Electronic Journal of Differential Equations,
Vol. 2014 (2014), No. 40, pp. 1-12.

Title: Sharp bounds of the number of zeros of Abelian integrals with parameters

Authors: Xianbo Sun (Guangxi Univ. of Finance and Economics, Nanning, China)
         Junmin Yang (Hebei Normal Univ., Shijiazhuang, China)

Abstract:
 In this article,  we study four Abelian integrals over compact level curves
 of four sixth-degree hyper-elliptic Hamiltonians with  parameters.
 We prove that the sharp bound of the number of zeros for each Abelian
 integral  is 2. The proofs rely mainly  on the Chebyshev criterion for Abelian
 integrals and  asymptotic expansions of
 Abelian integrals.

Submitted  November 11, 2013. Published February 05, 2014.
Math Subject Classifications: 34C05, 34C07, 34C08.
Key Words: Limit cycle;  Lienard system; Chebyshev system; bifurcation;
           heteroclinc loop.