Estimation Of The Lowest Upper Bounds For The Number Of Limit Cycles Of Bogdanov-Takens System Under Piecewise Polynomial Perturbation

Since the German mathematician Hilbert D.presented the 23 famous math problems at the Second International Congress of Mathematicians in 1900,many mathematicians who engaged in the study of ordinary differential equations and dynamic systems began to study Hilbert's 16th problem.One of the important extended research topics of the weak Hilbert's 16th problem is determining the lowest upper bound for the number of limit cycles of the Hamilton system under the piecewise polynomial perturbation.In this paper,the lowest upper bound for the number of limit cycles of Bogdanov-Takens system which under the piecewise n-degree polynomial perturbationis studied when the plane is divided into left and right regions.In which 0<?<<1,In the first chapter,the background and development on the research of ordinary differential equations and qualitative theory of differential equation are introduced,the research status of Bogdanov-Takens system and piecewise smooth differential system are summarized,and the main results of this paper are as follows:Under the non-continuous or continuous(i.e.Pn+0,y)?Pn-(0,y),Qn+0,y)?Qn(0,y))piece-wise n-degree polynomial perturbation,the lowest upper bound B2(n)or B2c(n)for the number of limit cycles of Bogdanov-Takens system satisfies[5n/2]?B2(n)?[7n/2](n?{1,2,3,4,5,6});B2(n)?16n+[n/2]-10(n?7);B2c(1)=1(n=1);[5n-3/2]?B2c(n)?[7n-3/2](n?{2,3,4,5,6});B2c(n)?16n+[n-3/2]-10(n?7).In the second chapter,the first-order Melnikov function of the near Bogdanov-Takens system is calculated and the lemmas and propositions related to this paper are introduced.In the third,fourth,fifth and sixth chapters,the theorem 1 and theorem 2 are proved when n=1,2,3,4,5,6 and n>7 respectively.