Emergent Behaviors of the Infinite Set of Lohe Hermitian Sphere Oscillators

We study the emergent behaviors of an infinite number of Lohe Hermitian sphere oscillators on the unit Hermitian sphere. For this, we propose an infinite analogue of the Lohe hermitian sphere model, and present sufficient frameworks leading to collective behaviors in terms of system parameters and initial data. Under some network topology, we show that practical synchronization emerges for a heterogeneous ensemble, whereas exponential synchronization can appear for a homogeneous ensemble. Furthermore we have also derived analogous results for the infinite swarm-sphere model. For the sender network topology in which coupling capacities depend only on the sender index number, we show that there are only two possible asymptotic states, namely complete phase synchrony or bi-cluster configuration for a homogeneous ensemble in a positive coupling regime.