焦点量与鞍点量的关系

对于一般情形,给出焦点量和鞍点量计算与约化的Maple算法,从而统一了焦点量和鞍点量的计算,并给出细焦点与细鞍点的变换,利用变换推导了焦点量和鞍点量的关系.     

具有细鞍点的二次系统

发散量为零的初等奇点,如果它是焦点,称它为细焦点;如果它是鞍点,称它为细鞍点。在二次系统的研究中。在某些场合,细鞍点与细焦点起到类似的作用。例如,具有两个细焦点(细鞍点)或一细焦点一细鞍点的二次系统必无极限环。若存在一个细焦点(细鞍点),则另外的细焦点至多是一阶的。本文进一步研究了具有细鞍点的二次系统,发现了与具有细焦点的二次系统有许多不同的性质。例如。具有细焦点的二次系统,其极限环未必集中分布,而本文证明:具有细鞍点的二次系统若存在极限环,则必集中分布(定理1)。我们还给出了点O外围存在极限环和不存在极     

一类四次系统极限环的个数与分布

本文研究一类四次系统的极限环,通过计算四次系统鞍点分界线之间的有向距离,计算一阶焦点量及二阶焦点量,判别同宿轨内外的稳定性,利用分支理论与定性分析技巧发现这类系统有六个极限环,并给出了它们的分布.     

一类四次系统极限环的个数与分布

本文研究一类四次系统的极限环,通过计算四次系统鞍点分界线之间的有向距离,计算一阶焦点量 及二阶焦点量,判别同宿轨内外的稳定性,利用分支理论与定性分析技巧发现这类系统有六个极限环, 并给出了它们的分布．     

奇点量公式的机器推导与一类三次系统的前10个鞍点量

本文给出了计算奇点量的两个递推公式，对一类三次系统用计算机推导出原点的前10个鞍点量，得到系统原点领域存在正则积分的充分必要条件。     

二次系统的细鞍点与分界线环

类似于扰动法,P.Joyal引进了细鞍点的鞍点量概念,并讨论了鞍点量与经过该鞍点的同宿轨道(即只经过一个鞍点的分界线环,记为S~((1))产生极限环的个数之间的关系.本文给出二次系统的鞍点量用系数表示的公式,为研究二次系统的分界线环在何种条件下能扰动出极限环及其个数提供了重要的工具.     

论复自治微分系统的奇点量

本文研究复自治微分系统(E),得到如下的主要结果:(ⅰ)引入奇点量概念,实现了实系统中焦点量和鞍点量的统一;(ⅱ)定义Lie不变量概念,得到奇点量结构定理和广义对称原理;(ⅲ)计算了(E₃)的全部120个基本Lie不变量,并应用奇点量结构定理得到(E₂)和(E₃⁽³⁾)的奇点量公式及可积性条件。     

一类三次系统的细奇点

本文研究了具有一个高阶奇点的系统（１,２）,通过在实系统中引入奇点量的概念使焦点量和鞍点量得到统一,并且计算了系统（１．２）的各阶奇点量,证明了这样的系统存在珐多三阶细奇点,而且同时存在细奇点的最大个数是四。当系统是四个细奇点时,它们可能是四个一阶细奇点或四个三阶细奇点。     

用范式理论研究常微分方程焦点量问题*

利用范式与焦点量之间关系的一个定理并拓展了矩阵表示法,借助于计算机代数语言Mathe-matica的帮助,本文给出了一种计算常微分方程焦点量问题的新方法,利用这种方法可以计算常微分方程的任意阶焦点量,并通过一个算例验证了本文所提出的方法的正确性。这种方法的优点是简捷、方便、只进行简单的代数运算。     

分界线环的稳定性和分支极限环的唯一性

本文证明若在鞍点处发散量保持为零,则在分界线环L_0分支出极限环的过程中发散量积分是连续的,因而当发散量沿L_0的积分不为零时,L_0产生的极限环是唯一的。本文还证明,仅由细鞍点的阶数和鞍点量的符号并不能给出判定过细鞍点的单叶(双叶)同宿分界线环的内侧(内外侧)稳定性的普适准则。最后证明具有以细鞍点为重点的不可约三次代数曲线解的二次微分系统必可积;二次系统的同宿分界线环因改变稳定性而生成的极限环是唯一的。     

焦点量算法与中心条件推导

对于一般情形, 基于后继函数法给出焦点量计算的递推公式;基于形式级数法给出焦点量计算和化简的Maple算法;给出了时间可逆条件的推导算法,给出了一类五次系统时间可逆的充要条件.     

