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.

本文提出了一种系统的算法算法，用于确定参数Liénard系统的极限环的最大数量，该极限环从对应的汉密尔顿系统的周期环面分叉。我们为考虑的系统具有切比雪夫性质的梅尔尼科夫函数提供了一个代数准则。通过使用该准则，我们将分析Chebyshev属性的问题简化为解决某些（参数）半代数系统的问题，并探索了使用多项式代数方法求解此类半代数系统的系统方法。几种具体的Liénard系统已经证明了该方法的可行性。