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

A gradient flow formulation of the Lohe matrix model with a high-order polynomial coupling