空间同宿环和异宿环的稳定性

关于平面同（异）宿环的稳定性已有不少文献讨论过,但关于空间同（异）宿环的稳定性尚没有任何结果．本文在可定义回复映射的条件下给出了同（异）宿环在其部分邻域中是渐近稳定的判据．这些结果在某种意义下是平面系统相应结果的推广,包括并推广了［２］,［３］的结果．本文最后讨论了Ｌｏｒｅｎｚ系统同宿环和三种群竞争系统异宿环的稳定性,所得结果和Ｓｐａｒｒｏｗ与Ｍａｙ等的数值结果相吻合．     

任意有限维空间中鞍焦点同宿环的稳定性

在改造和扩充空间同宿环的部分邻域稳定性定义的基础上,通过建立首次回归映射并考虑其压缩性和扩张性,对任意有限维空M系统连接双曲鞍焦点的同宿环的稳定性质作了深入的研究,包括在不同的主特征值条件下给出了在其管状邻域内稳定集合(流形)和不稳定集合(流形)的存在性及其维数.     

一类两点异宿环的扰动分支

设Lo为含两个双曲鞍点Si(i=1,2)的异宿环,Si的双曲比记为Tio(i=1,2),Mourtada在1994年证明如果未扰动系统与被扰系统均为C∞的,则当r10r20≠1时,Lo至多产生两个极限环.本文对C3系统证明了这一结论.     

一类两点异宿环的扰动分支

设 Ｌ０为含两个双曲鞍点 Ｓｉ（ｉ＝ １,２）的异宿环, Ｓｉ的双曲比记为ｒｉ０（ｉ＝ １,２）, Ｍｏｕｒｔａｄａ在１９９４年证明如果未扰动系统与被扰系统均为 Ｃ∞的,则当ｒ１０,２０≠１时, Ｌ０至多产生两个极限环．本文对 Ｃ３系统证明了这一结论．     

一类异宿环分支问题中极限环的唯一性

在具余维２奇点的四维系统的两参数开折的研究中出现一类三点异宿环的扰动分支,对此异宿环产生极限环的唯一性一直未得到完整的解决,本文圆满地解决了这一问题,并获得了全局分支中极限环的唯一性。     

一类异宿环分支极限环的唯一性

本文研究平面上一类两点或三点异宿环附近极限环的分支,在一简洁条件下证明了异宿环分支极限环的唯一性,并给出了极限环唯一存在的充要条件.作为对三维余维2分支的应用,解决了所出现的两点异宿环产生唯一极限环的问题.     

Banach空间中正则非退化异宿环的Lipschitz 扰动

本文研究Banach空间上离散动力系统的Lipschitz扰动.设f,g是Banach空间(X,||·||)上的连续自映射.如果f具有正则非退化返回排斥子或正则非退化异宿环且g是f的Lipschitz小扰动,则g也有正则非退化返回排斥子或正则非退化异宿环.另外,本文还证明完备度量空间中正则非退化异宿环蕴含正则非退化返回...     

同宿、异宿环分支出极限环的唯一性

本文在一简洁条件下证明任一含细鞍点的孤立同宿环及对称的双同宿环和两点异宿环至多产生一个极限环。     

连接两个具有一维不稳定流形的双曲鞍点异宿环的稳定性

本文研究任意有限维空间中连接两个具有一维不稳定流形的双曲鞍点异宿环的稳定性.借助适当的线性变换和坐标变换,将局部稳定流形和不稳定流形拉直,利用奇异流映射和正则流映射构造了Poincaré映射.通过技巧性地估计向量的模,给出了在横截面上Poincaré映射的初始点与首次回归点离异宿轨道与横截面交点的距离之比,得到了高维空间中连接两个带有一维不稳定流形的异宿环的非常简洁的稳定性判据.     

一类余维3的鞍-焦点异宿环分支

对余维3系统Xμ(x)具有包含一个双曲鞍-焦点O1和一个非双曲鞍-焦点O2的异宿环￡进行了研究.证明了在￡的邻域内有可数无穷条周期轨线和异宿轨线,当非粗糙异宿轨线ΓO破裂时Xμ(x)会产生同宿轨分支,并给出了相应的分支曲线和两种同宿环共存的参数值.在3参数扰动下ΓO破裂和O2点产生Hopf分支的情况下,在￡的邻域内有一条含O1点同宿环,可数无数多条的轨线同宿于O2点分支出的闭轨HO,一条或无穷多条(可数或连续统的)异宿轨线等.     

