Continuing the investigation for the piecewise polynomial perturbations of the linear center $\dot{x}=-y,\dot{y}=x$ from Buzzi et al. (2018) for the case where the switching boundary is a straight line, in this paper we allow that the switching boundary is non-regular, i.e. we consider a switching boundary which separates the plane into two angular sectors with angles $\alpha \in (0,\pi ]$ and $2\pi -\alpha$. Moreover, unlike the aforementioned work, we allow that the polynomial differential systems in the two sectors have different degrees. Depending on $\alpha$ and for arbitrary given degrees we provide an upper bound for the maximum number of limit cycles that bifurcate from the periodic orbits of the linear center using the averaging method up to any order. This upper bound is reached for the first two orders. On the other hand, we pay attention to the perturbation of the linear center inside this class of piecewise polynomial Liénard systems and give some better upper bounds in comparison with the one obtained in the general piecewise polynomial perturbations. Again our results imply that the non-regular switching boundary (i.e. when $\alpha \ne \pi$) of the piecewise polynomial perturbations usually leads to more limit cycles than the regular case (i.e. when $\alpha =\pi$) where the switching boundary is a straight line.

继续研究线性中心的分段多项式摄动 $\dot{X}=-ÿ，\dot{ÿ}=X$来自Buzzi等。（2018）对于切换边界是直线的情况，在本文中，我们允许切换边界是非规则的，即我们考虑将平面分为两个角度的两个角度扇区的切换边界$\alpha \in （0，\pi ]$ 和 $2\pi -\alpha$。此外，与前述工作不同，我们允许两个扇区中的多项式微分系统具有不同的度数。根据$\alpha$对于任意给定的度数，我们提供了使用平均法直至任意阶数的，从线性中心的周期性轨道分叉的最大极限环数的上限。前两个订单已达到此上限。另一方面，我们注意这类分段多项式Liénard系统内部的线性中心的摄动，并且与在一般分段多项式摄动中获得的上限相比，给出了一些更好的上限。同样，我们的结果表明非常规切换边界（即$\alpha \ne \pi$）的分段多项式扰动通常会导致比常规情况（例如，当 $\alpha =\pi$），其中切换边界是一条直线。