三维反转系统中余维2和3的异维环分支

研究了三维反转系统中具有2个鞍点的对称异维环分支问题.在此反转性意味着存在线性对合R,使得系统在R变换和时间逆向条件下仍保持不变.当R的不动点构成集合的维数dim Fix(R)=1时,我们研究了R-对称异维环,R-对称周期轨线,同宿环,重周期轨线和具有单参数族的无穷条周期轨线的存在性及它们的共存性.本文也明确得到了对称异维环的重同宿分支,且分支出的不可数无穷条周期轨道聚集在某条同宿轨道的小邻域内.进一步,作者也证明了相应的分支曲面及其存在区域.对于dim Fix(R)=2时的情形,本文得到了系统可分支出R-周期轨道和R-对称异宿环.     

四维空间中的异维环分支问题

利用局部活动坐标架法,讨论了四维空间中连接两个鞍点的异维环分支问题,在一些通有的假设下,分别得到了异维环保存、同宿环、周期轨存在的充分条件以及保存的异维环与分支出的周期轨共存(或不共存)的结果.     

四维空间中的异维环分支问题

利用局部活动也标架法,讨论了四维空间中连接两个鞍点的异维环分支问题,在一些通有的假设下,分别得到了异维环保存、同宿环、周期轨存在的允分条件以及保存的异维环与分支出的周期轨共存(或不共存)的结果.     

带有倾斜翻转的余维2反转异宿分支

本文运用由Zhu和Xia于1998年建立的方法,详细研究了一个四维反转系统中带有倾斜翻转的异宿环分支问题,取得了一系列有意义的结果．例如：R-对称同宿轨道的存在性、R-对称同宿轨道与R-对称异宿轨道、R-对称同宿轨道与R-对称周期轨道的共存性,并找到了反转异宿轨道分支中的R-对称倍同宿轨道分支（即：二重R-对称同宿分支）、收敛于同宿轨道的无穷多R-对称同宿轨道的存在性,最后给出了相关的分支曲面和存在区域．     

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.     

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。     

一类三次Kolmogorov系统的极限环分支

本文研究了一类三次Kolmogorov系统，得出了该系统可分支出三个极限环，且其中有两个是稳定的，同时给出了其中心条件.     

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.     

一类余维3的鞍-焦点异宿环分支

对余维3系统X_μ(x)具有包含一个双曲鞍-焦点O_1和一个非双曲鞍-焦点O_2的异宿环f进行了研究.证明了在f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Γ~0破裂时X_μ(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Γ~0破裂和O_2点产生Hopf分支的情况下,在f的邻域内有一条含O_1点同宿环,可数无效多条的轨线同宿于O_2点分支出的闭轨H_0,一条或无穷多条(可数或连续统的)异宿轨线等.     

具有轨道翻转和倾斜翻转的退化异维环分支

本文研究4 维系统中一类具有轨道翻转和倾斜翻转的退化异维环分支问题. 通过在未扰异维环的小管状邻域内建立局部活动坐标系, 本文建立Poincaré 映射, 确定分支方程. 由对分支方程的分析,本文讨论在小扰动下, 异宿环、同宿环和周期轨的存在性、不存在性和共存性, 且给出它们的分支曲面以及共存区域, 推广了已有结果.     

Global bifurcations near a degenerate heterodimensional cycle

This article is devoted to investigating the bifurcations of a heterodimensional cycle with orbit flip and inclination flip, which is a highly degenerate singular cycle. We show the persistence of the heterodimensional cycle and the existence of bifurcation surfaces for the homoclinic orbits or periodic orbits. It is worthy to mention that some new features produced by the degeneracies that the coexistence of heterodimensional cycles and multiple periodic orbits are presented as well, which is different from some known results in the literature. Moreover, an example is given to illustrate our results and clear up some doubts about the existence of the system which has a heterodimensional cycle with both orbit flip and inclination flip. Our strategy is based on moving frame, the fundamental solution matrix of linear variational system is chose to be an active local coordinate system along original heterodimensional cycle, which can clearly display the non-generic properties-``orbit flip" and ``inclination flip" for some sufficiently large time.     

