Codimension 3 nonresonant bifurcations of homoclinic orbits with two inclination flips

Homoclinic bifurcations in four-dimensional vector fields are investigated by setting up a local coordinate near a homoclinic orbit. This homoclinic orbit is principal but its stable and unstable foliations take inclination flip. The existence, nonexistence, and uniqueness of the 1-homoclinic orbit and 1-periodic orbit are studied. The existence of the two-fold 1 -periodic orbit and three-fold 1 -periodic orbit are also obtained. It is indicated that the number of periodic orbits bifurcated from this kind of homoclinic orbits depends heavily on the strength of the inclination flip.     

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.     

Codimension-4 resonant homoclinic bifurcations with orbit flips and inclination flips

The paper studies a codimension-4 resonant homoclinic bifurcation with one orbit flip and two inclination flips, where the resonance takes place in the tangent direction of homoclinic orbit.Local active coordinate system is introduced to construct the Poincar′e returning map, and also the associated successor functions. We prove the existence of the saddle-node bifurcation, the perioddoubling bifurcation and the homoclinic-doubling bifurcation, and also locate the corresponding 1-periodic orbit, 1-homoclinic orbit, double periodic orbits and some 2n-homoclinic orbits.     

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.     

Degenerate Orbit Flip Homoclinic Bifurcations with Higher Dimensions

Bifurcations of a degenerate homoclinic orbit with orbit flip in high dimensional system are studied. By establishing a local coordinate system and a Poincaré map near the homoclinic orbit, the existence and uniqueness of 1–homoclinic orbit and 1–periodic orbit are given. Also considered is the existence of 2–homoclinic orbit and 2–periodic orbit. In additon, the corresponding bifurcation surfaces are given. Project supported by the National Natural Science Foundation of China (No: 10171044), the Natural Science Foundation of Jiangsu Province (No: BK2001024), the Foundation for University Key Teachers of the Ministry of Education of China     

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.     

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．     

Codimension 3 Non-resonant Bifurcations of Rough Heteroclinic Loops with One Orbit Flip

Heteroclinic bifurcations in four dimensional vector fields are investigated by setting up a local coordinates near a rough heteroclinic loop. This heteroclinic loop has a principal heteroclinic orbit and a non-principal heteroclinic orbit that takes orbit flip. The existence, nonexistence, coexistence and uniqueness of the 1-heteroclinic loop, 1-homoclinic orbit and 1-periodic orbit are studied. The existence of the two-fold or three-fold 1-periodic orbit is also obtained.     

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 and Periodic Orbits Arising Near the Heteroclinic Cycle Connecting Saddle-focus and Saddle Under Reversible Condition

In this paper, we study the dynamical behavior for a 4-dimensional reversible system near its heteroclinic loop connecting a saddle-focus and a saddle. The existence of infinitely many reversible 1-homoclinic orbits to the saddle and 2-homoclinic orbits to the saddle-focus is shown. And it is also proved that, corresponding to each 1-homoclinic （resp. 2-homoclinic） orbit F, there is a spiral segment such that the associated orbits starting from the segment are all reversible 1-periodic （resp. 2-periodic） and accumulate onto F. Moreover, each 2-homoclinic orbit may be also accumulated by a sequence of reversible 4-homoclinic orbits.     

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.     

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.     

伴随超临界分支的通有同宿轨道分支

本文研究具有非双曲奇点的高维系统在小扰动下的同宿轨道分支问题,通过在未扰同宿轨道邻域建立局部坐标系,导出系统在新坐标系下的Poincare映射,对伴随超临界分支的通有同宿轨道的保存及分支出周期轨道的情况进行了讨论,推广和改进了一些文献的结果.     

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.     

Stability and uniqueness of periodic orbits produced during homoclinic bifurcation

Under a generic assumption, the existence and the uniqueness of the periodic orbit generating from a homoclinic bifurcation are shown, and the dimensions of its stable and unstable manifolds are given. In the case of a 3-dimensional system, our result revises the stability criterion given in [4,5].Supported by the National Natural Science Foundation of China.     

一类3维反转系统中的异维环分支

研究了一类3维反转系统中包含2个鞍点的对称异维环分支问题, 且仅限于研究系统的线性对合R的不变集维数为1的情形. 给出了R-对称异宿环与R-对称周期轨线存在和共存的条件, 同时也得到了R-对称的重周期轨线存在性. 其 次, 给出了异宿环、 同宿轨线、 重同宿轨线和单参数族周期轨线的存在性、 唯一性和共存性等结论, 并且发现不可数无穷条周期轨线聚集在某一同宿轨线的小邻域内. 最后给出了相应的分支图.     

Existence of periodic orbits and shift-invariant curve sequences near multiple homoclinics简

In this paper, the complicated dynamics is studied near a double homoclinic loops with bellows configuration for general systems. For the non-twisted multiple homoclinics, the existence of periodic orbit with the specified route and the existence of shift-invariant curve sequences defined on the cross sections of multiple homoclinics corresponding to any specified one-side infinite sequences are given. In addition, the existence regions are also located.     

Bifurcations of Fine 3-point-loop in Higher Dimensional Space

By using the linear independent fundamental solutions of the linear variational equation along the heteroclinic loop to establish a suitable local coordinate system in some small tubular neighborhood of the heteroclinic loop, the Poincaré map is constructed to study the bifurcation problems of a fine 3–point loop in higher dimensional space. Under some transversal conditions and the non–twisted condition, the existence, coexistence and incoexistence of 2–point–loop, 1–homoclinic orbit, simple 1–periodic orbit and 2–fold 1–periodic orbit, and the number of 1–periodic orbits are studied. Moreover, the bifurcation surfaces and existence regions are given. Lastly, the above bifurcation results are applied to a planar system and an inside stability criterion is obtained. This work is supported by the National Natural Science Foundation of China (10371040), the Shanghai Priority Academic Disciplines and the Scientific Research Foundation of Linyi Teacher’s University     

DEGENERATED HOMOCLINIC BIFURCATIONS WITH HIGHER DIMENSIONS

91.IntroductionandHypothesesInrecelltyears)withthedevelopmelltofnonlinearscienceandthedeepstudyof~icphenomena,anincreasinglylargenUInberofpapersared~edtothebifurcationProblemsofhomocliulcandheterocbocorbitsinhighdhansionalspace(see11--14]).Duetothedifficultyencountered,uofortunately,onlyafew(e.g.[1,13,14])areconcernedwiththeperiodicorbitsbifulcatedfromsingularloops.Papers[1,131discussedtheProblemofthehomoclinicloOPbifurcationinhighdimensionwithcodimension2,thatis,thesystemhasresonanteigenVa…