Emergent Behaviors of Lohe Tensor Flocks

摘要

We present a new aggregation model on the space of rank-m tensors with the same size, and study emergent dynamics of the proposed model. Our proposed new aggregation model is general enough to encompass Lohe type models such as the Kuramoto model, the Lohe sphere model and the Lohe matrix models for the ensemble of real rank-0, rank-1 and rank-2 tensors, respectively. In this regard, we call our proposed model as the Lohe tensor model for rank-m tensors with the same size. For emergent dynamics of the proposed model, we employ the ensemble diameter as a Lyapunov functional to derive a Riccati type differential inequality. As a direct application of these differential inequalities, we provide two sufficient frameworks leading to the emergent dynamics for homogeneous and heterogeneous ensembles in terms of system parameters and initial data.