具有一个细焦点和一个粗焦点的二次系统极限环的个数

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).     

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.     

具有细链双曲无穷远鞍点的二次系统

本文得到：具有细链双曲无穷远鞍点和一个细焦点的二次系统至多存在一个极限环,若有细无穷远分界线环S,则其内部不存在极限环,其稳定性与它包围的奇点的稳定性相反．     

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。     

THE MAXIMAL NUMBER OF LIMIT CYCLES FOR QUADRATIC DIFFERENTIAL SYSTEM  = -y + lx~2 + xy,  = x(l + ax + by)

In this paper, we investigate the maximal number of limit cycles surrounding a first order weak focus for the quadratic differential system. And proved such a system has at most two limit cycles under some certain conditions.     

二次系统(Ⅲ)n=0一阶细焦点外围极限环的惟一性

本文证明二次系统（Ⅲ）n=0方程当其细焦点的一阶细焦点量（w1)和三阶细焦点量（w3)的符号异号时,该细焦点外围至多有一个极限环;当ω1与ω3符号相同时,该细焦点外围可以出现二个极限环,并举出例子。ω     

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。     

具有二阶细焦点的二次系统极限环的唯一性

人们猜想,平面二次系统二阶细焦点外围至多存在一个极限环,但迄今未能证实.文[1,2]在某些参数取特定值之下证明了这一猜想.最近文[5]在具有零特征根奇点之下也证明了这一猜想.本文则在较一般的情况下证明了这一猜想,并使文[5]的结果成为本文的特例.此外,本文还给出了若干有环无环的条件.     

Ten limit cycles around a center-type singular point in a 3-d quadratic system with quadratic perturbation

In this paper, we show that perturbing a simple 3-d quadratic system with a center-type singular point can yield at least 10 small-amplitude limit cycles around a singular point. This result improves the 7 limit cycles obtained recently in a simple 3-d quadratic system around a Hopf singular point. Compared with Bautin’s result for quadratic planar vector fields, which can only have 3 small-amplitude limit cycles around an elementary center or focus, this result of 10 limit cycles is surprisingly high. The theory and methodology developed in this paper can be used to consider bifurcation of limit cycles in higher-dimensional systems.     

二次系统二阶细焦点外围极限环的唯一性

本文证明了平面二次系统二阶细焦点外围至多存在一个极限环这一猜想,并证明了若第二、第三焦点量的乘积大于零,则在二阶细焦点外围不存在极限环．     

Quadratic systems with a weak focus and a strong focus

It is proved that the quadratic system with a weak focus and a strong focus has a unique limit cycle around one of the two foci, if there exists simultaneously limit cycles around each of the two foci for the system.     

Invariant algebraic curves of large degree for quadratic system

In this paper we present for the first time examples of algebraic limit cycles and saddle loops of degree greater than 4 for planar quadratic systems. In particular, we give examples of algebraic limit cycles of degree 5 and 6, and algebraic saddle loops of degree 3 and 5 surrounding a strong focus. We also give an example of an invariant algebraic curve of degree 12 for which the quadratic system has no Darboux integrating factors or first integrals.     

Uniqueness of Limit Cycle for the Quadratic Systems with Weak Saddle and Focus

It is proved that the quadratic system with a weak saddle has at most one limit cycle,andthat if this system has a separatrix cycle passing through the weak saddle,then the stability of theseparatrix cycle is contrary to that of the singular point surrounded by it.     

Uniqueness of Limit Cycle for the Quadratic Systems with Weak Saddle and Focus

It is proved that the quadratic system with a weak saddle has at most one limit cycle, and that if this system has a separatrix cycle passing through the weak saddle, then the stability of the separatrix cycle is contrary to that of the singular point surrounded by it.     

一类具有不少于“3/8n~2”族极限环的平面n次系统

本文给出结果：（１）一类具有不少于［3/8n~2］个一阶细焦点的平面ｎ次系统．（２）一类具有不少于［３/8n~2］族权限环的平面ｎ次系统．     

Hopf Cyclicity of a Family of Generic Reversible Quadratic Systems with One Center

This paper is concerned with small quadratic perturbations to one parameter family of generic reversible quadratic vector fields with a simple center. The first objective is to show that this system exhibits two small amplitude limit cycles emerging from a Hopf bifurcation. The second one we prove that the system has no limit cycle around the weak focus of order two. The results may be viewed as a contribution to proving the conjecture on cyclicity proposed by Iliev (1998).     

Limit cycles of quadratic systems

In this paper, the global qualitative analysis of planar quadratic dynamical systems is established and a new geometric approach to solving Hilbert’s Sixteenth Problem in this special case of polynomial systems is suggested. Using geometric properties of four field rotation parameters of a new canonical system which is constructed in this paper, we present a proof of our earlier conjecture that the maximum number of limit cycles in a quadratic system is equal to four and their only possible distribution is (3:1) [V.A. Gaiko, Global Bifurcation Theory and Hilbert’s Sixteenth Problem, Kluwer, Boston, 2003]. Besides, applying the Wintner–Perko termination principle for multiple limit cycles to our canonical system, we prove in a different way that a quadratic system has at most three limit cycles around a singular point (focus) and give another proof of the same conjecture.     

A CLASS OF QUADRATIC DIFFERENTIAL SYSTEM WITH A THIRD ORDER WEAK FOCUS

This paper discusses the uniqueness of the limit cycle of quadratic differential system with a third order weak focus.     

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.     

BIFURCATION OF LIMIT CYCLE FOR QUADRATIC SYSTEM WITH INVARIANT PARABOLA

In a previous paper, we have proved that a planar quadratic system with invariant parabola Г has at most one limit cycle. In this paper, we use geometric characteristics to give necessary and sufficient conditions under which a PQSГ with three non-degenerate singular points can be transformed into two different definite forms. In this way, we obtain ail the bifurcations of such a system.