ON NUMBER OF LIMIT CYCLES FOR THE 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 at most 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).

On the(1,3)Distributions of Limit Cycles of Plane Quadratic Systems

In this paper, (a) we rerise Theorem 2 of Ref [1] omit the condition V_7>0 .(b) we discuss the relative positions of six curves M(s￣2, r)=0, J( s￣2, r)=0, L(s￣2,r)=0, T(s￣2,r)=0, Under the condition of the (1.3) distri-butions of limit cycles, we expand the variable regions of parameters ( s , r) and clearly. show them in figure, (c) we study the (1, 3) distributions of limit cycles of one kind quadratic systems with two singular points at the infinite: and (d) we give a generalmethod to discuss the ( 1 ,3) distibutions`of limit cycles of system (1.1) whatever there isone, two or three singular points at the infinite.

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.

Limit cycle problem for quadratic differential system x = -y + lx^2 + mxy, y =x(1 +ax +by)

The maximal number of limit cycles for a particular type Ⅲ system x = -y + lx2 + mxy, y =x(1 + ax + by) is studied and some errors that appeared in the paper by Suo Mingxia and Yue Xiting (Annals of Differential Equations, 2003,19(3):397-401) are corrected. By translating the system to be considered into the Lienard type and by using some related properties, we obtain several theorems with suitable conditions coefficients of the system, under which we prove that the system has at most two limit cycles. The conclusions improve the results given in Suo and Yue's paper mentioned above.

NECESSARY AND SUFFICIENT CONDITIONS OF EXISTENCE AND UNIQUENESS OF LIMIT CYCLES FOR A CLASS OF POLYNOMIAL SYSTEM

In this paper, we discuss the limit cycles of the systemdx/dt=y·[1+(A(x)]oy/dt=(-x+δy+α_1x^2+α_2xy+α_5x^2y)[1+B(x)] (1)where A(x)=sum form i=1 to n(a_ix~), B(x)=sum form j=1 to m(β_jx^j) and 1+B(x)>0. We prove that (1) possesses at most one limit cycle and give out the necessary and sufficient conditions of existence and uniqueness of limit cycles.

A METHOD PROVING THE UNIQUENESS OF THE LIMIT CYCLE OF THE QUADRATIC SYSTEM

We transform the quadratic system into the special system of Type (Ⅲ)a=0' and hence a string sufficient conditions are established to ensure that the considered system has at most one limit cycle.

HOPF BIFURCATION AND UNIQUENESS OF LIMIT CYCLE FOR A CLASS OF QUARTIC SYSTEM

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.

Uniqueness of limit cycles of quadratic system (Ⅲ) _(m=0)

IN ref.[1]of § 20,Ye Yanqian has investigated the impossibility of（2,2）distribution of lim-it cycles of quadratic systems,where the footnote 1)on p.553 gives the following conjecture:The quadratic system（Ⅲ）m=0

THE UNIQUENESS OF LIMIT CYCLE OF QUADRATIC SYSTEM(Ⅱ)_(m=0)

Without loss of generality, the quadratic system （Ⅱ）m=0 can be assumed as follows:Generally, system （1） has four singular points, focus （node） 0（0,0）, R（-1/a,y2）, saddle N（0, 1）, M（-1/a,y1）, where y1, 2=[a±（a2-4（l-aδ））1/2]/2a.

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.

UNIQUENESS AND DISTRIBUTION OF LIMIT CYCLES FOR BOUNDED QUADRATIC SYSTEM

In this paper, we first give a necessary and sufficient condition of a quadratic system with three finite critical points being bounded, and then, we use the methods and conclusions of [11] to provide some uniqueness theorems of limit cycles for bounded quadratic systems. As well, we prove that any bounded quadratic system can not have (2, 2)-distribution of limit cycles according to these uniqueness theorems.

UNIQUENESS OF LIMIT CYCLE FOR A CLASS OF QUARTIC SYSTEM ACCOMPANYING WITH QUADRATIC SYSTEM

In this paper,we consider a class of quartic system,which is more general and realistic than the quartic accompanying system. Consequently,we obtain sufficient conditions concerning the uniqueness of limit cycle as well as some other in-depth conclusions.

Existence of Limit Cycles for a Cubic Kolmogorov System with a Hyperbolic Solution

This paper is concerned with a cubic Kolmogorov system with a solution of central quadratic curve which neither contacts with the coordinate axes, nor passes through the origin. The conclusion is that such a system may possess limit cycles.

ON A CONJECTURE CONCERNING THE NON-EXISTENCE OF LIMIT CYCLES FOR QUADRATIC DIFFERENTIAL SYSTEMS

A conjecture on the non-existence of limit cycles for the quadratic differential system (1) under conditions (2) and iv) of (3) is discussed; interesting phenomena are revealed.

The Quadratic System Having a Parabola as Its Integral Curve Has at Most One Limit Cycle

In this note, we prove that the quadratic system having a parabola as its integralcurve has at most one limit cycle, and therefore the quadratic system havingquadratic curve as its integral curve has at most one limjt cycle. Considering Ref.[1], we have solved completely the problem of the bifurcations of limit cycle forsystem (1).

STUDY OF NON-EXISTENCE OF LIMIT CYCLE AROUND A WEAK FOCUS OF ORDER TWO OR THREE FOR QUADRATIC SYSTEMS

On the basis of[2—4], we only need to consider the case of n≠0. Without loss of generality, we can assume n=1, a】0. Hence the system（1）n,0 can be written as（1）1,0

On the Limit Cycles of Quadratic Differential Systems

In this paper we give the necessary and sufficient conditions for all finite critical points of quadratic differential systems to be weak foci, and solve an open problem proposed by Yanqian Ye.

STUDY ON THE UNIQUENESS OF LIMIT CYCLE OF THE QUADRATIC SYSTEM BY USING THE QUADRATIC CURVE WITHOUT CONTACT(CONTINUED 1)

In this paper, we continue to discuss the uniqueness of limit cycle of the quadratic system by using the quadratic curve without contact and several new criteria for the uniqueness have been obtained.

BIFURCATION OF LIMIT CYCLES FOR THE QUADRATIC DIFFERENTIAL SYSTEM (Ⅲ)l = n = 0

In this paper we will prove that limit cycles for the quadratic differential system (Ⅲ)l=n=o in Chinese classification are concentratedly distributed, and that the maximum number of limit cycles around O(0,0) is at least two.

ON THE NON-EXISTENCE OF LIMIT CYCLES OF CERTAIN QUADRATIC SYSTEMS

In §1 and §3, two conjectures mentioned by Ye Yanqian are studied. In §2, by use of elementary methods the author proves some non-existence theorems of limit cycles (LC, for abbreviation) for quadratic differential systems obtained recently by H.Giacomini, J. Llibre and M. Viano.