一类三次系统的极限环个数与奇点分支

给出二次系统I的一类相伴系统在奇点O(0,0)的焦点量公式,证明了O至多为2阶细焦点,δlmn=0时系统在O外围至多有一个极限环,从而说明了系统在细焦点外围至多有一个极限环。最后给出了各个奇点的分支情况及几何特征。     

二次微分系统(III)_n=0的极限环和分支现象

本文应用分支理论得到了二次系统（ＩＩ）ｎ＝０在Ｏ（０，０）外围极限环的存在和数目及分界线环和半稳定环分支曲线的所有可能的分支图进一步地，证明了该系统在Ｏ外围至多有三个极限环，且有以一个有限和两个无穷远鞍点或鞍结点为顶点的非单值多边环     

一类具细焦点的三次系统极限环的唯一性

继续相关文献的工作,给出与二次系统Ⅰ相伴的一类三次系统在奇点N(0,1/n)的焦点量公式,证明了系统在细焦点N外围至多有一个极限环,同时证明了当N或O为细焦点时,系统在另一个焦点外围无极限环,结合相关文献的结论,说明了具有细焦点的该系统在全平面至多有一个极限环.     

二次可积非哈密顿系统在三次扰动下的Hopf分支

研究含有中心的二次可积非哈密顿系统在三次扰动下的Hopf分支,证明了在中心附近可以出现且至多出现5个极限环.     

一类Z_2对称五次微分系统的中心条件和极限环分支

本文研究了一类Z2对称五次微分系统的中心条件和小振幅极限环分支.通过前6阶焦点量的计算,获得了原点为中心的充要条件,并证明系统从原点分支出的小振幅极限环的个数至多为6.最后通过构造后继函数,给出系统具有6个围绕原点的小振幅极限环的实例.     

一个三次等时中心在非光滑扰动下的极限环分支

本文研究了一个三次等时中心在非光滑扰动下的极限环分支问题.利用非光滑系统的一阶平均方法,获得了在任意小的分段三次多项式扰动下,从未扰动系统的周期环域中至多分支出7个极限环,而且此上界可以达到,推广了光滑扰动下的结果.     

两类一致等时系统的小振幅极限环分支

对于一类六次一致等时系统,得到了原点为中心的充要条件,并证明从细焦点至多可分支出7个小振幅极限环.对于一类五次一致等时系统,给出其具有6个小振幅极限环的具体实例.     

一类Leslie模型的定性分析

对一类Leslie模型进行定性分析，研究了其极限环的存在性，不存在性和唯一性．证明了该系统在细焦点外围至多有一个极限环，以及如果系统有奇数个极限环，则它恰有一个极限环．     

一类奇摄动系统奇异极限环的不变环面分支

对一类奇异摄动系统中由奇异极限环产生的不变环面分支进行了研究并利用不变环面的分支理论,讨论了由快系统的二重极限环和三重环分支出的不变环面的存在性.     

具三次曲线解的二次系统至多有一个极限环

本文研究具有三次曲线解x^3-x^2-y^2=0的二次系统,证明此类二次系统最多只有一个极限环,进而证明了具有三次的曲线解的二次系统至多有一个极限环。     

Cubic Lienard Equations with Quadratic Damping (II)

Abstract   Applying Hopf bifurcation theory and qualitative theory, we show that the general cubic Lienard equations with quadratic damping have at most three limit cycles. This implies that the guess in which the system has at most two limit cycles is false. We give the sufficient conditions for the system has at most three limit cycles or two limit cycles. We present two examples with three limit cycles or two limit cycles by using numerical simulation. Supported by the National Natural Science Foundation of China and National Key Basic Research Special Found (No. G1998020307).     

LIMIT CYCLE PROBLEM OF QUADRATIC SYSTEM OF TYPE （ Ⅲ ）m=0, （ Ⅲ ）

To continue the discussion in （Ⅰ ） and （ Ⅱ ）,and finish the study of the limit cycle problem for quadratic system （ Ⅲ ）m=0 in this paper. Since there is at most one limit cycle that may be created from critical point O by Hopf bifurcation,the number of limit cycles depends on the different situations of separatrix cycle to be formed around O. If it is a homoclinic cycle passing through saddle S1 on 1 ＋ax-y = 0,which has the same stability with the limit cycle created by Hopf bifurcation,then the uniqueness of limit cycles in such cases can be proved. If it is a homoclinic cycle passing through saddle N on x= 0,which has the different stability from the limit cycle created by Hopf bifurcation,then it will be a case of two limit cycles. For the case when the separatrix cycle is a heteroclinic cycle passing through two saddles at infinity,the discussion of the paper shows that the number of limit cycles will change from one to two depending on the different values of parameters of system.     

