Emergent Behavior of a Second-Order Lohe Matrix Model on the Unitary Group

We study a second-order extension to the first-order Lohe matrix model on the unitary group which can be reduced to the second-order Kuramoto model with inertia as a special case. For the proposed second-order model, we present several sufficient frameworks leading to the emergence of the complete and practical synchronizations in terms of the initial data and the system parameters. For the identical hamiltonians, we show that the complete synchronization emerges asymptotically. In contrast, for the non-identical hamiltonians, the practical synchronization occurs for some class of initial data when the product of the coupling strength and inertia is sufficiently small.