伴随超临界分支的非通有同宿轨道分支

考虑伴随超临界分支的高维非退化系统在非通有假设下的同宿轨道分支问题,通过在未扰同宿轨道邻域建立活动坐标架,导出系统在新坐标系下的全局Poincare映射,对伴随超临界分支的非通有同宿轨道的保存及分支周期轨道的情况导出相应的分支方程和分支图．     

高维同宿环扭曲分支与稳定性

利用沿同宿环的线性变分方程的线性独立解作为在同宿环的小管状邻域内的局部坐标系来建立Poincaré映射,研究了高维系统扭曲同宿环的分支问题．在非共振条件和共振条件下,获得了1-同宿环、 1-周期轨道、 2-同宿环、 2-周期轨道和两重2-同期轨道的存在性、 存在个数和存在区域．给出了相关的分支曲面的近似表示．同时,研究了高维系统同宿环和平面系统非扭曲同宿环的稳定性．     

快变量空间中的同宿轨道分支

本文考虑奇摄动问题的位于快变量空间中的奇异同宿轨道的保存和周期轨道分支问题文中关于奇异同宿轨道保存的结论推广了一些已知的结果,而周期轨道产生于奇异同宿轨道的分支则提供了一种新的分支类型     

快变量空间中的同宿轨道分支

本文考虑奇摄动问题的位于快变量空间中的奇异同宿轨道的保存和周期轨道分支问题.文中关于奇异同宿轨道保存的结论推广了一些已知的结果,而周期轨道产生于奇异同宿轨道的分支则提供了一种新的分支类型.     

非扭曲异宿环分支

考虑高维系统非扭曲细异宿环分支 ,给出了 1 同宿轨道和 1 周期轨道的存在性和存在域 ,并得到了 2 重周期轨道的分支曲面 .最后 ,这些分支结果被应用于平面系统细异宿环 ,获得了新的有趣的结论     

非扭曲异宿环分支

考虑高维系统非扭曲细异宿环分支,给出了1-同宿轨道和1-周期轨道的存在性和存在域,并得到了2-重周期轨道的分支曲面.最后,这些分支结果被应用于平面系统细异宿环,获得了新的有趣的结论.     

快变振荡下的同宿轨道分支

该文研究带有角变量的快变振荡系统的同宿轨道分支问题，通过建立完整的Poincaré映射，讨论其同宿轨道在小扰动下的保存性, 并进一步研究其分支出周期轨道的情况，推广和改进了一些文献的结果.     

具有共振特征值的轨道翻转双同宿环分支

该文研究了具有轨道翻转的双同宿环四维系统,在主特征值共振和沿轨道奇点处切方向共振下的两种分支．我们分别在系统奇点小邻域内利用规范型的解构造一个奇异映射,再在双同宿环的管状邻域内引起局部活动坐标架,利用系统线性变分方程的解定义了一个正则映射,通过复合两个映射而得到分支研究中一类重要的Poincaré映射,经过简单的计算最终得到后继函数的精确表达式．对分支方程细致地研究,我们给出了原双同宿环的保存性条件,并证明了“大” 1-同宿环分支曲面,2-重“大”1-周期轨分支曲面,“大”2-同宿环分支曲面的存在性、存在区域和近似表达式,及其分支出的“大”周期轨和“大”同宿轨的存在性区域和数量．     

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.     

高维同宿分支问题

本文通过应用指数二分性和将关于周期系统的Floquet方法推广到非周期系统，来构造适当的局部坐标系以建立Poincare映射，并用以解决各类余维为1和2的同宿分支问题．文中还给出了分支图和分支曲线．     

Homoclinic Flip Bifurcations Accompanied by Transcritical Bifurcation

The bifurcations of orbit flip homoclinic loop with nonhyperbolic equilibria are investigated. By constructing local coordinate systems near the unperturbed homoclinic orbit, Poincaré maps for the new system are established. Then the existence of homoclinic orbit and the periodic orbit is studied for the system accompanied with transcritical bifurcation.     

Bifurcation Analysis of the Multiple Flips Homoclinic Orbit

A high-codimension homoclinic bifurcation is considered with one orbit flip and two inclination flips accompanied by resonant principal eigenvalues. A local active coordinate system in a small neighborhood of homoclinic orbit is introduced. By analysis of the bifurcation equation, the authors obtain the conditions when the original flip homoclinic orbit is kept or broken. The existence and the existence regions of several double periodic orbits and one triple periodic orbit bifurcations are proved. Moreover, the complicated homoclinic-doubling bifurcations are found and expressed approximately.     

Bifurcation of Homoclinic Orbits with Saddle-Center Equilibrium

In this paper, the authors develop new global perturbation techniques for detecting the persistence of transversal homoclinic orbits in a more general nondegenerated system with action-angle variable. The unperturbed system is assumed to have saddle-center type equilibrium whose stable and unstable manifolds intersect in one dimensional manifold, and does not have to be completely integrable or near-integrable. By constructing local coordinate systems near the unperturbed homoclinic orbit, the conditions of existence of transversal homoclinic orbit are obtained, and the existence of periodic orbits bifurcated from homoclinic orbit is also considered.     

8-形同宿轨道附近的同宿轨分支性质

In this paper we investigate the homoclinic bifurcation properties near an eight-figure homoclinic orbit of co-dimension two of a planar dynamical system. The corresponding local bifurcation diagram is also illustrated by numerical computation.     

Homoclinic Bifurcation of Orbit Flip with Resonant Principal Eigenvalues

Codimension-3 bifurcations of an orbit-flip homoclinic orbit with resonant principal eigenvalues are studied for a four-dimensional system. The existence, number, co-existence and noncoexistence of 1-homoclinic orbit, 1-periodic orbit, 2n-homoclinic orbit and 2n-periodic orbit are obtained. The bifurcation surfaces and existence regions are also given.     

Homoclinic bifurcation with nonhyperbolic equilibria

The problem of homoclinic bifurcation is studied for a high dimensional system with nonhyperbolic equilibria. By constructing local coordinate systems near the unperturbed homoclinic orbit, Poincaré maps for the new system are established. Then the persistence of the homoclinic orbit and the bifurcation of the periodic orbit for the system accompanied with pitchfork bifurcation are obtained. Some known results are extended.     

Bifurcations of heteroclinic loops

By generalizing the Floquet method from periodic systems to systems with exponential dichotomy, a local coordinate system is established in a neighborhood of the heteroclinic loop Γ to study the bifurcation problems of homoclinic and periodic orbits. Asymptotic expressions of the bifurcation surfaces and their relative positions are given. The results obtained in literature concerned with the 1-hom bifurcation surfaces are improved and extended to the nontransversal case. Existence regions of the 1-per orbits bifurcated from Γ are described, and the uniqueness and incoexistence of the 1-hom and 1-per orbit and the inexistence of the 2-hom and 2-per orbit are also obtained. Project supported by the National Natural Science Foundation of China (Grant No. 19771037) and the National Science Foundation of America # 9357622. This paper was completed when the first author was visiting Northwestern University.     

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.