On Equilibria and Consensus of the Lohe Model with Identical Oscillators

Equilibria and consensus dynamics of the Lohe model with identical oscillators are investigated under more general interconnection topologies. In the existing literatures on the Lohe model, some equilibrium issues are discussed only for strongly connected directed graphs, and many consensus problems are solved only for undirected graphs. In this paper, many results on equilibria and consensus are generalized to the case of directed graphs containing directed spanning trees. To overcome a limitation of initial states of Lohe oscillators, a generalized form of the Lohe model is proposed which admits all the initial states except the origin. For the generalized Lohe model with identical oscillators, equilibria and global dynamical behaviors are investigated. Under undirected graph topologies, some sufficient conditions for almost global consensus are proposed. Finally, some numerical examples are given to validate the theoretical results.