Singular Initial Value Problems for Scalar Quasi-Linear Ordinary Differential Equations

关于(C^3,0)中的拟普通尖点叶状结构

On the quasi-ordinary cuspidal foliations in (C^3,0)

本研究探讨(C^3,0)中一类可借拟普通曲面消奇链消奇的全纯叶状结构，将二维尖点叶状结构的相关研究推广到三维维度。

Compactness and index of ordinary central configurations for the curved n-body problem

For the curved n-body problem, we show that the set of ordinary central configurations is away from most singular configurations in H^3, and away from a subset of singular configurations in S^3. We also show that each of the n!/2 geodesic ordinary central configurations for n masses has Morse index n-2. Then we get a direct corollary that there are at least (3n-4)(n-1)!/2 ordinary central configurations for given n masses if all ordinary central configurations of these masses are non-degenerate.

A Necessary Condition for existence of Lie Symmetries in Quasihomogeneous Systems of Ordinary Differential Equations

Lie symmetries for ordinary differential equations are studied. In systems of ordinary differential equations, there do not always exist non-trivial Lie symmetries around equilibrium points. We present a necessary condition for existence of Lie symmetries analytic in the neighbourhood of an equilibrium point. In addition, this result can be applied to a necessary condition for existence of a Lie symmetry in quasihomogeneous systems of ordinary differential equations. With the help of our main theorem, it is proved that several systems do not possess any analytic Lie symmetries.

Quasi-periodic solutions for perturbed generalized nonlinear vibrating string equation with singularities

{\bf Abstract}: The existence of 2-dimensional KAM tori is proved for the perturbed generalized nonlinear vibrating string equation with singularities $u_{tt}=((1-x^2)u_x)_x-mu-u^3$ subject to certain boundary conditions by means of infinite-dimensional KAM theory with the help of partial Birkhoff normal form, the characterization of the singular function space and the estimate of the integrals related to Legendre basis.

关于与奇异二阶常微分方程相关的全纯叶状结构的旗（flags）

On flags of holomorphic foliations associated with singular second-order ordinary differential equations

本文研究与二阶常微分方程相关的全纯叶状结构2-旗，分析其奇点集性质，给出容许该结构的二阶方程分类，提出高维2-旗芽构造方法，并证明二阶及以上方程形式上均对应$\boldsymbol{\rm \bf C}^3$原点处的2-旗芽。

"Life after death" in ordinary differential equations with a non-Lipschitz singularity

We consider a class of ordinary differential equations in $d$-dimensions featuring a non-Lipschitz singularity at the origin. Solutions of such systems exist globally and are unique up until the first time they hit the origin, $t = t_b$, which we term `blowup'. However, infinitely many solutions may exist for longer times. To study continuation past blowup, we introduce physically motivated regularizations: they consist of smoothing the vector field in a $ν$--ball around the origin and then removing the regularization in the limit $ν\to 0$. We show that this limit can be understood using a cer…