Synchronization of discrete-time dynamical networks with time-varying couplings

TL;DR

This paper investigates the conditions for local complete synchronization in discrete-time dynamical networks with time-varying couplings, extending classical tools like Hajnal diameter to analyze complex, evolving network structures.

Contribution

It introduces an extension of Hajnal diameter to infinite Jacobian matrix sequences and demonstrates its equivalence to other measures for verifying synchronization in time-varying networks.

Findings

Hajnal diameter effectively measures network synchronizability.

Synchronization depends on the existence of accessible vertices within fixed time intervals.

The approach applies to directed, weighted, and chaotic map networks.

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.