Convergence versus integrability in Poincare-Dulac normal form

Poincare' normal and renormalized forms

We briefly review the main aspects of (Poincaré-Dulac) normal forms; we have a look at the non-uniqueness problem, and discuss one of the proposed ways to ``further reduce'' the normal forms. We also mention some convergence results.

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。

Resonant normal forms as constrained linear systems

We show that a nonlinear dynamical system in Poincare'-Dulac normal form (in $\R^n$) can be seen as a constrained linear system; the constraints are given by the resonance conditions satisfied by the spectrum of (the linear part of) the system and identify a naturally invariant manifold for the flow of the ``parent'' linear system. The parent system is finite dimensional if the spectrum satisfies only a finite number of resonance conditions, as implied e.g. by the Poincare' condition. In this case our result can be used to integrate resonant normal forms, and sheds light on the geometry behind…

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

We present a geometric proof of the Poincaré-Dulac Normalization Theorem for analytic vector fields with singularities of Poincaré type. Our approach allows us to relate the size of the convergence domain of the linearizing transformation to the geometry of the complex foliation associated to the vector field. A similar construction is considered in the case of linearization of maps in a neighborhood of a hyperbolic fixed point.

常微分方程系统组的首次积分与不变量

First integrals and invariants of systems of ODEs

本文利用计算代数工具提出算法方法，识别线性部分含代数复特征值的自治常微分方程系统组的单项式与多项式首次积分代数生成元，并揭示其与 Poincaré-Dulac 规范形的关系。

一类三维多项式系统的可积性与线性化

Integrability and Linearizability of a Family of Three-Dimensional Polynomial Systems

本文研究了一类三维多项式系统的局部可积性与线性化，建立了Poincaré--Dulac范式收敛准则，并提出高效算法确定可积性必要条件，应用于二次子族分析，为三维多项式系统可积性提供新见解。

Geometry of Normal Forms for Dynamical Systems

We discuss several aspects of the geometry of vector fields in (Poincare'-Dulac) normal form. Our discussion relies substantially on Michel theory and aims at a constructive approach to simplify the analysis of normal forms via a splitting based on the action of certain groups. The case, common in Physics, of systems enjoying an a priori symmetry is also discussed in some detail.

Invariant ideals of local analytic and formal vector fields

The objective of this paper is to analyse analytic invariant sets of analytic ordinary differential equations (ODEs). For this purpose we introduce semi-invariants and invariant ideals as well as the notion of vector fields in Poincare- Dulac normal form (PDNF). We prove that all invariant ideals of a vector field in PDNF are already invariant for its semi-simple linear part. Additionally, this paper provides a natural characterization of invariant ideals via semi-invariants.

Bifurcation values for a family of planar vector fields of degree five

We study the number of limit cycles and the bifurcation diagram in the Poincaré sphere of a one-parameter family of planar differential equations of degree five $\dot {\bf x}=X_b({\bf x})$ which has been already considered in previous papers. We prove that there is a value $b^*>0$ such that the limit cycle exists only when $b\in(0,b^*)$ and that it is unique and hyperbolic by using a rational Dulac function. Moreover we provide an interval of length 27/1000 where $b^*$ lies. As far as we know the tools used to determine this interval are new and are based on the construction of algebraic curve…

Obstructions to Integrability of Nearly Integrable Dynamical Systems near Regular Level Sets

We study the existence of real-analytic first integrals and real-analytic integrability for perturbations of integrable systems in the sense of Bogoyavlenskij including non-Hamiltonian ones. We especially assume that there exists a family of periodic orbits on a regular level set of the first integrals having a connected and compact component and give sufficient conditions for nonexistence of the same number of real-analytic first integrals in the perturbed systems as the unperturbed ones and for their real-analytic nonintegrability near the level set such that the first integrals and commutat…

Non-integrability of the Armbruster-Guckenheimer-Kim quartic Hamiltonian through Morales-Ramis theory

We show the non-integrability of the three-parameter Armburster-Guckenheimer-Kim quartic Hamiltonian using Morales-Ramis theory, with the exception of the three already known integrable cases. We use Poincaré sections to illustrate the breakdown of regular motion for some parameter values.

On the integrability of polynomial fields in the plane by means of Picard-Vessiot theory

We study the integrability of polynomial vector fields using Galois theory of linear differential equations when the associated foliations is reduced to a Riccati type foliation. In particular we obtain integrability results for some families of quadratic vector fields, Liénard equations and equations related with special functions such as Hypergeometric and Heun ones. The Poincaré problem for some families is also approached.

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…

过扭接触结构的Bott可积性

Bott-integrability of overtwisted contact structures

研究过扭接触结构与Bott可积Reeb流的关系，证明其等价于欧拉类的Poincaré对偶由图链表示

Centralizers and normalizers of local analytic and formal vector fields

We investigate the structure of the centralizer and the normalizer of a local analytic or formal differential system at a nondegenerate stationary point, using the theory of Poincaré-Dulac normal forms. Our main results are concerned with the formal case. We obtain a description of the relation between centralizer and normalizer, sharp dimension estimates when the centralizer of the linearization has finite dimension, and lower estimates for the dimension of the centralizer in general. For a distinguished class of linear vector fields (which is sufficiently large to be of interest) we obtain a…

多参数系统的正规型计算

Computation of Normal Forms for Systems with Many Parameters

本文提出了计算多参数多项式ODE系统Poincaré-Dulac正规型的第三种方法，通过在参数空间而非传统向量场中求解同伦方程，实现了计算的并行化，从而支持更高阶的正规型计算。