Invariant difference schemes and their application to $SL(2,\mathbb{R})$ invariant ordinary differential equations
R. Rebelo, P. Winternitz
备注：
11 pages, 4 figures. Title, abstract, introduction and conclusion were rewritten for a better presentation and understanding of the article. Published: J. Phys. A: Math. Theor. 42 454016
We present an exposition of a method of discretizing ordinary differential equations while preserving their Lie point symmetries. This method is very general and can be applied to any ODE with a nontrivial symmetry group. The method is applied to obtain numerical slutions of second and third order ODEs invariant under two different realizations of $SL(2,\mathbb{R})$. The symmetry preserving method is shown to provide a better qualitative description of solutions than standard methods. In particular it provides solutions that are valid close to singularities and beyond them.
Lie group classification of first-order delay ordinary differential equations
给出了一阶时滞常微分方程的一种群分类，并给出了一阶时滞常微分方程的时滞参数（时滞关系）方程。这样的系统（延迟常微分系统或DODS）的一个子集，其中包括线性DODE和解决方案独立的延迟关系具有无限维对称代数，因为这样做的非线性可线性化的变量的可逆变换。非线性DODS具有维数为$n$，$0 \leq n \leq 3$的对称代数。它示出了如何精确的解析解的不变DODS可以使用对称性约简。