亏格一双中心的二次可逆Lotka-Volterra系统的二次扰动

该文研究在二次扰动下,亏格一双中心的二次可逆Lotka-Volterra系统周期环域产生极限环的个数问题.证明在二次扰动下,二次可逆Lotka-Volterra系统(rlv5)的周期环域产生极限环的个数不超过3.     

一类稀疏效应下捕食—被捕食系统的全局分析

本文讨论了稀疏效应下的一类捕食-被捕食系统得到了极限环不存在条件,极限环的存在性与唯一性定理和异宿轨道存在与唯一性定理。     

Degenerate bifurcations of nontwisted heterodimensional cycles with codimension 3

Bifurcations of heterodimensional cycles with highly degenerate conditions are studied by establishing a suitable local coordinate system in three-dimensional vector fields. The existence, coexistence and noncoexistence of the periodic orbit, homoclinic loop, heteroclinic loop and double periodic orbit are obtained under some generic hypotheses. The bifurcation surfaces and the existence regions are located; the number of the bifurcation surfaces exhibits variety and complexity of the bifurcation of degenerate heterodimensional cycles. The corresponding bifurcation graph is also drawn.     

一类可逆系统的Aubry-Mather集

本文利用推广的Aubry-Mather定理,获得了一类二阶可逆微分方程Aubry-Mather集存在的充分性条件.     

Bifurcation of Degenerate Homoclinic Orbits toSaddle-Center in Reversible Systems

The authors study the bifurcation of homoclinic orbits from a degenerate homoclinic orbit in reversible system. The unperturbed system is assumed to have saddle-center type equilibrium whose stable and unstable manifolds intersect in two-dimensional manifolds. A perturbation technique for the detection of symmetric and nonsymmetric homoctinic orbits near the primary homoclinic orbits is developed. Some known results are extended.     

Bifurcation of Degenerate Homoclinic Orbits to Saddle-Center in Reversible Systems

The authors study the bifurcation of homoclinic orbits from a degenerate homoclinic orbit in reversible system. The unperturbed system is assumed to have saddlecenter type equilibrium whose stable and unstable manifolds intersect in two-dimensional manifolds. A perturbation technique for the detection of symmetric and nonsymmetric homoclinic orbits near the primary homoclinic orbits is developed. Some known results are extended.     

On the Number of Limit Cycles in Small Perturbations of a Piecewise Linear Hamiltonian System with a Heteroclinic Loop

In this paper, the authors consider limit cycle bifurcations for a kind of nonsmooth polynomial differential systems by perturbing a piecewise linear Hamiltonian system with a center at the origin and a heteroclinic loop around the origin. When the degree of perturbing polynomial terms is n(n ≥ 1), it is obtained that n limit cycles can appear near the origin and the heteroclinic loop respectively by using the first Melnikov function of piecewise near-Hamiltonian systems, and that there are at most n + [(n+1)/2] limit cycles bifurcating from the periodic annulus between the center and the heteroclinic loop up to the first order in ε. Especially, for n = 1, 2, 3 and 4, a precise result on the maximal number of zeros of the first Melnikov function is derived.     

Limit Cycles Bifurcating from a Quadratic Reversible Lotka-Volterra System with a Center and Three Saddles

This paper is concerned with limit cycles which bifurcate from a period annulus of a quadratic reversible Lotka-Volterra system with sextic orbits. The authors apply the property of an extended complete Chebyshev system and prove that the cyclicity of the period annulus under quadratic perturbations is equal to two.     

Limit cycle bifurcations by perturbing a class of integrable systems with a polycycle

In this paper, we deal with the problem of limit cycle bifurcation near a 2-polycycle or 3-polycycle for a class of integrable systems by using the first order Melnikov function. We first get the formal expansion of the Melnikov function corresponding to the heteroclinic loop and then give some computational formulas for the first coefficients of the expansion. Based on the coefficients, we obtain a lower bound for the maximal number of limit cycles near the polycycle. As an application of our main results, we consider quadratic integrable polynomial systems, obtaining at least two limit cycles.