In this paper, by using Picard–Fuchs equations and Chebyshev criterion, we study the bifurcation of limit cycles for degenerate quadratic Hamilton systems with polycycles [Formula: see text] or [Formula: see text] under the perturbations of piecewise smooth polynomials with degree [Formula: see text]. Roughly speaking, for [Formula: see text], a polycycle [Formula: see text] is cyclically ordered collection of [Formula: see text] saddles together with orbits connecting them in specified order. The discontinuity is on the line [Formula: see text]. If the first order Melnikov function is not equal to zero identically, it is proved that the upper bounds of the number of limit cycles bifurcating from each of the period annuli with the boundary [Formula: see text] and [Formula: see text] are respectively [Formula: see text] and [Formula: see text] (taking into account the multiplicity).

中文翻译：

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分段光滑多项式扰动下具有多环S(2)或S(3)的退化二次哈密顿系统周期环的循环性

本文利用Picard-Fuchs方程和Chebyshev准则，研究了具有多环[公式：见正文]或[公式：见正文]的退化二次Hamilton系统在度数分段光滑多项式摄动下的极限环分岔。 [公式：见正文]。粗略地说，对于[公式：见文本]，多环[公式：见文本]是[公式：见文本]鞍座的循环有序集合，以及以指定顺序连接它们的轨道。不连续性位于 [公式：见正文] 线上。如果一阶 Melnikov 函数不完全等于 0，则证明从每个周期环以边界 [公式：见正文] 和 [公式：见正文] 分叉的极限环数的上界为分别是【公式：见正文】和【公式：