In this paper, the authors investigate the synchronization of an array of linearly coupled identical dynamical systems with a delayed coupling. Here the coupling matrix can be asymmetric and reducible. Some criteria e...In this paper, the authors investigate the synchronization of an array of linearly coupled identical dynamical systems with a delayed coupling. Here the coupling matrix can be asymmetric and reducible. Some criteria ensuring delay-independent and delay- dependent global synchronization are derived respectively. It is shown that if the coupling delay is less than a positive threshold, then the coupled network will be synchronized. On the other hand, with the increase of coupling delay, the synchronization stability of the network will be restrained, even eventually de-synchronized.展开更多
We determine the left eigenvector of a stochastic matrix M associated to the eigenvalue 1 in the commutative and the noncommutative cases. In the commutative case, we see that the eigenvector associated to the eigenva...We determine the left eigenvector of a stochastic matrix M associated to the eigenvalue 1 in the commutative and the noncommutative cases. In the commutative case, we see that the eigenvector associated to the eigenvalue 0 is (N1,Nn) , where Ni is the i–th iprincipal minor of N=M–In , where In is the identity matrix of dimension n. In the noncommutative case, this eigenvector is (P1-1,Pn-1) , where Pi is the sum in Q《αij》 of the corresponding labels of nonempty paths starting from i and not passing through i in the complete directed graph associated to M .展开更多
基金Project supported by the National Natural Science Poundation of China(Nos.60574044,60774074)the Graduate Student Innovation Fonndation of Fudan University.
文摘In this paper, the authors investigate the synchronization of an array of linearly coupled identical dynamical systems with a delayed coupling. Here the coupling matrix can be asymmetric and reducible. Some criteria ensuring delay-independent and delay- dependent global synchronization are derived respectively. It is shown that if the coupling delay is less than a positive threshold, then the coupled network will be synchronized. On the other hand, with the increase of coupling delay, the synchronization stability of the network will be restrained, even eventually de-synchronized.
文摘We determine the left eigenvector of a stochastic matrix M associated to the eigenvalue 1 in the commutative and the noncommutative cases. In the commutative case, we see that the eigenvector associated to the eigenvalue 0 is (N1,Nn) , where Ni is the i–th iprincipal minor of N=M–In , where In is the identity matrix of dimension n. In the noncommutative case, this eigenvector is (P1-1,Pn-1) , where Pi is the sum in Q《αij》 of the corresponding labels of nonempty paths starting from i and not passing through i in the complete directed graph associated to M .