模糊量及模糊数的表现特性

模糊集的表现定理是模糊数学的最基本理论.在表现定理的基础上,对各种模糊量:包括凸模糊量、正规模糊量、正规凸模糊量、凸有界模糊量、模糊数、有限模糊数、对称模糊数的表现定理进行了深入的研究,从而建立了不同类型模糊量与普通集合之间的联系.     

FOCAL QUANTITIES AND BIFURCATION OFLIMIT CYCLES FOR E~1_3 SYSTEM

1IntroductiollInthispapersweconsiderthenumberofthelimitcycleforthefollowingcubicpolynomialdifferentialsystemwiththeEdstyleFOrthepurposeofanalysingthenumberoflimitcycles,itisnecessarytocalculatethefocalquantities.FOrsystem(1))weperformfirstthefollowinghomeomorphictransformationThen,system(l)iscarriedintothefollowingsystemwhereIn[1],usingPoincaremethodandWuelimination,wedesignaspecialal-gorithmtocomputingthefocalvaluesforplanardynamicsystems.Byouralgorithm,weobtainthefollowingformularofcalcul…     

Conserved Quantities and Symmetries Related to Stochastic Dynamical Systems

The present article focuses on the three topics related to the notions of "conserved quantities" and "symmetries" in stochastic dynamical systems described by stochastic differential equations of Stratonovich type. The first topic is concerned with the relation between conserved quantities and symmetries in stochastic Hamilton dynamical systems, which is established in a way analogous to that in the deterministic Hamilton dynamical theory. In contrast with this, the second topic is devoted to investigate the procedures to derive conserved quantities from symmetries of stochastic dynamical systems without using either the Lagrangian or Hamiltonian structure. The results in these topics indicate that the notion of symmetries is useful for finding conserved quantities in various stochastic dynamical systems. As a further important application of symmetries, the third topic treats the similarity method to stochastic dynamical systems. That is, it is shown that the order of a stochastic system can be reduced, if the system admits symmetries. In each topic, some illustrative examples for stochastic dynamical systems and their conserved quantities and symmetries are given.     

变质量完整力学系统的Lie对称与守恒量

研究变质量完整系统的Lie对称和守恒量。利用常微分方程在无限小变换下的不变性建立系统Lie对称的确定方程。给出结构方程和守恒量。举例说明结果的应用。     

Parareal Algorithms Applied to Stochastic Differential Equations with Conserved Quantities

In this paper, we couple the parareal algorithm with projection methods of the trajectory on a specific manifold, defined by the preservation of some conserved quantities of stochastic differential equations. First, projection methods are introduced as the coarse and fine propagators. Second, we apply the projection methods for systems with conserved quantities in the correction step of original parareal algorithm. Finally, three numerical experiments are performed by different kinds of algorithms to show the property of convergence in iteration, and preservation in conserved quantities of model systems.     

Quantities at infinity in translational polynomial vector fields

In this paper, we study quantities at infinity and the appearance of limit cycles from the equator in polynomial vector fields with no singular points at infinity. We start by proving the algebraic equivalence of the corresponding quantities at infinity (also focal values at infinity) for the system and its translational system, then we obtain that the maximum number of limit cycles that can appear at infinity is invariant for the systems by translational transformation. Finally, we compute the singular point quantities of a class of cubic polynomial system and its translational system, reach with relative ease expressions of the first five quantities at infinity of the two systems, then we prove that the two cubic vector fields perturbed identically can have five limit cycles simultaneously in the neighborhood of infinity and construct two systems that allow the appearance of five limit cycles respectively. The positions of these limit cycles can be pointed out exactly without constructing Poincaré cycle fields. The technique employed in this work is essentially different from more usual ones, The calculation can be readily done with using computer symbol operation system such as Mathematics.     

一类7次微分自治系统的极限环分支

研究一类平面7次微分系统,通过作两个适当的变换以及焦点量的仔细计算,得出了系统的无穷远点与2个初等焦点能够同时成为广义细焦点的条件,进一步得出在一定条件下该系统能够分支出15个极限环的结论,其中5个大振幅极限环来自无穷远点,10个小振幅极限环来自2个初等焦点.     

On the equivalence of singular point quantities and the integrability of a fine critical singular point

In this paper, the equivalence of singular point quantities and the integrability of a fine critical singular point are discussed. For an extensive class of complex autonomous differential systems, the theoretical basis for the methods of using formal series and integrating factors to calculate the singular point quantities for the origin (a fine critical singular point) and judge its integrability is obtained. Two recursion formulas, which are much simpler than focal values for computation because the operation is rational, for computing singular point quantities are introduced.