一类三次多项式系统无穷远点的中心与等时中心

Centers of the Infinite Points and Isochronous Centers for the System of Cubic Polynomials of the First Kind

研究了一类含常数项和全二次项的三次多项式系统的无穷远点的中心与等时中心问题.通过同胚变换,三次实多项式系统的无穷远点转化为原点,研究三次系统的无穷远点的性质可以转化为研究系统原点的性质.通过复变换把实系统化为复系统,并运用计算机代数系统求出复系统原点的奇点量和周期常数,从而得到原点成为中心和等时中心的必要条件,并通过一系列方法证明了这些条件的充分性.

Investigated are the centers and isochronous centers at infinity of a cubic polynomials system with a constant term and full quadratic terms. Firstly, by means of homeomorphous transformation, the infinite of the cubic polynomial system is transformed into origin, hence the problem is transformed into study of the origin of the system. Then the real system was transformed into a complex system through complex transformation, and the singular point quantities and the period constants are computed respectively by computer algebra system, thus the necessary conditions for the center and isochronous center of the origin are worked out. At last, the sufficiency of these conditions are proven by a series of methods.