Normalization of Poincaré Singularities {\it via} Variation of Constants

Convergence versus integrability in Poincare-Dulac normal form

We show that, to find a Poincare-Dulac normalization for a vector field is the same as to find and linearize a torus action which preserves the vector field. Using this toric characterization and other geometrical arguments, we prove that any local analytic vector field which is integrable in the non-Hamiltonian sense admits a local convergent Poincare-Dulac normalization. These results generalize the main results of our previous paper from the Hamiltonian case to the non-Hamiltonian case. Similar results are presented for the case of isochore vector fields.

On blow-up solutions of differential equations with Poincaré-type compactifications

We provide explicit criteria for blow-up solutions of autonomous ordinary differential equations. Ideas are based on the quasi-homogeneous desingularization (blowing-up) of singularities and compactifications of phase spaces, which suitably desingularize singularities at infinity. We derive several type of compactifications and show that dynamics at infinity is qualitatively independent of the choice of such compactifications. We also show that hyperbolic invariant sets, such as equilibria and periodic orbits, at infinity induce blow-up solutions with specific blow-up rates. In particular, blo…

庞加莱-杜拉克正规形：形式与分析方面

Poincarè-Dulac Normal Forms: Formal and Analytic Aspects

本文探讨局部解析或形式向量场至庞加莱-杜拉克正规形的变换及收敛性，引入新变换方法，推广布鲁诺方法证明斯托洛维奇的归一化定理，还研究可积性与收敛性定理的关联。

Zero level perturbation of a certain third-order linear solvable ODE with an irregular singularity at the origin of Poincaré rank 1

We study an irregular singularity of Poincaré rank 1 at the origin of a certain third-order linear solvable homogeneous ODE. We perturb the equation by introducing a small parameter $\varepsilon\in (\mathbb{R}_+, 0)\,(\varepsilon < 1)$ which causes the splitting of the irregular singularity into two finite Fuchsian singularities. We show that when the solutions of the perturbed equation contain logarithmic terms, the Stokes matrices of the initial equation are limits of the part of the monodromy matrices around the finite resonant Fuchsian singularities of the perturbed equation.

二维紧流形上 Filippov 向量场的 Poincaré-Hopf 定理

Poincaré-Hopf Theorem for Filippov vector fields on 2-dimensional compact manifolds

本文提出涵盖伪平衡点与切向奇点的新指数定义，证明带光滑切换流形的二维紧流形上 Filippov 向量场满足 Poincaré-Hopf 定理，并推导 Hairy Ball 定理变体。

关于非线性Poincaré流

On the nonlinear Poincaré flow

本文开发了一种分析带奇点可微流动力的工具，为局部动力学提供抽象模型以控制不变流形大小，是[CY2]宣布结果的第一部分。

三维实流的分层 II：Poincaré平面扇形分解的推广

Stratification of three-dimensional real flows II: A generalization of Poincaré's planar sectorial decomposition

本文推广了Poincaré的平面扇形分解定理，为具有孤立奇点且约化后仅含双曲奇点的三维解析向量场建立了动力学分层定理。

On real center singularities of complex vector fields on surfaces

One of the various versions of the classical Lyapunov-Poincaré center theorem states that a nondegenerate real analytic center type planar vector field singularity admits an analytic first integral. In a more proof of this result, R. Moussu establishes important connection between this result and the theory of singularities of holomorphic foliations (\cite{moussu}). In this paper we consider generalizations for two main frameworks: (i) planar real analytic vector fields with "many" periodic orbits near the singularity and (ii) germs of holomorphic foliations having a suitable singularity in di…

带逆积分因子的单值奇点的主Bautin理想

Principal Bautin ideal of monodromic singularities with inverse integrating factors

该研究在两个假设下分析平面解析向量场族单值奇点的Poincaré映射结构，得出其渐近展开无对数项、具唯一广义Poincaré-Lyapunov量，且解析时Bautin理想为主理想、奇点环性为零的结论。

On C1-robust transitivity of volume-preserving flows

We prove that a divergence-free and C1-robustly transitive vector field has no singularities. Moreover, if the vector field is C4 then the linear Poincare flow associated to it admits a dominated splitting over M.

Normalization of bundle holomorphic contractions and applications to dynamics

We establish a Poincaré-Dulac theorem for sequences (G_n)_n of holomorphic contractions whose differentials d_0 G_n split regularly. The resonant relations determining the normal forms hold on the moduli of the exponential rates of contraction. Our results are actually stated in the framework of bundle maps. Such sequences of holomorphic contractions appear naturally as iterated inverse branches of endomorphisms of CP(k). In this context, our normalization result allows to precisely estimate the distortions of ellipsoids along typical orbits. As an application, we show how the Lyapunov exponen…

三次庞加莱中心问题的新解

New solutions of the Poincaré Center Problem in degree 3

本文针对平面自治系统，在代数积分曲线构型奇点为准齐次的条件下提出新的 Darboux 可积性准则，并据此构造了三次中心簇中此前未知的分支。

Index of Singularities of Real Vector Fields on Singular Hypersurfaces

Gómez-Mont, Seade and Verjovsky introduced an index, now called GSV-index, generalizing the Poincaré-Hopf index to complex vector fields tangent to singular hypersurfaces. The GSV-index extends to the real case. This is a survey paper on the joint research with Gómez-Mont and Giraldo about calculating the GSV-index $\Ind_{V_\pm,0}(X)$ of a real vector field $X$ tangent to a singular hypersurface $V=f^{-1}(0)$. The index $\Ind_{V_{\pm,0}}(X)$ is calculated as a combination of several terms. Each term is given as a signature of some bilinear form on a local algebra associated to $f$ and $X$. Mai…

Poincaré-Bendixson theorems for meromorphic connections and homogeneous vector fields

We first study the dynamics of the geodesic flow of a meromorphic connection on a Riemann surface, and prove a Poincaré-Bendixson theorem describing recurrence properties and $ω$-limit sets of geodesics for a meromorphic connection on $¶^1(\C)$. We then show how to associate to a homogeneous vector field $Q$ in ${\Bbb C}^n$ a rank 1 singular holomorphic foliation $\cal F$ of $¶^{n-1}(\C)$ and a (partial) meromorphic connection $\nabla^o$ along $\ca F$ so that integral curves of $Q$ are described by the geodesic flow of $\nabla^o$ along the leaves of $\ca F$, which are Riemann surfaces. The com…