Chaotic Synchronizaiton And Parameter Estimation Of Complex Networks

In this thesis we consider chaotic synchronization and parameter estimation of com-plex networks. Firstly, via linear stability analysis method, we discuss the stability of the synchronized state of coupled map networks (CMN). Some interesting theoretical results are obtained, including that the Jacobian matrix of the CMN can be diagonalized and the similarity transformation is independent of t; the eigenvalues of the matrix A range between 0 and 2, here A is the main body of the Jacobian matrix; if the local dynamics function is chaotic, the range of the coupling strength of CMN subject to synchronization is given with the maximum eigenvalue and the second minimal eigenvalue of A. These results lay a good foundation for the stability analysis of the synchronized state of CMN. Applying these results to a class of multi-layer center networks, some interesting and useful conclusions are induced. The numerical simulations are presented to verify the theoretical results.Secondly, by using adaptive control method we estimate the parameters of some discrete-time dynamical networks. Here we suppose that variables of the nodes can be ac-quired by some means, but the topological structure may be unknown to the researchers. Viewing original network as a driving system, we establish a responding system and an adaptive controlling system. By applying Lasalle's invariance principle and some results about matrix traces, we prove that the origin is an asymptotically stable equilibrium of the error system. Therefore we propose a method for estimating the topological struc-ture of some discrete-time dynamical networks based on the dynamical evolution of the networks. The network concerned can be 1-dimensional or N-dimensional, directed or undirected, weighted or unweighted, and the local dynamics of each node can be identical or nonidentical. The connections among the nodes can be all unknown or partially known.Via Frobenius matrix norm, we also propose a method for estimating topology of a discrete dynamical network based on the dynamical evolution of the network. Here the responding system and the adaptive controlling system are different from the former ones. The numerical simulations are presented to verify the theoretical results.Finally, we give an application of complex dynamical networks. In teaching practice we found that there are collaborative learning networks among students. The score of a student is not only related to himself (herself) but also to the scores of those students whom he (she) collaborates with. From investigation, we get the situation of collaborative learning in some classes and the scores of the students in several time-steps. And so we establish the complex networks of collaborative learning, and set up the models of student's score growth based on the complex networks. The numerical results show that in some extents the models accord with the actual situation, and can forecast students' scores.