The Rational Limit Cycle of a Special Case of Abel Differential Equations

Abstract

This work aims to study polynomial differential systems, focusing on their stability, limit cycles, and specific applications in Abel equations

A comprehensive study of planar polynomial differential systems is presented, including their definitions, characterizations of singular points, and the stability of these points. Nontrivial limit cycles in Abel equations are analyzed, including their existence, number, and multiplicity. We also discuss criteria such as Bendixson-Dulac to determine the presence of limit cycles. Key findings from the work of Mr. Changjian and others include: “On the Rational Limit Cycles of Abel Equations” The existence of nontrivial limit cycles for Abel equations under certain conditions on the degrees of the polynomials A(t) and B(t) is demonstrated. Then is shown that the multiplicity of nontrivial limit cycles can be unbounded, using the study of perturbed Abel equations.