Emergent behaviors of two attractively coupled swarm sphere models with frustration

We study the collective behaviors of an attractively coupled system of swarm sphere models. The swarm sphere model is a prototype particle model describing the aggregate dynamics of particles on a sphere under suitable conditions on system parameters and initial data. In this paper, we investigate emergent dynamics resulting from the attractive coupling of two swarm sphere models with heterogeneous system parameters such as natural frequency matrices, frustration matrices, and inter and intra coupling strengths. Due to the intricate interplay between system parameters, several collective behaviors of the whole mixed ensemble emerge. To this end, we provide several sufficient frameworks leading to the local and global aggregation dynamics that can be achieved in both complete and practical senses. The proposed frameworks are formulated in terms of system parameters and initial data, and we provide local and global aggregation estimates within the proposed frameworks. As a direct application of emergent dynamics, we show that the swarm sphere model with a uniform frustration is uniformly l(2)-stable with respect to initial data.