  We present a generalized Lohe matrix model for a homogeneous ensemble with
higher order couplings via the gradient flow approach. For the homogeneous free
flow with the same hamiltonian, it is well known that the Lohe matrix model
with cubic couplings can recast as a gradient system with a potential which is
a squared Frobenius norm of of averaged state. In this paper, we further derive
a generalized Lohe matrix model with higher-order couplings via gradient flow
approach for a polynomial potential. For the proposed model, we also provide a
sufficient framework in terms of coupling strengths and initial data, which
leads to the emergent dynamics of the homogeneous ensemble.
