

Preprint 40/2007
Adaptive dynamical networks via neighborhood information: synchronization and pinning control
Wenlian Lu
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Submission date: 17. Apr. 2007
Pages: 36
published in: Chaos, 17 (2007) 2, art-no. 023122
DOI number (of the published article): 10.1063/1.2737829
Bibtex
PACS-Numbers: 05.45.Gg, 05.45.Xt, 02.30.Hq
Keywords and phrases: adaptive complex networks, synchronization, pinning control
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Abstract:
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.