Collective behaviors of the Lohe hermitian sphere model with inertia

TL;DR

This paper extends the Lohe hermitian sphere model to include inertia, analyzing how it affects collective behavior and aggregation, with results showing conditions for synchronization in both homogeneous and heterogeneous systems.

Contribution

It introduces a second-order inertial extension of the Lohe hermitian sphere model and provides conditions for emergent synchronization considering inertia effects.

Findings

Inertia causes oscillatory trajectories in initial phases.

Synchronization occurs under specific parameter and initial data conditions.

Increasing coupling strength promotes aggregation in heterogeneous systems.

Abstract

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.