Linear Estimate of the Number of Zeros of Abelian Integrals for a Kind of Quartic Hamiltonians

The Monotonicity of Period Function for Codimension Four Quadratic System Q4

“…As the period function TðhÞ is given by Abelian integral (1.2), it is natural that the initial problem can be reduced to estimate the number of zeros of Abelian integral dTðhÞ=dh: A simple but important fact is that dTðhÞ=dh can be expressed as a linear combination of two basic integrals which satisfy a Picard-Fuchs equation. Some results concerned with the study of the number of zeros of Abelian integrals can be found in [CsH,F1,Gl3,I1,I2,NY,P,RZh,Zh,ZLL,ZS,ZZ2] and references therein.…”

Perturbations from a kind of quartic Hamiltonians under general cubic polynomials

“…(iv) Zhao and Zhang [5] proved that the number of isolated zeros of Abelian integrals determined by vector fields (X 0 ) under perturbations of polynomials of degree n is not more than 7n + 5 for any n ∈ N . Liu [6] studied the upper bound of the number of zeros of the related abelian integral for the case (E).…”

Poincare bifurcation of a kind of Hamiltonian system under polynomial perturbation

“…Horozov & Iliev [3] concluded Z(2, n) ≤ 5n+15 for the Abelian integrals corresponding to cubic Hamiltonians. Zhao & Zhang [12] …”