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

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.

Keywords

• Limit cycle
• Lienard system
• Chebyshev system
• bifurcation
• heteroclinc loop