关于一类平面n+2次系统极限环唯一性的一个注记

本文继续完善文[1]和[2]的工作,利用广义Lienard方程和张芷芬唯一性定 理证明了,当n≥3时一类n+2次生化反应系统极限环的唯一性.至此,该系统极 限环唯一性问题得到完整解决.     

平面2n＋1次微分系统的极限环

本研究一类2n 1次微分方程的极限环，得到了系统极限环存在与不存在的若干充分条件．     

Liénard方程在有限区间上极限环的存在唯n性定理

Liénard方程是工程技术中常见的一个重要方程,因此引起数学工作者的重视。1946年H.J.Eckweiler提出方程有无限多个环的猜想。后来H.S.Hochstadt与B.Stephon,R.N.D''Heedene等人在μ的特定限制下证明上述方程在|x|<(n+1)π中至少存在n个极限环。1980年张芷芬教授在文献[1]中证明了对任何非零常数μ,上述方程在|x|<(n+1)π中恰有n个极限环,使此项工作大进一步。本文提出一类极为广泛的条件保证在一定区域内Liénard方程恰好存在n个极限环,并以过去的各条件作为特例。     

一类具有n~2/2(n为偶数)或n~2+1/2(n为奇数)个细焦点的(E_n)系统

通过对一类特殊的平面n次多项式微分系统进行参数小扰动,得到了一类具有(n~2)/2(n为偶数)或(n~2+1)/2(n为奇数)个2阶细焦点的(E_n)系统.进一步证明了该系统具有(n~2)/2(n为偶数)或(n~2+1)/2(n为奇数)族极限环,得到了S(n)≥(n~2)/2(n为偶数)或S(n)(n~2+1)/2(n为奇数),改进了已有文献的结果.     

一类平面多项式系统极限环的存在唯一性

研究一类平面2n 1次多项式微分系统的极限环问题,利用Hopf分枝理论得到了该系统极限环存在性与稳定性的若干充分条件,利用Cherkas和Zheilevych的唯一性定理得到了极限环唯一性的若干充分条件.     

一类E13系统极限环的惟一性

研究一类E13系统x=y,y=-x+δy+nx2+mxy+ly2+bxy2,求出奇点O的焦点量W0=δ,W1=m(n+l),W2=-mnb.证明了W0=W1=W2=0时O为中心.其次证明了W0=0,W1W2≥0时系统无极限环;W0=0,W1W2＜0时系统至多有一个极限环.最后讨论了n=0,b＞0的情况.证明了存在δ0,0＜δ0≤-l/m,当0＜δ＜δ0时系统存在惟一极限环,δ=δ0时系统存在无穷远分界线环,δ≤0或δ＞δ0时系统无闭轨与奇闭轨.     

一类和式极限问题的初等解法及推广

在高等数学学习中 ,我们求和式极限 :limn→∞ Σni=1fi( n)的途径大致有这么几种 :( 1 )先求和 :Σni=1fi( n) ,再求极限 ;( 2 )利用夹逼准则 ;( 3 )利用定积分的定义 ,把和式极限表示成定积分 ,通过计算定积分 ,求得和式的极限 ;( 4)综合运用 ( 1 )、( 2 )、( 3 )求出和式的极限。现在 ,我们考虑如下一类和式的极限问题 :例 1　求 limn→∞sin πnn+1 +sin2πnn+12+… +sinπn+1n;例 2　求 limn→∞cosπ2 n2 n+12+cos2π2 n2 n+14+… +cosπ22 n+12 n;例 3　求 limn→∞sin πnn+1n+sin2πnn+1n2+… +sinπn+1nn.当然 ,与此类似的题目 ,…     

一类二维系统的极限环的个数

本文利用Dulac函数方法讨论一类二维系统在环城上的包围多个奇点的极限环的唯一性及在n连城上极限环的唯n-1性，并给出了两个多项式的例子，讨论了极限环的唯一性和唯二性．     

平面2n+1次系统极限环的唯一性

讨论了微分方程组 dx/dt=-y(1 -ax2 n) +bx-cx2 n+ 1,dy/dt=x(1 -ax2 n) ,并且给出了其极限环存在唯一的条件 .     

一类具有二虚不变直线的三次系统的极限环

研究一类具有二虚不变直线的三次系统:X′=y(1+X2),y′=-x+δy+nx2+mxy+ly2+bxy2,分析奇点的性态并求出奇点O的焦点量w0=δ,w1=m(n+l),w2=-mn(b-1).证明了w0=w1=w2=0时O为中心,并证明了w0=0,w1w2≥0时系统无极限环;w0=0,w1w2<0时系统至多有一个极限环.     

一类可积非哈密顿系统的极限环个数的上界

In this paper, we consider the perturbations of two non-Hamiltonian integrable systems(1.3)μ, (4.1)μ. For the former,it is proved that the system under the polynomial perturbations hasat most f-n/2] limit cycles in the finite plane and the upper bound is sharp. The proof relies on acareful analysis of a related Abelian integral. For the latter, we obtain an estimate number of isolatedzeros of the corresponding Abelian integral.     

