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。     

同宿、异宿环分支出极限环的唯一性

本文在一简洁条件下证明任一含细鞍点的孤立同宿环及对称的双同宿环和两点异宿环至多产生一个极限环。     

一类两点异宿环的扰动分支

设Lo为含两个双曲鞍点Si(i=1,2)的异宿环,Si的双曲比记为Tio(i=1,2),Mourtada在1994年证明如果未扰动系统与被扰系统均为C∞的,则当r10r20≠1时,Lo至多产生两个极限环.本文对C3系统证明了这一结论.     

一类两点异宿环的扰动分支

设 Ｌ０为含两个双曲鞍点 Ｓｉ（ｉ＝ １,２）的异宿环, Ｓｉ的双曲比记为ｒｉ０（ｉ＝ １,２）, Ｍｏｕｒｔａｄａ在１９９４年证明如果未扰动系统与被扰系统均为 Ｃ∞的,则当ｒ１０,２０≠１时, Ｌ０至多产生两个极限环．本文对 Ｃ３系统证明了这一结论．     

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.     

三点粗异宿环分支

本文研究高维系统连接三个鞍点的粗异宿环的分支问题.在一些横截性条件和非扭曲条件下,获得了Γ附近的1-异宿三点环, 1-异宿两点环、 1-同宿环和1-周期轨的存在性,唯一性和不共存性.同时给出了分支曲面和存在域.上述结果被进一步推广到连接l个鞍点的异宿环的情况,其中l≥2.     

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.     

高维系统极限环的分支

本文利用分支函数方法,讨论高维系统极限环的局部分支及一类全局分支,推广了 Hopf 分支定理.     

奇闭轨分支出极限环的唯一性(Ⅰ)

本文得到了含一个细鞍点的奇闭轨产生极限环的唯一性定理,给出了作为闭轨族边界的奇闭轨产生唯一极限环的简洁判据.     

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.     

Bifurcation of limit cycles from a heteroclinic loop with a cusp

In this article, we study the expansion of the first Melnikov function of a near-Hamiltonian system near a heteroclinic loop with a cusp and a saddle or two cusps, obtaining formulas to compute the first coefficients of the expansion. Then we use the results to study the problem of limit cycle bifurcation for two polynomial systems.     

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.     

几类非线性微分系统极限环的存在性和唯一性

给出系统(E1),(E2)和(E3)等非线性微分系统存在闭轨的一些新的判定条件,推广了非线性微分系统极限环的存在性和唯一性许多这方面研究的结果,并大大改进了它们的某些条件．在这个基础上,还给出了系统(E1)和(E2)恰有一个极限环的一组充分条件．     

BIFURCATIONS OF LIMIT CYCLES FROM A HETEROCLINIC CYCLE OF HAMILTONIAN SYSTEMS

§１．ＮｏｒｍａｌＦｏｒｍｓｏｆＤｉｓｐｌａｃｅｍｅｎｔＦｕｎｃｔｉｏｎｓＣｏｎｓｉｄｅｒａｐｌａｎａｒＣ∞ｓｙｓｔｅｍｏｆｔｈｅｆｏｒｍｘ＝ｆ（ｘ）＋λｆ０（ｘ，δ，λ）≡ｆ（ｘ，δ，λ），（１．１）ｗｈｅｒｅｘ∈Ｒ２，λ∈Ｒ，δ∈Ｒｍ，ａｎｄｔｒ...     

Global Bifurcation of a Perturbed Double-Homoclinic Loop

This paper deals with a kind of fourth degree systems with perturbations. By using the method of multi-parameter perturbation theory and qualitative analysis, it is proved that the system can have six limit cycles.     

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.     

Limit cycle bifurcations of some Liénard systems

In this paper we first give some general theorems on the limit cycle bifurcation for near-Hamiltonian systems near a double homoclinic loop or a center as a preliminary. Then we use these theorems to study some polynomial Liénard systems with perturbations and give new lower bounds for the maximal number of limit cycles of these systems.     

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.