论文标题
异质一阶聚集模型的渐近收敛性：从球体到统一组
Asymptotic convergence of heterogeneous first-order aggregation models: from the sphere to the unitary group
论文作者
论文摘要
我们为异质振荡器的一阶聚集模型提供详细的渐近行为。由于固有频率的差异，人们可以期望所有相对距离都会融合到确定的正值，并且每个振荡器都会收敛到可能不同的固定点。为了建立所需的结果，我们引入了一种新的方法，称为缩小方法，当固有频率的自由度为一种时，可以应用于特定情况。通过这种方式，我们要说的是，尽管允许使用小的扰动，但仍然可以保证朝向梯度流的平衡。通过使用降低方法研究平衡结构，并进行了数值模拟以支持理论结果，将几种一阶聚集模型作为具体示例。
We provide the detailed asymptotic behavior for first-order aggregation models of heterogeneous oscillators. Due to the dissimilarity of natural frequencies, one could expect that all relative distances converge to definite positive value and furthermore that each oscillator converges to a possibly different stationary point. In order to establish the desired results, we introduce a novel method, called dimension reduction method that can be applied to a specific situation when the degree of freedom of the natural frequency is one. 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 by using the dimension reduction method to study the structure of the equilibrium, and numerical simulations are conducted to support theoretical results.