A kind of bifurcation of limit cycle from a nilpotent critical point

In this paper, an interesting and new bifurcation phenomenon that limit cycles could be bifurcated from nilpotent node (focus) by changing its stability is investigated. It is different from lowing its multiplicity in order to get limit cycles. We prove that $n^2+n-1$ limit cycles could be bifurcated by this way for $2n+1$ degree systems. Moreover, this upper bound could be reached. At last, we give two examples to show that $N(3)=1$ and $N(5)=5$ respectively. Here, $N(n)$ denotes the number of small-amplitude limit cycles around a nilpotent node (focus) with $n$ being the degree of polynomials in the vector field.     

Bifurcation of limit cycles from the global center of a class of integrable non-Hamilton systems

In this paper, we consider the bifurcation of limit cycles for system $\dot{x}=-y(x^2+a^2)^m,~\dot{y}=x(x^2+a^2)^m$ under perturbations of polynomials with degree n, where $a\neq0$, $m\in \mathbb{N}$. By using the averaging method of first order, we bound the number of limit cycles that can bifurcate from periodic orbits of the center of the unperturbed system. Particularly, if $m=2, n=5$, the sharp bound is 5.     

Linear estimate of the number of limit cycles for a class of non-linear systems

A dynamic system has a finite number of limit cycles. However, finding the upper bound of the number of limit cycles is an open problem for general non-linear dynamical systems. In this paper, we investigated a class of non-linear systems under perturbations. We proved that the upper bound of the number of zeros of the related elliptic integrals of the given system is 7n + 5 including multiple zeros, which also gives the upper bound of the number of limit cycles for the given system.     

Bifurcation of limit cycles of the nongeneric quadratic reversible system with discontinuous perturbations

By using the Picard-Fuchs equation and the property of the Chebyshev space to the discontinuous differential system, we obtain an upper bound of the number of limit cycles for the nongeneric quadratic reversible system when it is perturbed inside all discontinuous polynomials with degree n.     

On the cyclicity of weight-homogeneous centers

Let W be a weight-homogeneous planar polynomial differential system with a center. We find an upper bound of the number of limit cycles which bifurcate from the period annulus of W under a generic polynomial perturbation. We apply this result to a particular family of planar polynomial systems having a nilpotent center without meromorphic first integral.     

Abelian integrals and limit cycles

The paper deals with generic perturbations from a Hamiltonian planar vector field and more precisely with the number and bifurcation pattern of the limit cycles. In this paper we show that near a 2-saddle cycle, the number of limit cycles produced in unfoldings with one unbroken connection, can exceed the number of zeros of the related Abelian integral, even if the latter represents a stable elementary catastrophe. We however also show that in general, finite codimension of the Abelian integral leads to a finite upper bound on the local cyclicity. In the treatment, we introduce the notion of simple asymptotic scale deformation.     

Poincar{\'e} Bifurcation from an Elliptic Hamiltonian of Degree Four with Two-saddle Cycle

In this paper, we consider Poincar{\''e} bifurcation from an elliptic Hamiltonian of degree four with two-saddle cycle. Based on the Chebyshev criterion, not only one case in the Li{\''e}nard equations of type $(3, 2)$ is discussed again in a different way from the previous ones, but also its two extended cases are investigated, where the perturbations are given respectively by adding $\varepsilon y(d_0 + d_2 v^{2n})\frac{\partial }{{\partial y}}$ with $ n\in \mathbb{N^+}$ and $\varepsilon y(d_0 + d_4 {v^4}+ d_2 v^{2n+4})\frac{\partial }{{\partial y}}$ with $n=-1$ or $ n\in \mathbb{N^+}$, for small $\varepsilon > 0$. For the above cases, we obtain all the sharp upper bound of the number of zeros for Abelian integrals, from which the existence of limit cycles at most via the first-order Melnikov functions is determined. Finally, one example of double limit cycles for the latter case is given.     

Estimating the number of limit cycles in polynomials systems

We describe a method based on algorithms of computational algebra for obtaining an upper bound for the number of limit cycles bifurcating from a center or a focus of polynomial vector field. We apply it to a cubic system depending on six parameters and prove that in the generic case at most six limit cycles can bifurcate from any center or focus at the origin of the system.     

Quadratic perturbations of quadratic codimension-four centers

We study the stratum in the set of all quadratic differential systems , with a center, known as the codimension-four case Q₄. It has a center and a node and a rational first integral. The limit cycles under small quadratic perturbations in the system are determined by the zeros of the first Poincaré-Pontryagin-Melnikov integral I. We show that the orbits of the unperturbed system are elliptic curves, and I is a complete elliptic integral. Then using Picard-Fuchs equations and the Petrov's method (based on the argument principle), we set an upper bound of eight for the number of limit cycles produced from the period annulus around the center.