Most studies of the time-reversibility are limited to a linear or an affine involution. In this paper, the authors consider the case of a quadratic involution. For a polynomial differential system with a linear part in the standard form (−y, x) in ℝ², by using the method of regular chains in a computer algebraic system, the computational procedure for finding the necessary and sufficient conditions of the system to be time-reversible with respect to a quadratic involution is given. When the system is quadratic, the necessary and sufficient conditions can be completely obtained by this procedure. For some cubic systems, the necessary and sufficient conditions for these systems to be time-reversible with respect to a quadratic involution are also obtained. These conditions can guarantee the corresponding systems to have a center. Meanwhile, a property of a center-focus system is discovered that if the system is time-reversible with respect to a quadratic involution, then its phase diagram is symmetric about a parabola.

大多数时间可逆性研究仅限于线性或仿射对合。在本文中，作者考虑了二次对合的情况。对于在 ℝ ²中具有标准形式 (− y, x )的线性部分的多项式微分系统，通过在计算机代数系统中使用正则链的方法，给出了寻找系统关于二次对合时间可逆的充分必要条件的计算过程。当系统是二次方程时，通过这个过程可以完全得到充要条件。对于某些立方系统，还获得了这些系统相对于二次对合时间可逆的充分必要条件。这些条件可以保证相应的系统有一个中心。同时，发现了中心聚焦系统的一个特性，如果该系统相对于二次对合是时间可逆的，则其相图关于抛物线对称。