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

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.

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• Chaos synchronization in networks of coupled maps with time-varying topologies
• Consensus and synchronization in discrete-time networks of multi-agents with stochastically switching topologies and time delays
• Synchronization in discrete-time networks with general pairwise coupling
• Stability of phase difference trajectories of networks of Kuramoto oscillators with time-varying couplings and intrinsic frequencies
• Synchronized chaos in networks of simple units
• Symbolic dynamics and synchronization of coupled map networks with multiple delays
• Achieving synchronization in arrays of coupled differential systems with time-varying couplings
• Controlling chaos in some laser systems via variable coupling and feedback time delay
• Averaging plus Learning Models and Their Asymptotics
• On the stability and convergence of a class of consensus systems with a nonlinear input
• Condensation versus independence in weakly interacting CMLs