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We provide the detailed asymptotic behavior for first-order aggregationmodels of heterogeneous oscillators. Due to the dissimilarity of naturalfrequencies, one could expect that all relative distances converge to definitepositive value and furthermore that each oscillator converges to a possiblydifferent stationary point. In order to establish the desired results, weintroduce a novel method, called dimension reduction method that can be appliedto a specific situation when the degree of freedom of the natural frequency isone. In this way, we would say that although a small perturbation is allowed,convergence toward an equilibrium of the gradient flow is still guaranteed.Several first-order aggregation models are provided as concrete examples byusing the dimension reduction method to study the structure of the equilibrium,and numerical simulations are conducted to support theoretical results.

Asymptotic convergence of heterogeneous first-order aggregation models: from the sphere to the unitary group