Bounding the number of limit cycles for parametric Liénard systems using symbolic computation methods

This paper presents a systematical and algorithmic approach for determining the maximum number of limit cycles of parametric Liénard system that bifurcate from the period annulus of the corresponding Hamiltonian system. We provide an algebraic criterion for the Melnikov function of the considered system to have Chebyshev property. By using this criterion, we reduce the problem of analyzing the Chebyshev property to that of solving some (parametric) semi-algebraic systems, and a systematical approach with polynomial algebra methods to solve such semi-algebraic systems is explored. The feasibility of the proposed approach has been shown by several concrete Liénard systems.