Chebyshev criterion is a powerful tool in the study of limit cycle bifurcations in dynamical systems based on Abelian integrals, but it is difficult when the Abelian integrals involve parameters. In this paper, we consider the Abelian integrals on the periodic annuli of a Hamiltonian with one parameter, arising from the generalized Liénard system, and identify the parameter values such that the Abelian integrals have Chebyshev property. In particular, the bounds on the number of zeros of the Abelian integrals are derived for different parameter intervals. The main mathematical tools are transformations and polynomial boundary theory, which overcome the difficulties in symbolic computations and analysis, arising from large parametric-semi-algebraic systems.

中文翻译：

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阿贝尔积分参数辨识实现切比雪夫性质

Chebyshev 准则是研究基于阿贝尔积分的动力系统极限环分岔的有力工具，但当阿贝尔积分涉及参数时，就比较困难。在本文中，我们考虑由广义 Liénard 系统产生的具有一个参数的哈密顿量的周期环上的阿贝尔积分，并确定参数值，使得阿贝尔积分具有切比雪夫性质。特别是，阿贝尔积分的零点数的界限是针对不同的参数区间推导出来的。主要的数学工具是变换和多项式边界理论，它们克服了大型参数半代数系统在符号计算和分析方面的困难。