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We present constants of motion for the finite-dimensional Lohe typeaggregation models with frustration and we apply them to analyze the emergenceof collective behaviors. The Lohe type models have been proposed as possiblenon-abelian and higher-dimensional generalizations of the Kuramoto model, whichis a prototype phase model for synchronization. The aim of this paper is tostudy the emergent collective dynamics of these models under the effect of(interaction) frustration, which generalizes phase-shift frustrations in theKuramoto model. To this end, we present constants of motion, i.e., conservedquantities along the flow generated by the models under consideration, and,from the perspective of the low-dimensional dynamics thus so obtained, deriveseveral results concerning the emergent asymptotic patterns of the Kuramoto andLohe sphere models.

Constants of motion for the finite-dimensional Lohe type models with frustration and applications to emergent dynamics