For two families of planar piecewise smooth polynomial differential systems, whose unperturbed system has a degenerate center at the origin, we study the biggest lower bound for the maximum number of limit cycles bifurcating from the periodic orbits of the center. These results are extensions of the known ones on unperturbed nondegenerate Σ-center, derived from a nonsmooth harmonic oscillator model, to degenerate Σ-center. Our study involves some new computational treatments. The main tools are the generalized polar coordinate change and the generalized Lyapunov polar coordinate change together with an averaging theory for one-dimensional piecewise smooth differential equations. Finally, we present two Maple programs for computing the averaging functions and consequently the biggest lower bound on the maximum number of limit cycles of degenerate (2k, 2l)-center under general polynomial perturbations of degree n.

中文翻译：

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从分段光滑微分系统的退化中心分岔的极限环数

对于两个平面分段光滑多项式微分系统族，其原点具有退化中心，我们研究了从中心的周期轨道分叉的最大极限环数的最大下界。这些结果是对未受扰非退化的已知结果的扩展Σ-center，源自非光滑谐振子模型，退化Σ-中央。我们的研究涉及一些新的计算处理。主要工具是广义极坐标变化和广义 Lyapunov 极坐标变化以及一维分段光滑微分方程的平均理论。最后，我们提出了两个 Maple 程序来计算平均函数，从而计算退化的最大极限环数的最大下限(2ķ, 2l)-在度的一般多项式扰动下的中心n.