具有惯性的Lohe厄米球模型的集体行为
Collective behaviors of the Lohe hermitian sphere model with inertia
摘要：
提出了一个一阶Lohe厄米球模型的二阶推广，并研究了它的突现渐近动力学。我们提出的模型采用了惯性效应作为二阶扩展。惯性项会使粒子轨迹在很小的时间间隔（初始层）内产生振荡行为，这给基于单调性的参数的应用带来了技术上的困难。对于紧急估计，我们采用两点关联函数，它被定义为粒子位置之间的内积。对于具有相同频率矩阵的齐次系综，我们给出了两个充分的系统参数和初始数据框架，证明了两点关联函数趋于1，这与完全聚集完全相同.相比之下，对于具有不同频率矩阵的异质系综，我们提供了一个充分的框架，在系统参数和初始数据，这使得两点关联函数接近统一，通过增加主耦合强度。
英文摘要：
We present a second-order extension of the first-order Lohe hermitian sphere(LHS) model and study its emergent asymptotic dynamics. Our proposed model incorporates an inertial effect as a second-order extension. The inertia term can generate an oscillatory behavior of particle trajectory in a small time interval(initial layer) which causes a technical difficulty for the application of monotonicity-based arguments. For emergent estimates, we employ two-point correlation function which is defined as an inner product between positions of particles. For a homogeneous ensemble with the same frequency matrix, we provide two sufficient frameworks in terms of system parameters and initial data to show that two-point correlation functions tend to the unity which is exactly the same as the complete aggregation. In contrast, for a heterogeneous ensemble with distinct frequency matrices, we provide a sufficient framework in terms of system parameters and initial data, which makes two-point correlation functions close to unity by increasing the principal coupling strength.
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