An Exactly Solvable Planar Differential System Of Degree Six With An Explicit Algebraic Limit Cycle∗

Abstract

A multi-parameter and planar first-order autonomous differential system of degree six is shown to be integrable. More specifically, we show that it possesses a time-dependent first integral and an autonomous one. This enables us to analytically obtain the solutions. Moreover, under some appropriate conditions, we prove that this system admits an explicit hyperbolic algebraic limit cycle. Finally, we draw the possible phase portraits on the Poincaré disk and illustrate the geometric behavior of the trajectories of the system.