Weak Centers and Local Bifurcation of Critical Periods in a Z2-Equivariant Vector Field of Degree 5

With the help of algebraic manipulator-Mathematica, we identify the order of weak centers at [Formula: see text] and the origin as well as the number of local critical periods in a [Formula: see text]-equivariant vector field of degree 5. We show that [Formula: see text] 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.