Zeros of Abelian integrals for a quartic Hamiltonian with figure-of-eight loop through a nilpotent saddle. (February 2016)

• Record Type: Journal Article
• Authors: Yang, Jihua; Zhao, Liqin
• Is Part Of: Nonlinear analysis. Volume 27(2016)
• Page Start: 350
• Page End: 365
• Publication Date: 2016-02
• Subjects: Hamiltonian system -- Abelian integral -- Weakened Hilbert's 16th problem -- Picard–Fuchs equation -- Chebyshev space; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2015.08.005
• Languages: English
• ISSNs: 1468-1218

Abstract

Abstract: In this paper, we give the upper bound of the number of zeros of Abelian integral I ( h ) = ∮ Γ h g ( x, y ) d y − f ( x, y ) d x, where Γ h is the closed orbit defined by H ( x, y ) = − x 2 + x 4 + y 4 + r x 2 y 2 = h, r ≥ 0, r ≠ 2, h ∈ Σ, Σ is the maximal open interval on which the ovals { Γ h } exist, f ( x, y ) and g ( x, y ) are real polynomials in x and y of degree at most n .