Abstract
This paper investigates the synchronization of directed networks whose coupling matrices are reducible and asymmetrical by pinning-controlled schemes. A strong sufficient condition is obtained to guarantee that the synchronization of the kind of networks can be achieved. For the weakly connected network, a method is presented in detail to solve two challenging fundamental problems arising in pinning control of complex networks: (1) How many nodes should be pinned? (2) How large should the coupling strength be used in a fixed complex network to realize synchronization? Then, we show the answer to the question that why all the diagonal block matrices of Perron-Frobenius normal matrices should be pinned? Besides, we find out the relation between the Perron-Frobenius normal form of coupling matrix and the differences of two synchronization conditions for strongly connected networks and weakly connected ones with linear coupling configuration. Moreover, we propose adaptive feedback algorithms to make the coupling strength as small as possible. Finally, numerical simulations are given to verify our theoretical analysis.
Highlights
Complex dynamical networks are found to be common systems in our real world [1,2,3,4,5,6], such as genetic regulatory networks, biological neural networks, telephone graphs, etc
For the weakly connected network, a method is presented in detail to solve two challenging fundamental problems arising in pinning control of complex networks: 1) How many nodes should be pinned? 2) How large should the coupling strength be used in a fixed complex network to realize synchronization? we show the answer to the question that why all the diagonal block matrices of Perron-Frobenius normal matrices should be pinned? Besides, we find out the relation between the Perron-Frobenius normal form of coupling matrix and the differences of two synchronization conditions for strongly connected networks and weakly connected ones with linear coupling configuration
This paper considered the globally synchronization for a class of linearly coupled complex networks with reducible and asymmetrical coupling configuration
Summary
Complex dynamical networks are found to be common systems in our real world [1,2,3,4,5,6], such as genetic regulatory networks, biological neural networks, telephone graphs, etc. Lyapunov stability theories and algebraic graph theories, as two essential tools, are used to study dynamic behaviors of the complex networks. In [8], two profound problems were solved: one is how to choose suitable pinning schemes for a given complex network, the other is how large the coupling strength should be used in a complex network to achieve synchronization. [9] showed the number of node which should be pinned in a complex network in order to reach synchronization. In [16], the authors investigated the synchronization of nonlinearly coupled networks through an innovative local adaptive approach. The crucial problem that how to select an optimal combination between the number of pinned nodes and the feedback control gain is studied in [19]
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