Bifurcations of limit cycles in a Z 4-equivariant quintic planar vector field

In this paper, a Z4-equivariant quintic planar vector field is studied. The Hopf bifurcation method and polycycle bifurcation method are combined to study the limit cycles bifurcated from the compounded cycle with 4 hyperbolic saddle points. It is found that this special quintic planar polynomial system has at least four large limit cycles which surround all singular points. By applying the double homoclinic loops bifurcation method and Hopf bifurcation method, we conclude that 28 limit cycles with two different configurations exist in this special planar polynomial system. The results acquired in this paper are useful for studying the weakened 16th Hilbert＇s Problem.     

Bifurcation of limit cycles in small perturbations of a class of hyper-elliptic Hamiltonian systems of degree 5 with a cusp

This paper deals with small perturbations of a class of hyper-elliptic Hamiltonian system, which is a Li é nard system of the form \(\dot{x}=y,\)  \(\;\dot{y}=Q_1(x)+\varepsilon yQ_2(x)\) with \(Q_1\) and \(Q_2\) polynomials of degree 4 and 3, respectively. It is shown that this system can undergo degenerated Hopf bifurcation and Poincar é bifurcation, which emerge at most three limit cycles for \(\varepsilon\) sufficiently small.     

Bifurcation analysis of a Leslie-type predator–prey system with simplified Holling type IV functional response and strong Allee effect on prey

In this paper, a Leslie-type predator–prey system with simplified Holling type IV functional response and strong Allee effect on prey is proposed. The dissipativity of the system and the existence of all possible equilibria are investigated. The investigation emphasizes the exploring of bifurcation. It is shown that the system exists several non-hyperbolic positive equilibria, such as a weak focus of multiplicities one and two, (degenerate) saddle–nodes and Bogdanov–Takens singularities (cusp case) of codimensions 2 and 3. At these equilibria, it is proved that the system undergoes various kinds of bifurcations, such as saddle–node bifurcation, Hopf bifurcation, degenerate Hopf bifurcation and Bogdanov–Takens bifurcation of codimensions 2 and 3. With the parameters selected properly, there exhibits a limit cycle, a homoclinic loop, two limit cycles, a semistable limit cycle, or the simultaneous occurrence of a homoclinic loop and a limit cycle in the system. Moreover, it is also proved that the system has a cusp of codimension at least 4. Hence, there may exist three limit cycles generated from Hopf bifurcation of codimension 3. Numerical simulations are done to support the theoretical results.     

On the number of limit cycles in small perturbations of a class of hyper-elliptic Hamiltonian systems with one nilpotent saddle

In this paper, we make a complete study on small perturbations of Hamiltonian vector field with a hyper-elliptic Hamiltonian of degree five, which is a Liénard system of the form x^′=y, y^′=Q₁(x)+εyQ₂(x) with Q₁ and Q₂ polynomials of degree respectively 4 and 3. It is shown that this system can undergo degenerated Hopf bifurcation and Poincaré bifurcation, which emerges at most three limit cycles in the plane for sufficiently small positive ε. And the limit cycles can encompass only an equilibrium inside, i.e. the configuration (3,0) of limit cycles can appear for some values of parameters, where (3,0) stands for three limit cycles surrounding an equilibrium and no limit cycles surrounding two equilibria.     

Two limit cycles in a Leslie–Gower predator–prey model with additive Allee effect

In this work, a bidimensional continuous-time differential equations system is analyzed which is derived from Leslie type predator–prey schemes by considering a nonmonotonic functional response and Allee effect on population prey. For ecological reason, we describe the bifurcation diagram of limit cycles that appear only at the first quadrant in the system obtained. We also show that under certain conditions over the parameters, the system allows the existence of a stable limit cycle surrounding an unstable limit cycle generated by Hopf bifurcation. Furthermore, we give conditions over the parameters such that the model allows long-term extinction or survival of both populations.     

The planar discontinuous piecewise linear refracting systems have at most one limit cycle

In this paper we investigate the limit cycles of planar piecewise linear differential systems with two zones separated by a straight line. It is well known that when these systems are continuous they can exhibit at most one limit cycle, while when they are discontinuous the question about maximum number of limit cycles that they can exhibit is still open. For these last systems there are examples exhibiting three limit cycles.The aim of this paper is to study the number of limit cycles for a special kind of planar discontinuous piecewise linear differential systems with two zones separated by a straight line which are known as refracting systems. First we obtain the existence and uniqueness of limit cycles for refracting systems of focus-node type. Second we prove that refracting systems of focus–focus type have at most one limit cycle, thus we give a positive answer to a conjecture on the uniqueness of limit cycle stated by Freire, Ponce and Torres in Freire et al. (2013). These two results complete the proof that any refracting system has at most one limit cycle.