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 Equa...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.展开更多
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 ha...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).展开更多
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.
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.
As a continuation of,the author studies the limit cycle bifurcation around the focus S_(1)other than O(0,0)for the system(1)asδvaries.A conjecture on the mon-existence of limit cycles around S_(1),and another one on ...As a continuation of,the author studies the limit cycle bifurcation around the focus S_(1)other than O(0,0)for the system(1)asδvaries.A conjecture on the mon-existence of limit cycles around S_(1),and another one on the non-coexistence of limit cycles ariund both O and S_(1)are given,together with some numerical examples.展开更多
A class of quartic and quintic differential system is introduced. We show that under suitable assumptions, one, two or four algebraic limit cycles can occur. These limit cycles are analytically given.
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 quadrat...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.展开更多
Qualitative properties of critical points, integral lines and limit cycles are studied. Interesting relations between quantities characterizing local properties and those characterizing global properties are obtained.
In this paper we study the variation of limit cycles around different foci when a coefficient in the equation of the quadratic differential system varies.
文摘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.
文摘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).
基金the National Natural Science Foundation of China.
文摘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.
基金The project is partially supported by the National Natural Foundation of China with the grant number 10901013.
文摘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.
文摘As a continuation of,the author studies the limit cycle bifurcation around the focus S_(1)other than O(0,0)for the system(1)asδvaries.A conjecture on the mon-existence of limit cycles around S_(1),and another one on the non-coexistence of limit cycles ariund both O and S_(1)are given,together with some numerical examples.
文摘A class of quartic and quintic differential system is introduced. We show that under suitable assumptions, one, two or four algebraic limit cycles can occur. These limit cycles are analytically given.
文摘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.
文摘Qualitative properties of critical points, integral lines and limit cycles are studied. Interesting relations between quantities characterizing local properties and those characterizing global properties are obtained.
文摘In this paper we study the variation of limit cycles around different foci when a coefficient in the equation of the quadratic differential system varies.
