On The Number Of Limit Cycles For Bogdanov-Takens System Under A Class Of Piecewise Polynomial Perturbation

Determining the lowest upper bounds for the number of limit cycles of the Hamilton system under piecewise polynomial perturbation,is one of the important extensions of the research field of the Weakening Hilbert’s 16th problem.In this paper,the number of limit cycles of the Bogdanov-Takens system （?） under the piecewise n-degree polynomial perturbation is studied when the plane is divided into upper and lower regions.In the first chapter,the research background related to this topic,such as Hilbert’s 16th problem and the Weakening Hilbert’s 16th problem,are introduced.The research progress of Hilbert’s 16th problem and the research status of the number of limit cycles for the Bogdanov-Takens system under polynomial perturbation are summarized briefly,and also introducing some achievements in the research field of segmented smooth system.The main research results of this paper are as follows:The lowest upper bounds for the number of limit cycles of Bogdanov-Takens system under the perturbation of discontinuous piecewise polynomials of degree n（n ∈ N+）and continuous piecewise polynomials of degree 1 and 2 are B2（n）,B2c（1）,B2c（2）,respectively.Satisfing（1）When n=1,2 ≤ B2（1）≤4;When n ∈ {2,3,4},2n ≤ B2（n）≤ 7n+[n-1/2];When n≥5,B2（n）≤ 7n+[n-1/2].（2）B2c（1）=0,3 ≤B2c（2）≤8.In the second chapter,the Picard-Fuchs system of equations,generalized power series solutions and other contents are satisfied with the generators I00+,I10+,I01+,I11+ of the first-order Melnikov function expressions.And other lemmas related to estimating the number of the zero point of the Melkinov function are introduced,such as first-order and second-order differential operators,Descartes sign rule etc.In the third chapter,the first-order Melnikov function M1（h）of the Bogdanov-Takens system under the perturbation of discontinuous piecewise polynomials of degree n is calculated,and the independence of the coefficient polynomial of the generator with respect to the perturbation polynomial is demonstrated.In the fourth and fifth chapter,the conclusions（1）and conclusions（2）are certified,respectively.