The linear relationship of different elements in N2 is studied. A conjecture is put forward as follows. For any α∈N2 satisfying α>3002, there exists a linear relationship which includes 3 or 4 elements and takes...The linear relationship of different elements in N2 is studied. A conjecture is put forward as follows. For any α∈N2 satisfying α>3002, there exists a linear relationship which includes 3 or 4 elements and takes α as the maximal element.展开更多
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.
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.
The works of Ecalle, Martinet, Moussu and Ramis in France and Ilyashenko in Russia on the finiteness of limit cycles (LC, for abbreviation) of planar polynomial differential systems are well-known. But there is still...The works of Ecalle, Martinet, Moussu and Ramis in France and Ilyashenko in Russia on the finiteness of limit cycles (LC, for abbreviation) of planar polynomial differential systems are well-known. But there is still a distance between these results and the answer of Hilbert's 16th problem even in the case n =2. Through the contributions of mathematicians in the past 40 years on the qualitative theory of quadratic differential systems (QDS), it seems to us that this case is especially simple, maybe we can slove the Hilbert problem without using the results of Ecalle et al.展开更多
Abstract:;In the paper we will examine PS having 4 finite elementary CP and at most 3 CP at infinity, to see which of them can actually be QS or PS having at most 2 contact points on any straight line;and we will see,...Abstract:;In the paper we will examine PS having 4 finite elementary CP and at most 3 CP at infinity, to see which of them can actually be QS or PS having at most 2 contact points on any straight line;and we will see,may be,these PS can all be quartic systems.展开更多
文摘The linear relationship of different elements in N2 is studied. A conjecture is put forward as follows. For any α∈N2 satisfying α>3002, there exists a linear relationship which includes 3 or 4 elements and takes α as the maximal element.
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.
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The works of Ecalle, Martinet, Moussu and Ramis in France and Ilyashenko in Russia on the finiteness of limit cycles (LC, for abbreviation) of planar polynomial differential systems are well-known. But there is still a distance between these results and the answer of Hilbert's 16th problem even in the case n =2. Through the contributions of mathematicians in the past 40 years on the qualitative theory of quadratic differential systems (QDS), it seems to us that this case is especially simple, maybe we can slove the Hilbert problem without using the results of Ecalle et al.
文摘Abstract:;In the paper we will examine PS having 4 finite elementary CP and at most 3 CP at infinity, to see which of them can actually be QS or PS having at most 2 contact points on any straight line;and we will see,may be,these PS can all be quartic systems.
