Perturbation of coupling matrices and its effect on the synchronizability in arrays of coupled chaotic systems

In a recent paper, wavelet analysis was used to perturb the coupling matrix in
an array of identical chaotic systems in order to improve its synchronization. As
the synchronization criterion is determined by the second smallest eigenvalue X2 of
the coupling matrix, the problem is equivalent to studying how X2 of the coupling
matrix changes with perturbation. In the aforementioned paper, a small percentage
of the wavelet coefficients are modified. However, this result in a perturbed matrix
where every element is modified and nonzero. The purpose of this paper is to present
some results on the change of X2 due to perturbation. In particular, we show that
as the number of systems n-->x , perturbations which only add local coupling
will not change X2. On the other hand, we show that there exists perturbations
which affect an arbitrarily small percentage of matrix elements, each of which is
changed by an arbitrarily small amount and yet can make X2 arbitrarily large. These
results give conditions on what the perturbation should be in order to improve the
synchronizability in an array of coupled chaotic systems. This analysis allows us to
prove and explain some of the synchronization phenomena observed in a recently
studied network where random coupling are added to a locally connected array.
Finally we classify various classes of coupling matrices such as small world networks
and scale free networks according to their synchronizability in the limit.