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

Abstract

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 λ2 of the coupling matrix, the problem is equivalent to studying how λ2 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 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 affect 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 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.

Performance of pinning-controlled synchronization

“…in which λ min is the second lowest eigenvalue of matrix M, which is in accordance with the results in [15] (the first lowest eigenvalue is always the zero one). From Eq.…”

Obtaining certificates for complete synchronisation of coupled oscillators

“…For κ = 1.1: • Additional increases in polynomial order fail to lower the value of κ for which (23) holds. This was expected since κN < 1 corresponds to the case when the fixed point of the van der Pol oscillator withẋ 2 = (−k…”

“…, n. Then condition (31) holds for a constant P if sup{λ i + λ j } < 0, which clearly holds if (23) holds for a constant P but also if there exists a λ i > 0 such that λ i < −λ j for all j, j ̸ = i. Thus, inequality (31) connotes a more relaxed requirement than inequality (23).…”

Synchronizability and Synchronization Dynamics of Weighed and Unweighed Scale Free Networks With Degree Mixing

“…Sofar, the effects of the network topology on its synchronizability have been studied mainly with respect to the presence of scale free topologies or patterns in the network; see for example, [T. Nishikawa et al 2003], [Motter, Zhou & J.Kurths 2005b], [Fan & Wang 2005], [Lu, Chen & Cheng 2004], [Wu 2003 Scale free networks, which are common in nature, were found to show better synchronizability for increasing values of the power law exponent in [T. Nishikawa et al 2003], [Motter et al 2005b]. Specifically, the relationship was analyzed between the network structure and its synchronizability.…”