Sufficient Conditions for Fast Switching Synchronization in Time-Varying Network Topologies

Abstract

In previous work [1], empirical evidence indicated that a time-varying network could propagate sufficient information to allow synchronization of the sometimes coupled oscillators, despite an instantaneously disconnected topology. We prove here that if the network of oscillators synchronizes for the static time-average of the topology, then the network will synchronize with the time-varying topology if the time-average is achieved sufficiently fast. Fast switching, fast on the time-scale of the coupled oscillators, overcomes the descychnronizing decoherence suggested by disconnected instantaneous networks. This result agrees in spirit with that of [1] where empirical evidence suggested that a moving averaged graph Laplacian could be used in the master-stability function analysis [52]. A new fast switching stability criterion here-in gives sufficiency of a fast-switching network leading to synchronization. Although this sufficient condition appears to be very conservative, it provides new insights about the requirements for synchronization when the network topology is time-varying. In particular, it can be shown that networks of oscillators can synchronize even if at every point in time the frozen-time network topology is insufficiently connected to achieve synchronization.MCS: 37A25, 37D45, 37N25, 93B05, 94C15, 05C80, 92D30

Network synchronization: Spectral versus statistical properties

“…Figure 1 illustrates the idea in intuitive terms. Suppose in the graph G we identify a huge part H and a much smaller part S. (Alternatively, we can imagine the possibility of appending a small set of nodes S to an existing graph, which is a realistic scenario if one considers time-varying connections which might come on and off [21,27,28]). By (13), the value of λ 2 is constrained by the properties of S. However, all the gross statistical properties of G are determined by H. If H is any graph which is claimed to have good synchronizability, we can force G to have poor synchronizability by appending S to H. In other words, for large networks, the synchronizability of G and H can be very different, although many of their statistical properties are essentially the same.…”

Robust Synchronization via Homogeneous Parameter-Dependent Polynomial Contraction Matrix

“…Similarly, homogeneous parameter-dependent polynomial asymptotical contraction matrix (HPD-PACM) can be defined by using condition (12). Let R(y, θ,θ, γ) be a matrix of polynomial as…”

“…In [11], impulsive synchronization criteria is proposed for uncertain dynamical network where the network coupling functions are unknown but bounded, under assumptions that both the intrinsic nonlinear function and the coupling nonlinear function satisfy Lipschitz-like conditions. In [12], for fast-switching topology, a local synchronization criteria is given at a sufficiently large switching rate. Also for local synchronization, conditions are proposed by using the time-average topology to approximate the time-varying topology [13].…”

Controle de Pinning Adaptativo de Sincronização com Aplicações a Rede de Veículos Autônomos

“…Em redes complexas reais ligações entre agentes são criadas ou desfeitas, variando assim a topologia da rede. Estudos mostram que, sob determinadas condições de rápida alternância entre topologias, redes complexas cuja topologia evolui ao longo do tempo também podem sincronizar [2]. Neste trabalho analisamos a aplicação da estratégia de controle pinning adaptativo descentralizado [3] aplicado a redes complexas com topologia variante no tempo; particularmente a redes de veículos autônomos cuja topologia é determinada pelo movimento dos agentes, que segue um modelo aleatório (random walk model) [4,5] Em sistemas multiagentes, muitas vezes necessita-se que processos que são realizados nos agentes ocorram de forma síncrona, por exemplo, num sistema multiveicular autônomo pode-se necessitar comunicação síncrona dos agentes com um receptor comum.…”