Exponential synchronization of Kuramoto oscillators with time delayed coupling

Abstract

We discuss the asymptotic frequency synchronization for the non-identical Kuramoto oscillators with time delayed interactions. We provide explicit lower bound on the coupling strength and upper bound on the time delay in terms of initial configurations ensuring exponential synchronization. This generalizes not only the frequency synchronization estimate by Choi et al. [Physica D, 241, (2012), 735-754] for the non-identical Kuramoto oscillators without time delays but also improves previous result by Schmidt et al. [Automatica, 48, (2012), 3008-3017] in the case of homogeneous time delays where the initial phase diameter is assumed to be less than π/2. The proof relies on a Lyapunov functional approach.

Convergence to consensus for a Hegselmann-Krause-type model with distributed time delay

“…In addition, it is reasonable to introduce a delay in the model as a reaction time or simply as a time to receive the information from outside, in order to let the dynamics more realistic. For first order models, we refer to [5,8,10], while for delayed Cucker-Smale-type models we mention [6,7,15,25]. In particular, in very recent papers (see [9,18,23]), the authors analyzed modified Cucker-Smale models with distributed time delay, thanks to which agents are influenced by the other ones on a time interval [t − τ (t), t].…”