Complete aggregation of the Lohe tensor model with the same free flow

Abstract

The Lohe tensor model is a first-order tensor-valued continuous-time dynamical system for the aggregation of tensors with the same rank and size. It reduces to the well-known aggregation models such as the Kuramoto model, the Lohe sphere model, and the Lohe matrix model as special cases for low-rank tensors. We present conditions for the solution splitting property, asymptotic convergence toward a set of equilibria, and analyze two possible asymptotic states (completely aggregate state and bi-polar state), which can emerge from a set of initial data. Moreover, we provide a sufficient framework leading to the aforementioned two asymptotic states in terms of initial data and system parameters.