重整化群方法是处理微分方程奇异摄动问题的有效工具。该方法通过构造重整化群方程消去时间共振产生的长期项，得到系统的长时间近似解。在本项目中，我们将围绕“微分方程的重整化群方法”这一主题展开研究。首先，构建几类非线性生物模型的二阶重整化群近似解，应用推广的吉洪诺夫奇异摄动理论研究近似解的长时间行为，并进行数值模拟。其次，发展时空共振系统的重整化群方法，克服时空共振集为零测集的困难，构建三维和二维空间中二次非线性Schrödinger方程的重整化群近似解，并在适当的能量空间比较近似解与真实解的误差。

The renormalization group (RG) method is an effective tool to deal with the singular perturbation problem of differential equations. The long-time approximate solution of the system is obtained by constructing the RG equation to eliminate the secular term generated by the time resonance.In this project, we will focus on the topic of "renormalization group method of differential equations".Firstly, the second-order renormalization group approximate solutions of several nonlinear biological models are constructed, and the extended singular perturbation theory is applied to study the long time behavior of the approximate solutions, and the numerical simulation is carried out.Secondly, the renormalization group method of the time-space resonance system was developed to overcome the difficulty of zero measurement set of the time-space resonance set, and the approximate solution of the quadratic nonlinear Schrodinger equation in the three-dimensional and two-dimensional space was constructed, and the error between the approximate solution and the real solution was compared in the appropriate energy space.