Some basic inequalities in higher dimensional non-Euclid space

In this paper, the concept of a finite mass-points system∑N(H(A))(N>n) being in a sphere in an n-dimensional hyperbolic space Hn and a finite mass-points system∑N(S(A))(N>n) being in a hyperplane in an n-dimensional spherical space Sn is introduced, then, the rank of the Cayley-Menger matrix AN(H)(or a AN(S)) of the finite mass-points system∑∑N(S(A))(or∑N(S(A))) in an n-dimensional hyperbolic space Hn (or spherical space Sn) is no more than n 2 when∑N(H(A))(N>n) (or∑N(S(A))(N>n)) are in a sphere (or hyperplane). On the one hand, the Yang-Zhang's inequalities, the Neuberg-Pedoe's inequalities and the inequality of the metric addition in an n-dimensional hyperbolic space Hn and in an n-dimensional spherical space Sn are established by the method of characteristic roots. These are basic inequalities in hyperbolic geometry and spherical geometry. On the other hand, some relative problems and conjectures are brought.     

Multiple shock fronts for hyperbolic conservation laws in higher dimensional space

The local existence of multiple shock fronts for hyperbolic conservation laws in higher dimensional space is established under the assumption that its frozen problem produces multiple uniformly stable planar shock fronts.     

Three dimensional linear motion transformation in a higher order dimensional space

The objective of the work presented in this paper is an attempt at solving and transforming of the known from the classical mechanics three dimensional – single mass mathematical and mechanical vibration models in a higher order dimensional space with any virtual sectional curvature – positive or negative, constant or variable. The object of the investigation is a class of three dimensional surfaces. The aims of the work presented in this paper are to illustrate the performance of the common algorithm in three dimensional linear motion transformation, that means to transform 3D space in a higher order dimensional space and a comparison is derived on the behavior of the common algorithm depending on the surface properties. A characterization of the Riemannian Manifolds is performed by means of curvature operators in the three dimensional solution. The computer codes Mathematica and MATLAB are used in the numerical simulation. The system motion is investigated in a 3-D qualitative aspect in time and frequency domain. The application can be in topology when geodesists make snap shots of the surface profile, then the curved lines can be analyzed and transformed in the desired space dimension. Any kind of a trajectory of motion can be transformed successfully in a higher order dimensional space and vice verse by means of applying of the common algorithm.     

Motion transformation in a higher order dimensional space

The objective of the work presented in this paper is an attempt at solving and transforming of the known from the classical mechanics two dimensional-plane single mass mechanical and mathematical vibration models in a higher order dimensional space with any virtual sectional curvature-positive or negative, constant or variable. A characterization of the Riemannian manifolds is performed by means of curvature operators. The computer codes Mathematica and MATLAB are used in the numerical simulation. The objects of the investigation are a sphere – with a positive constant sectional curvature, a cylinder-with a zero constant sectional curvature, helicoid-with a negative variable sectional curvature, a torus-with a variable (±) sectional curvature, any virtual surface of second order-with a variable (±) sectional curvature, pseudo-sphere – with a negative constant sectional curvature and a saddle-with a negative variable sectional curvature. The system motion is investigated in a qualitative aspect in time and frequency domain on the cited surfaces. The common algorithm derived in the paper can transform any motion from 3D space to curved manifold. We can derive the trajectory in an explicit form on the curved manifold. We can change the trajectory by a suitable variation of the curved manifold.     

Cyclicity of planar polycycles and ensembles with codimension 3 degeneration through a saddle-node and two hyperbolic saddles

The cyclicity of four classes of codimension 3 plnnar polycycles and ensembles containing a saddle-node and two hyperbdic saddles is dealt with. The exnct cyclicity or cyclicity bound of them is obtained by finitely-smooth normal form theory.     

Stability estimate for the hyperbolic inverse boundary value problem by local Dirichlet-to-Neumann map

In this paper we consider the stability of the inverse problem of determining a function q(x) in a wave equation in a bounded smooth domain in Rn from boundary observations. This information is enclosed in the hyperbolic (dynamic) Dirichlet-to-Neumann map associated to the solutions to the wave equation. We prove in the case of n?2 that q(x) is uniquely determined by the range restricted to a subboundary of the Dirichlet-to-Neumann map whose stability is a type of double logarithm.     

高阶拟线性双曲型方程的精确边界能控性

By means of the existence and uniqueness of semi-global C^1 solution to the mixed initial-boundary value problem with general nonlinear boundary conditions for first order quasilinear hyperbolic systems with zero eigenvalues ,the local exact boundary controllability for higher order quasilinear hyperbolic equations is established.     

X-调和映射的稳定性

在一般调和映射基础上定义了X-调和映射和次椭圆调和映射，得到了X-调和映射的稳定性定理，它是Leung一般调和映射及其稳定性定理的推广．