

Preprint 48/2007
Synchronization of discrete-time dynamical networks with time-varying couplings
Wenlian Lu, Fatihcan M. Atay, and Jürgen Jost
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Submission date: 10. May. 2007
Pages: 31
published in: SIAM journal on mathematical analysis, 39 (2007) 4, p. 1231-1259
DOI number (of the published article): 10.1137/060657935
Bibtex
MSC-Numbers: 37C60, 15A51, 94C15
Keywords and phrases: synchronization, dynamical network, time-varying coupling, Hajnal diameter, Lyapunov exponent, spanning tree
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Abstract:
We study the local complete synchronization of discrete-time
dynamical networks with time-varying couplings. Our conditions for
the temporal variation of the couplings are rather general and
include both variations in the network structure and in the
reaction dynamics; the reactions could, for example, be driven by
a random dynamical system. A basic tool is the concept of Hajnal
diameter which we extend to infinite Jacobian matrix sequences.
The Hajnal diameter can be used to verify synchronization and we
show that it is equivalent to other quantities which have been
extended to time-varying cases, such as the projection radius,
projection Lyapunov exponents, and transverse Lyapunov exponents.
Furthermore, these results are used to investigate the
synchronization problem in coupled map networks with time-varying
topologies and possibly directed and weighted edges. In this
case, the Hajnal diameter of the infinite coupling matrices can be
used to measure the synchronizability of the network process. As
we show, the network is capable of synchronizing some chaotic map
if and only if there exists an integer T>0 such that for any
time interval of length T, there exists a vertex which can
access other vertices by directed paths in that time interval.