Bound the number of limit cycles bifurcating from center of polynomial Hamiltonian system via interval analysis

•Bound the number of limit cycles by algebraic criterion.•Estimate the number of isolated zeroes of Abelian integral by interval analysis.•A Lienard system of type (4, 3) could have at most six limit cycles at finite plane.•The drawbacks in some works are commented.

The algebraic criterion for Abelian integral was posed in (Grau et al. Trans Amer Math Soc 2011) and (Mañosas et al. J Differ Equat 2011) to bound the number of limit cycles bifurcating from the center of polynomial Hamiltonian system. Thisapproach reduces the estimation to the number of the limit cycle bifurcating from the center to solve the associated semi-algebraic systems (the system consists of polynomial equations, inequations and polynomial inequalities). In this paper, a systematic procedure with interval analysis has been explored to solve the SASs. In this application, we proved a hyperelliptic Hamiltonian system of degree five with a pair of conjugate complex critical points that could give rise to at most six limit cycles at finite plane under perturbations ɛ(a+bx+cx3+x4)y∂∂x. Moreover we comment the results of some related works that are not reliable by using numerical approximation.