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Adaptive dynamical networks via neighborhood information: synchronization and pinning control
In this paper, we introduce a model of adaptive dynamical network by integrating the complex network model and adaptive technique. This model is characterized by that the adaptive updating laws for each vertex in the network depend only on the state information of its neighborhood besides itself and external controllers. This suggests that adaptive technique be added to a complex network without breaking its intrinsic existing network topology. The core of adaptive dynamical networks is to design suitable adaptive updating laws to attain certain aims. Here, we propose two series of adaptive laws to synchronize and pin a complex network respectively. Based on the Lyapunov function method, we can prove that under several mild conditions, with the adaptive technique, a connected network topology is sufficient to synchronize or stabilize any chaotic dynamics of the uncoupled system. This implies that these adaptive updating laws actually enhance synchronizability and stabilizability respectively. We find out that even though these adaptive methods can success for all networks with connectivity, the underlying network topology can affect the convergent rate and the terminal average coupling and pinning strength. And, this influence can be measured by the smallest nonzero eigenvalue of the corresponding Laplacian. Moreover, we detailed study the influence of the prior parameters in this adaptive laws and present several numerical examples to verify our theoretical results and further discussions.