The Computation Of The Lower Upper Bound For The Number Of Zeros Of A Kind Of Integrable System Through The Generalized Roll Theorem

Finding the lower upper bound for the number of zeros of Abelian integrals is an important problem of bifurcation theory of limit-cycles of a perturbated polynominal integrable system on the plane.The number of zeros of Abelian integrals of a kind of integrable system is caculated in this article.Consider the system:where Pn(x,y)and Qn(X,y)are polynomials of degrees not greater than n,G(x,y)＝(?)are nonzeros real number(s=1,...,K1;t=(?)is a small parameter.The aim of this work is to provide upper bounds for the number of limit cycles bifurcating from period annulus(?)1,...,K2),eis a small parameter.The aim of this work is to provide upper bounds for the number of limit cycles bifurcating from period annulus#12(?)In the progress of the caculation of Abelian integrals,we mainly use means of polynomial division、partial fraction deposition、parameter derivation and mathmatical induction.The Abelian integrals are expressed as the sum of radical、fraction and polynomial.Then we can caculate the upper bound of zeros,that is the upper bound of limit-cycles bifurcating from the period annulus.The main result of this article is that the number Z(I)of Abelian integrals of I(r)in(0,r0)satisfies:...