Emergent dynamics of the Lohe matrix ensemble on a network under time-delayed interactions

Abstract

We study emergent dynamics of the Lohe matrix model on the unitary group in the presence of several external factors. First, we consider how the static network structure can affect the emergent dynamics for the identical ensemble. To be more precise, we present a sufficient framework leading to the completely aggregated state and improve asymptotic dynamics on the all-to-all network case. Second, we investigate the uniform time-delay structure effect on the emergent dynamics of the Lohe matrix model and show that complete aggregation can occur when the size of the time-delay is sufficiently small. Finally, we study the interplay between the network structure and time-delayed interactions in the asymptotic dynamics of the Lohe matrix model. In this case, we cannot expect complete synchronous behavior. Instead, we show that the system exhibits weak aggregation, namely, practical aggregation, when the coupling strength is sufficiently large.