### The Center Problem and Time-Reversibility with Respect to a Quadratic Involution for a Class of Polynomial Differential Systems with Order 2 or 3

YANG Jing1,2, YANG Ming1,2, LU Zhengyi3

1. 1. Chengdu Institute of Computer Application, Chinese Academy of Sciences, Chengdu 610041, China;
2. University of Chinese Academy of Sciences, Beijing 100049, China;
3. School of Mathematical Sciences, Sichuan Normal University, Chengdu 610068, China
• Received:2020-03-06 Revised:2020-12-06 Online:2022-08-25 Published:2022-08-02
• Contact: LU Zhengyi,Email:zhengyilu@hotmail.com
• Supported by:
The work presented in this paper was partially supported by the Specialized Research Fund for the Doctoral Program of Higher Education (SRFDP, China) under Grant No. 20115134110001.

YANG Jing, YANG Ming, LU Zhengyi. The Center Problem and Time-Reversibility with Respect to a Quadratic Involution for a Class of Polynomial Differential Systems with Order 2 or 3[J]. Journal of Systems Science and Complexity, 2022, 35(4): 1608-1636.

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 $\mathbb{R}^2$, 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.
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