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.
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
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.
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.
The resolution of differential games often concerns the difficult problem of two points border value （TPBV）, then ascribe linear quadratic differential game to Hamilton system. To Hamilton system, the algorithm of s...The resolution of differential games often concerns the difficult problem of two points border value （TPBV）, then ascribe linear quadratic differential game to Hamilton system. To Hamilton system, the algorithm of symplectic geometry has the merits of being able to copy the dynamic structure of Hamilton system and keep the measure of phase plane. From the viewpoint of Hamilton system, the symplectic characters of linear quadratic differential game were probed; as a try, Symplectic-Runge-Kutta algorithm was presented for the resolution of infinite horizon linear quadratic differential game. An example of numerical calculation was given, and the result can illuminate the feasibility of this method. At the same time, it embodies the fine conservation characteristics of symplectic algorithm to system energy.展开更多
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
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
In this paper we study the relation of trajectories and limit cycles between index-inverse differential systems, especially, index-inverse quadratic differential systems.
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.
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 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.
基金Project supported by the National Aeronautics Base Science Foundation of China (No.2000CB080601)the National Defence Key Pre-research Program of China during the 10th Five-Year Plan Period (No.2002BK080602)
文摘The resolution of differential games often concerns the difficult problem of two points border value （TPBV）, then ascribe linear quadratic differential game to Hamilton system. To Hamilton system, the algorithm of symplectic geometry has the merits of being able to copy the dynamic structure of Hamilton system and keep the measure of phase plane. From the viewpoint of Hamilton system, the symplectic characters of linear quadratic differential game were probed; as a try, Symplectic-Runge-Kutta algorithm was presented for the resolution of infinite horizon linear quadratic differential game. An example of numerical calculation was given, and the result can illuminate the feasibility of this method. At the same time, it embodies the fine conservation characteristics of symplectic algorithm to system energy.
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In this paper we study the relation of trajectories and limit cycles between index-inverse differential systems, especially, index-inverse quadratic differential systems.
