共查询到19条相似文献,搜索用时 150 毫秒
1.
Codimension 3 nonresonant bifurcations of homoclinic orbits with two inclination flips 总被引:5,自引:0,他引:5
SHUI Shuliang & ZHU Deming College of Mathematics Physics Zhejiang Normal University Jinhua China Department of Mathematics East China Normal University Shanghai China 《中国科学A辑(英文版)》2005,48(2):248-260
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. 相似文献
2.
《Annals of Differential Equations》2008,(1):52-64
The homoclinic bifurcations under resonant conditions are considered in the ho- moclinic manifold consisting of a series of homoclinic orbits for the fourth-dimensional system.The existence,coexistence and uniqueness of 1-homoclinic orbit,1-periodic orbit and 2-fold 1-periodic orbit are obtained under resonant condition,the correspon- ding bifurcation surfaces and existing regions are also given. 相似文献
3.
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. 相似文献
4.
Bifurcations of rough heteroclinic loop with two saddle points 总被引:7,自引:0,他引:7
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. 相似文献
5.
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. 相似文献
6.
The homoclinic bifurcations in four dimensional vector fields are investigated by setting up a local coordinates near the homoclinic orbit. This homoclinic orbit is non-principal in the meanings that its positive semi-orbit takes orbit flip and its unstable foliation takes inclination flip. The existence, nonexistence, uniqueness and coexistence of the 1-homoclinic orbit and the 1-periodic orbit are studied. The existence of the twofold periodic orbit and three-fold periodic orbit are also obtained. 相似文献
7.
Zai-hong WANG School of Mathematical Sciences Capital Normal University Beijing China 《中国科学A辑(英文版)》2007,50(8):1205-1216
In this paper, we deal with the existence of unbounded orbits of the mapping {θ1 = θ 2nπ 1/ρμ(θ) o(ρ-1),ρ1=ρ c-μ′(θ) o(1), ρ→∞,where n is a positive integer, c is a constant and μ(θ) is a 2π-periodic function. We prove that if c > 0 and μ(θ) ≠ 0, θ∈ [0, 2π], then every orbit of the given mapping goes to infinity in the future for ρ large enough; if c < 0 and μ(θ) ≠ 0, θ∈ [0, 2π], then every orbit of the given mapping goes to infinity in the past for ρ large enough. By using this result, we prove that the equation x″ f(x)x′ ax -bx- φ(x) =p(t) has unbounded solutions provided that a, b satisfy 1/√a 1/√b = 2/n and F(x)(= ∫x0 f(s)ds),and φ(x) satisfies some limit conditions. At the same time, we obtain the existence of 2π-periodic solutions of this equation. 相似文献
8.
Qianqian Ji Weipeng Zhang Qiuying Lu Xiaodong Li 《Journal of Nonlinear Modeling and Analysis》2020,2(1):25-44
In this work, bifurcation analysis near double homoclinic loops with Ws inclination ?ip of Γ1 and nonresonant eigenvalues is presented in a four-dimensional system. We establish a Poincar´e map by constructing local active coordinates approach in some tubular neighborhood of unperturbed double homoclinic loops. Through studying the bifurcation equations, we obtain the condition that the original double homoclinic loops are persistent, and get the existence or the nonexistence regions of the large 1-homoclinic orbit and the large 1-periodic orbit. At last, an analytical example is given to illustrate our main results. 相似文献
9.
The authors study the bifurcation problems of rough heteroclinic loop connecting three saddle points for the case β1 > 1, β2 > 1, β3 < 1 and β1β2β3 < 1. The existence, number, coexistence and incoexistence of 2-point-loop, 1-homoclinic orbit and 1-periodic orbit are studied. Meanwhile, the bifurcation surfaces and existence regions are given. 相似文献
10.
Li-qin ZHAO Department of Mathematics Beijing Normal University Beijing China 《中国科学A辑(英文版)》2007,50(3):334-344
In this paper, we investigate the number and the distribution of the limit cycles bifurcated from a kind of degenerate planar polycycles through three singular points: a saddle-node P0, a fine saddle P1 with finite order m∈N, a contractive (attracting) saddle P2 with the hyperbolicity ratio q2(0)■Q. The connection between P0 and P1 is of hh-type and the connection between P0 and P2 is of hp-type. It is assumed that the connections between P0 to P2 and P0 to P1 keep unbroken. We obtain that the cyclicity of this polycycle is not more than 3m 1, which is linearly dependent on the order of the resonant saddle P1 We also show that the cyclicity is not more than m 3 if q2(0)>m, and that the nearer q2(0)is close to 1, the more the limit cycles are bifurcated. 相似文献
11.
Bifurcations of heteroclinic loops 总被引:14,自引:0,他引:14
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. 相似文献
12.
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 相似文献
13.
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. 相似文献
14.
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 相似文献
15.
Li Qin ZHAO 《数学学报(英文版)》2005,21(3):507-516
This paper deals with the cyclicity of a kind of degenerate planar polycycles through a saddle-node and two hyperbolic saddles, where the hyperbolicity ratio of the saddle (which connects the saddle-node with hp-connection) is equal to 1 and that of the other saddle is irrational. It is obtained that the cyclicity of this kind of polycycle is no more than 5 if the hp-connection keeps unbroken under the C^∞ perturbations. 相似文献
16.
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. 相似文献
17.
André Vanderbauwhede Bernold Fiedler 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》1992,43(2):292-318
We show that in conservative systems each non-degenerate homoclinic orbit asymptotic to a hyperbolic equilibrium possesses an associated family of periodic orbits. The family is parametrized by the period, and the periodic orbits accumulate on the homoclinic orbit as the period tends to infinity. A similar result holds for symmetric homoclinic orbits in reversible systems. Our results extend earlier work by Devaney and Henrard, and provide a positive answer to a conjecture of Strömgren. We present a unified approach to both the conservative and the reversible case, based on a technique introduced recently by X.-B. Lin.Dedicated to Prof. Klaus Kirchgässner on the occasion of his sixtieth birthday 相似文献
18.
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. 相似文献
19.
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) 相似文献