从连续到离散动力系统的重整化方法：渐近解、约化和不变流形
The renormalization method from continuous to discrete dynamical systems: asymptotic solutions, reductions and invariant manifolds
摘要：
将基于泰勒展开的微分方程渐近分析的重整化方法推广到差分方程。提出的重整化方法是基于牛顿-麦克劳林展开。证明了重整化方法的几个基本定理。给出了一些有趣的应用，包括量子非谐振子和离散边界层的渐近解、某些离散动力学系统的约化和不变流形。在此基础上，提出了基于Newton-Maclaurin展开的同伦重整化方法，并将其应用于不含小参数的差分方程。
英文摘要：
The renormalization method based on the Taylor expansion for asymptotic analysis of differential equations is generalized to difference equations. The proposed renormalization method is based on the Newton-Maclaurin expansion. Several basic theorems on the renormalization method are proven. Some interesting applications are given, including asymptotic solutions of quantum anharmonic oscillator and discrete boundary layer, the reductions and invariant manifolds of some discrete dynamics systems. Furthermore, the homotopy renormalization method based on the Newton-Maclaurin expansion is proposed and applied to those difference equations including no a small parameter.
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