Monotonicity Of The Ratio Of Abelian Integrals For Several Classes Hyper-elliptic Hamiltonian Systems

This thesis focuses on investigating the features of the ratio of two Abelian integrals P(h) under first-order polynomial perturbation of three classes of quarticHamiltonian systems, which are unfolding of a degenerate equilibrium with codimension 5 (cf. [64]). The first hyper-elliptic Hamiltonian function is H(x, y) = (?), its Hamilton system has two equilibria a saddle and a nilpotentcenter, a homoclinic loop ;The second hyper-elliptic Hamiltonian function is H(x, y) =(?), its Hamilton system has two equilibria a saddle and a center ,a homoclinic loop; The third hyper-elliptic Hamiltonian function is H(x,y) = (?), its Hamilton system has three equilibria a center a saddle and a nilpotent cusp ,a nilpotent cusp loop and a homoclinic loop. By using the idea of reference (cf. [54]),and the method of theoretical analysis,numerical analysis,we have finally come to the conclusion that P(h) is monotone with the help of some mathematical tools such as Maple 11.0.Therefore,the number of isolated zeros of Abelian integrals about the three systems is no more than one, giving the exceptional case of non-chebyshev vectorfield (cf. [36]),ie,the three class of quartic Hamiltonian systems with degenerateequilibrium point is Chebyshev from first-order polynomial perturbation. Besides these,we write the Picard-Fuchs equations of the three classes of quarticpolynomially perturbed Hamiltonian systems. Particularly, we study the first class system with nilpotent center.By scaling,the studies on the number of isolated zeros of Abelian integrals can be converted to study the number of intersection points of the curve (P(h),Q(h)) with a straight line(cf. [25], [53], etc). Also,we have investigated the nature of Abelian integrals near the two end points of Hamiltonian values.