On the Equivalence and Qualitative Behavior of the Rational Differential Equations

Abstract

Recent study has shown great interest in the existence of center for planar differential systems. Suffcient conditions have been given for a critical point of some polynomial systems to be a center. However, it is challenging to generalize these results to higher-order or non-polynomial systems by traditional methods. In this paper, we tackle the problem by studying the equivalence of differential systems using the Mironekos method. Our method includes the construction of quasi integrals (which were first introduced in this paper) for the rational differential equations, We give some new criterions for two differential equations to be equivalent. Applying these results to the study of center of planar differential systems, we generalize the conclusions made in existing literature for a specific polynomial system to a wide
range of higher-order polynomial or non-polynomial systems.