The star-coupled linear underdamped system subject to multiplicative quadratic noise and periodic sinusoidal excitation are considered. The multiplicative noise is modeled as a quadratic polynomial function of symmetric trichotomous noise. Firstly, the stability conditions of the system and the stationary solution of the mean field are deduced by establishing the differential equations of the mean field. Then the synchronization conditions of the principal and general particles are obtained respectively. These conditions are beneficial for studying the collective behavior of the system and the motion characteristics of particles in different situations. Then the output signal-to-noise ratio of the coupled system is derived in detail for the first time. Moreover, the dynamic behaviors of the system are analyzed by analytical results and numerical simulations, including the collective behavior under different conditions, and the concrete influence of parameters on the output signal-to-noise ratio.

考虑了受到乘性二次噪声和周期性正弦激励的星形耦合线性欠阻尼系统。乘法噪声被建模为对称三分噪声的二次多项式函数。首先，通过建立平均场的微分方程，推导出系统的稳定性条件和平均场的平稳解。然后分别得到主粒子和一般粒子的同步条件。这些条件有利于研究系统的集体行为和粒子在不同情况下的运动特性。然后首次详细推导出耦合系统的输出信噪比。而且，