = 1, . . ., ), which means that the group synchronous manifold  is globally asymptotically stable.Thus the proof is completed.Remark 11.In Theorem 10, the adaptive constant  can be large enough such that inequality (20) holds.Therefore, the group synchronization can be easily guaranteed.

Numerical Simulations
In this section, an illustrative example is provided to verify the above theoretical analysis.We consider a network (see Figure 1) which consists of three groups with 4 Lorenz chaotic oscillators [20] as the group, 5 Chua oscillators [21] as the -group, and 6 Chen oscillators [22] as the -group, respectively.The Lorenz, Chua, and Chen chaotic attractors are displayed in Figure 2. The network as an example of the controlled network (1) is described by where  12) and (18), respectively.
Figures 3 and 4 display the numerical simulation results of network (23) under the linear feedback and adaptive feedback pinning control, respectively.They both show that a set of nodes belonging to each group synchronizes to the same behavior; that is, the group synchronization has been achieved.
Remark 12.By the above theoretical analysis and instance, it is concluded that the pinning control strategy can be applied to large networks with tens of thousands of nodes based on the accurate network topology.In fact, this control strategy depends on the structure information rather than the size of the network.

Conclusions
In this paper, we have investigated the group synchronization in a complex dynamical network consisting of different groups of coupled chaotic oscillators.The linear feedback and adaptive feedback pinning schemes are presented, respectively.The corresponding sufficient conditions are derived to ensure each group synchronizes to the same states.Our study shows that the global asymptotic stability of the group synchronous manifold can be guaranteed by controlling a few nodes which have external connections.For the large networks with obvious cluster property, our pinning control schemes can save much cost in practical.