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1.
We construct a planar cubic system and demonstrate that it has at least 13 limit cycles. The construction is essentially based on counting the number of zeros of some Abelian integrals.  相似文献   

2.
The finite generators of Abelian integral are obtained, where Γh is a family of closed ovals defined by H(x,y)=x2+y2+ax4+bx2y2+cy4=h, hΣ, ac(4acb2)≠0, Σ=(0,h1) is the open interval on which Γh is defined, f(x,y), g(x,y) are real polynomials in x and y with degree 2n+1 (n?2). And an upper bound of the number of zeros of Abelian integral I(h) is given by its algebraic structure for a special case a>0, b=0, c=1.  相似文献   

3.
A concrete numerical example of Z6-equivariant planar perturbed Hamiltonian polynomial vector fields of degree 5 having at least 24 limit cycles and the configurations of compound eyes are given by using the bifurcation theory of planar dynamical systems and the method of detection functions. There is reason to conjecture that the Hilbert number H(2k + 1) ≥ (2k + 1)2 - 1 for the perturbed Hamiltonian systems.  相似文献   

4.
This paper concerns with the number of limit cycles from an asymmetric Hamiltonian of degree three under cubic perturbation. Eleven limit cycles are found and three different distributions are given by using the methods of bifurcation theory and qualitative analysis, two of which are new.  相似文献   

5.
研究一类五次系统无穷远点的中心、拟等时中心条件与极限环分支问题.首先通过同胚变换将系统无穷远点转化成原点,然后求出该原点的前8个奇点量,从而导出无穷远点成为中心和最高阶细焦点的条件,在此基础上给出了五次多项式系统在无穷远点分支出8个极限环的实例.同时通过一种最新算法求出无穷远点为中心时的周期常数,得到了拟等时中心的必要条件,并利用一些有效途径一一证明了条件的充分性.  相似文献   

6.
In this paper, by using qualitative analysis, we investigate the number of limit cycles of perturbed cubic Hamiltonian system with perturbation in the form of (2n+2m) or (2n+2m+1)th degree polynomials . We show that the perturbed systems has at most (n+m) limit cycles, and has at most n limit cycles if m=1. If m=1, n=1 and m=1, n=2, the general conditions for the number of existing limit cycles and the stability of the limit cycles will be established, respectively. Such conditions depend on the coefficients of the perturbed terms. In order to illustrate our results, two numerical examples on the location and stability of the limit cycles are given.  相似文献   

7.
This paper concerns with limit cycles through Hopf and homoclinic bifurcations for near-Hamiltonian systems. By using the coefficients appeared in Melnikov functions at the centers and homoclinic loops, some sufficient conditions are obtained to find limit cycles.  相似文献   

8.
Hopf bifurcation which produces oscillations is a very important phenomena in the theory and application of dynamical systems. Almost all works available about Hopf bifurcations are related to a non-degenerate focus or center. For the case of a degenerate focus or center, the study of the bifurcations becomes challenge. In this paper, we consider the bifurcation of limit cycles for a quartic near-Hamiltonian system by perturbing a nilpotent center. We take coefficients as parameters, then we can get six limit cycles.  相似文献   

9.
On the number of limit cycles in double homoclinic bifurcations   总被引:7,自引:0,他引:7  
LetL be a double homoclinic loop of a Hamiltonian system on the plane. We obtain a condition under whichL generates at most two large limit cycles by perturbations. We also give conditions for the existence of at most five or six limit cycles which appear nearL under perturbations.  相似文献   

10.
In this paper, bifurcations of limit cycles at three fine focuses for a class of Z 2-equivariant non-analytic cubic planar differential systems are studied. By a transformation, we first transform nonanalytic systems into analytic systems. Then sufficient and necessary conditions for critical points of the systems being centers are obtained. The fact that there exist 12 small amplitude limit cycles created from the critical points is also proved. Henceforth we give a lower bound of cyclicity of Z 2-equivariant non-analytic cubic differential systems.  相似文献   

11.
In this paper we study some equivariant systems on the plane. We first give some criteria for the outer or inner stability of compound cycles of these systems. Then we investigate the number of limit cycles which appear near a compound cycle of a Hamiltonian equivariant system under equivariant perturbations. In the last part of the paper we present an application of our general theory to show that a Z3 equivariant system can have 13 limit cycles.  相似文献   

12.
The center conditions and bifurcation of limit cycles for a class of fifth degree systems are investigated. Two recursive formulas to compute singular quantities at infinity and at the origin are given. The first nine singular point quantities at infinity and first seven singular point quantities at the origin for the system are given in order to get center conditions and study bifurcation of limit cycles. Two fifth degree systems are constructed. One allows the appearance of eight limit cycles in the neighborhood of infinity,which is the first example that a polynomial differential system bifurcates eight limit cycles at infinity. The other perturbs six limit cycles at the origin.  相似文献   

13.
Abstract. It is proved that the quadratic system with a weak focus and a strong focus has atmost one limit cycle around the strong focus, and as the weak focus is a 2nd -order (or 3rd-order ) weak focus the quadratic system has at most two (one) limit cycles which have (1,1)-distribution ((0,1)-distribution).  相似文献   

14.
证明了具有三次曲线解y=-x(x-1)2 4/24的Kolmogorov三次系统是有存在极限环可能的.  相似文献   

15.
16.
This paper deals with Liénard equations of the form , , with P and Q polynomials of degree 5 and 4 respectively. Attention goes to perturbations of the Hamiltonian vector fields with an elliptic Hamiltonian of degree six, exhibiting a double figure eight loop. The number of limit cycles and their distributions are given by using the methods of bifurcation theory and qualitative analysis.  相似文献   

17.
This paper studies a class of quartic system which is more general and realistic than the quartic accompanying system. x'=-y+ex+lx^2+mxy+ny^2,y'=x(1-Ay)(1+Cy^2),(*) where C 〉 0. Sufficient conditions are obtained for the uniqueness of limit cycle of system (*) and some more in-depth conclusion such as Hopf bifurcation.  相似文献   

18.
弱化希尔伯特第16问题及其研究现状   总被引:2,自引:0,他引:2  
V.I.Arnold多次提出如下问题:对于给定的自然数n≥2,所有n次多项式1-形式,沿一切可能的m≥3次闭代数曲线族的阿贝尔积分的孤立零点的最大个数Z(m,n)=?由Poincare-Pontryagin定理可知,当阿贝尔积分不恒为零时,A(n)=Z(n+1,n)给出n次Hamilton系统在n次多项式扰动下从原有周期环域分支出极限环的最大个数,因此Arnold把这个问题称为弱化的希尔伯特第16问题.30多年来,对此问题的研究取得了一定进展,也遇到了很大困难.本文拟对这个问题和相关研究工作做一个粗浅的介绍.  相似文献   

19.
This paper investigates both homoclinic bifurcation and Hopf bifurcation which occur concurrently in a class of planar perturbed discontinuous systems of Filippov type. Firstly, based on a geometrical interpretation and a new analysis of the so-called successive function, sufficient conditions are proposed for the existence and stability of homoclinic orbit of unperturbed systems. Then, with the discussion about Poincaré map, bifurcation analyses of homoclinic orbit and parabolic–parabolic (PP) type pseudo-focus are presented. It is shown that two limit cycles can appear from the two different kinds of bifurcation in planar Filippov systems.  相似文献   

20.
In this paper we will prove that limit cycles for the quadratic differential system (III)l=n=0 in Chinese classification are concentratedly distributed, and that the maximum number of limit cycles around O (0,0) is at least two. This project is supported by the Natural Science Foundation of Educational Committee of Jiangsu Province.  相似文献   

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