With the help of algebraic manipulator-Mathematica, we identify the order of weak centers at $(\pm 1,0)$ and the origin as well as the number of local critical periods in a $Z_2$-equivariant vector field of degree 5. We show that $(\pm 1,0)$ and the origin can be weak centers of infinite order (i.e. isochronous center) and at most fourth-order weak centers of finite order. Furthermore, we prove that at most four local critical periods bifurcate from the bicenter and the origin, respectively. Our approach is a combination of computational algebraic techniques.

中文翻译：

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5 次 Z2 等变向量场中关键周期的弱中心和局部分叉

在代数操纵器Mathematica的帮助下，我们确定了弱中心的顺序$(\pm \mathrm{1个},0)$和起源以及局部关键时期的数量$Z_{\mathrm{2个}}$-5 次等变向量场。我们表明$(\pm \mathrm{1个},0)$原点可以是无限阶弱中心（即等时中心），最多可以是四阶有限阶弱中心。此外，我们证明最多四个局部临界期分别从双中心和原点分叉。我们的方法是计算代数技术的组合。