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

## Abstract

In a recent paper, wavelet analysis is used to perturb the coupling matrix in an array of identical chaotic systems in order to improve its synchronization. When the coupling matrix is symmetric, the synchronization criterion is determined by the second smallest eigenvalue λ2 of the coupling matrix and the problem is reduced to studying how λ 2 of the coupling matrix changes with perturbation. In the aforementioned paper, a small percentage of the wavelet coefficients are modified. However, this results in a perturbed matrix where every element is modified and nonzero. The purpose of this Letter is to present some results on the change of λ2 due to perturbation. In particular, we show that as the number of systems n → ∞, perturbations which only add local coupling will not change λ2. On the other hand, we show that there exists perturbations which modify an arbitrarily small percentage of matrix elements, each of which is changed by an arbitrarily small amount and yet can make λ2 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 justify and explain some of the synchronization phenomena observed in a recently studied network where random coupling is added to a locally connected array. We propose to classify various classes of coupling matrices such as small world networks and scale free networks according to their synchronizability in the limit. Finally, we briefly discuss the case of time-varying coupling. © 2003 Elsevier B.V. All rights reserved.