The Cyclicity For A Class Of Cubic Hamiltonians With Symmetrical Figure-of-eight Loop

In this article, our main purpose is to study the small perturbation problem ofHamiltonian system with fgure-of-eight loop. More special, we give the upper boundand the lower bound of the number of zeros of abelian integral, respectively. First,whencalculating the lower bound, we get the expansion of I(h) at h=0. Then, using theimplicit function theorem, we get the lower bound of the number of zeros of abelianintegral is5[n+12]5. Second,when calculating the upper bound, we complicate theabelian integral. Then, we use the argument principle in some complex domains toestimate the number of zeros of abelian integral. We get the upper bound of thenumber of zeros of abelian integral is5[n+12]+5.