Bifurcations of nontwisted heteroclinic loop

Bifurcations of nontwisted and fine heteroclinic loops are studied for higher dimensional systems. The existence and its associated existing regions are given for the 1-hom orbit and the 1-per orbit, respectively, and bifurcation surfaces of the two-fold periodic orbit are also obtained. At last, these bifurcation results are applied to the fine heteroclinic loop for the planar system, which leads to some new and interesting results.     

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     

扭曲退化同宿分支

§ 1　 HypothesesConsider the following system:z.=f(z) , (1 .1 )and its perturbed systemz.=f(z) +g(z,μ) (1 .2 )where z∈ Rm+n,μ∈ Rk,k≥ 3,0≤ |μ| 1 ,f,g∈ Cr,r≥ 4 ,g(z,0 ) =0 .For simplicity,we sup-pose thatf(p) =0 ,g(p,μ) =0 .Moreover,for(1 .1 ) we assume(H1 ) The stable manifold Wspand the unstable manifold Wupof z=p are m-dimension-al and n-dimensional,respectively.The linearization Df(p) atthe equilibrium z=p has realmultiple-2 eigenvaluesλ1 and -ρ1 ,such thatany remaining eige…     

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…     

Codimension 2 reversible heteroclinic bifurcations with inclination flips

In this paper, the heteroclinic bifurcation problem with real eigenvalues and two inclination-flips is investigated in a four-dimensional reversible system. We perform a detailed study of this case by using the method originally established in the papers “Problems in Homoclinic Bifurcation with Higher Dimensions” and “Bifurcation of Heteroclinic Loops,” and obtain fruitful results, such as the existence and coexistence of R-symmetric homoclinic orbit and R-symmetric heteroclinic loops, R-symmetric homoclinic orbit and R-symmetric periodic orbit. The double R-symmetric homoclinic bifurcation (i.e., two-fold R-symmetric homoclinic bifurcation) for reversible heteroclinic loops is found, and the existence of infinitely many R-symmetric periodic orbits accumulating onto a homoclinic orbit is demonstrated. The relevant bifurcation surfaces and the existence regions are also located. This work was supported by National Natural Science Foundation of China (Grant No. 10671069)     

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 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.     

EXISTENCE OF PERIODIC SOLUTIONS TO A PERTURBED FOUR-DIMENSIONAL SYSTEM

Consider a k multiple closed orbit on an invariant surface of a four dimensional system, after a suitable perturbation, the closed orbit can generate periodic orbits and double-period orbits. Using bifurcation methods and techniques, sufficient conditions for the existence of periodic solutions to the perturbed four dimensional system are obtained, and the period-doubling bifurcations is discussed.     

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.     

Bifurcations of Rough Heteroclinic Loops with Three Saddle Points

In this paper, we study the bifurcation problems of rough heteroclinic loops connecting three saddle points for a higher-dimensional system. Under some transversal conditions and the nontwisted condition, the existence, uniqueness, and incoexistence of the 1-heteroclinic loop with three or two saddle points, 1-homoclinic orbit and 1-periodic orbit near Γ are obtained. Meanwhile, the bifurcation surfaces and existence regions are also given. Moreover, the above bifurcation results are extended to the case for heteroclinic loop with l saddle points. Received January 4, 2001, Accepted July 3, 2001.     

Codimension 2 bifurcations of double homoclinic loops

In this work, the double-homoclinic-loop bifurcations in four dimensional vector fields are investigated by setting up local coordinates near the double homoclinic loops. We get the existence, uniqueness and incoexistence of the large 1-hom and large 1-per orbit, and their corresponding existence regions are located. Furthermore, the inexistence of the large 2-hom and large 2-per orbit are also demonstrated.     

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.     

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.     

E1ementary Bifurcations of Non—Critical but N0n—Hyperbolic Invariant Tori

Consider the time-periodic peturbations of n-dimensional autonomous systems with non-hyperbolic but non-critical closed orbits in the phase space.The elementary bifurcations,such as the saddle-node,transcritical,pitchfork bifurcation to a non-hyperbolic but non-critical invariant torus of the unperturbed systems in the extended phase space(x,t),are sutdied.Some conditions which depend only on ithe original systems and can be used to determine the bifurcation structures of these problems are obtained.The theory is applied to two concrete examples.     

Bifurcations of rough heteroclinic loop with two saddle points

The bifurcation problems of rough 2-point-loop are studied for the case p11 > λ11, p21 < λ21, P11p21 <λ111λ21. where - pi1 < 0 and λi1 > 0 are the pair of principal eigenvalues of unperturbed system at saddle point pi, i = 1,2. Under the transversal and nontwisted conditions, the authors obtain some results of the existence of one 1-periodic orbit, one 1-periodic and one 1-homoclinic loop, two 1-periodic orbits and one 2-fold 1-periodic orbit. Moreover, the bifurcation surfaces and the existence regions are given, and the corresponding bifurcation graph is drawn.     

The Unfolding of Equivariant Bifurcation Problems with Two Types of State Variables in the Presence of Parameter Symmetry

The unfolding of equivariant bifurcation problems with two types of state variables under the action of group K（Г, △） is discussed by using DA-algebraic tools. One of the main results is the equivariant versal unfolding theorem.     

Bifurcation of Periodic Orbits and Chaos in Duffing Equation

Duffing equation with fifth nonlinear-restoring force, one external forcing and a phase shift is investigated, The conditions of existences for primary resonance, second-order, third-order subharmonics, morder subharmonics and chaos are given by using second-averaging method, Melnikov methods and bifurcation theory. Numerical simulations including bifurcation diagrams, bifurcation surfaces, phase portraits, not only show the consistence with the theoretical analysis, but also exhibit the new dynamical behaviors. We show the onset of chaos, chaos suddenly disappearing to period orbit, one-band and double-band chaos, period-doubling bifurcations from period 1, 2, and 3 orbits, period-windows （period-2, 3 and 5） in chaotic regions.     

EXISTENCE AND STABILITY OF PERIODIC ORBITS FOR A PERTURBED FOUR-DIMENSIONAL SYSTEM

The purpose of this paper is to study periodic orbits of a perturbed four- dimensional system.Using bifurcation methods and the integral manifold theory,sufficient conditions for the existence and stability of periodic orbits of the perturbed four-dimensional system are obtained.     

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.     

Homoclinic Bifurcations in Symmetric Unfoldings of a Singularity with Three-fold Zero Eigenvalue

In this paper we study the singularity at the origin with three-fold zero eigenvalue forsymmetric vector fields with nilpotent linear part and 3-jet C^∞-equivalent to y δ/δx zδ/δy ax^2yδ/δz with a≠0. We first obtain several subfamilies of the symmetric versal unfoldings of this singularityby using the normal form and blow-up methods under some conditions, and derive the local and global bifurcation behavior, then prove analytically the existence of the Sil‘nikov homoclinic bifurcation for some subfamilies of the symmetric versal unfoldings of this singularity, by using the generalized Mel‘nikov methods of a homoclinic orbit to a hyperbolic or non-hyperbolic equilibrium in a highdimensional space.